Docstoc

Fuzzy Sets and Fuzzy Techniques - Lecture 14 -- Repetition

Document Sample
Fuzzy Sets and Fuzzy Techniques - Lecture 14 -- Repetition Powered By Docstoc
					 Fuzzy Sets                                                                                          Fuzzy Sets
 and Fuzzy                                                                                           and Fuzzy
 Techniques                                                                                          Techniques
                                                                                                                                                                        Lecture notes
    Joakim                                                                                              Joakim
   Lindblad                                                                                            Lindblad

Outline                                                                                             Outline

L1: Intro                                                                                           L1: Intro

L1–3: Basics
                                 Fuzzy Sets and Fuzzy Techniques                                    L1–3: Basics

L4: Constr.
and
                                                 Lecture 14 – Repetition                            L4: Constr.
                                                                                                    and
uncertainty                                                                                         uncertainty

L5: Features                                                                                        L5: Features
                                                                                                                    www.cb.uu.se/~joakim/course/fuzzy/vt10/lectures.html
L6 Features                                                                                         L6 Features
                                                         Joakim Lindblad
L7: Distances                                                                                       L7: Distances
                                                         joakim@cb.uu.se
L8: Set                                                                                             L8: Set
operations                                                                                          operations

L9: Fuzzy                                           Centre for Image Analysis                       L9: Fuzzy
numbers                                                                                             numbers
                                                      Uppsala University
L10: Fuzzy                                                                                          L10: Fuzzy
logic                                                                                               logic

L11: Fuzzy                                                 2010-03-16                               L11: Fuzzy
control                                                                                             control

L12: Fuzzy                                                                                          L12: Fuzzy
segmentation                                                                                        segmentation

L13: Defuzzi-                                                                                       L13: Defuzzi-
                 Joakim Lindblad, 2010-03-16   (1/146)                                                              Joakim Lindblad, 2010-03-16   (2/146)
fication                                                                                             fication




 Fuzzy Sets                                                                                          Fuzzy Sets
 and Fuzzy                                                                                           and Fuzzy
 Techniques
                                                                    Topics of today                  Techniques
                                                                                                                                                  L1: Introduction, motivation
    Joakim                                                                                              Joakim
   Lindblad                                                                                            Lindblad

Outline
                  L1: Introduction, motivation                                                      Outline

L1: Intro       L1–3: Basics of fuzzy sets, cutworthiness, the extension principle                  L1: Intro

L1–3: Basics                                                                                        L1–3: Basics
                  L4: Constructing fuzzy sets, uncertainty measures
L4: Constr.                                                                                         L4: Constr.
and
uncertainty
                  L5: Fuzzy thresholding, Fuzzy c-means clustering                                  and
                                                                                                    uncertainty

L5: Features      L6: Features of fuzzy sets                                                        L5: Features

L6 Features       L7: Distances on and between fuzzy sets                                           L6 Features

L7: Distances                                                                                       L7: Distances
                  L8: Operations on fuzzy sets
L8: Set                                                                                             L8: Set
operations
                  L9: Fuzzy numbers and fuzzy arithmetics                                           operations

L9: Fuzzy                                                                                           L9: Fuzzy
numbers          L10: Fuzzy logic and approximate reasoning                                         numbers

L10: Fuzzy                                                                                          L10: Fuzzy
logic            L11: Fuzzy control                                                                 logic

L11: Fuzzy       L12: Fuzzy segmentation                                                            L11: Fuzzy
control                                                                                             control

L12: Fuzzy       L13: Defuzzification                                                                L12: Fuzzy
segmentation                                                                                        segmentation

L13: Defuzzi-                                                                                       L13: Defuzzi-
                 Joakim Lindblad, 2010-03-16   (3/146)                                                              Joakim Lindblad, 2010-03-16   (4/146)
fication                                                                                             fication




 Fuzzy Sets                                                                                          Fuzzy Sets
 and Fuzzy                                                                                           and Fuzzy
 Techniques
                                                                 About the course                    Techniques
                                                                                                                                     What will we learn in this course?
    Joakim                                                                                              Joakim
   Lindblad                                                Fuzzy Sets and Fuzzy Techniques             Lindblad                                             Fuzzy Sets and Fuzzy Techniques

Outline                                                                                             Outline

L1: Intro                                                                                           L1: Intro

L1–3: Basics                                                                                        L1–3: Basics       • The basics of fuzzy sets
L4: Constr.
                 http://www.cb.uu.se/~joakim/course/fuzzy/vt10/                                     L4: Constr.
and                                                                                                 and
                                                                                                                           • How to define fuzzy sets
uncertainty         • 13 lectures + repetition                                                      uncertainty            • How to perform operations on fuzzy sets
L5: Features                                                                                        L5: Features           • How to extend crisp concepts to fuzzy ones
                    • 2 computer exercises (giving up to 4 bonus points on the
L6 Features                                                                                         L6 Features            • How to extract information from fuzzy sets
L7: Distances
                        exam)                                                                       L7: Distances
                                                                                                                       • The very basics of fuzzy logic and fuzzy reasoning
L8: Set             • 1 small project work + presentation (written option                           L8: Set
operations                                                                                          operations         • We will look at some applications of fuzzy in
                        possible)
L9: Fuzzy                                                                                           L9: Fuzzy              • Image processing
numbers             • Written exam (2nd exam, what is a good date for you?)                         numbers
                                                                                                                           • Control systems
L10: Fuzzy                                                                                          L10: Fuzzy
logic                                                                                               logic                  • Machine intelligence / expert systems
L11: Fuzzy                                                                                          L11: Fuzzy
control                                                                                             control

L12: Fuzzy                                                                                          L12: Fuzzy
segmentation                                                                                        segmentation

L13: Defuzzi-                                                                                       L13: Defuzzi-
                 Joakim Lindblad, 2010-03-16   (5/146)                                                              Joakim Lindblad, 2010-03-16   (6/146)
fication                                                                                             fication




 Fuzzy Sets                                                                                          Fuzzy Sets
 and Fuzzy                                                                                           and Fuzzy
 Techniques
                                                             What is a fuzzy set?                    Techniques
                                                                                                                                                                         Why Fuzzy?
    Joakim                                                                                              Joakim
   Lindblad                                                                                            Lindblad

Outline                                                                                             Outline
                                                                                                                    Precision is not truth.
L1: Intro                                                                                           L1: Intro
                                                                                                                                                                                    - Henri Matisse
L1–3: Basics
                 Btw., what is a set?                                   “... to be an element...”   L1–3: Basics

L4: Constr.                                                                                         L4: Constr.
and                                                                                                 and
uncertainty                                                                                         uncertainty
                 A set is a collection of its members.                                                              So far as the laws of mathematics refer to reality, they are not
L5: Features                                                                                        L5: Features
                                                                                                                    certain. And so far as they are certain, they do not refer to
L6 Features                                                                                         L6 Features
                                  The notion of fuzzy sets is an extension                                          reality.
L7: Distances                                                                                       L7: Distances
                                  of the most fundamental property of sets.                                                                                         - Albert Einstein
L8: Set                                                                                             L8: Set
operations
                            Fuzzy sets allows a grading of to what extent                           operations

L9: Fuzzy                                                                                           L9: Fuzzy
numbers                   an element of a set belongs to that specific set.                          numbers

L10: Fuzzy                                                                                          L10: Fuzzy
                                                                                                                    As complexity rises, precise statements lose meaning and
logic                                                                                               logic
                                                                                                                    meaningful statements lose precision.
L11: Fuzzy                                                                                          L11: Fuzzy
control                                                                                             control
                                                                                                                                                                    - Lotfi Zadeh
L12: Fuzzy                                                                                          L12: Fuzzy
segmentation                                                                                        segmentation

L13: Defuzzi-                                                                                       L13: Defuzzi-
                 Joakim Lindblad, 2010-03-16   (7/146)                                                              Joakim Lindblad, 2010-03-16   (8/146)
fication                                                                                             fication
 Fuzzy Sets                                                                            Fuzzy Sets
 and Fuzzy                                                                             and Fuzzy
 Techniques
                                                         What is a fuzzy set?          Techniques
                                                                                                                                                  What is a fuzzy set?
    Joakim                                                                                Joakim
   Lindblad                                                                              Lindblad                                                      Randomness vs. Fuzziness

Outline                                                                               Outline

L1: Intro       Fuzzy is not just another name for probability.                       L1: Intro

L1–3: Basics                                                                          L1–3: Basics

L4: Constr.     The number 10 is not probably big!                                    L4: Constr.
and                                                                                   and                Randomness refers to an event that may or may not occur.
uncertainty     ...and number 2 is not probably not big.                              uncertainty

L5: Features                                                                          L5: Features
                                                                                                               Randomness: frequency of car accidents.
L6 Features                        Uncertainty is a consequence of                    L6 Features
                                                                                                        Fuzziness refers to the boundary of a set that is not precise.
L7: Distances             non-sharp boundaries between the notions/objects,           L7: Distances
                                                                                                                 Fuzziness: seriousness of a car accident.
L8: Set
operations
                                and not caused by lack of information.                L8: Set
                                                                                      operations
                                                                                                                                                                    Prof. George J. Klir
L9: Fuzzy
numbers
                Statistical models deal with random events and outcomes;              L9: Fuzzy
                                                                                      numbers

L10: Fuzzy
                fuzzy models attempt to capture and quantify nonrandom                L10: Fuzzy
logic           imprecision.                                                          logic

L11: Fuzzy                                                                            L11: Fuzzy
control                                                                               control

L12: Fuzzy                                                                            L12: Fuzzy
segmentation                                                                          segmentation

L13: Defuzzi-                                                                         L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (9/146)                                                 Joakim Lindblad, 2010-03-16    (10/146)
fication                                                                               fication




 Fuzzy Sets                                                                            Fuzzy Sets
 and Fuzzy                                                                             and Fuzzy
 Techniques
                                                               What is Fuzzy?          Techniques
                                                                                                                                        L1–3: Basics of fuzzy sets
    Joakim                                                                                Joakim
   Lindblad                                                                              Lindblad

Outline                                                                               Outline

L1: Intro                                                                             L1: Intro

L1–3: Basics    Using fuzzy techniques is                                             L1–3: Basics

L4: Constr.                                                                           L4: Constr.
and                                                                                   and
uncertainty                                                                           uncertainty

L5: Features                                                                          L5: Features

L6 Features       to avoid throwing away data early (by crisp, possibly false,        L6 Features

L7: Distances   decisions).                                                           L7: Distances

L8: Set                                                                               L8: Set
operations                                                                            operations

L9: Fuzzy                                                                             L9: Fuzzy
numbers                                                                               numbers

L10: Fuzzy                                                                            L10: Fuzzy
logic                                                                                 logic

L11: Fuzzy                                                                            L11: Fuzzy
control                                                                               control

L12: Fuzzy                                                                            L12: Fuzzy
segmentation                                                                          segmentation

L13: Defuzzi-                                                                         L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (11/146)                                                Joakim Lindblad, 2010-03-16    (12/146)
fication                                                                               fication




 Fuzzy Sets                                                                            Fuzzy Sets
 and Fuzzy                                                                             and Fuzzy
 Techniques
                                                                         Fuzzy sets    Techniques
                                                                                                                                                                  Fuzzy sets
    Joakim                                                                                Joakim
   Lindblad                                                                              Lindblad
                                                                                                      Continuous (analog) fuzzy sets
Outline                                                                               Outline
                A fuzzy set of a reference set is a set of ordered pairs
L1: Intro                                                                             L1: Intro                                      A : X → [0, 1] , X is dense
L1–3: Basics                                                                          L1–3: Basics
                                              F = { x, µF (x) | x ∈ X },
L4: Constr.
and
                                                                                      L4: Constr.
                                                                                      and
                                                                                                      Discrete fuzzy sets
uncertainty     where µF : X → [0, 1].                                                uncertainty

L5: Features                                                                          L5: Features                                  A : {x1 , x2 , x3 , ..., xs } → [0, 1]
L6 Features     Where there is no risk for confusion, we use the same symbol          L6 Features

L7: Distances   for the fuzzy set, as for its membership function.                    L7: Distances
                                                                                                      Digital fuzzy sets
L8: Set
operations
                Thus                                                                  L8: Set
                                                                                      operations      If a discrete-universal membership function can take only a
L9: Fuzzy                         F = { x, F (x) | x ∈ X },                           L9: Fuzzy       finite number n ≥ 2 of distinct values, then we call this fuzzy
numbers                                                                               numbers
                where F : X → [0, 1].                                                                 set a digital fuzzy set.
L10: Fuzzy                                                                            L10: Fuzzy
logic                                                                                 logic

L11: Fuzzy
                To define a fuzzy set ⇔ To define a membership function                 L11: Fuzzy                                                   1     2     3
control                                                                               control                 A : {x1 , x2 , x3 , ..., xs } → {0, n−1 , n−1 , n−1 , ..., n−2 , 1}
                                                                                                                                                                         n−1
L12: Fuzzy                                                                            L12: Fuzzy
segmentation                                                                          segmentation

L13: Defuzzi-                                                                         L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (13/146)                                                Joakim Lindblad, 2010-03-16    (14/146)
fication                                                                               fication




 Fuzzy Sets                                                                            Fuzzy Sets
 and Fuzzy                                                                             and Fuzzy
 Techniques
                                    Fuzzy sets of different types and                   Techniques
                                                                                                                           Basic concepts and terminology
    Joakim                                                                                Joakim
   Lindblad                                                    levels                    Lindblad     The support of a fuzzy set
Outline         The membership function may be vague in itself.                       Outline
                                                                                                                                supp(A) = {x ∈ X | A(x) > 0}
L1: Intro       Fuzzy sets of type 2: A : X → F([0, 1])                               L1: Intro

L1–3: Basics                                                                          L1–3: Basics

L4: Constr.                                                                           L4: Constr.
and                                                                                   and
                                                                                                      A crossover point of a fuzzy set
uncertainty                                                                           uncertainty

L5: Features                                                                          L5: Features                                                     ¯
                                                                                                                                                A(x) = A(x)
L6 Features                                                                           L6 Features

L7: Distances                                                                         L7: Distances

L8: Set                                                                               L8: Set         The height, h(A) of a fuzzy set
operations                                                                            operations

L9: Fuzzy
numbers
                                                                                      L9: Fuzzy
                                                                                      numbers
                                                                                                                                            h(A) = max A(x)
                Also the domain of the membership function may be fuzzy.                                                                              x∈X
L10: Fuzzy                                                                            L10: Fuzzy
logic           Fuzzy sets defined so that the elements of the universal set are       logic

L11: Fuzzy
                themselves fuzzy sets are called level 2 fuzzy sets.                  L11: Fuzzy      A normal fuzzy set
control                                                                               control

L12: Fuzzy                                                                            L12: Fuzzy
                                                                                                                                                 h(A) = 1
segmentation                                                                          segmentation

L13: Defuzzi-
                                                    A : F(X ) → [0, 1]                L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (15/146)                                                Joakim Lindblad, 2010-03-16    (16/146)
fication                                                                               fication
 Fuzzy Sets                                                                                  Fuzzy Sets
 and Fuzzy                                                                                   and Fuzzy
 Techniques
                                     Basic concepts and terminology                          Techniques
                                                                                                                                 Basic concepts and terminology
    Joakim                                                                                      Joakim
   Lindblad                                                                                    Lindblad
                An α-cut of a fuzzy set A is a crisp set αA that contains all
Outline                                                                                     Outline
                the elements in X that have membership value in A greater
L1: Intro                                                                                   L1: Intro
                than or equal to α.                                                                         For any fuzzy set A and α1 < α2 it holds that                         α2A   ⊆α1 A.
L1–3: Basics                                                                                L1–3: Basics

L4: Constr.                                                                                 L4: Constr.     All α-cuts and all strong α-cuts for two distinct families of
and                                               α                                         and
uncertainty                                        A = {x | A(x) ≥ α}                       uncertainty     nested crisp sets.
L5: Features                                                                                L5: Features

L6 Features     A strong α-cut of a fuzzy set A is a crisp set α+A that                     L6 Features     The set of all levels α ∈ [0, 1] that represent distinct α-cuts of
L7: Distances
                contains all the elements in X that have membership value in A              L7: Distances   a given fuzzy set A is called the level set of A.
L8: Set                                                                                     L8: Set
operations      strictly greater than α.                                                    operations

L9: Fuzzy                                                                                   L9: Fuzzy
                                                                                                                              Λ(A) = {α | A(x) = α for some x ∈ X }.
numbers                                                                                     numbers
                                                 α+
L10: Fuzzy
                                                      A = {x | A(x) > α}                    L10: Fuzzy
logic                                                                                       logic

L11: Fuzzy
control         We observe that the strong α-cut 0+A is equivalent to the                   L11: Fuzzy
                                                                                            control

L12: Fuzzy      support supp(A). The 1-cut 1A is often called the core of A.                L12: Fuzzy
segmentation                                                                                segmentation

L13: Defuzzi-                                                                               L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (17/146)                                                      Joakim Lindblad, 2010-03-16   (18/146)
fication                                                                                     fication




 Fuzzy Sets                                                                                  Fuzzy Sets
 and Fuzzy                                                                                   and Fuzzy
 Techniques
                                     Basic concepts and terminology                          Techniques
                                                                                                                                 Basic concepts and terminology
    Joakim                                                                                      Joakim
   Lindblad                                                                                    Lindblad

Outline                                                                                     Outline

L1: Intro
                A fuzzy set A defined on Rn is convex iff                                     L1: Intro

L1–3: Basics                                                                                L1–3: Basics

L4: Constr.
                                 A(λx1 + (1 − λ)x2 ) ≥ min (A(x1 ), A(x2 )) ,               L4: Constr.
and                                                                                         and
uncertainty                                                                                 uncertainty
                for all λ ∈ [0, 1], x1 , x2 ∈ Rn and all α ∈ [0, 1].                                        Don’t forget to read in the book here!
L5: Features                                                                                L5: Features

L6 Features

L7: Distances
                Or, equivalently, A is convex if and only if all its α-cuts     αA,   for
                                                                                            L6 Features

                                                                                            L7: Distances
                                                                                                            Chapter 1.4 and Chapter 2.
L8: Set         any α in the interval α ∈ (0, 1], are convex sets.                          L8: Set
operations                                                                                  operations

L9: Fuzzy                                                                                   L9: Fuzzy
numbers         Any property that is generalized from classical set theory into             numbers

L10: Fuzzy      the domain of fuzzy set theory by requiring that it holds in all            L10: Fuzzy
logic                                                                                       logic

L11: Fuzzy
                α-cuts in the classical sense is called a cutworthy property.               L11: Fuzzy
control                                                                                     control

L12: Fuzzy                                                                                  L12: Fuzzy
segmentation                                                                                segmentation

L13: Defuzzi-                                                                               L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (19/146)                                                      Joakim Lindblad, 2010-03-16   (20/146)
fication                                                                                     fication




 Fuzzy Sets                                                                                  Fuzzy Sets
 and Fuzzy                                                                                   and Fuzzy
 Techniques
                                         Standard fuzzy set operations                       Techniques
                                                                                                                                 Basic concepts and terminology
    Joakim                                                                                      Joakim
   Lindblad                                                                                    Lindblad

Outline                                                                                     Outline

L1: Intro             ¯
                      A(x) = 1 − A(x)              − fuzzy complement                       L1: Intro
                                                                                                            Set inclusion
L1–3: Basics          (A ∩ B)(x) = min[A(x), B(x)] − fuzzy intersection                     L1–3: Basics

L4: Constr.
and
                      (A ∪ B)(x) = max[A(x), B(x)] − fuzzy union                            L4: Constr.
                                                                                            and                                    A⊆B           iff     A(x) ≤ B(x) ∀x ∈ X
uncertainty                                                                                 uncertainty

L5: Features
                The standard fuzzy set operations form a De Morgan algebra                  L5: Features
                                                                                                            Equality
L6 Features                                                                                 L6 Features
                For standard fuzzy set operations, the law of contradiction                                                        A=B           iff     A(x) = B(x) ∀x ∈ X
L7: Distances                                                                               L7: Distances

L8: Set
operations
                                                           ¯
                                                         A∩A=∅
                                                                                            L8: Set
                                                                                            operations
                                                                                                            Scalar cardinality
L9: Fuzzy                                                                                   L9: Fuzzy                                                |A| =         A(x)
numbers                                                                                     numbers
                and the law of excluded middle, are violated.                                                                                                x∈X
L10: Fuzzy                                                                                  L10: Fuzzy
logic                                                                                       logic
                                                           ¯
                                                         A∪A=X
L11: Fuzzy                                                                                  L11: Fuzzy
control                                                                                     control

L12: Fuzzy                                                                                  L12: Fuzzy
segmentation                                                                                segmentation

L13: Defuzzi-                                                                               L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (21/146)                                                      Joakim Lindblad, 2010-03-16   (22/146)
fication                                                                                     fication




 Fuzzy Sets                                                                                  Fuzzy Sets
 and Fuzzy                                                                                   and Fuzzy
 Techniques
                                     Basic concepts and terminology                          Techniques
                                                                                                                                                             Extension principle
    Joakim                                                                                      Joakim
   Lindblad                                                                                    Lindblad     Any given function f : X → Y induces two functions,
Outline                                                                                     Outline

L1: Intro                                                                                   L1: Intro
                                                                                                                                                             f : F(X ) → Y
L1–3: Basics    Standard fuzzy intersection and fuzzy union of two fuzzy sets               L1–3: Basics                                   [f (A)](y ) =        sup        A(x)
L4: Constr.     are cutworthy and strong cutworty.                                          L4: Constr.
                                                                                                                                                              x|y =f (x)
and                                                                                         and
uncertainty     Due to associativity of min and max, any finite                              uncertainty
                                                                                                            and
L5: Features    intersection/union. However, take caution with infinitely many               L5: Features

L6 Features     intersections/unions.                                                       L6 Features
                                                                                                                                                      f −1 : F(Y ) → X
L7: Distances                                                                               L7: Distances
                                                                                                                                            [f −1 (B)](x) = B(f (x))
L8: Set
operations
                Decomposition theorems                                                      L8: Set
                                                                                            operations

L9: Fuzzy       Each standard fuzzy set is uniquely represented by the family of            L9: Fuzzy
                                                                                                            Strong cutworthiness
numbers                                                                                     numbers
                all its α-cuts, or by the family of all its strong α-cuts.                                  For any A ∈ F(X ), and a function f : X → Y , it holds that
L10: Fuzzy                                                                                  L10: Fuzzy
logic                                                                                       logic

L11: Fuzzy                                                                                  L11: Fuzzy
control                                                                                     control                                           f (A) =              f (α+ A)
L12: Fuzzy                                                                                  L12: Fuzzy                                                   α∈[0,1]
segmentation                                                                                segmentation

L13: Defuzzi-                                                                               L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (23/146)                                                      Joakim Lindblad, 2010-03-16   (24/146)
fication                                                                                     fication
 Fuzzy Sets                                                                                        Fuzzy Sets
 and Fuzzy                                                                                         and Fuzzy
 Techniques
                                              L4: Constructing fuzzy sets,                         Techniques
                                                                                                                                                L4: Constructing fuzzy sets,
    Joakim                                                                                            Joakim
   Lindblad                                         Uncertainty measures                             Lindblad                                         Uncertainty measures
Outline                                                                                           Outline                                                           Methods of construction
L1: Intro                                                                                         L1: Intro

L1–3: Basics                                                                                      L1–3: Basics

L4: Constr.                                                                                       L4: Constr.
and                                                                                               and
uncertainty                                                                                       uncertainty

L5: Features                                                                                      L5: Features

L6 Features                                                                                       L6 Features        • Direct methods and indirect methods
L7: Distances                                                                                     L7: Distances
                                                                                                                     • One expert and multiple experts
L8: Set                                                                                           L8: Set
operations                                                                                        operations

L9: Fuzzy                                                                                         L9: Fuzzy
numbers                                                                                           numbers

L10: Fuzzy                                                                                        L10: Fuzzy
logic                                                                                             logic

L11: Fuzzy                                                                                        L11: Fuzzy
control                                                                                           control

L12: Fuzzy                                                                                        L12: Fuzzy
segmentation                                                                                      segmentation

L13: Defuzzi-                                                                                     L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (25/146)                                                            Joakim Lindblad, 2010-03-16   (26/146)
fication                                                                                           fication




 Fuzzy Sets                                                                                        Fuzzy Sets
 and Fuzzy                                                                                         and Fuzzy
 Techniques
                                     Direct methods with one expert                                Techniques
                                                                                                                                            Direct methods with multiple
    Joakim                                                                                            Joakim
   Lindblad                                                                                          Lindblad                                                   experts
Outline                                                                                           Outline

L1: Intro                                                                                         L1: Intro

L1–3: Basics                                                                                      L1–3: Basics
                   • Define the complete membership function based on a
L4: Constr.                                                                                       L4: Constr.
and                    justifiable mathematical formula                                            and
                                                                                                                  The opinions of several experts need to be aggregated.
uncertainty                                                                                       uncertainty
                           • Often based on mapping of directly measurable features of
L5: Features                                                                                      L5: Features    Example: Average (Probabilistic interpretation)
                               the elements of X
L6 Features                                                                                       L6 Features
                                                                                                                                                                     n
L7: Distances
                   • Exemplifying it for some selected elements of X and                          L7: Distances                                                 1
                                                                                                                                                     A(x) =               ai (x)
L8: Set
operations
                       interpolate (/extrapolate) MF in some way.                                 L8: Set
                                                                                                  operations
                                                                                                                                                                n
                                                                                                                                                                    i=1
L9: Fuzzy                  • Expert of some kind                                                  L9: Fuzzy
numbers                                                                                           numbers

L10: Fuzzy                                                                                        L10: Fuzzy
logic                                                                                             logic

L11: Fuzzy                                                                                        L11: Fuzzy
control                                                                                           control

L12: Fuzzy                                                                                        L12: Fuzzy
segmentation                                                                                      segmentation

L13: Defuzzi-                                                                                     L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (27/146)                                                            Joakim Lindblad, 2010-03-16   (28/146)
fication                                                                                           fication




 Fuzzy Sets                                                                                        Fuzzy Sets
 and Fuzzy                                                                                         and Fuzzy
 Techniques
                                                                    Indirect methods               Techniques
                                                                                                                                                             Uncertainty measures
    Joakim                                                                                            Joakim
   Lindblad                                                                                          Lindblad

Outline                                                                                           Outline

L1: Intro                                                                                         L1: Intro

L1–3: Basics                                                                                      L1–3: Basics
                It may be easier/more objective to ask simpler questions to the
L4: Constr.                                                                                       L4: Constr.
and             experts, than the membership directly.                                            and
uncertainty                                                                                       uncertainty
                                                                                                                     • Nonspecificity of crisp sets
L5: Features
                Example: Pairwise comparisons                                                     L5: Features
                                                                                                                     • Nonspecificity of fuzzy sets
L6 Features                                                                                       L6 Features
                   • Problem: Determine membership ai = A(xi )
L7: Distances                                                                                     L7: Distances      • Fuzziness of fuzzy sets
L8: Set            • Extracted information: Pairwise relative belongingness,                      L8: Set
operations                                                ai                                      operations
                       matrix P with pij ≈                aj
L9: Fuzzy                                                                                         L9: Fuzzy
numbers                                                                                           numbers

L10: Fuzzy                                                                                        L10: Fuzzy
logic                                                                                             logic

L11: Fuzzy                                                                                        L11: Fuzzy
control                                                                                           control

L12: Fuzzy                                                                                        L12: Fuzzy
segmentation                                                                                      segmentation

L13: Defuzzi-                                                                                     L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (29/146)                                                            Joakim Lindblad, 2010-03-16   (30/146)
fication                                                                                           fication




 Fuzzy Sets                                                                                        Fuzzy Sets
 and Fuzzy                                                                                         and Fuzzy
 Techniques
                                                                           Nonspecificity           Techniques
                                                                                                                                                            Fuzziness of fuzzy sets
    Joakim                                                                                            Joakim
   Lindblad                                                                    Hartley function      Lindblad

Outline         Hartley [1928]:                                                                   Outline
                                                                                                                  A measure of fuzziness is a function
L1: Intro                                                                                         L1: Intro
                The amount of uncertainty (measure in bits) associated with                                                                                f : F(X ) → R+
L1–3: Basics                                                                                      L1–3: Basics
                a finite set of possible alternatives is
L4: Constr.                                                                                       L4: Constr.
and                                                                                               and             that expresses the degree to which the boundary of a set is
uncertainty                                              U(A) = log2 |A|                          uncertainty
                                                                                                                  non-sharp.
L5: Features                                                                                      L5: Features

L6 Features     Relates to the nonspecificity inherent in each set.                                L6 Features     Essential requirements:
L7: Distances                                                                                     L7: Distances
                Generalized Hartley function:                                                                        1 f (A) = 0 iff A is a crisp set
L8: Set                                                                                           L8: Set
operations
                                                                    h(A)
                                                                                                  operations         2 f (A) attains its maximum iff A(x) = 0.5 for all x ∈ X
                                                           1
                                                                           log2 |αA| dα
L9: Fuzzy                                                                                         L9: Fuzzy
numbers                                  U(A) =                                                   numbers            3 f (A) ≤ f (B) when set A is “undoubtedly” sharper than
                                                         h(A)   0
L10: Fuzzy                                                                                        L10: Fuzzy           set B
logic                                                                                             logic

L11: Fuzzy
                Average of the α-cuts of the normalized counterpart of A.                         L11: Fuzzy                a) A(x) ≤ B(x) when B(x) ≤ 0.5
control
                Fuzzy sets that are equal when normalized have the same
                                                                                                  control                   b) A(x) ≥ B(x) when B(x) ≥ 0.5
L12: Fuzzy                                                                                        L12: Fuzzy
segmentation    nonspecificity.                                                                    segmentation

L13: Defuzzi-                                                                                     L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (31/146)                                                            Joakim Lindblad, 2010-03-16   (32/146)
fication                                                                                           fication
 Fuzzy Sets                                                                         Fuzzy Sets
 and Fuzzy                                                                          and Fuzzy
 Techniques
                                                         Fuzziness of fuzzy sets    Techniques
                                                                                                                                            Fuzziness of fuzzy sets
    Joakim                                                                             Joakim
   Lindblad                                                                           Lindblad

Outline                                                                            Outline
                                                                                                   A simple and intuitive distance measure is the Hamming
L1: Intro                                                                          L1: Intro

L1–3: Basics                                                                       L1–3: Basics
                                                                                                   distance.
L4: Constr.
                One way to measure fuzziness of a set A is to measure the          L4: Constr.
                                                                                                                    d(A, B) =      |A(x) − B(x)|
and
uncertainty
                distance between A and the nearest crisp set.                      and
                                                                                   uncertainty
                                                                                                   The measure of fuzziness as the distance to the complement,
L5: Features                                                                       L5: Features
                Another way is to view the fuzziness of a set as the lack of                       then becomes
L6 Features                                                                        L6 Features

L7: Distances   distinction between the set and its complement.                    L7: Distances
                                                                                                                                     ¯          ¯
                                                                                                                       f (A) = d(X , X ) − d(A, A)
L8: Set                                                                            L8: Set
operations                                                                         operations
                Both views require a distance measure.                                                                           =          (1 − |A(x) − (1 − A(x))|)
L9: Fuzzy                                                                          L9: Fuzzy
numbers                                                                            numbers

L10: Fuzzy                                                                         L10: Fuzzy
                                                                                                                                 =          (1 − |2A(x) − 1|)
logic                                                                              logic

L11: Fuzzy                                                                         L11: Fuzzy
control                                                                            control

L12: Fuzzy                                                                         L12: Fuzzy
segmentation                                                                       segmentation

L13: Defuzzi-                                                                      L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (33/146)                                             Joakim Lindblad, 2010-03-16   (34/146)
fication                                                                            fication




 Fuzzy Sets                                                                         Fuzzy Sets
 and Fuzzy                                                                          and Fuzzy
 Techniques
                                                              Information gain?     Techniques
                                                                                                                            L5: Fuzzy thresholding, Fuzzy
    Joakim                                                                             Joakim
   Lindblad                                                                           Lindblad                                         c-means clustering
Outline                                                                            Outline

L1: Intro                                                                          L1: Intro

L1–3: Basics    Fuzziness and nonspecificity are distinct types of uncertainty      L1–3: Basics

L4: Constr.     and totally independent of each other.                             L4: Constr.
and                                                                                and
uncertainty     They are also totally different in their connections to             uncertainty

L5: Features    information. When nonspecificity is reduced, we view this as a      L5: Features

L6 Features     gain in information, regardless of any associated change in        L6 Features

L7: Distances                                                                      L7: Distances
                fuzziness. The opposite, however, is not true.
L8: Set                                                                            L8: Set
operations      A reduction of fuzziness is reasonable to consider as a gain of    operations

L9: Fuzzy
numbers
                information only if the nonspecificity also decreases or remains    L9: Fuzzy
                                                                                   numbers

L10: Fuzzy
                the same.                                                          L10: Fuzzy
logic                                                                              logic

L11: Fuzzy                                                                         L11: Fuzzy
control                                                                            control

L12: Fuzzy                                                                         L12: Fuzzy
segmentation                                                                       segmentation

L13: Defuzzi-                                                                      L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (35/146)                                             Joakim Lindblad, 2010-03-16   (36/146)
fication                                                                            fication




 Fuzzy Sets                                                                         Fuzzy Sets
 and Fuzzy                                                                          and Fuzzy
 Techniques
                                                                  Thresholding      Techniques
                                                                                                                                      Fuzzy c-means clustering
    Joakim                                                                             Joakim
   Lindblad                                                                           Lindblad                                                                    Bezdek
                Thresholding and fuzzy thresholding of fuzzy sets, based on
Outline                                                                            Outline

L1: Intro
                different ways of measuring and minimizing fuzziness.               L1: Intro

L1–3: Basics                                                                       L1–3: Basics

L4: Constr.                                                                        L4: Constr.
                                                                                                   Chapter 13.2
and                                                                                and
uncertainty                                                                        uncertainty     Algorithm
L5: Features                                                                       L5: Features

L6 Features                                                                        L6 Features
                                                                                                      • make initial guess for cluster means
L7: Distances                                                                      L7: Distances      • iteratively
L8: Set                                                                            L8: Set                 • use the estimated means to assign samples to clusters
operations                                                                         operations
                                                                                                           • update means
L9: Fuzzy       Membership distributions assigned using                            L9: Fuzzy
numbers                                                                            numbers            • until there are no changes in means
L10: Fuzzy        a) Pal and Rosenfeld (1988)                                      L10: Fuzzy
logic                                                                              logic

L11: Fuzzy
                  b) Huang and Wang (1995)                                         L11: Fuzzy
control                                                                            control
                  c) Fuzzy c-means (Bezdek 1981) algorithms.
L12: Fuzzy                                                                         L12: Fuzzy
segmentation                                                                       segmentation

L13: Defuzzi-                                                                      L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (37/146)                                             Joakim Lindblad, 2010-03-16   (38/146)
fication                                                                            fication




 Fuzzy Sets                                                                         Fuzzy Sets
 and Fuzzy                                                                          and Fuzzy
 Techniques
                                                   Fuzzy c-means clustering         Techniques
                                                                                                                                     L6: Features of fuzzy sets
    Joakim                                                                             Joakim
   Lindblad                                                                           Lindblad

Outline                                                                            Outline

L1: Intro                                                                          L1: Intro

L1–3: Basics
                   • a partition of the observed set is represented by a c × n     L1–3: Basics

L4: Constr.            matrix U = [uik ], where uik corresponds to the             L4: Constr.
and                                                                                and
uncertainty            membership value (anything between 0 and 1!) of the kth     uncertainty

L5: Features           element (out of n), to the ith cluster (out of c)           L5: Features

L6 Features        • boundaries between subgroups are not crisp                    L6 Features

L7: Distances                                                                      L7: Distances
                   • each element may belong to more than one cluster - its
L8: Set                                                                            L8: Set
operations             ”overall” membership equals one                             operations

L9: Fuzzy                                                                          L9: Fuzzy
numbers            • objective function includes parameter controlling degree of   numbers

L10: Fuzzy             fuzziness                                                   L10: Fuzzy
logic                                                                              logic

L11: Fuzzy                                                                         L11: Fuzzy
control                                                                            control

L12: Fuzzy                                                                         L12: Fuzzy
segmentation                                                                       segmentation

L13: Defuzzi-                                                                      L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (39/146)                                             Joakim Lindblad, 2010-03-16   (40/146)
fication                                                                            fication
 Fuzzy Sets                                                                                 Fuzzy Sets
 and Fuzzy                                                                                  and Fuzzy
 Techniques
                                                   L6: Features of fuzzy sets               Techniques
                                                                                                                                                                   L6: Features of fuzzy sets
    Joakim                                                                                     Joakim
   Lindblad                                                                                   Lindblad                                                                                                                    Aggregation over α-cuts

Outline                                                                                    Outline

L1: Intro                                                                                  L1: Intro

L1–3: Basics       • Spatial fuzzy sets                                                    L1–3: Basics    Given a function f : P(X ) → R.
L4: Constr.        • Scalar descriptors of (spatial) fuzzy sets                            L4: Constr.
                                                                                                           We can extends this function to f : F(X ) → R,
and                                                                                        and
uncertainty            • Definitions                                                        uncertainty
                                                                                                           using one of the following equations
L5: Features           • Inter-relations                                                   L5: Features

L6 Features
                   • Other shape descriptors                                               L6 Features                                                                                           1
L7: Distances                                                                              L7: Distances                                                    f (A) =                                                        f (αA) dα,                            (1)
                       • Vector-valued and non-numerical                                                                                                                              0
L8: Set                                                                                    L8: Set
operations         • Feature estimation                                                    operations                                                       f (A) =                   sup [αf (αA)]                                                              (2)
L9: Fuzzy              • Area and higher order moments                                     L9: Fuzzy                                                                               α∈(0,1]
numbers                                                                                    numbers
                       • Perimeter
L10: Fuzzy                                                                                 L10: Fuzzy
logic                                                                                      logic           Both these definitions provide consistency for the crisp case.
L11: Fuzzy                                                                                 L11: Fuzzy
control                                                                                    control

L12: Fuzzy                                                                                 L12: Fuzzy
segmentation                                                                               segmentation

L13: Defuzzi-                                                                              L13: Defuzzi-
                Joakim Lindblad, 2010-03-16    (41/146)                                                    Joakim Lindblad, 2010-03-16                    (42/146)
fication                                                                                    fication




 Fuzzy Sets                                                                                 Fuzzy Sets
 and Fuzzy                                                                                  and Fuzzy
 Techniques
                                                   L6: Features of fuzzy sets               Techniques
                                                                                                                                                                   L6: Features of fuzzy sets
    Joakim                                                                                     Joakim
   Lindblad     The area of a fuzzy set A on X ⊆ R is                                         Lindblad

Outline                                                                                    Outline         Geometric moments:
L1: Intro                                      area(A) =          A(x) dx                  L1: Intro
                                                                                                           The moment mp,q (A) of a fuzzy set A defined on X ⊂ R2 , is
L1–3: Basics                                                  X                            L1–3: Basics
                                                               1
L4: Constr.                                                                                L4: Constr.
and                                                       =           area(αA) dα          and                                                   mp,q (A) =                        A(x, y ) x p y q dxdy .
uncertainty                                                                                uncertainty
                                                              0
L5: Features                                                                               L5: Features                                                                    X
L6 Features     For a discrete fuzzy set, the area is equal to the cardinality of          L6 Features
                                                                                                            for integers p, q ≥ 0.
L7: Distances   the set                                                                    L7: Distances
                                                                                                           Remark: The area of a set is the m0,0 moment.
L8: Set
operations
                                    area(A) = |A| =      A(x)                              L8: Set
                                                                                           operations
                                                                         X                                 Remark: The centroid (centre of gravity) of a set is
L9: Fuzzy                                                                                  L9: Fuzzy
numbers                                                                                    numbers

L10: Fuzzy
                The perimeter of a fuzzy set A                                             L10: Fuzzy
                                                                                                                                                                               m1,0 (A) m0,1 (A)
logic                                                                                      logic                                                   (xc , yc ) =                        ,
                                                                  1                                                                                                            m0,0 (A) m0,0 (A)
L11: Fuzzy                                                                                 L11: Fuzzy
control                                       perim(A) =              perim(αA) dα         control

L12: Fuzzy                                                    0                            L12: Fuzzy
segmentation                                                                               segmentation

L13: Defuzzi-                                                                              L13: Defuzzi-
                Joakim Lindblad, 2010-03-16    (43/146)                                                    Joakim Lindblad, 2010-03-16                    (44/146)
fication                                                                                    fication




 Fuzzy Sets                                                                                 Fuzzy Sets
 and Fuzzy                                                                                  and Fuzzy
 Techniques
                                                                         Inter-relations    Techniques
                                                                                                                                                                           Estimation of features
    Joakim                                                                                     Joakim
   Lindblad                                                                                   Lindblad

Outline                                                                                    Outline

L1: Intro                                                                                  L1: Intro
                All the definitions listed above reduce to the corresponding                                Features of a continuous spatial shape S, can be estimated
L1–3: Basics                                                                               L1–3: Basics

L4: Constr.
                customary definitions for crisp sets. However, some                         L4: Constr.     from features of its digitization D(S).
and             inter-relations which these notions satisfy in the crisp case, do          and
                                                                                                           The precision of such estimates is limited by the spatial
uncertainty                                                                                uncertainty

L5: Features
                not hold for the generalized (fuzzified) definitions.                        L5: Features    resolution of the digital representation.
L6 Features     For example: The isoperimetric inequality,                                 L6 Features
                                                                                                           For object represented by digital spatial fuzzy sets, where the
L7: Distances                                                                              L7: Distances
                                                                             2                             membership of a point indicates to what extent the pixel/voxel
L8: Set                                        4π · area(A) ≤ perim (A),                   L8: Set
operations                                                                                 operations      is covered by the imaged object, significant improvements in
L9: Fuzzy                                                                                  L9: Fuzzy       precision of feature estimates can be obtained. Especially for
numbers         Bogomolny proposed (1987) modified definitions. However,                     numbers
                                                                                                           small objects/limited resolution.
L10: Fuzzy      these definitions are often seen as less intuitive.                         L10: Fuzzy
logic                                                                                      logic

L11: Fuzzy                                                                                 L11: Fuzzy
control                                                                                    control

L12: Fuzzy                                                                                 L12: Fuzzy
segmentation                                                                               segmentation

L13: Defuzzi-                                                                              L13: Defuzzi-
                Joakim Lindblad, 2010-03-16    (45/146)                                                    Joakim Lindblad, 2010-03-16                    (46/146)
fication                                                                                    fication




 Fuzzy Sets                                                                                 Fuzzy Sets
 and Fuzzy                                                                                  and Fuzzy
 Techniques
                                                          Estimation of features            Techniques
                                                                                                                                                                           Estimation of features
    Joakim                                                                                     Joakim
   Lindblad                                                                                   Lindblad                                                                                 Pixel coverage digitization
                                                                                                           If the membership function correponds to pixel/voxel coverage
Outline                                                                                    Outline
                                                                                                           then it is possible to derive very precise estimates.
L1: Intro                                                                                  L1: Intro

L1–3: Basics                                                                               L1–3: Basics    Trade-off between spatial and grey-level resolution
L4: Constr.     Significant improvement in the precision of                                 L4: Constr.
and                                                                                        and
uncertainty
                feature estimates can be achieved using a fuzzy                            uncertainty

L5: Features                                                                               L5: Features                               2

                approach.                                                                                                                                                                                                 10
                                                                                                                                                                                      Absolute value of rel. error in %




L6 Features                                                                                L6 Features
                                                                                                               Relative error in %




                                                                                                                                      0
L7: Distances                                                                              L7: Distances

L8: Set         Exploiting fuzzy can provide an alternative to                             L8: Set                                   −2                                                                                    1
operations                                                                                 operations

L9: Fuzzy
                increasing the spatial resolution of the image.                            L9: Fuzzy                                 −4
                                                                                                                                                                            n= 1
                                                                                                                                                                            n= 2
                                                                                                                                                                            n= 3
                                                                                                                                                                                                                                 n= 1
                                                                                                                                                                                                                                 n= 2
                                                                                                                                                                                                                                 n= 3
numbers                                                                                    numbers                                                                          n= 5                                                 n= 5
                                                                                                                                                                            n=10                                                 n=10
                                                                                                                                                                            n= ∞                                                 n= ∞
L10: Fuzzy                                                                                 L10: Fuzzy                                −6                                                                                   0.1
logic                                                                                      logic                                       0   200   400        600      800       1000                                         10               100          1000
                                                                                                                                                 Grid resolution                                                                        Grid resolution

L11: Fuzzy                                                                                 L11: Fuzzy
control                                                                                    control
                                                                                                           Figure: Relative error of perimeter estimates for different membership
L12: Fuzzy                                                                                 L12: Fuzzy
segmentation                                                                               segmentation
                                                                                                           resolutions.
L13: Defuzzi-                                                                              L13: Defuzzi-
                Joakim Lindblad, 2010-03-16    (47/146)                                                    Joakim Lindblad, 2010-03-16                    (48/146)
fication                                                                                    fication
 Fuzzy Sets                                                                                          Fuzzy Sets
 and Fuzzy                                                                                           and Fuzzy
 Techniques
                                                          Estimation of features                     Techniques
                                                                                                                                            L7: Distances on and between
    Joakim                                                                                              Joakim
   Lindblad                                                                                            Lindblad                                                fuzzy sets
Outline                                                                                             Outline

L1: Intro                                                                                           L1: Intro

L1–3: Basics                                                                                        L1–3: Basics
                   • Spatial fuzzy sets are of a particular interest in image
L4: Constr.                                                                                         L4: Constr.
and                    analysis.                                                                    and
uncertainty                                                                                         uncertainty

L5: Features
                   • Features of spatial fuzzy sets → shape descriptors.                            L5: Features

L6 Features        • “Horizontal” and “vertical” approach in definitions.                            L6 Features

L7: Distances                                                                                       L7: Distances
                   • Particular membership functions → High precision
L8: Set                                                                                             L8: Set
operations             estimates.                                                                   operations

L9: Fuzzy
numbers
                   • Fuzzy feature values? (Still relatively unexplored)                            L9: Fuzzy
                                                                                                    numbers

L10: Fuzzy                                                                                          L10: Fuzzy
logic                                                                                               logic

L11: Fuzzy                                                                                          L11: Fuzzy
control                                                                                             control

L12: Fuzzy                                                                                          L12: Fuzzy
segmentation                                                                                        segmentation

L13: Defuzzi-                                                                                       L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (49/146)                                                              Joakim Lindblad, 2010-03-16    (50/146)
fication                                                                                             fication




 Fuzzy Sets                                                                                          Fuzzy Sets
 and Fuzzy                                                                                           and Fuzzy
 Techniques
                                        L7: Distances on and between                                 Techniques
                                                                                                                                                                    Set to set distances
    Joakim                                                                                              Joakim
   Lindblad                                                fuzzy sets                                  Lindblad

Outline                                                                                             Outline

L1: Intro          • Set to set distances                                                           L1: Intro

L1–3: Basics                                                                                        L1–3: Basics
                   • (Point to set distances)                                                                       Distances between fuzzy sets
L4: Constr.                                                                                         L4: Constr.
and                • Point to point distances                                                       and
uncertainty                                                                                         uncertainty       a) Membership focused (vertical)
L5: Features
                A mix of notions                                                                    L5: Features
                                                                                                                      b) Spatially focused (horizontal)
L6 Features                                                                                         L6 Features

L7: Distances
                   • The objects that the distance is measured between (start                       L7: Distances
                                                                                                                      c) Mix of spatial and membership (tolerance)
L8: Set
                       and stop)                                                                    L8: Set           d) Feature distances (low or high dimensional representations)
operations                                                                                          operations
                       - crisp or fuzzy, point or set
L9: Fuzzy                                                                                           L9: Fuzzy
                                                                                                                      e) Morphological (mixed focus)
numbers            • The space where a path between start and stop is                               numbers

L10: Fuzzy             embedded (spatial cost function)                                             L10: Fuzzy
logic                                                                                               logic
                            - Unconstrained (Euclidean)
L11: Fuzzy                                                                                          L11: Fuzzy
control                     - Constrained (geodesic/cost function)                                  control

L12: Fuzzy         • Output: Crisp (number) or fuzzy                                                L12: Fuzzy
segmentation                                                                                        segmentation

L13: Defuzzi-                                                                                       L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (51/146)                                                              Joakim Lindblad, 2010-03-16    (52/146)
fication                                                                                             fication




 Fuzzy Sets                                                                                          Fuzzy Sets
 and Fuzzy                                                                                           and Fuzzy
 Techniques
                                                            Membership focused                       Techniques
                                                                                                                                                                  Membership focused
    Joakim                                                                                              Joakim
   Lindblad                                                                      Lp norm               Lindblad                                                                              Lp norm

Outline                                                                                             Outline         Discrete version:
L1: Intro
                “The functional approach”                                                           L1: Intro

L1–3: Basics                                                                                        L1–3: Basics

L4: Constr.
                The most common:                                                                    L4: Constr.                                                                        1/p
                                                                                                                                                       n
and             Based on the family of Minkowski distances                                          and
uncertainty                                                                                         uncertainty               dp (A, B) =                     |µA (xi ) − µB (xi )|p         ,   p ≥ 1,
L5: Features                                                                                        L5: Features                                      i=1
L6 Features                                                                                         L6 Features
                                                                                 1/p                                         d∞ (A, B) = max (|µA (xi ) − µB (xi )|) .
                                                                                                                                                   i=1...n
L7: Distances
                            dp (A, B) =                  |µA (x) − µB (x)|p dx         ,   p ≥ 1,   L7: Distances

L8: Set                                             X                                               L8: Set
operations
                     dEssSup (A, B) = lim dp (A, B)
                                                                                                    operations      dp for p ≥ 1 are all metrics in the discrete case.
L9: Fuzzy                                       p→∞                                                 L9: Fuzzy
numbers                                                                                             numbers
                           d∞ (A, B) = sup |µA (x) − µB (x)| .                                                      L4: Hamming distance.
L10: Fuzzy                                                                                          L10: Fuzzy
logic                                           x∈X                                                 logic

L11: Fuzzy                                                                                          L11: Fuzzy                                    d1 (A, B) =           |A(x) − B(x)|
control                                                                                             control

L12: Fuzzy                                                                                          L12: Fuzzy
segmentation                                                                                        segmentation

L13: Defuzzi-                                                                                       L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (53/146)                                                              Joakim Lindblad, 2010-03-16    (54/146)
fication                                                                                             fication




 Fuzzy Sets                                                                                          Fuzzy Sets
 and Fuzzy                                                                                           and Fuzzy
 Techniques
                                                            Membership focused                       Techniques
                                                                                                                                                                        Spatially focused
    Joakim                                                                                              Joakim
   Lindblad                                                       Set operations approach              Lindblad
                Tversky 1977, et al. defines a measure of similarity based on
Outline                                                                                             Outline
                three components as follows:
L1: Intro                                                                                           L1: Intro

L1–3: Basics
                            S(a, b) = θf (A ∩ B) − αf (A − B) − βf (B − A)                          L1–3: Basics       • Nearest point
L4: Constr.
and
                                                                                                    L4: Constr.
                                                                                                    and
                                                                                                                       • Mean distance
uncertainty                                                                                         uncertainty
                                                                                                                       • Hausdorff
L5: Features                                                                                        L5: Features

L6 Features                                                                                         L6 Features

L7: Distances                                                                                       L7: Distances
                                                                                                                    Three (four) approaches:
L8: Set
operations
                                                                                                    L8: Set
                                                                                                    operations
                                                                                                                       • fuzzy distance
L9: Fuzzy                                                                                           L9: Fuzzy          • weighting with membership
numbers                                                                                             numbers

L10: Fuzzy                                                                                          L10: Fuzzy
                                                                                                                       • morphological and integration of alpha-cuts
logic                                                                                               logic

L11: Fuzzy                                                                                          L11: Fuzzy
control         Figure: Representation of two objects that each contains its own                    control

L12: Fuzzy      unique features and also contains common features.                                  L12: Fuzzy
segmentation                                                                                        segmentation

L13: Defuzzi-                                                                                       L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (55/146)                                                              Joakim Lindblad, 2010-03-16    (56/146)
fication                                                                                             fication
 Fuzzy Sets                                                                                          Fuzzy Sets
 and Fuzzy                                                                                           and Fuzzy
 Techniques
                                                                     Spatially focused               Techniques
                                                                                                                                                                        Spatially focused
    Joakim                                                                                              Joakim
   Lindblad                                                                              Hausdorff      Lindblad                                                                             Hausdorff

Outline         Crisp:                                                                              Outline

L1: Intro                                                                                           L1: Intro
                                                                                                                    Ralescu and Ralescu (1984)
L1–3: Basics             dH (A, B) = max{sup inf d(x, y ), sup inf d(x, y )}                        L1–3: Basics

L4: Constr.                                               x∈A y ∈B             y ∈A x∈B             L4: Constr.                                                         1
and
uncertainty
                                                               +
                                         = inf{r ∈ R | A ⊆ Dr (B) ∧ B ⊆ Dr (A)}
                                                                                                    and
                                                                                                    uncertainty                               dH1 (A, B) =                  dH (αA,α B) dα,
                                                                                                                                                                    0
L5: Features                                                                                        L5: Features
                                                                                                                                            dH∞ (A, B) = sup dH (αA,α B),
L6 Features     where Dr (A) is the dilation of the set A by a ball of radius r                     L6 Features
                                                                                                                                                                   α>0
L7: Distances                                                                                       L7: Distances
                           Dr (A) = {y ∈ X | ∃x ∈ A : d(x, y ) ≤ r }
L8: Set                                                                                             L8: Set         where dH is the Hausdorff distance between two crisp sets,
operations                                                                                          operations

L9: Fuzzy       The Hausdorff distance between A and B is the smallest                               L9: Fuzzy
                                                                                                                    A serious problem is that the distance between two fuzzy sets A
numbers                                                                                             numbers

L10: Fuzzy
                amount that A must be expanded to contain B and vice versa.                         L10: Fuzzy      and B is infinite if height(A) = height(B).
logic                                                                                               logic
                Is a metric on the set of nonempty compact sets.
L11: Fuzzy                                                                                          L11: Fuzzy      No good solution to that problem is found.
control         Remark:                                                                             control

L12: Fuzzy
segmentation
                Usually extended with: dH (A, ∅) = ∞ and dH (∅, ∅) = 0                              L12: Fuzzy
                                                                                                    segmentation

L13: Defuzzi-                                                                                       L13: Defuzzi-
                Joakim Lindblad, 2010-03-16    (57/146)                                                             Joakim Lindblad, 2010-03-16   (58/146)
fication                                                                                             fication




 Fuzzy Sets                                                                                          Fuzzy Sets
 and Fuzzy                                                                                           and Fuzzy
 Techniques
                                                                     Feature distances               Techniques
                                                                                                                                                         Point to point distances
    Joakim                                                                                              Joakim
   Lindblad                                                   “Pattern recognition approach”           Lindblad

Outline                                                                                             Outline

L1: Intro                                                                                           L1: Intro

L1–3: Basics                                                                                        L1–3: Basics

L4: Constr.                                                                                         L4: Constr.
and             Use of a feature representation of the observed sets as an                          and
uncertainty                                                                                         uncertainty
                intermediate step in the distance calculations.                                                     Distances between points in a fuzzy set
L5: Features                                                                                        L5: Features

L6 Features     The distance between sets A and B is then given in terms of                         L6 Features     Defining the cost of traveling along a path
L7: Distances   the distance between their feature vectors.                                         L7: Distances

L8: Set
operations
                Often global shape features are used (think shape matching,                         L8: Set
                                                                                                    operations

L9: Fuzzy       image retrieval).                                                                   L9: Fuzzy
numbers                                                                                             numbers

L10: Fuzzy                                                                                          L10: Fuzzy
logic                                                                                               logic

L11: Fuzzy                                                                                          L11: Fuzzy
control                                                                                             control

L12: Fuzzy                                                                                          L12: Fuzzy
segmentation                                                                                        segmentation

L13: Defuzzi-                                                                                       L13: Defuzzi-
                Joakim Lindblad, 2010-03-16    (59/146)                                                             Joakim Lindblad, 2010-03-16   (60/146)
fication                                                                                             fication




 Fuzzy Sets                                                                                          Fuzzy Sets
 and Fuzzy                                                                                           and Fuzzy
 Techniques
                                                                           Cost function             Techniques
                                                                                                                                                                               Cost function
    Joakim                                                                                              Joakim
   Lindblad                                                                    “Snow shoveling”        Lindblad                                                                             variations

Outline                                                                                             Outline

L1: Intro                                                                                           L1: Intro

L1–3: Basics
                Define the distance along a path πi between points x and y in                        L1–3: Basics
                the fuzzy set A                                                                                     Membership as another dimension
L4: Constr.                                                                                         L4: Constr.
and                                                                                                 and             integrate the arc-length
uncertainty                                                                                         uncertainty

L5: Features                                   dA (πi (x, y )) =              A(t) dt               L5: Features    Bloch 1995, Toivanen 1996:
                                                                        s∈π
L6 Features                                                                                         L6 Features
                                                                                                                                                                                        2
L7: Distances                                                                                       L7: Distances                                                               dA(t)
                The distance between points x and y in A is                                                                                dA (π) =                  1+                     dt
L8: Set
operations      the distance along the shortest path
                                                                                                    L8: Set
                                                                                                    operations                                               s∈π                 dt
L9: Fuzzy                                                                                           L9: Fuzzy
numbers
                                                dA (x, y ) =         inf      dA (π)                numbers         Problem: How to relate scale of membership to spatial distance
L10: Fuzzy                                                       π∈Π(x,y )                          L10: Fuzzy
                                                                                                                    in the domain?
logic                                                                                               logic

L11: Fuzzy                                                                                          L11: Fuzzy
control                                                                                             control

L12: Fuzzy                                                                                          L12: Fuzzy
segmentation                                                                                        segmentation

L13: Defuzzi-                                                                                       L13: Defuzzi-
                Joakim Lindblad, 2010-03-16    (61/146)                                                             Joakim Lindblad, 2010-03-16   (62/146)
fication                                                                                             fication




 Fuzzy Sets                                                                                          Fuzzy Sets
 and Fuzzy                                                                                           and Fuzzy
 Techniques
                                                              Constrained distance                   Techniques
                                                                                                                                                                              Connectedness
    Joakim                                                                                              Joakim
   Lindblad                                                                                            Lindblad

Outline                                                                                             Outline

L1: Intro       Geodesic distance – shortest path within the set; not allowed to                    L1: Intro
                                                                                                                    Connectedness, Rosenfeld 1979
L1–3: Basics    go out of the set – a path that descends the least in terms of                      L1–3: Basics
                                                                                                                    Strength of a path – the strength of its weakest link
L4: Constr.
and
                membership.                                                                         L4: Constr.
                                                                                                    and
uncertainty                                                                                         uncertainty     Strength of a link between two points defined by the
L5: Features                ıtre
                Bloch and Maˆ 1995:                                                                 L5: Features    membership function.
L6 Features                                                                                         L6 Features
                                                                                    ds                              The connectedness of two points x and y in A –
                                                                                π
L7: Distances                                 d(x, y ) =          inf                               L7: Distances
                                                                                                                    the strength of the strongest path between x and y
L8: Set                                                       π∈ΠcA (x,y ) cA (x, y )               L8: Set
operations                                                                                          operations

L9: Fuzzy
numbers
                where cA (x, y ) is the strength of connectedness of points x                       L9: Fuzzy
                                                                                                    numbers
                                                                                                                                                  cA (x, y ) =       sup        inf A(t)
                and y , and ΠcA (x, y ) is the set of path contained within the                                                                                    π∈Π(x,y ) t∈π
L10: Fuzzy                                                                                          L10: Fuzzy
logic           α-cut cAA.                                                                          logic

L11: Fuzzy                                                                                          L11: Fuzzy
control                                                                                             control

L12: Fuzzy                                                                                          L12: Fuzzy
segmentation                                                                                        segmentation

L13: Defuzzi-                                                                                       L13: Defuzzi-
                Joakim Lindblad, 2010-03-16    (63/146)                                                             Joakim Lindblad, 2010-03-16   (64/146)
fication                                                                                             fication
 Fuzzy Sets                                                                                    Fuzzy Sets
 and Fuzzy                                                                                     and Fuzzy
 Techniques
                                                L8: Operations on fuzzy sets                   Techniques
                                                                                                                                                 Standard fuzzy operations
    Joakim                                                                                        Joakim
   Lindblad                                                                                      Lindblad

Outline                                                                                       Outline

L1: Intro                                                                                     L1: Intro

L1–3: Basics                                                                                  L1–3: Basics

L4: Constr.                                                                                   L4: Constr.
and                                                                                           and
uncertainty                                                                                   uncertainty

L5: Features                                                                                  L5: Features
                                                                                                                      ¯
                                                                                                                      A(x) = 1 − A(x)              − fuzzy complement
L6 Features                                                                                   L6 Features             (A ∩ B)(x) = min[A(x), B(x)] − fuzzy intersection
L7: Distances                                                                                 L7: Distances           (A ∪ B)(x) = max[A(x), B(x)] − fuzzy union
L8: Set                                                                                       L8: Set
operations                                                                                    operations

L9: Fuzzy                                                                                     L9: Fuzzy
numbers                                                                                       numbers

L10: Fuzzy                                                                                    L10: Fuzzy
logic                                                                                         logic

L11: Fuzzy                                                                                    L11: Fuzzy
control                                                                                       control

L12: Fuzzy                                                                                    L12: Fuzzy
segmentation                                                                                  segmentation

L13: Defuzzi-                                                                                 L13: Defuzzi-
                  Joakim Lindblad, 2010-03-16   (65/146)                                                        Joakim Lindblad, 2010-03-16   (66/146)
fication                                                                                       fication




 Fuzzy Sets                                                                                    Fuzzy Sets
 and Fuzzy                                                                                     and Fuzzy
 Techniques
                                                   Properties of the standard                  Techniques
                                                                                                                                                           Aggregation operators
    Joakim                                                                                        Joakim
   Lindblad                                                        operations                    Lindblad

Outline                                                                                       Outline

L1: Intro                                                                                     L1: Intro

L1–3: Basics                                                                                  L1–3: Basics      Aggregation operators are used to combine several fuzzy sets in
L4: Constr.          • They are generalizations of the corresponding (uniquely                L4: Constr.       order to produce a single fuzzy set.
and                                                                                           and
uncertainty              defined!) classical set operations.                                   uncertainty
                                                                                                                Associative aggregation operations
L5: Features         • They satisfy the cutworthy and strong cutworthy                        L5: Features

L6 Features                                                                                   L6 Features          • (general) fuzzy intersections - t-norms
                         properties. They are the only ones that do.
L7: Distances                                                                                 L7: Distances        • (general) fuzzy unions - t-conorms
L8: Set
                     • The standard fuzzy intersection of two sets contains (is               L8: Set
operations               bigger than) all other fuzzy intersections of those sets.            operations        Non-associative aggregation operations
L9: Fuzzy                                                                                     L9: Fuzzy
numbers              • The standard fuzzy union of two sets is contained in (is               numbers              • averaging operations - idempotent aggregation operations
L10: Fuzzy               smaller than) all other fuzzy unions of those sets.                  L10: Fuzzy
logic                                                                                         logic

L11: Fuzzy                                                                                    L11: Fuzzy
control                                                                                       control

L12: Fuzzy                                                                                    L12: Fuzzy
segmentation                                                                                  segmentation

L13: Defuzzi-                                                                                 L13: Defuzzi-
                  Joakim Lindblad, 2010-03-16   (67/146)                                                        Joakim Lindblad, 2010-03-16   (68/146)
fication                                                                                       fication




 Fuzzy Sets                                                                                    Fuzzy Sets
 and Fuzzy                                                                                     and Fuzzy
 Techniques
                                                             Fuzzy complements                 Techniques
                                                                                                                                                                            Generators
    Joakim                                                                                        Joakim
   Lindblad                                                          Axiomatic requirements      Lindblad                                                               Increasing generators

Outline                                                                                       Outline              • Increasing generator
L1: Intro                                                                                     L1: Intro              is a strictly increasing continuous function g : [0, 1] → R,
L1–3: Basics                                                                                  L1–3: Basics           such that g (0) = 0.
L4: Constr.                                                                                   L4: Constr.          • A pseudo-inverse of increasing generator g is defined as
and                                                                                           and
uncertainty     Ax c1. c(0) = 1 and c(1) = 0. boundary condition                              uncertainty                                                
                                                                                                                                                          0           for a ∈ (−∞, 0)
L5: Features    Ax c2. For all a, b ∈ [0, 1], if a ≤ b, then c(a) ≥ c(b). monotonicity        L5: Features
                                                                                                                                         g (−1) =          g −1 (a)    for a ∈ [0, g (1)]
L6 Features              c1 and c2 are called axiomatic skeleton for fuzzy complements        L6 Features
                                                                                                                                                           1           for a ∈ (g (1), ∞)
                                                                                                                                                         
L7: Distances                                                                                 L7: Distances
                Ax c3. c is a continuous function.
L8: Set                                                                                       L8: Set              • An example:
operations      Ax c4. c is involutive, i.e., c(c(a)) = a, for each a ∈ [0, 1].               operations

L9: Fuzzy
numbers
                                                                                              L9: Fuzzy
                                                                                              numbers
                                                                                                                                                   = ap , p > 0
                                                                                                                                               g (a)
                                                                                                                                                     
L10: Fuzzy                                                                                    L10: Fuzzy                                              0     for a ∈ (−∞, 0)
                                                                                                                                                           1
logic                                                                                         logic
                                                                                                                                        g (−1) (a) =     a p for a ∈ [0, 1]
L11: Fuzzy                                                                                    L11: Fuzzy
                                                                                                                                                         1   for a ∈ (1, ∞)
                                                                                                                                                     
control                                                                                       control

L12: Fuzzy                                                                                    L12: Fuzzy
segmentation                                                                                  segmentation

L13: Defuzzi-
                  Joakim Lindblad, 2010-03-16   (69/146)
                                                                                              L13: Defuzzi-     Similarly for Decreasing generators
                                                                                                                                  (70/146)
                                                                                                                Joakim Lindblad, 2010-03-16
fication                                                                                       fication




 Fuzzy Sets                                                                                    Fuzzy Sets
 and Fuzzy                                                                                     and Fuzzy
 Techniques
                                         Generating fuzzy complements                          Techniques
                                                                                                                                                               Fuzzy intersections
    Joakim                                                                                        Joakim
   Lindblad                                                                                      Lindblad                                                             Axiomatic requirements

Outline
                  Theorem                                                                     Outline

L1: Intro         (First Characterization Theorem of Fuzzy Complements.)                      L1: Intro

L1–3: Basics      Let c be a function from [0, 1] to [0, 1]. Then c is a                      L1–3: Basics      For all a, b, d ∈ [0, 1],
L4: Constr.
and
                  (involutive) fuzzy complement iff there exists an increasing                 L4: Constr.
                                                                                              and
                                                                                                              Ax i1. i(a, 1) = a. boundary condition
uncertainty       generator g such that, for all a ∈ [0, 1]                                   uncertainty     Ax i2. b ≤ d implies i(a, b) ≤ i(a, d). monotonicity
L5: Features                                                                                  L5: Features
                                                                                                              Ax i3. i(a, b) = i(b, a). commutativity
                                                            −1
L6 Features                                      c(a) = g        (g (1) − g (a)).             L6 Features
                                                                                                              Ax i4. i(a, i(b, d)) = i(i(a, b), d). associativity
L7: Distances                                                                                 L7: Distances

L8: Set                                                                                       L8: Set
operations        Theorem                                                                     operations
                                                                                                                    Axioms i1 - i4 are called axiomatic skeleton for fuzzy
L9: Fuzzy
numbers           (Second Characterization Theorem of Fuzzy Complements.)
                                                                                              L9: Fuzzy
                                                                                              numbers
                                                                                                                intersections.
L10: Fuzzy        Let c be a function from [0, 1] to [0, 1]. Then c is a                      L10: Fuzzy            If the sets are crisp, i becomes the classical (crisp)
logic                                                                                         logic
                  (involutive) fuzzy complement iff there exists an decreasing                                   intersection.
L11: Fuzzy                                                                                    L11: Fuzzy
control           generator f such that, for all a ∈ [0, 1]                                   control

L12: Fuzzy                                                                                    L12: Fuzzy
segmentation
                                                 c(a) = f −1 (f (0) − f (a)).                 segmentation

L13: Defuzzi-                                                                                 L13: Defuzzi-
                  Joakim Lindblad, 2010-03-16   (71/146)                                                        Joakim Lindblad, 2010-03-16   (72/146)
fication                                                                                       fication
 Fuzzy Sets                                                                                        Fuzzy Sets
 and Fuzzy                                                                                         and Fuzzy
 Techniques
                                                                Fuzzy intersections                Techniques
                                                                                                                                                                Fuzzy intersections
    Joakim                                                                                            Joakim
   Lindblad                                                Additional (optional) requirements        Lindblad                                            Examples of t-norms frequently used

Outline                                                                                           Outline

L1: Intro                                                                                         L1: Intro
                                                                                                                                                                       • Drastic intersection
L1–3: Basics      For all a, b, d ∈ [0, 1],                                                       L1–3: Basics
                                                                                                                                                                                  8
                                                                                                                                                                                      < a   if b = 1
L4: Constr.                                                                                       L4: Constr.                                                             i(a, b) =     b   if a = 1
and             Ax i5. i is a continuous function. continuity                                     and
                                                                                                                                                                                        0   otherwise
                                                                                                                                                                                      :
uncertainty                                                                                       uncertainty
                Ax i6. i(a, a) ≤ a. subidempotency
L5: Features                                                                                      L5: Features                                                         • Bounded difference
                Ax i7. a1 < a2 and b1 < b2 implies i(a1 , b1 ) < i(a2 , b2 ).                                                                                             i(a, b) = max[0, a + b − 1]
L6 Features                                                                                       L6 Features
                       strict monotonicity
L7: Distances                                                                                     L7: Distances                                                        • Algebraic product
L8: Set                                                                                           L8: Set                                                                 i(a, b) = ab
operations                                                                                        operations
                  Note:                                                                                                                                                • Standard intersection
L9: Fuzzy                                                                                         L9: Fuzzy                                                               i(a, b) = min[a, b]
numbers              The standard fuzzy intersection, i(a, b) = min[a, b], is the only            numbers

L10: Fuzzy        idempotent t-norm.                                                              L10: Fuzzy
logic                                                                                             logic

L11: Fuzzy                                                                                        L11: Fuzzy
control                                                                                           control              • imin (a, b) ≤ max(0, a + b − 1) ≤ ab ≤ min(a, b).
L12: Fuzzy                                                                                        L12: Fuzzy
segmentation                                                                                      segmentation         • For all a, b ∈ [0, 1],              imin (a, b) ≤ i(a, b) ≤ min[a, b].
L13: Defuzzi-                                                                                     L13: Defuzzi-
                  Joakim Lindblad, 2010-03-16   (73/146)                                                            Joakim Lindblad, 2010-03-16   (74/146)
fication                                                                                           fication




 Fuzzy Sets                                                                                        Fuzzy Sets
 and Fuzzy                                                                                         and Fuzzy
 Techniques
                                                                Fuzzy intersections                Techniques
                                                                                                                                                                          Fuzzy unions
    Joakim                                                                                            Joakim
   Lindblad                                                         How to generate t-norms          Lindblad                                                         Axiomatic requirements

Outline                                                                                           Outline

L1: Intro                                                                                         L1: Intro

L1–3: Basics                                                                                      L1–3: Basics      For all a, b, d ∈ [0, 1],
L4: Constr.                                                                                       L4: Constr.
                                                                                                                  Ax u1. u(a, 0) = a. boundary condition
and
uncertainty
                  Theorem                                                                         and
                                                                                                  uncertainty
                                                                                                                  Ax u2. b ≤ d implies u(a, b) ≤ u(a, d). monotonicity
L5: Features      (Characterization Theorem of t-norms) Let i be a binary                         L5: Features
                                                                                                                  Ax u3. u(a, b) = u(b, a). commutativity
L6 Features       operation on the unit interval. Then, i is an Archimedean                       L6 Features
                                                                                                                  Ax u4. u(a, u(b, d)) = u(u(a, b), d). associativity
L7: Distances     t-norm iff there exists a decreasing generator f such that                       L7: Distances

L8: Set                                                                                           L8: Set
operations
                            i(a, b) = f         (−1)
                                                       (f (a) + f (b)),      for a, b ∈ [0, 1].
                                                                                                  operations          Axioms u1 - u4 are called axiomatic skeleton for fuzzy unions.
L9: Fuzzy                                                                                         L9: Fuzzy           They differ from the axiomatic skeleton of fuzzy intersections only
numbers                                                                                           numbers
                                                                                                                    in boundary condition.
L10: Fuzzy                                                                                        L10: Fuzzy
logic                                                                                             logic               For crisp sets, u behaves like a classical (crisp) union.
L11: Fuzzy                                                                                        L11: Fuzzy
control                                                                                           control

L12: Fuzzy                                                                                        L12: Fuzzy
segmentation                                                                                      segmentation

L13: Defuzzi-                                                                                     L13: Defuzzi-
                  Joakim Lindblad, 2010-03-16   (75/146)                                                            Joakim Lindblad, 2010-03-16   (76/146)
fication                                                                                           fication




 Fuzzy Sets                                                                                        Fuzzy Sets
 and Fuzzy                                                                                         and Fuzzy
 Techniques
                                                                          Fuzzy unions             Techniques
                                                                                                                                                                          Fuzzy unions
    Joakim                                                                                            Joakim
   Lindblad                                                Additional (optional) requirements        Lindblad                                         Examples of t-conorms frequently used

Outline                                                                                           Outline

L1: Intro                                                                                         L1: Intro
                                                                                                                                                                       • Drastic union
                  For all a, b, d ∈ [0, 1],
L1–3: Basics                                                                                      L1–3: Basics
                                                                                                                                                                                  8
                                                                                                                                                                                      < a    if b = 0
L4: Constr.
                Ax u5. u is a continuous function. continuity
                                                                                                  L4: Constr.                                                             u(a, b) =     b    if a = 0
and                                                                                               and
                                                                                                                                                                                        1    otherwise
                                                                                                                                                                                      :
uncertainty                                                                                       uncertainty
                Ax u6. u(a, a) ≥ a. superidempotency
L5: Features                                                                                      L5: Features                                                         • Bounded sum
                Ax u7. a1 < a2 and b1 < b2 implies u(a1 , b1 ) < u(a2 , b2 ).                                                                                             u(a, b) = min[1, a + b]
L6 Features                                                                                       L6 Features
                       strict monotonicity
L7: Distances                                                                                     L7: Distances                                                        • Algebraic sum
L8: Set                                                                                           L8: Set
                                                                                                                                                                          u(a, b) = a + b − ab
                  Note:
operations                                                                                        operations
                                                                                                                                                                       • Standard intersection
L9: Fuzzy            Requirements u5 - u7 are analogous to Axioms i5 - i7.                        L9: Fuzzy                                                               u(a, b) = max[a, b]
numbers                                                                                           numbers
                     The standard fuzzy union, u(a, b) = max[a, b], is the only idempotent
L10: Fuzzy                                                                                        L10: Fuzzy
logic             t-conorm.                                                                       logic

L11: Fuzzy
control
                                                                                                  L11: Fuzzy
                                                                                                  control
                                                                                                                       • max[a, b] ≤ a + b − ab ≤ min(1, a + b) ≤ umax (a, b).
L12: Fuzzy                                                                                        L12: Fuzzy           • For all a, b ∈ [0, 1],              max[a, b] ≤ u(a, b) ≤ umax (a, b).
segmentation                                                                                      segmentation

L13: Defuzzi-                                                                                     L13: Defuzzi-
                  Joakim Lindblad, 2010-03-16   (77/146)                                                            Joakim Lindblad, 2010-03-16   (78/146)
fication                                                                                           fication




 Fuzzy Sets                                                                                        Fuzzy Sets
 and Fuzzy                                                                                         and Fuzzy
 Techniques
                                        Combinations of set operations                             Techniques
                                                                                                                                           Dual triples - Six theorems (1)
    Joakim                                                                                            Joakim
   Lindblad                            De Morgan laws and duality of fuzzy operations                Lindblad
                                                                                                                    Theorem
Outline                                                                                           Outline           The triples min, max, c            and   imin , umax , c are dual with respect to any
L1: Intro                                                                                         L1: Intro         fuzzy complement c.
                  De Morgan laws in classical set theory:
L1–3: Basics                                                                                      L1–3: Basics

L4: Constr.
                                      ¯ ¯                                         ¯ ¯             L4: Constr.       Theorem
and                              A∩B =A∪B                       and       A ∪ B = A ∩ B.          and
uncertainty                                                                                       uncertainty       Given a t-norm i and an involutive fuzzy complement c, the binary
L5: Features                                                                                      L5: Features      operation u on [0, 1], defined for all a, b ∈ [0, 1] by
                  The union and intersection operation are dual with respect to
L6 Features                                                                                       L6 Features
                  the complement.                                                                                                                     u(a, b) = c(i(c(a), c(b)))
L7: Distances                                                                                     L7: Distances

L8: Set                                                                                           L8: Set            is a t-conorm such that i, u, c is a dual triple.
operations        De Morgan laws for fuzzy sets:                                                  operations

L9: Fuzzy                                                                                         L9: Fuzzy
numbers
                   c(i(A, B)) = u(c(A), c(B))                   and c(u(A, B)) = i(c(A), c(B))
                                                                                                  numbers           Theorem
L10: Fuzzy                                                                                        L10: Fuzzy        Given a t-conorm u and an involutive fuzzy complement c, the binary
logic                                                                                             logic
                                                                                                                    operation i on [0, 1], defined for all a, b ∈ [0, 1] by
L11: Fuzzy        for a t-norm i, a t-conorm u, and fuzzy complement c.                           L11: Fuzzy
control                                                                                           control
                                                                                                                                                      i(a, b) = c(u(c(a), c(b)))
L12: Fuzzy        Notation: i, u, c denotes a dual triple.                                        L12: Fuzzy
segmentation                                                                                      segmentation

L13: Defuzzi-                                                                                     L13: Defuzzi-     is a t-norm such that i, u, c is a dual triple.
                  Joakim Lindblad, 2010-03-16   (79/146)                                                            Joakim Lindblad, 2010-03-16   (80/146)
fication                                                                                           fication
 Fuzzy Sets                                                                                                          Fuzzy Sets
 and Fuzzy                                                                                                           and Fuzzy
 Techniques
                                         Dual triples - Six theorems (2)                                             Techniques
                                                                                                                                                                             Aggregation operations
    Joakim                                                                                                              Joakim
   Lindblad                                                                                                            Lindblad                                                                          Definition

Outline                                                                                                             Outline

L1: Intro         Theorem                                                                                           L1: Intro

L1–3: Basics      Given an involutive fuzzy complement c and an increasing generator g of                           L1–3: Basics    Aggregations on fuzzy sets are operations by which several
L4: Constr.       c, the t-norm and the t-conorm generated by g are dual with respect to c.                         L4: Constr.     fuzzy sets are combined in a desirable way to produce a single
and                                                                                                                 and
uncertainty                                                                                                         uncertainty     fuzzy set.
L5: Features      Theorem                                                                                           L5: Features

L6 Features
                  Let i, u, c be a dual triple generated by an increasing generator g of the                        L6 Features
                                                                                                                                    Definition
L7: Distances     involutive fuzzy complement c. Then the fuzzy operations i, u, c satisfy the                      L7: Distances
                                                                                                                                    Aggregation operation on n fuzzy sets (n ≥ 2) is a function
L8: Set           law of excluded middle, and the law of contradiction.                                             L8: Set
operations                                                                                                          operations                  h : [0, 1]n → [0, 1].
L9: Fuzzy                                                                                                           L9: Fuzzy
numbers           Theorem                                                                                           numbers
                                                                                                                                    Applied to fuzzy sets A1 , A2 , . . . , An , function h produces an aggregate
L10: Fuzzy                                                                                                          L10: Fuzzy
logic
                  Let i, u, c be a dual triple that satisfies the law of excluded middle and                         logic           fuzzy set A, by operating on membership grades to these sets for each
                  the law of contradiction. Then i, u, c does not satisfy the distributive                                          x ∈ X:
L11: Fuzzy                                                                                                          L11: Fuzzy
control           laws.                                                                                             control                             A(x) = h(A1 (x), A2 (x), . . . , An (x)).
L12: Fuzzy                                                                                                          L12: Fuzzy
segmentation                                                                                                        segmentation

L13: Defuzzi-                                                                                                       L13: Defuzzi-
                  Joakim Lindblad, 2010-03-16   (81/146)                                                                            Joakim Lindblad, 2010-03-16   (82/146)
fication                                                                                                             fication




 Fuzzy Sets                                                                                                          Fuzzy Sets
 and Fuzzy                                                                                                           and Fuzzy
 Techniques
                                                           Axiomatic requirements                                    Techniques
                                                                                                                                                                               Averaging operations
    Joakim                                                                                                              Joakim
   Lindblad                                                                                                            Lindblad

Outline                                                                                                             Outline

L1: Intro                                                                                                           L1: Intro
                Ax h1 h(0, 0, . . . , 0) = 0 and h(1, 1, . . . , 1) = 1.                    boundary conditions                        • If an aggregation operator h is monotonic and idempotent (Ax
L1–3: Basics                                                                                                        L1–3: Basics

L4: Constr.
                Ax h2 For any (a1 , a2 , . . . , an ) and (b1 , b2 , . . . , bn ), such that ai , bi ∈ [0, 1]       L4: Constr.
                                                                                                                                         h2 and Ax h5), then for all (a1 , a2 , . . . , an ) ∈ [0, 1]n
and                   and ai ≤ bi for i = 1, . . . , n,                                                             and
uncertainty                                                                                                         uncertainty
                                                h(a1 , a2 , . . . , an ) ≤ h(b1 , b2 , . . . , bn ).                                          min(a1 , a2 , . . . , an ) ≤ h(a1 , a2 , . . . , an ) ≤ max(a1 , a2 , . . . , an ).
L5: Features                                                                                                        L5: Features

L6 Features
                         h is monotonic increasing in all its arguments.                                            L6 Features
                                                                                                                                       • All aggregation operators between the standard fuzzy
L7: Distances
                Ax h3 h is continuous.                                                                              L7: Distances
                                                                                                                                         intersection and the standard fuzzy union are idempotent.
L8: Set         Ax h4 h is a symmetric function in all its arguments; for any permutation p                         L8: Set
operations            on {1, 2, . . . , n}                                                                          operations         • The only idempotent aggregation operators are those between
L9: Fuzzy
                                                h(a1 , a2 , . . . , an ) = h(ap(1) , ap(2) , . . . , ap(n) ).       L9: Fuzzy            standard fuzzy intersection and standard fuzzy union.
numbers                                                                                                             numbers

L10: Fuzzy      Ax h5 h is an idempotent function; for all a ∈ [0, 1]                                               L10: Fuzzy
logic
                                                h(a, a, . . . , a) = a.                                             logic           Idempotent aggregation operators are called averaging
L11: Fuzzy                                                                                                          L11: Fuzzy      operations.
control                                                                                                             control

L12: Fuzzy                                                                                                          L12: Fuzzy
segmentation                                                                                                        segmentation

L13: Defuzzi-                                                                                                       L13: Defuzzi-
                  Joakim Lindblad, 2010-03-16   (83/146)                                                                            Joakim Lindblad, 2010-03-16   (84/146)
fication                                                                                                             fication




 Fuzzy Sets                                                                                                          Fuzzy Sets
 and Fuzzy                                                                                                           and Fuzzy
 Techniques
                                                                 Averaging operations                                Techniques
                                                                                                                                                        Do we need more than standard
    Joakim                                                                                                              Joakim
   Lindblad                                                                                                            Lindblad                                           operations?
Outline           Generalized means:                                                                                Outline

L1: Intro                                                                                                  1        L1: Intro

L1–3: Basics
                                                                         α    α            α
                                                                        a1 + a2 + · · · + an               α
                                                                                                                    L1–3: Basics
                                                                                                                                           Intersection: No positive compensation (trade-off)
                               hα (a1 , a2 , . . . , an ) =                                                     ,                          between the memberships of the fuzzy sets observed.
L4: Constr.                                                                     n                                   L4: Constr.
and                                                                                                                 and
uncertainty                                                                                                         uncertainty            Union: Full compensation of lower degrees of membership
L5: Features
                  for α ∈ R, and α = 0, and for α < 0 ai = 0.                                                       L5: Features           by the maximal membership.
L6 Features           • Geometric mean: For α → 0,                                                                  L6 Features
                                                                                                    1
L7: Distances                      lim hα (a1 , a2 , . . . , an ) = (a1 · a2 · · · · · an ) n ;
                                  α→0
                                                                                                                    L7: Distances   In reality of decision making, rarely either happens.
L8: Set                                                                                                             L8: Set
operations            • Harmonic mean: For α = −1,                                                                  operations

L9: Fuzzy                         h−1 (a1 , a2 , . . . , an ) =
                                                                                 n
                                                                                                ;                   L9: Fuzzy
                                                                                                                                        (non-verbal) “merging connectives” → (language) connectives
                                                                   1        1               1
numbers
                                                                   a1
                                                                        +   a2
                                                                                 + ··· +   an
                                                                                                                    numbers
                                                                                                                                                               {’and’, ’or’,...,}.
L10: Fuzzy
logic
                      • Arithmetic mean: For α = 1,                                                                 L10: Fuzzy
                                                                                                                    logic
                                                                 1
L11: Fuzzy                        h1 (a1 , a2 , . . . , an ) =     (a1 + a2 + . . . an ).                           L11: Fuzzy      Aggregation operations called compensatory and are needed
control                                                          n                                                  control

L12: Fuzzy                                                                                                          L12: Fuzzy
                                                                                                                                    to model fuzzy sets representing to, e.g., managerial decisions.
segmentation      Functions hα satisfy axioms Ax h1 - Ax h5.                                                        segmentation

L13: Defuzzi-                                                                                                       L13: Defuzzi-
                  Joakim Lindblad, 2010-03-16   (85/146)                                                                            Joakim Lindblad, 2010-03-16   (86/146)
fication                                                                                                             fication




 Fuzzy Sets                                                                                                          Fuzzy Sets
 and Fuzzy                                                                                                           and Fuzzy
 Techniques
                                                             An Application: Fuzzy                                   Techniques
                                                                                                                                                             L9: Fuzzy numbers and fuzzy
    Joakim                                                                                                              Joakim
   Lindblad                                                          morphologies                                      Lindblad                                              arithmetics
Outline                                                                     Morphological operations                Outline

L1: Intro                                                                                                           L1: Intro

L1–3: Basics
                      • Mathematical morphology is completely based on set                                          L1–3: Basics

L4: Constr.                                                                                                         L4: Constr.
and                      theory. Fuzzification started in 1980s.                                                     and
uncertainty                                                                                                         uncertainty
                      • Basic morphological operations are dilation and erosion.
L5: Features                                                                                                        L5: Features

L6 Features
                         Many others can be derived from them.                                                      L6 Features

L7: Distances         • Dilation and erosion are, in crisp case, dual operations                                    L7: Distances

L8: Set                  with respect to the complementation: D(A) = c(E (cA)).                                     L8: Set
operations                                                                                                          operations

L9: Fuzzy
                      • In crisp case, dilation and erosion fulfil a certain number of                               L9: Fuzzy
numbers                                                                                                             numbers
                         properties.
L10: Fuzzy                                                                                                          L10: Fuzzy
logic                                                                                                               logic
                  Main construction principles:
L11: Fuzzy                                                                                                          L11: Fuzzy
control                 α-cut decomposition;                                                                        control

L12: Fuzzy
segmentation
                        fuzzification of set operations.                                                             L12: Fuzzy
                                                                                                                    segmentation

L13: Defuzzi-                                                                                                       L13: Defuzzi-
                  Joakim Lindblad, 2010-03-16   (87/146)                                                                            Joakim Lindblad, 2010-03-16   (88/146)
fication                                                                                                             fication
 Fuzzy Sets                                                                                       Fuzzy Sets
 and Fuzzy                                                                                        and Fuzzy
 Techniques
                                                                 Interval numbers                 Techniques
                                                                                                                                                                 Interval numbers
    Joakim                                                                                           Joakim
   Lindblad                                                                                         Lindblad
                                                                                                                 For closed intervals A = [a1 , a2 ] and B = [b1 , b2 ], the four
Outline                                                                                          Outline
                                                                                                                 arithmetic operations are defined as follows (equivalent with
L1: Intro       An interval number, representing an uncertain real number                        L1: Intro

L1–3: Basics                                                                                     L1–3: Basics
                                                                                                                 definition on previous slide)
L4: Constr.                          A = [a1 , a2 ] = {x | a1 ≤ x ≤ a2 , x ∈ R}                  L4: Constr.
and                                                                                              and
uncertainty                                                                                      uncertainty
                                                                                                                            A+B                    = [a1 + b1 , a2 + b2 ]
L5: Features    For intervals A and B, and operator ∗ ∈ {+, −, ·, /}                             L5: Features

L6 Features     we define                                                                         L6 Features
                                                                                                                   A−B =A+               B−        = [a1 − b2 , a2 − b1 ]
L7: Distances                    A ∗ B = {a ∗ b | a ∈ A, b ∈ B}                                  L7: Distances

L8: Set                                                                                          L8: Set                     A·B                   = [min(a1 b1 , a1 b2 , a2 b1 , a2 b2 ),
operations
                Division, A/B, is not defined when 0 ∈ B.                                         operations

L9: Fuzzy                                                                                        L9: Fuzzy
                                                                                                                                                      max(a1 b1 , a1 b2 , a2 b1 , a2 b2 )]
numbers                                                                                          numbers
                The result of an arithmetic operation on closed intervals is                                      and, if 0 ∈ [b1 , b2 ]
                                                                                                                            /
L10: Fuzzy                                                                                       L10: Fuzzy
                                                                                                                                                          1 1
logic           again a closed interval.                                                         logic             A/B = A · B −1        = [a1 , a2 ] · [ b2 , b1 ]
                                                                                                                                                   a a a a                 a a a a
L11: Fuzzy
control
                                                                                                 L11: Fuzzy
                                                                                                 control
                                                                                                                                         = [min( b1 , b1 , b2 , b2 ), max( b1 , b1 , b2 , b2 )].
                                                                                                                                                     1    2    1    2       1    2    1    2
L12: Fuzzy                                                                                       L12: Fuzzy
segmentation                                                                                     segmentation

L13: Defuzzi-                                                                                    L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (89/146)                                                           Joakim Lindblad, 2010-03-16   (90/146)
fication                                                                                          fication




 Fuzzy Sets                                                                                       Fuzzy Sets
 and Fuzzy                                                                                        and Fuzzy
 Techniques
                                 Fuzzy numbers and fuzzy intervals                                Techniques
                                                                                                                                 Fuzzy numbers and fuzzy intervals
    Joakim                                                                                           Joakim
   Lindblad                                                                                         Lindblad

Outline                                                                                          Outline

L1: Intro                                                                                        L1: Intro
                                                                                                                 Theorem (4.1)
L1–3: Basics    A fuzzy number is a fuzzy set on R                                               L1–3: Basics    Let A ∈ F(R). Then, A is a fuzzy number iff there exists a
L4: Constr.                                                                                      L4: Constr.
                                                                                                                 closed interval [a, b] = ∅ such that
and
uncertainty
                                                         A : R → [0, 1]                          and
                                                                                                 uncertainty
                                                                                                                                         
L5: Features                                                                                     L5: Features                             1       for x ∈ [a, b]
L6 Features     such that                                                                        L6 Features                    A(x) =      l(x) for x ∈ (−∞, a)
L7: Distances                                                                                    L7: Distances                              r (x) for x ∈ (b, ∞)
                                                                                                                                         
                 (i) A is normal (height(A) = 1)
L8: Set                                                                                          L8: Set
operations
                (ii) αA is a closed interval for all α ∈ (0, 1]                                  operations
                                                                                                                 where l : (−∞, a) → [0, 1] is monotonic non-decreasing,
L9: Fuzzy                                                                                        L9: Fuzzy
                (iii) The support of A, Supp(A) =                     0+A,   is bounded
numbers                                                                                          numbers         continuous from the right, and l(x) = 0 for x < ω1
L10: Fuzzy
logic
                                                                                                 L10: Fuzzy
                                                                                                 logic
                                                                                                                 and r : (b, ∞) → [0, 1] is monotonic non-increasing, continuous
L11: Fuzzy                                                                                       L11: Fuzzy
                                                                                                                 from the left, and r (x) = 0 for x > ω2 .
control                                                                                          control

L12: Fuzzy                                                                                       L12: Fuzzy
segmentation                                                                                     segmentation

L13: Defuzzi-                                                                                    L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (91/146)                                                           Joakim Lindblad, 2010-03-16   (92/146)
fication                                                                                          fication




 Fuzzy Sets                                                                                       Fuzzy Sets
 and Fuzzy                                                                                        and Fuzzy
 Techniques
                                         Arithmetics on fuzzy numbers                             Techniques
                                                                                                                                          Arithmetics on fuzzy numbers
    Joakim                                                                                           Joakim
   Lindblad                                                                                         Lindblad

Outline
                Moving from interval numbers, we can define arithmetics on                        Outline

L1: Intro
                fuzzy numbers based on two principles:                                           L1: Intro
                                                                                                                 Theorem (4.2)
L1–3: Basics
                   1   Cutworthiness (thanks to inclusion monotonicity of                        L1–3: Basics

L4: Constr.
                       intervals)                                                                L4: Constr.     Let ∗ ∈ {+, −, ·, /}, and let A, B denote continuous fuzzy
and                                                                                              and
uncertainty                              α
                                          (A ∗ B) =αA ∗αB                                        uncertainty     numbers. Then, the fuzzy set A ∗ B defined by the extension
L5: Features                                                                                     L5: Features    principle (prev. slide) is a continuous fuzzy number.
L6 Features             in combination with                                                      L6 Features

L7: Distances                          A∗B =                              α (A   ∗ B)            L7: Distances   Lemma
L8: Set
operations
                                                               α∈(0,1]                           L8: Set
                                                                                                 operations
                                                                                                                 (A ∗ B)(z) = sup min [A(x), B(x)] ⇒ α(A ∗ B) =αA ∗αB
                                                                                                                                       z=x∗y
L9: Fuzzy                                                                                        L9: Fuzzy
numbers
                   2   or the extension principle
                                                                                                 numbers         So the two definitions are equivalent for continuous fuzzy
L10: Fuzzy
logic
                                                                                                 L10: Fuzzy
                                                                                                 logic
                                                                                                                 numbers. (The proof is built on continuity.)
L11: Fuzzy                               (A ∗ B)(z) = sup min [A(x), B(x)]                       L11: Fuzzy
control                                                       z=x∗y                              control

L12: Fuzzy                                                                                       L12: Fuzzy
segmentation                                                                                     segmentation

L13: Defuzzi-                                                                                    L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (93/146)                                                           Joakim Lindblad, 2010-03-16   (94/146)
fication                                                                                          fication




 Fuzzy Sets                                                                                       Fuzzy Sets
 and Fuzzy                                                                                        and Fuzzy
 Techniques
                                                   MIN and MAX operators                          Techniques
                                                                                                                                                    MIN and MAX operators
    Joakim                                                                                           Joakim
   Lindblad                                                                                         Lindblad

Outline                          MIN(A, B)(z) =               sup        min [A(x), B(x)] ,      Outline

L1: Intro                                                  z=min(x,y )                           L1: Intro

L1–3: Basics                  MAX(A, B)(z) =                  sup         min [A(x), B(x)]       L1–3: Basics

L4: Constr.                                                z=max(x,y )                           L4: Constr.
and                                                                                              and
uncertainty                                                                                      uncertainty
                Again, for continuous fuzzy numbers, this is equivalent with a
L5: Features                                                                                     L5: Features
                definition based on cutworthiness.
L6 Features                                                                                      L6 Features
                                 α                               α    α
L7: Distances                        (MIN(A, B)) = MIN( A, B),                                   L7: Distances
                             α
L8: Set
operations
                                 (MAX(A, B)) = MAX(αA,α B),                       ∀α ∈ (0, 1].   L8: Set
                                                                                                 operations

L9: Fuzzy                                                                                        L9: Fuzzy
numbers                                                                                          numbers

L10: Fuzzy
logic
                Where, for intervals [a1 , a2 ], [b1 , b2 ]                                      L10: Fuzzy
                                                                                                 logic

L11: Fuzzy
                      MIN([a1 , a2 ], [b1 , b2 ]) = [min(a1 , b1 ), min(a2 , b2 )],              L11: Fuzzy
control                                                                                          control
                         MAX([a1 , a2 ], [b1 , b2 ]) = [max(a1 , b1 ), max(a2 , b2 )].
L12: Fuzzy                                                                                       L12: Fuzzy         Figure: Comparison of the operators MIN, min, MAX, and max.
segmentation                                                                                     segmentation

L13: Defuzzi-                                                                                    L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (95/146)                                                           Joakim Lindblad, 2010-03-16   (96/146)
fication                                                                                          fication
 Fuzzy Sets                                                                                                 Fuzzy Sets
 and Fuzzy                                                                                                  and Fuzzy
 Techniques
                                         Arithmetics on fuzzy numbers                                       Techniques
                                                                                                                                                                            Linguistic variables
    Joakim                                                                                                     Joakim
   Lindblad                                                                                                   Lindblad     When fuzzy numbers are connected to linguistic concepts, such
                                                                                                                           as very small, small, medium, and interpreted in a particular
Outline
                We can define a partial ordering                          on the set of fuzzy numbers       Outline
                                                                                                                           context, the resulting constructs are usually called linguistic
L1: Intro                                                                                                  L1: Intro
                R by                                                                                                       variables.
L1–3: Basics                                                                                               L1–3: Basics

L4: Constr.
and
                               A       B ⇔ MIN(A, B) = A or, alternatively                                 L4: Constr.
                                                                                                           and
uncertainty                                                                                                uncertainty
                               A       B ⇔ MAX(A, B) = B
L5: Features                                                                                               L5: Features

L6 Features                                                                                                L6 Features

L7: Distances
                Not all fuzzy numbers are comparable (only partial order).                                 L7: Distances

L8: Set
                However, values of linguistic variables are often defined by                                L8: Set
operations
                fuzzy numbers that are comparable.                                                         operations

L9: Fuzzy                                                                                                  L9: Fuzzy
numbers         For example:                                                                               numbers

L10: Fuzzy                                                                                                 L10: Fuzzy
logic                                                                                                      logic
                         very small             small      medium              large   very large
L11: Fuzzy                                                                                                 L11: Fuzzy
control                                                                                                    control

L12: Fuzzy                                                                                                 L12: Fuzzy                         Figure: An example of a linguistic variable.
segmentation                                                                                               segmentation

L13: Defuzzi-                                                                                              L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (97/146)                                                                     Joakim Lindblad, 2010-03-16   (98/146)
fication                                                                                                    fication




 Fuzzy Sets                                                                                                 Fuzzy Sets
 and Fuzzy                                                                                                  and Fuzzy
 Techniques
                                                                 Interval equations                         Techniques
                                                                                                                                                                                Fuzzy equations
    Joakim                                                                                                     Joakim
   Lindblad                                                               Equation A + X = B                  Lindblad
                                                                                                                           The solution to a fuzzy equation can be obtained by solving a
Outline                                                                                                    Outline
                                                                                                                           set of interval equations, one for each nonzero α in the level set
L1: Intro                                                                                                  L1: Intro

L1–3: Basics                                                                                               L1–3: Basics
                                                                                                                           Λ(A) ∪ Λ(B).
L4: Constr.
                A+X =B                                                                                     L4: Constr.
and                                                                                                        and             The equation A + X = B has a solution iff
uncertainty     Let X = [x1 , x2 ].                                                                        uncertainty

L5: Features
                Then [a1 + x1 , a2 + x2 ] = [b1 , b2 ] follows immediately.                                L5: Features     (i) αb1 −α a1 ≤α b2 −α a2 for every α ∈ (0, 1], and
L6 Features
                Clearly: x1 = b1 − a1 and x2 = b2 − a2 .
                                                                                                           L6 Features     (ii) α ≤ β implies
L7: Distances                                                                                              L7: Distances        αb −α a ≤β b −β a ≤β b −β a ≤α b −α a .
                                                                                                                                  1    1      1  1    2    2    2    2
L8: Set         Since X must be an interval, it is required that x1 ≤ x2 .                                 L8: Set
operations
                That is, the equation has a solution iff b1 − a1 ≤ b2 − a2 .
                                                                                                           operations
                                                                                                                           If a solution αX exists for every α ∈ (0, 1] (property (i)),
L9: Fuzzy                                                                                                  L9: Fuzzy
numbers                                                                                                    numbers         and property (ii) is satisfied, then the solution X is given by
                Then X = [b1 − a1 , b2 − a2 ] is the solution.
L10: Fuzzy                                                                                                 L10: Fuzzy
logic                                                                                                      logic
                                                                                                                                                                    X =              αX
L11: Fuzzy                                                                                                 L11: Fuzzy
control                                                                                                    control                                                         α∈(0,1]
L12: Fuzzy                                                                                                 L12: Fuzzy
segmentation                                                                                               segmentation

L13: Defuzzi-                                                                                              L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (99/146)                                                                     Joakim Lindblad, 2010-03-16   (100/146)
fication                                                                                                    fication




 Fuzzy Sets                                                                                                 Fuzzy Sets
 and Fuzzy                                                                                                  and Fuzzy
 Techniques
                                                                     Fuzzy equations                        Techniques
                                                                                                                                            L10: Fuzzy logic and approximate
    Joakim                                                                                                     Joakim
   Lindblad                                                                Equation A · X = B                 Lindblad                                             reasoning
Outline                                                                                                    Outline

L1: Intro       Similarly as A + X = B                                                                     L1: Intro

L1–3: Basics                                                                                               L1–3: Basics
                The equation A · X = B has a solution iff
L4: Constr.                                                                                                L4: Constr.
and                    αb /αa                                                                              and
uncertainty      (i)     1   1      ≤αb2 /αa2 for every α ∈ (0, 1], and                                    uncertainty

L5: Features    (ii) α ≤ β         implies αb1 /αa1 ≤βb1 /βa1 ≤βb2 /βa2 ≤αb2 /αa2 .                        L5: Features

L6 Features                                                                                                L6 Features

L7: Distances   If the solution exists, it has the form                                                    L7: Distances

L8: Set                                                                                                    L8: Set
operations                                               X =              αX                               operations

L9: Fuzzy                                                                                                  L9: Fuzzy
numbers
                                                               α∈(0,1]                                     numbers

L10: Fuzzy                                                                                                 L10: Fuzzy
logic           where αX = [αb1 /αa1 ,α b2 /αa2 ].                                                         logic

L11: Fuzzy                                                                                                 L11: Fuzzy
control         Again, X = B/A is not a solution of the equation.                                          control

L12: Fuzzy                                                                                                 L12: Fuzzy
segmentation                                                                                               segmentation

L13: Defuzzi-                                                                                              L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (101/146)                                                                    Joakim Lindblad, 2010-03-16   (102/146)
fication                                                                                                    fication




 Fuzzy Sets                                                                                                 Fuzzy Sets
 and Fuzzy                                                                                                  and Fuzzy
 Techniques
                                     Classical logic: A brief overview                                      Techniques
                                                                                                                                                Classical logic: A brief overview
    Joakim                                                                                                     Joakim
   Lindblad                                                                     Logic functions               Lindblad                                                   Logic functions of two variables

Outline                                                                                                    Outline
                                                                                                                                          v2       1      1 0        0     Function             Adopted
                                                                                                                                          v1       1      0 1        0     name                 symbol
L1: Intro                                                                                                  L1: Intro
                                                                                                                                          ω1 0            0 0        0     Zero function            0
L1–3: Basics                                                                                               L1–3: Basics
                                                                                                                                          ω2 0            0 0        1     NOR function          v1 ↓ v2
L4: Constr.                                                                                                L4: Constr.
and                                                                                                        and                            ω3 0            0 1        0     Inhibition           v1 > v2
uncertainty     Logic function assigns a truth value to a combination of truth                             uncertainty
                                                                                                                                          ω4 0            0 1        1     Negation                 ¯
                                                                                                                                                                                                   v2
L5: Features    values of its variables:                                                                   L5: Features
                                                                                                                                          ω5 0            1 0        0     Inhibition           v1 < v2
L6 Features                                                                                                L6 Features
                                                                                                                                          ω6 0            1 0        1     Negation                 ¯
                                                                                                                                                                                                   v1
                                                                n
L7: Distances                           f : {true, false} → {true, false}                                  L7: Distances
                                                                                                                                          ω7 0            1 1        0     Exclusive OR         v1 ⊕ v2
L8: Set                                                                                                    L8: Set
                                                                                                                                          ω8 0            1 1        1     NAND function          v1 |v2
operations       n                                                  2n                                     operations
                2 choices of n arguments → 2                             logic functions of n variables.                                  ω9 1            0 0        0     Conjunction          v1 ∧ v2
L9: Fuzzy                                                                                                  L9: Fuzzy
numbers                                                                                                    numbers                       ω10 1            0 0        1     Equivalence          v1 ⇔ v2
L10: Fuzzy                                                                                                 L10: Fuzzy                    ω11 1            0 1        0     Assertion               v1
logic                                                                                                      logic
                                                                                                                                         ω12 1            0 1        1     Implication          v1 ⇐ v2
L11: Fuzzy                                                                                                 L11: Fuzzy
control                                                                                                    control                       ω13 1            1 0        0     Assertion               v2
L12: Fuzzy                                                                                                 L12: Fuzzy                    ω14 1            1 0        1     Implication          v1 ⇒ v2
segmentation                                                                                               segmentation
                                                                                                                                         ω15 1            1 1        0     Disjunction          v1 ∨ v2
L13: Defuzzi-
fication
                Joakim Lindblad, 2010-03-16   (103/146)
                                                                                                           L13: Defuzzi-
                                                                                                           fication
                                                                                                                                         ω16 1
                                                                                                                           Joakim Lindblad, 2010-03-16    1 1
                                                                                                                                                         (104/146)   1     One function             1
 Fuzzy Sets                                                                                   Fuzzy Sets
 and Fuzzy                                                                                    and Fuzzy
 Techniques
                                       Classical logic: A brief overview                      Techniques
                                                                                                                                  Classical logic: A brief overview
    Joakim                                                                                       Joakim
   Lindblad                                                           Logic primitives          Lindblad                                                             Logic formulae

Outline                                                                                      Outline

L1: Intro                                                                                    L1: Intro

L1–3: Basics                                                                                 L1–3: Basics

L4: Constr.                                                                                  L4: Constr.
and                                                                                          and             Definition
uncertainty     We can express all the logic functions of n variables by using               uncertainty

L5: Features    only a small number of simple logic functions. Such a set is a               L5: Features      1. If v is a logic variable, then v and v are logic formulae;
                                                                                                                                                       ¯
L6 Features     complete set of logic primitives.                                            L6 Features
                                                                                                               2. If v1 and v2 are logic formulae, then v1 ∧ v2 and v1 ∨ v2
L7: Distances                                                                                L7: Distances
                Examples:                                                                                         are also logic formulae;
L8: Set                                                                                      L8: Set
operations                {negation, conjunction, disjunction},                              operations
                                                                                                               3. Logic formulae are only those defined (obtained) by the
L9: Fuzzy                 {negation, implication}.                                           L9: Fuzzy
numbers                                                                                      numbers              two previous rules.
L10: Fuzzy                                                                                   L10: Fuzzy
logic                                                                                        logic

L11: Fuzzy                                                                                   L11: Fuzzy
control                                                                                      control

L12: Fuzzy                                                                                   L12: Fuzzy
segmentation                                                                                 segmentation

L13: Defuzzi-                                                                                L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (105/146)                                                      Joakim Lindblad, 2010-03-16   (106/146)
fication                                                                                      fication




 Fuzzy Sets                                                                                   Fuzzy Sets
 and Fuzzy                                                                                    and Fuzzy
 Techniques
                                       Classical logic: A brief overview                      Techniques
                                                                                                                                                          Fuzzy propositions
    Joakim                                                                                       Joakim
   Lindblad                                                               Inference rules       Lindblad

Outline                                                                                      Outline
                                                                                                             The range of truth values of fuzzy propositions is not only
L1: Intro       Tautology is (any) logic formula that corresponds to a logic                 L1: Intro
                                                                                                             {0, 1}, but [0, 1].
L1–3: Basics    function one.                                                                L1–3: Basics
                                                                                                             The truth of a fuzzy proposition is a matter of degree.
L4: Constr.
and
                Contradiction is (any) logic formula that corresponds to a                   L4: Constr.
                                                                                             and
uncertainty     logic function zero.                                                         uncertainty     Classification of fuzzy propositions:
L5: Features                                                                                 L5: Features
                                                                                                                • Unconditional and unqualified propositions
L6 Features     Inference rules are tautologies used for making deductive                    L6 Features
                                                                                                                    “The temperature is high.”
L7: Distances                                                                                L7: Distances
                inferences.                                                                                     • Unconditional and qualified propositions
L8: Set                                                                                      L8: Set
operations
                Examples:                                                                    operations
                                                                                                                    “The temperature is high is very true.”
L9: Fuzzy                                                                                    L9: Fuzzy
numbers
                   • (a ∧ (a ⇒ b)) ⇒ b                    modus ponens                       numbers            • Conditional and unqualified propositions
L10: Fuzzy
                      ¯                                                                      L10: Fuzzy             “If the temperature is high, then it is hot.”
logic              • (b ∧ (a ⇒ b)) ⇒ ¯
                                     a                    modus tollens                      logic

L11: Fuzzy                                                                                   L11: Fuzzy         • Conditional and qualified propositions
control
                   • (a ⇒ b) ∧ (b ⇒ c)) ⇒ (a ⇒ c)                   hypothetical syllogism   control
                                                                                                                    “If the temperature is high, then it is hot is true.”
L12: Fuzzy                                                                                   L12: Fuzzy
segmentation                                                                                 segmentation

L13: Defuzzi-                                                                                L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (107/146)                                                      Joakim Lindblad, 2010-03-16   (108/146)
fication                                                                                      fication




 Fuzzy Sets                                                                                   Fuzzy Sets
 and Fuzzy                                                                                    and Fuzzy
 Techniques
                                              Linguistic hedges (modifiers)                    Techniques
                                                                                                                                                                      Modifiers
    Joakim                                                                                       Joakim
   Lindblad                                                                                     Lindblad

Outline                                                                                      Outline
                                                                                                             Strong modifier reduces the truth value of a proposition.
L1: Intro                                                                                    L1: Intro
                                                                                                             Weak modifier increases the truth value of a proposition (by
L1–3: Basics       • Linguistic hedges are linguistic terms by which other                   L1–3: Basics
                                                                                                             weakening the proposition).
L4: Constr.
and
                       linguistic terms are modified.                                         L4: Constr.
                                                                                             and
uncertainty                                                                                  uncertainty     One commonly used class of modifiers is
                       “Tina      is   young is true.”
L5: Features                                                                                 L5: Features
                       “Tina      is   very young is true.”                                                                 hα (a) = aα ,              for α ∈ R + and a ∈ [0, 1].
L6 Features                                                                                  L6 Features
                       “Tina      is   young is very true.”
L7: Distances                                                                                L7: Distances
                       “Tina      is   very young is very true.”                                             For α < 1, hα is a weak modifier.
L8: Set
operations
                                                                                             L8: Set
                                                                                             operations
                                                                                                                                               √
                   • Fuzzy predicates and fuzzy truth values can be modified.                                       Example: H : fairly ↔ h(a) = a.
L9: Fuzzy                                                                                    L9: Fuzzy
numbers                Crisp predicates cannot be modified.                                   numbers
                                                                                                             For α > 1, hα is a strong modifier.
L10: Fuzzy         • Examples of hedges: very, fairly, extremely.                            L10: Fuzzy
                                                                                                                   Example: H : very ↔ h(a) = a2 .
logic                                                                                        logic

L11: Fuzzy                                                                                   L11: Fuzzy
control                                                                                      control         h1 is the identity modifier.
L12: Fuzzy                                                                                   L12: Fuzzy
segmentation                                                                                 segmentation

L13: Defuzzi-                                                                                L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (109/146)                                                      Joakim Lindblad, 2010-03-16   (110/146)
fication                                                                                      fication




 Fuzzy Sets                                                                                   Fuzzy Sets
 and Fuzzy                                                                                    and Fuzzy
 Techniques
                                                               Fuzzy quantifiers               Techniques
                                                                                                                                                          Fuzzy propositions
    Joakim                                                                                       Joakim
   Lindblad                                                                                     Lindblad                                   Unconditional and unqualified propositions

Outline                                                                                      Outline
                                                                                                             The canonical form
L1: Intro                                                                                    L1: Intro

L1–3: Basics                                                                                 L1–3: Basics
                                                                                                                                                       p : ν is F
L4: Constr.
and
                   • Absolute quantifiers:                                                    L4: Constr.
                                                                                             and
                                                                                                             ν is a variable on some universal set V
uncertainty                       “about 10”; “much more than 100”, ...                      uncertainty     F is a fuzzy set on V that represents a fuzzy predicate
L5: Features
                   • Relative quantifiers:
                                                                                             L5: Features        (e.g., low, tall, young, expensive...)
L6 Features                                                                                  L6 Features

L7: Distances
                                  “almost all”; “about half”, ...                            L7: Distances
                                                                                                             The degree of truth of p is
L8: Set         Examples:                                                                    L8: Set
operations
                p: “There are about 3 high-fluent students in the group.”
                                                                                             operations                                    T (p) = F (v ),      for v ∈ ν.
L9: Fuzzy                                                                                    L9: Fuzzy
numbers         q: “Almost all students in the group are high-fluent.”                        numbers
                                                                                                             T is a fuzzy set on V . Its membership function is derived form the
L10: Fuzzy                                                                                   L10: Fuzzy
logic                                                                                        logic           membership function of a fuzzy predicate F .
L11: Fuzzy                                                                                   L11: Fuzzy      The role of a function T is to connect fuzzy sets and fuzzy propositions.
control                                                                                      control

L12: Fuzzy                                                                                   L12: Fuzzy
                                                                                                             In case of unconditional and unqualified propositions, the identity function
segmentation                                                                                 segmentation    is used.
L13: Defuzzi-                                                                                L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (111/146)                                                      Joakim Lindblad, 2010-03-16   (112/146)
fication                                                                                      fication
 Fuzzy Sets                                                                                                Fuzzy Sets
 and Fuzzy                                                                                                 and Fuzzy
 Techniques
                                                                   Fuzzy propositions                      Techniques
                                                                                                                                                                          Fuzzy propositions
    Joakim                                                                                                    Joakim
   Lindblad                                      Unconditional and qualified propositions                     Lindblad                                      Conditional and unqualified propositions

Outline                                                                                                   Outline

L1: Intro                                                                                                 L1: Intro

L1–3: Basics
                The canonical form                                                                        L1–3: Basics
                                                                                                                          The canonical form
L4: Constr.                                                                                               L4: Constr.
and                                                                                                       and
uncertainty                 p : ν is F is S                       (truth qualified proposition)            uncertainty                                   p : If X is A, then Y is B,
L5: Features                                                                                              L5: Features

L6 Features     where ν is a variable on some universal set V ,                                           L6 Features     where X , Y are variables on X , Y respectively,
L7: Distances   F is a fuzzy set on V that represents a fuzzy predicate,                                  L7: Distances   and A, B are fuzzy sets on X , Y respectively.
L8: Set
operations
                and S is a fuzzy truth qualifier.                                                          L8: Set
                                                                                                          operations      Alternative form:
L9: Fuzzy       To calculate the degree of truth T (p) of the proposition p, we                           L9: Fuzzy                                                  X,Y      is R
numbers                                                                                                   numbers
                use:
L10: Fuzzy                                                                                                L10: Fuzzy      where R(x, y ) = J (A(x), B(x)) is a fuzzy set on X × Y
logic                                  T (p) = S(F (v ))                                                  logic
                                                                                                                          representing a suitable fuzzy implication.
L11: Fuzzy                                                                                                L11: Fuzzy
control                                                                                                   control

L12: Fuzzy                                                                                                L12: Fuzzy
segmentation                                                                                              segmentation

L13: Defuzzi-                                                                                             L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (113/146)                                                                   Joakim Lindblad, 2010-03-16   (114/146)
fication                                                                                                   fication




 Fuzzy Sets                                                                                                Fuzzy Sets
 and Fuzzy                                                                                                 and Fuzzy
 Techniques
                                                                   Fuzzy propositions                      Techniques
                                                                                                                                                                           Fuzzy implications
    Joakim                                                                                                    Joakim
   Lindblad                                          Conditional and qualified propositions                   Lindblad                                                                         Definition(s)

Outline                                                                                                   Outline

L1: Intro                                                                                                 L1: Intro       A fuzzy implication J of two fuzzy propositions p and q is a
L1–3: Basics                                                                                              L1–3: Basics    function of the form
L4: Constr.                                                                                               L4: Constr.
and                                                                                                       and
uncertainty     The canonical form                                                                        uncertainty                                    J : [0, 1] × [0, 1] → [0, 1],
L5: Features                                                                                              L5: Features

L6 Features
                                         p : If X is A, then Y is B is S                                  L6 Features     which for any truth values a = T (p) and b = T (q) defines the
L7: Distances                                                                                             L7: Distances   truth value J (a, b) of the conditional proposition
                where X , Y are variables on X , Y respectively,
L8: Set                                                                                                   L8: Set                     “if p, then q”.
operations      A, B are fuzzy sets on X , Y respectively,                                                operations
                                                                                                                          Fuzzy implications as extensions of the classical logic implication:
L9: Fuzzy
numbers
                and S is a truth qualifier.                                                                L9: Fuzzy
                                                                                                          numbers
                                                                                                                                               Crisp implication a ⇒ b             Fuzzy implication J (a, b)
L10: Fuzzy                                                                                                L10: Fuzzy                  (S)      ¯∨b
                                                                                                                                               a                                   u(c(a), b)
logic                                                                                                     logic
                                                                                                                                      (R)      max{x ∈ {0, 1} | a ∧ x ≤ b}         sup{x ∈ [0, 1] | i(a, x) ≤ b}
L11: Fuzzy                                                                                                L11: Fuzzy                  (QL)     ¯ ∨ (a ∧ b)
                                                                                                                                               a                                   u(c(a), i(a, b))
control                                                                                                   control                     (QL)      a ¯
                                                                                                                                               (¯ ∧ b) ∨ b                         u(i(c(a), c(b)), b)
L12: Fuzzy                                                                                                L12: Fuzzy
segmentation                                                                                              segmentation

L13: Defuzzi-                                                                                             L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (115/146)                                                                   Joakim Lindblad, 2010-03-16   (116/146)
fication                                                                                                   fication




 Fuzzy Sets                                                                                                Fuzzy Sets
 and Fuzzy                                                                                                 and Fuzzy
 Techniques
                                                                    Fuzzy implications                     Techniques
                                                                                                                                                                     Binary fuzzy relations
    Joakim                                                                                                    Joakim
   Lindblad                                                  How to select fuzzy implication                 Lindblad
                                                                                                                             • A crisp binary relation R on sets X , Y is any (crisp)
Outline                                                                                                   Outline
                                                                                                                                 subset of X × Y .
L1: Intro                                                                                                 L1: Intro

L1–3: Basics                                                                                              L1–3: Basics
                                                                                                                             • xRy
L4: Constr.     Look at Table 11.2 , Table 11.3, and Table 11.4                                           L4: Constr.
                                                                                                                                          ( x ∈ X is in relation R with y ∈ Y ) iff (x, y ) ∈ R
and                                                                                                       and
uncertainty
                (pp. 315-317).                                                                            uncertainty        • A fuzzy binary relation R on sets X , Y is any fuzzy subset
L5: Features    One good choice:                                                                          L5: Features           of X × Y .
L6 Features                                                              1   a≤b                          L6 Features        • Elements x ∈ X and y ∈ Y are in relation R up to some
                                                 Js (a, b) =
L7: Distances                                                            0   a>b                          L7: Distances
                                                                                                                                 extent.
L8: Set                                                                                                   L8: Set
operations      One frequently used implication: Lukasiewicz                                              operations
                                                                                                                          The standard composition of two fuzzy relations, P(X , Y ) and Q(Y , Z ),
L9: Fuzzy                                                                                                 L9: Fuzzy       is a binary relation R(X , Z ) defined by
numbers                                       Ja (a, b) = min[1, 1 − a + b]                               numbers

L10: Fuzzy                                                                                                L10: Fuzzy                     R(x, z) = [P ◦ Q](x, z) = max min[P(x, y ), Q(y , z)]
logic                                                                                                     logic                                                             y ∈Y
L11: Fuzzy                                                                                                L11: Fuzzy
control                                                                                                   control                             for all x ∈ X and all z ∈ Z .
L12: Fuzzy                                                                                                L12: Fuzzy
segmentation                                                                                              segmentation    This composition is based on standard t-norm, and standard t-conorm. It
L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (117/146)
                                                                                                          L13: Defuzzi-   is also referred to as max-min composition.
                                                                                                                          Joakim Lindblad, 2010-03-16 (118/146)
fication                                                                                                   fication




 Fuzzy Sets                                                                                                Fuzzy Sets
 and Fuzzy                                                                                                 and Fuzzy
 Techniques
                                                            Binary fuzzy relations                         Techniques
                                                                                                                                                                                   Inference rules
    Joakim                                                                                                    Joakim
   Lindblad                                                                                                  Lindblad
                To represent (fuzzy) binary relations, membership matrices
Outline         are convenient.                                                                           Outline         Fuzzy inference rules are basis for approximate reasoning.
L1: Intro                                                                                                 L1: Intro
                                                                                                                          As an example, three classical inference rules
L1–3: Basics                             R = [rxy ],         where rxy = R(x, y ).                        L1–3: Basics
                                                                                                                                   (Modus ponens, Modus Tollens, Hypothetical syllogism)
L4: Constr.                                                                                               L4: Constr.
and
                An example:                                                                               and             are generalized by using compositional rule of inference
uncertainty                                                                                               uncertainty
                Determine R = P ◦ Q = [rij ] = [pik ] ◦ [qkj ] = [maxk min(pik , qkj )]
L5: Features                                                                                              L5: Features    For a given fuzzy relation R on X × Y , and a given fuzzy set A′
L6 Features
                                               2
                                             0.3            0.5     0.8
                                                                        3 2
                                                                              0.9   0.5   0.7     0.7
                                                                                                      3   L6 Features     on X , a fuzzy set B ′ on Y can be derived for all y ∈ Y , so that
L7: Distances           R    =     P ◦ Q = 4 0.0            0.7     1.0 5 ◦ 4 0.3   0.2   0.0     0.9 5   L7: Distances
                                             0.4            0.6     0.5       1.0   0.0   0.5     0.5
L8: Set                                                                                                   L8: Set                                 B ′ (y ) = sup min[A′ (x), R(x, y )].
operations                                                                                                operations
                                   2                                 3                                                                                              x∈X
L9: Fuzzy                            0.8       0.3    0.5     0.5                                         L9: Fuzzy
numbers                      =     4 1.0       0.2    0.5     0.7 5 .                                     numbers
                                     0.5       0.4    0.5     0.6
                                                                                                                            In matrix form, compositional rule of inference is
L10: Fuzzy                                                                                                L10: Fuzzy
logic                                                                                                     logic

L11: Fuzzy      For example                                                                               L11: Fuzzy                                                 B′ = A′ ◦ R
control                                                                                                   control

L12: Fuzzy                         r23    =    max[min(0.0, 0.7), min(0.7, 0.0), min(1.0, 0.5)]           L12: Fuzzy
segmentation                              =    max[0.0, 0.0, 0.5] = 0.5.                                  segmentation

L13: Defuzzi-                                                                                             L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (119/146)                                                                   Joakim Lindblad, 2010-03-16   (120/146)
fication                                                                                                   fication
 Fuzzy Sets                                                                                         Fuzzy Sets
 and Fuzzy                                                                                          and Fuzzy
 Techniques
                                                                        Inference rules             Techniques
                                                                                                                                             Multiconditional approximate
    Joakim                                                                                             Joakim
   Lindblad                                          Example: Generalized modus ponens                Lindblad                                                  reasoning
Outline                                                                                            Outline

L1: Intro                                                                                          L1: Intro

L1–3: Basics                                                                                       L1–3: Basics

L4: Constr.       Rule:               If X is A, then Y is B                                       L4: Constr.     General schema is of the form:
and                                                                                                and
uncertainty       Fact:               X is A′                                                      uncertainty       Rule 1:             If X is A1 , then Y is B1
L5: Features      Conclusion:         Y is B ′                                                     L5: Features      Rule 2:             If X is A2 , then Y is B2
L6 Features                                                                                        L6 Features                           ...
L7: Distances
                In this case,                                                                      L7: Distances     Rule n:             If X is An , then Y is Bn
                                                  R(x, y ) = J [A(x), B(y )]
L8: Set                                                                                            L8: Set           Fact:               X is A′
operations                                                                                         operations
                and                                                                                                  Conclusion:         Y is B ′
L9: Fuzzy
numbers
                                              B ′ (y ) = sup min[A′ (x), R(x, y )].                L9: Fuzzy
                                                                                                   numbers
                                                          x∈X
L10: Fuzzy                                                                                         L10: Fuzzy      A′ , Aj are fuzzy sets on X ,
logic                                                                                              logic

L11: Fuzzy                                                                                         L11: Fuzzy
                                                                                                                   B ′ , Bj are fuzzy sets on Y ,            for all j.
control                                                                                            control

L12: Fuzzy                                                                                         L12: Fuzzy
segmentation                                                                                       segmentation

L13: Defuzzi-                                                                                      L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (121/146)                                                            Joakim Lindblad, 2010-03-16   (122/146)
fication                                                                                            fication




 Fuzzy Sets                                                                                         Fuzzy Sets
 and Fuzzy                                                                                          and Fuzzy
 Techniques
                                                          Approximate reasoning                     Techniques
                                                                                                                                             Multiconditional approximate
    Joakim                                                                                             Joakim
   Lindblad                                                        Method of interpolation            Lindblad                                                  reasoning
                Most common way to determine B ′ is by using                                                                                                  Method of interpolation-Example
Outline                                                                                            Outline
                       method of interpolation.
L1: Intro                                                                                          L1: Intro

L1–3: Basics
                Step 1. Calculate the degree of consistency between the given fact                 L1–3: Basics

L4: Constr.
                and the antecedent of each rule.                                                   L4: Constr.
and             Use height of intersection of the associated sets:                                 and
uncertainty                                                                                        uncertainty

L5: Features
                                  rj (A′ ) = h(A′ ∧ Aj ) = sup min[A′ (x), Aj (x)].                L5: Features

L6 Features                                                       x∈X                              L6 Features

L7: Distances                                                                                      L7: Distances
                                                                             ′                 ′
L8: Set
                  Step 2. Truncate each Bj by the value rj (A ) and determine B as                 L8: Set
operations      the union of truncated sets:                                                       operations

L9: Fuzzy                                                                                          L9: Fuzzy
numbers
                               B ′ (y ) = sup min[rj (A′ ), Bj (y )],        for all y ∈ Y .       numbers

L10: Fuzzy                                    j∈Nn                                                 L10: Fuzzy
logic                                                                                              logic

L11: Fuzzy        A special case of the composition rule of inference, with                        L11: Fuzzy
control                                                                                            control
                                      R(x, y ) = sup min[Aj (x), Bj (y )]
L12: Fuzzy                                                 j∈Nn                                    L12: Fuzzy
segmentation                                                                                       segmentation

L13: Defuzzi-   whereLindblad,B ′ (y ) = supx∈X min[A′ (x), R(x, y )] = (A′ ◦ R)(y ).
                Joakim
                       then 2010-03-16 (123/146)                                                   L13: Defuzzi-
                                                                                                                   Joakim Lindblad, 2010-03-16   (124/146)
fication                                                                                            fication




 Fuzzy Sets                                                                                         Fuzzy Sets
 and Fuzzy                                                                                          and Fuzzy
 Techniques
                                                                L11: Fuzzy control                  Techniques
                                                                                                                                                 Conventional control system
    Joakim                                                                                             Joakim
   Lindblad                                                                                           Lindblad

Outline                                                                                            Outline

L1: Intro                                                                                          L1: Intro

L1–3: Basics                                                                                       L1–3: Basics

L4: Constr.                                                                                        L4: Constr.
and                                                                                                and
uncertainty                                                                                        uncertainty

L5: Features                                                                                       L5: Features

L6 Features                                                                                        L6 Features

L7: Distances                                                                                      L7: Distances

L8: Set                                                                                            L8: Set
operations                                                                                         operations
                                                                                                                   PID Control (Proportional-Integral-Derivative)
L9: Fuzzy                                                                                          L9: Fuzzy
numbers                                                                                            numbers
                                                                                                                   The PID controller is the workhorse of the process industries.
L10: Fuzzy                                                                                         L10: Fuzzy
logic                                                                                              logic
                                                                                                                                                                                  t
L11: Fuzzy                                                                                         L11: Fuzzy                                                                                     dε
control                                                                                            control                        Output = bias + KP ε + KI                           ε dt + KD
L12: Fuzzy                                                                                         L12: Fuzzy                                                                 0                   dt
segmentation                                                                                       segmentation

L13: Defuzzi-                                                                                      L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (125/146)                                                            Joakim Lindblad, 2010-03-16   (126/146)
fication                                                                                            fication




 Fuzzy Sets                                                                                         Fuzzy Sets
 and Fuzzy                                                                                          and Fuzzy
 Techniques
                                                                Fuzzy logic control                 Techniques
                                                                                                                                                                   L11: Fuzzy control
    Joakim
   Lindblad
                                                                                                       Joakim
                                                                                                      Lindblad
                                                                                                                   Useful cases
                Methodology first developed by Mamdani in 1975 used to                                                 1 The control processes are too complex to analyze by conventional
Outline                                                                                            Outline
                control a steam plant. Based on work by Zadeh (1973) on                                                   quantitative techniques.
L1: Intro                                                                                          L1: Intro
                fuzzy algorithms for complex systems and decision processes.                                          2 The available sources of information are interpreted qualitatively,
L1–3: Basics                                                                                       L1–3: Basics
                                                                                                                          inexactly, or uncertainly.
L4: Constr.
and
                In a manner analogous to conventional control systems,                             L4: Constr.
                                                                                                   and             Advantage of Fuzzy logic control
uncertainty     inputs of a system are mapped to outputs using fuzzy                               uncertainty

L5: Features    logic rather than differential equations.                                           L5: Features       • Flexible
L6 Features                                                                                        L6 Features                •   Universal approximator
L7: Distances      • Can be used for systems that are difficult or impossible to                     L7: Distances      • Easy to understand
L8: Set                model mathematically.                                                       L8: Set                    •   Powerful – yet simple
operations                                                                                         operations

L9: Fuzzy          • Can also be applied to processes that are too complex or                      L9: Fuzzy          • Linguistic control
numbers                                                                                            numbers
                       nonlinear to be controlled with traditional strategies.                                                •   linguistic terms – human knowledge
L10: Fuzzy                                                                                         L10: Fuzzy
logic              • Human operators often are capable of managing complex                         logic              • Tolerant of imprecision / Robust control
                                                                                                                              •   more than 1 control rules - an error of a rule is not fatal
L11: Fuzzy
control
                       situations of a plant without knowing anything about                        L11: Fuzzy
                                                                                                   control                    •   limited trust in input data
L12: Fuzzy
                       differential equations.                                                      L12: Fuzzy         • Parallel or distributed control
segmentation                                                                                       segmentation
                                                                                                                              •   multiple fuzzy rules - complex nonlinear system
L13: Defuzzi-                                                                                      L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (127/146)                                                            Joakim Lindblad, 2010-03-16   (128/146)
fication                                                                                            fication
 Fuzzy Sets                                                                              Fuzzy Sets
 and Fuzzy                                                                               and Fuzzy
 Techniques
                                                             L11: Fuzzy control          Techniques
                                                                                                                                                        Fuzzy control
    Joakim                                                                                  Joakim
   Lindblad                                                                                Lindblad

Outline                                                                                 Outline

L1: Intro                                                                               L1: Intro

L1–3: Basics                                                                            L1–3: Basics

L4: Constr.                                                                             L4: Constr.
and             Disadvantages                                                           and
uncertainty                                                                             uncertainty                                                   Four main components
L5: Features       • More complex than PID                                              L5: Features

L6 Features                                                                             L6 Features
                                                                                                           1   The fuzzification interface : transforms input crisp values
                   • More parameters to tune                                                                   into fuzzy values
L7: Distances                                                                           L7: Distances

L8: Set
                   • Difficult to analyze mathematically (stability?)                     L8: Set            2   The knowledge base : contains a knowledge of the
operations                                                                              operations
                                                                                                               application domain and the control goals.
L9: Fuzzy                                                                               L9: Fuzzy
numbers                                                                                 numbers
                                                                                                           3   The decision-making logic : performs inference for fuzzy
L10: Fuzzy                                                                              L10: Fuzzy
logic                                                                                   logic                  control actions
L11: Fuzzy                                                                              L11: Fuzzy         4   The defuzzification interface : provides a crisp control
control                                                                                 control

L12: Fuzzy                                                                              L12: Fuzzy
                                                                                                               action out
segmentation                                                                            segmentation

L13: Defuzzi-                                                                           L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (129/146)                                                 Joakim Lindblad, 2010-03-16   (130/146)
fication                                                                                 fication




 Fuzzy Sets                                                                              Fuzzy Sets
 and Fuzzy                                                                               and Fuzzy
 Techniques
                                                                        Five steps...    Techniques
                                                                                                                                                          An example
    Joakim                                                                                  Joakim
   Lindblad                                                                                Lindblad

Outline
                How to build a fuzzy controller in five easy steps. . .                  Outline

L1: Intro                                                                               L1: Intro
                   1   Partition input and output spaces:
L1–3: Basics                                                                            L1–3: Basics

L4: Constr.
                       Select meaningful linguistic states for each variable and        L4: Constr.
and
uncertainty
                       express them as appropriate fuzzy sets.                          and
                                                                                        uncertainty

L5: Features       2   Fuzzification of input:                                           L5: Features

L6 Features            Introduce a fuzzification function for each input variable        L6 Features

L7: Distances          to express the associated measurement uncertainty.               L7: Distances

L8: Set                                                                                 L8: Set
operations         3   Formulate a set of inference rules:                              operations

L9: Fuzzy              If ǫ = A and dǫ = B, then C .
                                    dt                                                  L9: Fuzzy
numbers                                                                                 numbers

L10: Fuzzy
                   4   Design an inference engine:                                      L10: Fuzzy
logic                  Use method of interpolation (Lecture 10).                        logic

L11: Fuzzy                                                                              L11: Fuzzy
control            5   Select a suitable defuzzification method (Lecture 13).            control

L12: Fuzzy                                                                              L12: Fuzzy
segmentation                                                                            segmentation

L13: Defuzzi-                                                                           L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (131/146)                                                 Joakim Lindblad, 2010-03-16   (132/146)
fication                                                                                 fication




 Fuzzy Sets                                                                              Fuzzy Sets
 and Fuzzy                                                                               and Fuzzy
 Techniques
                                                   L12: Fuzzy segmentation               Techniques
                                                                                                                                                  Fuzzy connectedness
    Joakim                                                                                  Joakim
   Lindblad                                                                                Lindblad

Outline                                                                                 Outline

L1: Intro                                                                               L1: Intro

L1–3: Basics                                                                            L1–3: Basics

L4: Constr.                                                                             L4: Constr.     Hanging-togetherness natural grouping of voxels constituting
and                                                                                     and
uncertainty                                                                             uncertainty     an object a human viewer readily sees in a display of the scene
L5: Features                                                                            L5: Features    as a Gestalt in spite of intensity heterogeneity.
L6 Features                                                                             L6 Features
                                                                                                        Basic idea:
L7: Distances                                                                           L7: Distances

L8: Set                                                                                 L8: Set
                                                                                                        Compute global hanging-togetherness from local
operations                                                                              operations      hanging-togetherness.
L9: Fuzzy                                                                               L9: Fuzzy
numbers                                                                                 numbers

L10: Fuzzy                                                                              L10: Fuzzy
logic                                                                                   logic

L11: Fuzzy                                                                              L11: Fuzzy
control                                                                                 control

L12: Fuzzy                                                                              L12: Fuzzy
segmentation                                                                            segmentation

L13: Defuzzi-                                                                           L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (133/146)                                                 Joakim Lindblad, 2010-03-16   (134/146)
fication                                                                                 fication




 Fuzzy Sets                                                                              Fuzzy Sets
 and Fuzzy                                                                               and Fuzzy
 Techniques
                                                          Fuzzy connectedness            Techniques
                                                                                                                                                  Fuzzy connectedness
    Joakim                                                                                  Joakim
   Lindblad                                                                                Lindblad

Outline                                                                                 Outline
                                                                                                        Fuzzy spel adjacency is a reflexive and symmetric fuzzy
                Strength of a path – the strength of its weakest link                                   relation α in Zn and assigns a value to a pair of spels (c, d)
L1: Intro                                                                               L1: Intro

L1–3: Basics    A. Rosenfeld 1979                                                       L1–3: Basics    based on how close they are spatially.
L4: Constr.     Strength of a link between two points defined by the                     L4: Constr.     Fuzzy spel affinity is a reflexive and symmetric fuzzy relation κ
and                                                                                     and
uncertainty     membership function.                                                    uncertainty     in Zn and assigns a value to a pair of spels (c, d) based on how
L5: Features    J. K. Udupa and S. Samarasekera 1996                                    L5: Features    close they are spatially and intensity-based-property-wise (local
L6 Features
                Strength of a link between two points defined by affinity                  L6 Features
                                                                                                        hanging-togetherness).
L7: Distances                                                                           L7: Distances

L8: Set
                The connectedness of two points x and y in A –                          L8: Set
operations      the strength of the strongest path between x and y                      operations                         µκ (c, d) = h(µα (c, d), µ(c), µ(d), c, d)
L9: Fuzzy                                                                               L9: Fuzzy
numbers                                                                                 numbers
                                                                                                        The fuzzy κ-connectedness assigns a value to a pair of spels (c,
L10: Fuzzy                                    cA (x, y ) =     sup    inf A(t)          L10: Fuzzy
logic                                                        π∈Π(x,y ) t∈π              logic           d) that is the maximum of the strengths of connectedness
L11: Fuzzy                                                                              L11: Fuzzy      assigned to all possible paths from c to d (global
control                                                                                 control
                                                                                                        hanging-togetherness).
L12: Fuzzy                                                                              L12: Fuzzy
segmentation                                                                            segmentation

L13: Defuzzi-                                                                           L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (135/146)                                                 Joakim Lindblad, 2010-03-16   (136/146)
fication                                                                                 fication
 Fuzzy Sets                                                                                         Fuzzy Sets
 and Fuzzy                                                                                          and Fuzzy
 Techniques
                                              Components of fuzzy affinity                            Techniques
                                                                                                                                                               Fuzzy connectedness
    Joakim                                                                                             Joakim
   Lindblad                                                                                           Lindblad

Outline                                                                                            Outline

L1: Intro       Fuzzy spel adjacency µα (c, d) indicates the degree of spatial                     L1: Intro
                                                                                                                   Computation – A graph search problem
L1–3: Basics    adjacency of spels The homogeneity-based component                                 L1–3: Basics

L4: Constr.
                µψ (c, d) indicates the degree of local hanging-togetherness of                    L4: Constr.     Dynamic programming solution (think distance transform or
and                                                                                                and
uncertainty     spels due to their similarities of intensities The                                 uncertainty     level sets computation)
L5: Features                                                                                       L5: Features
                object-feature-based component µϕ (c, d) indicates the                                             Practical usage examples:
L6 Features                                                                                        L6 Features
                degree of local hanging-togetherness of spels with respect to
L7: Distances                                                                                      L7: Distances      • Seed foreground (one or multiple seeds), threshold at some
                some given object feature
L8: Set                                                                                            L8: Set                level of fuzzy connectedness.
operations      Example:                                                                           operations

L9: Fuzzy                                                 1                                        L9: Fuzzy
                                                                                                                      • Seed different regions and let them compete (relative fc,
numbers                                               µκ = µα (µψ + µϕ )                           numbers
                                                                                                                          iterated relative fc).
L10: Fuzzy
                                                          2                                        L10: Fuzzy
logic                                                                                              logic

L11: Fuzzy                                                                                         L11: Fuzzy
control                                                                                            control

L12: Fuzzy                                                                                         L12: Fuzzy
segmentation                                                                                       segmentation

L13: Defuzzi-                                                                                      L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (137/146)                                                            Joakim Lindblad, 2010-03-16   (138/146)
fication                                                                                            fication




 Fuzzy Sets                                                                                         Fuzzy Sets
 and Fuzzy                                                                                          and Fuzzy
 Techniques
                                         Pixel coverage representations                             Techniques
                                                                                                                                            Pixel coverage representations
    Joakim                                                                                             Joakim
   Lindblad                                                                                           Lindblad
                Coverage representation – constrained fuzziness
Outline                                                                                            Outline

L1: Intro          • Restrict to one specific meaning of memberships.                               L1: Intro

L1–3: Basics                                                                                       L1–3: Basics
                                                                                                                      • Pixel coverage representations are shown to be superior to
                   • Restrict to crisp imaged objects (no clouds or flames).
L4: Constr.                                                                                        L4: Constr.            crisp image object representations for many reasons.
and                                                                                                and
uncertainty                                                                                        uncertainty

L5: Features
                                                                Pixel coverage digitization        L5: Features
                                                                                                                      • By suitably utilizing information available in images it is
L6 Features
                                   ⇒                            Let the value of a pixel/voxel     L6 Features
                                                                                                                          possible to perform pixel coverage segmentations.
L7: Distances                                                   be equal to the part of it being   L7: Distances
                                                                                                                      • We observe that for reasonable noise levels, the achieved
L8: Set                                                         covered by the object.             L8: Set
operations                                                                                         operations             pixel coverage representation provides a more accurate
L9: Fuzzy          • Keep the good sides of fuzzy (robust, precise features,                       L9: Fuzzy              representation of image objects than a perfect, noise free,
numbers                                                                                            numbers
                       information rich representation).                                                                  crisp representation.
L10: Fuzzy                                                                                         L10: Fuzzy
logic
                   • Enable strong theoretical results.                                            logic

L11: Fuzzy                                                                                         L11: Fuzzy
control            • Clear meaning of feature values – features describe the                       control

L12: Fuzzy             continuous crisp object.                                                    L12: Fuzzy
segmentation                                                                                       segmentation

L13: Defuzzi-                                                                                      L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (139/146)                                                            Joakim Lindblad, 2010-03-16   (140/146)
fication                                                                                            fication




 Fuzzy Sets                                                                                         Fuzzy Sets
 and Fuzzy                                                                                          and Fuzzy
 Techniques
                                                               L13: Defuzzification                  Techniques
                                                                                                                                                                L13: Defuzzification
    Joakim                                                                                             Joakim
   Lindblad                                                                                           Lindblad

Outline                                                                                            Outline

L1: Intro                                                                                          L1: Intro

L1–3: Basics                                                                                       L1–3: Basics    Defuzzification is a process that maps a fuzzy set to a crisp set.
L4: Constr.
and
                                                                                                   L4: Constr.
                                                                                                   and
                                                                                                                   Approaches:
uncertainty                                                                                        uncertainty

L5: Features                                                                                       L5: Features       • Defuzzification to a point.
L6 Features                                                                                        L6 Features
                                                                                                                      • Defuzzification to a set.
L7: Distances                                                                                      L7: Distances

L8: Set                                                                                            L8: Set
operations                                                                                         operations         • Generating a good representative of a fuzzy set.
L9: Fuzzy                                                                                          L9: Fuzzy
numbers                                                                                            numbers
                                                                                                                      • Recovering a crisp original set.
L10: Fuzzy                                                                                         L10: Fuzzy
logic                                                                                              logic

L11: Fuzzy                                                                                         L11: Fuzzy
control                                                                                            control

L12: Fuzzy                                                                                         L12: Fuzzy
segmentation                                                                                       segmentation

L13: Defuzzi-                                                                                      L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (141/146)                                                            Joakim Lindblad, 2010-03-16   (142/146)
fication                                                                                            fication




 Fuzzy Sets                                                                                         Fuzzy Sets
 and Fuzzy                                                                                          and Fuzzy
 Techniques
                                                      Defuzzification to a point                     Techniques
                                                                                                                                                             Defuzzification to a set
    Joakim                                                                                             Joakim
   Lindblad                                                    Examples of common methods             Lindblad                                                   A.k.a. Averaging procedures

Outline                                                                                            Outline

L1: Intro          • Centre of gravity (Set of real numbers)                                       L1: Intro

L1–3: Basics                                                         Pxmax                         L1–3: Basics
                                                                       xmin   x · A(x)
L4: Constr.                                            COG (A) =       Pxmax                 .     L4: Constr.
                                                                                                                      • α-cuts
and
                                                                           xmin   A(x)             and
uncertainty                                                                                        uncertainty
                                                                                                                                     chosen at various levels α (often α = 0.5 or α = 1).
L5: Features                                                                                       L5: Features
                   • Mean of maxima (Set of real numbers)
L6 Features                                                                                        L6 Features
                                                                                                                      • Average α-cuts
                                                                                                                               based on an integration of set-valued function,
                                                                        P
L7: Distances                                                               x∈core(A)    x         L7: Distances
                                                       MeOM(A) =                             .
L8: Set                                                                   |core(A)|                L8: Set                     called Kudo-Aumann integration.
operations                                                                                         operations
                                                                                                                      • Feature distance minimization
L9: Fuzzy          • Centre of area (COA)                                                          L9: Fuzzy
numbers
                       COA(A) is the value that minimizes the expression
                                                                                                   numbers                     find the crisp set at the minimal feature distance to
L10: Fuzzy
logic                                             ŕ                                          ŕ
                                                                                                   L10: Fuzzy
                                                                                                   logic
                                                                                                                               the given fuzzy set.
                                                  ŕ   COA(A)              sup(X )       ŕ
L11: Fuzzy                                        ŕ    X                   X              ŕ        L11: Fuzzy
control                                           ŕ
                                                    ŕ             A(x) −          A(x)ŕŕ .         control
                                                      ŕ x=inf(X )        x=COA(A)           ŕ
L12: Fuzzy                                                                                         L12: Fuzzy
segmentation                                                                                       segmentation

L13: Defuzzi-                                                                                      L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (143/146)                                                            Joakim Lindblad, 2010-03-16   (144/146)
fication                                                                                            fication
 Fuzzy Sets                                                                                       Fuzzy Sets
 and Fuzzy                                                                                        and Fuzzy
 Techniques
                                                          Defuzzification to a set                 Techniques
                                                                                                                                Defuzzification by feature distance
    Joakim                                                                                           Joakim
   Lindblad                                                                     Average α-cuts      Lindblad                                         minimization
Outline         Let a fuzzy set A be given by a membership function                              Outline

L1: Intro       µ : R → [0, 1].                                                                  L1: Intro

L1–3: Basics
                   • Sets F (w ) are α cuts, Aα of the fuzzy set A, for α ∈ [0, 1];              L1–3: Basics    Definition
L4: Constr.
and                • Selectors are ϕ(α) = inf Aα and ϕ(α) = sup Aα .
                                                                                                 L4: Constr.
                                                                                                 and
                                                                                                                 An optimal defuzzification D(A) of a fuzzy set A on a reference
uncertainty                                                                                      uncertainty
                                                                                                                 set X , with respect to the distance d, is
L5: Features    Then, the average α-cut of A is                                                  L5: Features

L6 Features                                                                                      L6 Features           D(A) ∈ {C ∈ P(X ) | d(A, C ) = min [d(A, B)]} .           (3)
L7: Distances                     Aµ =                    inf Aα dα,           sup Aα dα .       L7: Distances                                             B∈P(X )
L8: Set                                          [0,1]                 [0,1]                     L8: Set
operations                                                                                       operations

L9: Fuzzy                                                                                        L9: Fuzzy       I.e., the crisp set that is at smallest distance to the fuzzy set.
numbers                                                                                          numbers

L10: Fuzzy                                                                                       L10: Fuzzy
                                                                                                                 Use a feature distance containing both local and global
logic                                                                                            logic           features. Find a minimum using a suitable optimization
L11: Fuzzy                                                                                       L11: Fuzzy
control                                                                                          control
                                                                                                                 method.
L12: Fuzzy                                                                                       L12: Fuzzy
segmentation                                                                                     segmentation

L13: Defuzzi-                                                                                    L13: Defuzzi-
                Joakim Lindblad, 2010-03-16   (145/146)                                                          Joakim Lindblad, 2010-03-16   (146/146)
fication                                                                                          fication

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:22
posted:8/30/2011
language:English
pages:19