# Fuzzy Sets and Fuzzy Techniques - Lecture 14 -- Repetition

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```					 Fuzzy Sets                                                                                          Fuzzy Sets
and Fuzzy                                                                                           and Fuzzy
Techniques                                                                                          Techniques
Lecture notes
Joakim                                                                                              Joakim

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
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-
ﬁcation                                                                                             ﬁcation

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

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-
ﬁcation                                                                                             ﬁcation

Fuzzy Sets                                                                                          Fuzzy Sets
and Fuzzy                                                                                           and Fuzzy
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-
ﬁcation                                                                                             ﬁcation

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

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
L12: Fuzzy                                                                                          L12: Fuzzy
segmentation                                                                                        segmentation

L13: Defuzzi-                                                                                       L13: Defuzzi-
ﬁcation                                                                                             ﬁcation
Fuzzy Sets                                                                            Fuzzy Sets
and Fuzzy                                                                             and Fuzzy
Techniques
What is a fuzzy set?          Techniques
What is a fuzzy set?
Joakim                                                                                Joakim

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-
ﬁcation                                                                               ﬁcation

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

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-
ﬁcation                                                                               ﬁcation

Fuzzy Sets                                                                            Fuzzy Sets
and Fuzzy                                                                             and Fuzzy
Techniques
Fuzzy sets    Techniques
Fuzzy sets
Joakim                                                                                Joakim
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-
ﬁcation                                                                               ﬁcation

Fuzzy Sets                                                                            Fuzzy Sets
and Fuzzy                                                                             and Fuzzy
Techniques
Fuzzy sets of diﬀerent types and                   Techniques
Basic concepts and terminology
Joakim                                                                                Joakim
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-
ﬁcation                                                                               ﬁcation
Fuzzy Sets                                                                                  Fuzzy Sets
and Fuzzy                                                                                   and Fuzzy
Techniques
Basic concepts and terminology                          Techniques
Basic concepts and terminology
Joakim                                                                                      Joakim
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-
ﬁcation                                                                                     ﬁcation

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

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-
ﬁcation                                                                                     ﬁcation

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

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-
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-
ﬁcation                                                                                     ﬁcation
Fuzzy Sets                                                                                        Fuzzy Sets
and Fuzzy                                                                                         and Fuzzy
Techniques
L4: Constructing fuzzy sets,                         Techniques
L4: Constructing fuzzy sets,
Joakim                                                                                            Joakim
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-
ﬁcation                                                                                           ﬁcation

Fuzzy Sets                                                                                        Fuzzy Sets
and Fuzzy                                                                                         and Fuzzy
Techniques
Direct methods with one expert                                Techniques
Direct methods with multiple
Joakim                                                                                            Joakim
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-
ﬁcation                                                                                           ﬁcation

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

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-
ﬁcation                                                                                           ﬁcation

Fuzzy Sets                                                                                        Fuzzy Sets
and Fuzzy                                                                                         and Fuzzy
Techniques
Nonspeciﬁcity           Techniques
Fuzziness of fuzzy sets
Joakim                                                                                            Joakim

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-
ﬁcation                                                                                           ﬁcation
Fuzzy Sets                                                                         Fuzzy Sets
and Fuzzy                                                                          and Fuzzy
Techniques
Fuzziness of fuzzy sets    Techniques
Fuzziness of fuzzy sets
Joakim                                                                             Joakim

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-
ﬁcation                                                                            ﬁcation

Fuzzy Sets                                                                         Fuzzy Sets
and Fuzzy                                                                          and Fuzzy
Techniques
Information gain?     Techniques
L5: Fuzzy thresholding, Fuzzy
Joakim                                                                             Joakim
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-
ﬁcation                                                                            ﬁcation

Fuzzy Sets                                                                         Fuzzy Sets
and Fuzzy                                                                          and Fuzzy
Techniques
Thresholding      Techniques
Fuzzy c-means clustering
Joakim                                                                             Joakim
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-
ﬁcation                                                                            ﬁcation

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

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-
ﬁcation                                                                            ﬁcation
Fuzzy Sets                                                                                 Fuzzy Sets
and Fuzzy                                                                                  and Fuzzy
Techniques
L6: Features of fuzzy sets               Techniques
L6: Features of fuzzy sets
Joakim                                                                                     Joakim

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-
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-
ﬁcation                                                                                    ﬁcation

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

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-
ﬁcation                                                                                    ﬁcation

Fuzzy Sets                                                                                 Fuzzy Sets
and Fuzzy                                                                                  and Fuzzy
Techniques
Estimation of features            Techniques
Estimation of features
Joakim                                                                                     Joakim
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-
ﬁcation                                                                                    ﬁcation
Fuzzy Sets                                                                                          Fuzzy Sets
and Fuzzy                                                                                           and Fuzzy
Techniques
Estimation of features                     Techniques
L7: Distances on and between
Joakim                                                                                              Joakim
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-
ﬁcation                                                                                             ﬁcation

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

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-
ﬁcation                                                                                             ﬁcation

Fuzzy Sets                                                                                          Fuzzy Sets
and Fuzzy                                                                                           and Fuzzy
Techniques
Membership focused                       Techniques
Membership focused
Joakim                                                                                              Joakim

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-
ﬁcation                                                                                             ﬁcation

Fuzzy Sets                                                                                          Fuzzy Sets
and Fuzzy                                                                                           and Fuzzy
Techniques
Membership focused                       Techniques
Spatially focused
Joakim                                                                                              Joakim
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-
ﬁcation                                                                                             ﬁcation
Fuzzy Sets                                                                                          Fuzzy Sets
and Fuzzy                                                                                           and Fuzzy
Techniques
Spatially focused               Techniques
Spatially focused
Joakim                                                                                              Joakim

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-
ﬁcation                                                                                             ﬁcation

Fuzzy Sets                                                                                          Fuzzy Sets
and Fuzzy                                                                                           and Fuzzy
Techniques
Feature distances               Techniques
Point to point distances
Joakim                                                                                              Joakim

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-
ﬁcation                                                                                             ﬁcation

Fuzzy Sets                                                                                          Fuzzy Sets
and Fuzzy                                                                                           and Fuzzy
Techniques
Cost function             Techniques
Cost function
Joakim                                                                                              Joakim

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-
ﬁcation                                                                                             ﬁcation

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

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-
ﬁcation                                                                                             ﬁcation
Fuzzy Sets                                                                                    Fuzzy Sets
and Fuzzy                                                                                     and Fuzzy
Techniques
L8: Operations on fuzzy sets                   Techniques
Standard fuzzy operations
Joakim                                                                                        Joakim

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-
ﬁcation                                                                                       ﬁcation

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

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-
ﬁcation                                                                                       ﬁcation

Fuzzy Sets                                                                                    Fuzzy Sets
and Fuzzy                                                                                     and Fuzzy
Techniques
Fuzzy complements                 Techniques
Generators
Joakim                                                                                        Joakim

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-
L13: Defuzzi-     Similarly for Decreasing generators
(70/146)
ﬁcation                                                                                       ﬁcation

Fuzzy Sets                                                                                    Fuzzy Sets
and Fuzzy                                                                                     and Fuzzy
Techniques
Generating fuzzy complements                          Techniques
Fuzzy intersections
Joakim                                                                                        Joakim

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-
ﬁcation                                                                                       ﬁcation
Fuzzy Sets                                                                                        Fuzzy Sets
and Fuzzy                                                                                         and Fuzzy
Techniques
Fuzzy intersections                Techniques
Fuzzy intersections
Joakim                                                                                            Joakim

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-
ﬁcation                                                                                           ﬁcation

Fuzzy Sets                                                                                        Fuzzy Sets
and Fuzzy                                                                                         and Fuzzy
Techniques
Fuzzy intersections                Techniques
Fuzzy unions
Joakim                                                                                            Joakim

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-
ﬁcation                                                                                           ﬁcation

Fuzzy Sets                                                                                        Fuzzy Sets
and Fuzzy                                                                                         and Fuzzy
Techniques
Fuzzy unions             Techniques
Fuzzy unions
Joakim                                                                                            Joakim

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-
ﬁcation                                                                                           ﬁcation

Fuzzy Sets                                                                                        Fuzzy Sets
and Fuzzy                                                                                         and Fuzzy
Techniques
Combinations of set operations                             Techniques
Dual triples - Six theorems (1)
Joakim                                                                                            Joakim
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.
ﬁcation                                                                                           ﬁcation
Fuzzy Sets                                                                                                          Fuzzy Sets
and Fuzzy                                                                                                           and Fuzzy
Techniques
Dual triples - Six theorems (2)                                             Techniques
Aggregation operations
Joakim                                                                                                              Joakim

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-
ﬁcation                                                                                                             ﬁcation

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

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-
ﬁcation                                                                                                             ﬁcation

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

L1: Intro                                                                                                  1        L1: Intro

L1–3: Basics
α    α            α
a1 + a2 + · · · + an               α
L1–3: Basics
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-
ﬁcation                                                                                                             ﬁcation

Fuzzy Sets                                                                                                          Fuzzy Sets
and Fuzzy                                                                                                           and Fuzzy
Techniques
An Application: Fuzzy                                   Techniques
L9: Fuzzy numbers and fuzzy
Joakim                                                                                                              Joakim
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-
ﬁcation                                                                                                             ﬁcation
Fuzzy Sets                                                                                       Fuzzy Sets
and Fuzzy                                                                                        and Fuzzy
Techniques
Interval numbers                 Techniques
Interval numbers
Joakim                                                                                           Joakim
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-
ﬁcation                                                                                          ﬁcation

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

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-
ﬁcation                                                                                          ﬁcation

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

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-
ﬁcation                                                                                          ﬁcation

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

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-
ﬁcation                                                                                          ﬁcation
Fuzzy Sets                                                                                                 Fuzzy Sets
and Fuzzy                                                                                                  and Fuzzy
Techniques
Arithmetics on fuzzy numbers                                       Techniques
Linguistic variables
Joakim                                                                                                     Joakim
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-
ﬁcation                                                                                                    ﬁcation

Fuzzy Sets                                                                                                 Fuzzy Sets
and Fuzzy                                                                                                  and Fuzzy
Techniques
Interval equations                         Techniques
Fuzzy equations
Joakim                                                                                                     Joakim
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-
ﬁcation                                                                                                    ﬁcation

Fuzzy Sets                                                                                                 Fuzzy Sets
and Fuzzy                                                                                                  and Fuzzy
Techniques
Fuzzy equations                        Techniques
L10: Fuzzy logic and approximate
Joakim                                                                                                     Joakim
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-
ﬁcation                                                                                                    ﬁcation

Fuzzy Sets                                                                                                 Fuzzy Sets
and Fuzzy                                                                                                  and Fuzzy
Techniques
Classical logic: A brief overview                                      Techniques
Classical logic: A brief overview
Joakim                                                                                                     Joakim

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
L13: Defuzzi-
ﬁcation
ω16 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

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-
ﬁcation                                                                                      ﬁcation

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

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-
ﬁcation                                                                                      ﬁcation

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

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-
ﬁcation                                                                                      ﬁcation

Fuzzy Sets                                                                                   Fuzzy Sets
and Fuzzy                                                                                    and Fuzzy
Techniques
Fuzzy quantiﬁers               Techniques
Fuzzy propositions
Joakim                                                                                       Joakim

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-
ﬁcation                                                                                      ﬁcation
Fuzzy Sets                                                                                                Fuzzy Sets
and Fuzzy                                                                                                 and Fuzzy
Techniques
Fuzzy propositions                      Techniques
Fuzzy propositions
Joakim                                                                                                    Joakim

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-
ﬁcation                                                                                                   ﬁcation

Fuzzy Sets                                                                                                Fuzzy Sets
and Fuzzy                                                                                                 and Fuzzy
Techniques
Fuzzy propositions                      Techniques
Fuzzy implications
Joakim                                                                                                    Joakim

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-
ﬁcation                                                                                                   ﬁcation

Fuzzy Sets                                                                                                Fuzzy Sets
and Fuzzy                                                                                                 and Fuzzy
Techniques
Fuzzy implications                     Techniques
Binary fuzzy relations
Joakim                                                                                                    Joakim
• 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-
L13: Defuzzi-   is also referred to as max-min composition.
ﬁcation                                                                                                   ﬁcation

Fuzzy Sets                                                                                                Fuzzy Sets
and Fuzzy                                                                                                 and Fuzzy
Techniques
Binary fuzzy relations                         Techniques
Inference rules
Joakim                                                                                                    Joakim
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-
ﬁcation                                                                                                   ﬁcation
Fuzzy Sets                                                                                         Fuzzy Sets
and Fuzzy                                                                                          and Fuzzy
Techniques
Inference rules             Techniques
Multiconditional approximate
Joakim                                                                                             Joakim
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-
ﬁcation                                                                                            ﬁcation

Fuzzy Sets                                                                                         Fuzzy Sets
and Fuzzy                                                                                          and Fuzzy
Techniques
Approximate reasoning                     Techniques
Multiconditional approximate
Joakim                                                                                             Joakim
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-
ﬁcation                                                                                            ﬁcation

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

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-
ﬁcation                                                                                            ﬁcation

Fuzzy Sets                                                                                         Fuzzy Sets
and Fuzzy                                                                                          and Fuzzy
Techniques
Fuzzy logic control                 Techniques
L11: Fuzzy control
Joakim
Joakim
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-
ﬁcation                                                                                            ﬁcation
Fuzzy Sets                                                                              Fuzzy Sets
and Fuzzy                                                                               and Fuzzy
Techniques
L11: Fuzzy control          Techniques
Fuzzy control
Joakim                                                                                  Joakim

Outline                                                                                 Outline

L1: Intro                                                                               L1: Intro

L1–3: Basics                                                                            L1–3: Basics

L4: Constr.                                                                             L4: Constr.
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-
ﬁcation                                                                                 ﬁcation

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

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-
ﬁcation                                                                                 ﬁcation

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

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-
ﬁcation                                                                                 ﬁcation

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

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-
ﬁcation                                                                                 ﬁcation
Fuzzy Sets                                                                                         Fuzzy Sets
and Fuzzy                                                                                          and Fuzzy
Techniques
Components of fuzzy aﬃnity                            Techniques
Fuzzy connectedness
Joakim                                                                                             Joakim

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-
ﬁcation                                                                                            ﬁcation

Fuzzy Sets                                                                                         Fuzzy Sets
and Fuzzy                                                                                          and Fuzzy
Techniques
Pixel coverage representations                             Techniques
Pixel coverage representations
Joakim                                                                                             Joakim
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-
ﬁcation                                                                                            ﬁcation

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

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-
ﬁcation                                                                                            ﬁcation

Fuzzy Sets                                                                                         Fuzzy Sets
and Fuzzy                                                                                          and Fuzzy
Techniques
Defuzziﬁcation to a point                     Techniques
Defuzziﬁcation to a set
Joakim                                                                                             Joakim

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-
ﬁcation                                                                                            ﬁcation
Fuzzy Sets                                                                                       Fuzzy Sets
and Fuzzy                                                                                        and Fuzzy
Techniques
Defuzziﬁcation to a set                 Techniques
Defuzziﬁcation by feature distance
Joakim                                                                                           Joakim
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-