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Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Techniques Lecture notes Joakim Joakim Lindblad Lindblad Outline Outline L1: Intro L1: Intro L1–3: Basics Fuzzy Sets and Fuzzy Techniques L1–3: Basics L4: Constr. and Lecture 14 – Repetition L4: Constr. and uncertainty uncertainty L5: Features L5: Features www.cb.uu.se/~joakim/course/fuzzy/vt10/lectures.html L6 Features L6 Features Joakim Lindblad L7: Distances L7: Distances joakim@cb.uu.se L8: Set L8: Set operations operations L9: Fuzzy Centre for Image Analysis L9: Fuzzy numbers numbers Uppsala University L10: Fuzzy L10: Fuzzy logic logic L11: Fuzzy 2010-03-16 L11: Fuzzy control control L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (1/146) Joakim Lindblad, 2010-03-16 (2/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Topics of today Techniques L1: Introduction, motivation Joakim Joakim Lindblad Lindblad Outline L1: Introduction, motivation Outline L1: Intro L1–3: Basics of fuzzy sets, cutworthiness, the extension principle L1: Intro L1–3: Basics L1–3: Basics L4: Constructing fuzzy sets, uncertainty measures L4: Constr. L4: Constr. and uncertainty L5: Fuzzy thresholding, Fuzzy c-means clustering and uncertainty L5: Features L6: Features of fuzzy sets L5: Features L6 Features L7: Distances on and between fuzzy sets L6 Features L7: Distances L7: Distances L8: Operations on fuzzy sets L8: Set L8: Set operations L9: Fuzzy numbers and fuzzy arithmetics operations L9: Fuzzy L9: Fuzzy numbers L10: Fuzzy logic and approximate reasoning numbers L10: Fuzzy L10: Fuzzy logic L11: Fuzzy control logic L11: Fuzzy L12: Fuzzy segmentation L11: Fuzzy control control L12: Fuzzy L13: Defuzziﬁcation L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (3/146) Joakim Lindblad, 2010-03-16 (4/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques About the course Techniques What will we learn in this course? Joakim Joakim Lindblad Fuzzy Sets and Fuzzy Techniques Lindblad Fuzzy Sets and Fuzzy Techniques Outline Outline L1: Intro L1: Intro L1–3: Basics L1–3: Basics • The basics of fuzzy sets L4: Constr. http://www.cb.uu.se/~joakim/course/fuzzy/vt10/ L4: Constr. and and • How to deﬁne fuzzy sets uncertainty • 13 lectures + repetition uncertainty • How to perform operations on fuzzy sets L5: Features L5: Features • How to extend crisp concepts to fuzzy ones • 2 computer exercises (giving up to 4 bonus points on the L6 Features L6 Features • How to extract information from fuzzy sets L7: Distances exam) L7: Distances • The very basics of fuzzy logic and fuzzy reasoning L8: Set • 1 small project work + presentation (written option L8: Set operations operations • We will look at some applications of fuzzy in possible) L9: Fuzzy L9: Fuzzy • Image processing numbers • Written exam (2nd exam, what is a good date for you?) numbers • Control systems L10: Fuzzy L10: Fuzzy logic logic • Machine intelligence / expert systems L11: Fuzzy L11: Fuzzy control control L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (5/146) Joakim Lindblad, 2010-03-16 (6/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques What is a fuzzy set? Techniques Why Fuzzy? Joakim Joakim Lindblad Lindblad Outline Outline Precision is not truth. L1: Intro L1: Intro - Henri Matisse L1–3: Basics Btw., what is a set? “... to be an element...” L1–3: Basics L4: Constr. L4: Constr. and and uncertainty uncertainty A set is a collection of its members. So far as the laws of mathematics refer to reality, they are not L5: Features L5: Features certain. And so far as they are certain, they do not refer to L6 Features L6 Features The notion of fuzzy sets is an extension reality. L7: Distances L7: Distances of the most fundamental property of sets. - Albert Einstein L8: Set L8: Set operations Fuzzy sets allows a grading of to what extent operations L9: Fuzzy L9: Fuzzy numbers an element of a set belongs to that speciﬁc set. numbers L10: Fuzzy L10: Fuzzy As complexity rises, precise statements lose meaning and logic logic meaningful statements lose precision. L11: Fuzzy L11: Fuzzy control control - Lotﬁ Zadeh L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (7/146) Joakim Lindblad, 2010-03-16 (8/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques What is a fuzzy set? Techniques What is a fuzzy set? Joakim Joakim Lindblad Lindblad Randomness vs. Fuzziness Outline Outline L1: Intro Fuzzy is not just another name for probability. L1: Intro L1–3: Basics L1–3: Basics L4: Constr. The number 10 is not probably big! L4: Constr. and and Randomness refers to an event that may or may not occur. uncertainty ...and number 2 is not probably not big. uncertainty L5: Features L5: Features Randomness: frequency of car accidents. L6 Features Uncertainty is a consequence of L6 Features Fuzziness refers to the boundary of a set that is not precise. L7: Distances non-sharp boundaries between the notions/objects, L7: Distances Fuzziness: seriousness of a car accident. L8: Set operations and not caused by lack of information. L8: Set operations Prof. George J. Klir L9: Fuzzy numbers Statistical models deal with random events and outcomes; L9: Fuzzy numbers L10: Fuzzy fuzzy models attempt to capture and quantify nonrandom L10: Fuzzy logic imprecision. logic L11: Fuzzy L11: Fuzzy control control L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (9/146) Joakim Lindblad, 2010-03-16 (10/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques What is Fuzzy? Techniques L1–3: Basics of fuzzy sets Joakim Joakim Lindblad Lindblad Outline Outline L1: Intro L1: Intro L1–3: Basics Using fuzzy techniques is L1–3: Basics L4: Constr. L4: Constr. and and uncertainty uncertainty L5: Features L5: Features L6 Features to avoid throwing away data early (by crisp, possibly false, L6 Features L7: Distances decisions). L7: Distances L8: Set L8: Set operations operations L9: Fuzzy L9: Fuzzy numbers numbers L10: Fuzzy L10: Fuzzy logic logic L11: Fuzzy L11: Fuzzy control control L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (11/146) Joakim Lindblad, 2010-03-16 (12/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Fuzzy sets Techniques Fuzzy sets Joakim Joakim Lindblad Lindblad Continuous (analog) fuzzy sets Outline Outline A fuzzy set of a reference set is a set of ordered pairs L1: Intro L1: Intro A : X → [0, 1] , X is dense L1–3: Basics L1–3: Basics F = { x, µF (x) | x ∈ X }, L4: Constr. and L4: Constr. and Discrete fuzzy sets uncertainty where µF : X → [0, 1]. uncertainty L5: Features L5: Features A : {x1 , x2 , x3 , ..., xs } → [0, 1] L6 Features Where there is no risk for confusion, we use the same symbol L6 Features L7: Distances for the fuzzy set, as for its membership function. L7: Distances Digital fuzzy sets L8: Set operations Thus L8: Set operations If a discrete-universal membership function can take only a L9: Fuzzy F = { x, F (x) | x ∈ X }, L9: Fuzzy ﬁnite number n ≥ 2 of distinct values, then we call this fuzzy numbers numbers where F : X → [0, 1]. set a digital fuzzy set. L10: Fuzzy L10: Fuzzy logic logic L11: Fuzzy To deﬁne a fuzzy set ⇔ To deﬁne a membership function L11: Fuzzy 1 2 3 control control A : {x1 , x2 , x3 , ..., xs } → {0, n−1 , n−1 , n−1 , ..., n−2 , 1} n−1 L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (13/146) Joakim Lindblad, 2010-03-16 (14/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Fuzzy sets of diﬀerent types and Techniques Basic concepts and terminology Joakim Joakim Lindblad levels Lindblad The support of a fuzzy set Outline The membership function may be vague in itself. Outline supp(A) = {x ∈ X | A(x) > 0} L1: Intro Fuzzy sets of type 2: A : X → F([0, 1]) L1: Intro L1–3: Basics L1–3: Basics L4: Constr. L4: Constr. and and A crossover point of a fuzzy set uncertainty uncertainty L5: Features L5: Features ¯ A(x) = A(x) L6 Features L6 Features L7: Distances L7: Distances L8: Set L8: Set The height, h(A) of a fuzzy set operations operations L9: Fuzzy numbers L9: Fuzzy numbers h(A) = max A(x) Also the domain of the membership function may be fuzzy. x∈X L10: Fuzzy L10: Fuzzy logic Fuzzy sets deﬁned so that the elements of the universal set are logic L11: Fuzzy themselves fuzzy sets are called level 2 fuzzy sets. L11: Fuzzy A normal fuzzy set control control L12: Fuzzy L12: Fuzzy h(A) = 1 segmentation segmentation L13: Defuzzi- A : F(X ) → [0, 1] L13: Defuzzi- Joakim Lindblad, 2010-03-16 (15/146) Joakim Lindblad, 2010-03-16 (16/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Basic concepts and terminology Techniques Basic concepts and terminology Joakim Joakim Lindblad Lindblad An α-cut of a fuzzy set A is a crisp set αA that contains all Outline Outline the elements in X that have membership value in A greater L1: Intro L1: Intro than or equal to α. For any fuzzy set A and α1 < α2 it holds that α2A ⊆α1 A. L1–3: Basics L1–3: Basics L4: Constr. L4: Constr. All α-cuts and all strong α-cuts for two distinct families of and α and uncertainty A = {x | A(x) ≥ α} uncertainty nested crisp sets. L5: Features L5: Features L6 Features A strong α-cut of a fuzzy set A is a crisp set α+A that L6 Features The set of all levels α ∈ [0, 1] that represent distinct α-cuts of L7: Distances contains all the elements in X that have membership value in A L7: Distances a given fuzzy set A is called the level set of A. L8: Set L8: Set operations strictly greater than α. operations L9: Fuzzy L9: Fuzzy Λ(A) = {α | A(x) = α for some x ∈ X }. numbers numbers α+ L10: Fuzzy A = {x | A(x) > α} L10: Fuzzy logic logic L11: Fuzzy control We observe that the strong α-cut 0+A is equivalent to the L11: Fuzzy control L12: Fuzzy support supp(A). The 1-cut 1A is often called the core of A. L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (17/146) Joakim Lindblad, 2010-03-16 (18/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Basic concepts and terminology Techniques Basic concepts and terminology Joakim Joakim Lindblad Lindblad Outline Outline L1: Intro A fuzzy set A deﬁned on Rn is convex iﬀ L1: Intro L1–3: Basics L1–3: Basics L4: Constr. A(λx1 + (1 − λ)x2 ) ≥ min (A(x1 ), A(x2 )) , L4: Constr. and and uncertainty uncertainty for all λ ∈ [0, 1], x1 , x2 ∈ Rn and all α ∈ [0, 1]. Don’t forget to read in the book here! L5: Features L5: Features L6 Features L7: Distances Or, equivalently, A is convex if and only if all its α-cuts αA, for L6 Features L7: Distances Chapter 1.4 and Chapter 2. L8: Set any α in the interval α ∈ (0, 1], are convex sets. L8: Set operations operations L9: Fuzzy L9: Fuzzy numbers Any property that is generalized from classical set theory into numbers L10: Fuzzy the domain of fuzzy set theory by requiring that it holds in all L10: Fuzzy logic logic L11: Fuzzy α-cuts in the classical sense is called a cutworthy property. L11: Fuzzy control control L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (19/146) Joakim Lindblad, 2010-03-16 (20/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Standard fuzzy set operations Techniques Basic concepts and terminology Joakim Joakim Lindblad Lindblad Outline Outline L1: Intro ¯ A(x) = 1 − A(x) − fuzzy complement L1: Intro Set inclusion L1–3: Basics (A ∩ B)(x) = min[A(x), B(x)] − fuzzy intersection L1–3: Basics L4: Constr. and (A ∪ B)(x) = max[A(x), B(x)] − fuzzy union L4: Constr. and A⊆B iﬀ A(x) ≤ B(x) ∀x ∈ X uncertainty uncertainty L5: Features The standard fuzzy set operations form a De Morgan algebra L5: Features Equality L6 Features L6 Features For standard fuzzy set operations, the law of contradiction A=B iﬀ A(x) = B(x) ∀x ∈ X L7: Distances L7: Distances L8: Set operations ¯ A∩A=∅ L8: Set operations Scalar cardinality L9: Fuzzy L9: Fuzzy |A| = A(x) numbers numbers and the law of excluded middle, are violated. x∈X L10: Fuzzy L10: Fuzzy logic logic ¯ A∪A=X L11: Fuzzy L11: Fuzzy control control L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (21/146) Joakim Lindblad, 2010-03-16 (22/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Basic concepts and terminology Techniques Extension principle Joakim Joakim Lindblad Lindblad Any given function f : X → Y induces two functions, Outline Outline L1: Intro L1: Intro f : F(X ) → Y L1–3: Basics Standard fuzzy intersection and fuzzy union of two fuzzy sets L1–3: Basics [f (A)](y ) = sup A(x) L4: Constr. are cutworthy and strong cutworty. L4: Constr. x|y =f (x) and and uncertainty Due to associativity of min and max, any ﬁnite uncertainty and L5: Features intersection/union. However, take caution with inﬁnitely many L5: Features L6 Features intersections/unions. L6 Features f −1 : F(Y ) → X L7: Distances L7: Distances [f −1 (B)](x) = B(f (x)) L8: Set operations Decomposition theorems L8: Set operations L9: Fuzzy Each standard fuzzy set is uniquely represented by the family of L9: Fuzzy Strong cutworthiness numbers numbers all its α-cuts, or by the family of all its strong α-cuts. For any A ∈ F(X ), and a function f : X → Y , it holds that L10: Fuzzy L10: Fuzzy logic logic L11: Fuzzy L11: Fuzzy control control f (A) = f (α+ A) L12: Fuzzy L12: Fuzzy α∈[0,1] segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (23/146) Joakim Lindblad, 2010-03-16 (24/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques L4: Constructing fuzzy sets, Techniques L4: Constructing fuzzy sets, Joakim Joakim Lindblad Uncertainty measures Lindblad Uncertainty measures Outline Outline Methods of construction L1: Intro L1: Intro L1–3: Basics L1–3: Basics L4: Constr. L4: Constr. and and uncertainty uncertainty L5: Features L5: Features L6 Features L6 Features • Direct methods and indirect methods L7: Distances L7: Distances • One expert and multiple experts L8: Set L8: Set operations operations L9: Fuzzy L9: Fuzzy numbers numbers L10: Fuzzy L10: Fuzzy logic logic L11: Fuzzy L11: Fuzzy control control L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (25/146) Joakim Lindblad, 2010-03-16 (26/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Direct methods with one expert Techniques Direct methods with multiple Joakim Joakim Lindblad Lindblad experts Outline Outline L1: Intro L1: Intro L1–3: Basics L1–3: Basics • Deﬁne the complete membership function based on a L4: Constr. L4: Constr. and justiﬁable mathematical formula and The opinions of several experts need to be aggregated. uncertainty uncertainty • Often based on mapping of directly measurable features of L5: Features L5: Features Example: Average (Probabilistic interpretation) the elements of X L6 Features L6 Features n L7: Distances • Exemplifying it for some selected elements of X and L7: Distances 1 A(x) = ai (x) L8: Set operations interpolate (/extrapolate) MF in some way. L8: Set operations n i=1 L9: Fuzzy • Expert of some kind L9: Fuzzy numbers numbers L10: Fuzzy L10: Fuzzy logic logic L11: Fuzzy L11: Fuzzy control control L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (27/146) Joakim Lindblad, 2010-03-16 (28/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Indirect methods Techniques Uncertainty measures Joakim Joakim Lindblad Lindblad Outline Outline L1: Intro L1: Intro L1–3: Basics L1–3: Basics It may be easier/more objective to ask simpler questions to the L4: Constr. L4: Constr. and experts, than the membership directly. and uncertainty uncertainty • Nonspeciﬁcity of crisp sets L5: Features Example: Pairwise comparisons L5: Features • Nonspeciﬁcity of fuzzy sets L6 Features L6 Features • Problem: Determine membership ai = A(xi ) L7: Distances L7: Distances • Fuzziness of fuzzy sets L8: Set • Extracted information: Pairwise relative belongingness, L8: Set operations ai operations matrix P with pij ≈ aj L9: Fuzzy L9: Fuzzy numbers numbers L10: Fuzzy L10: Fuzzy logic logic L11: Fuzzy L11: Fuzzy control control L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (29/146) Joakim Lindblad, 2010-03-16 (30/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Nonspeciﬁcity Techniques Fuzziness of fuzzy sets Joakim Joakim Lindblad Hartley function Lindblad Outline Hartley [1928]: Outline A measure of fuzziness is a function L1: Intro L1: Intro The amount of uncertainty (measure in bits) associated with f : F(X ) → R+ L1–3: Basics L1–3: Basics a ﬁnite set of possible alternatives is L4: Constr. L4: Constr. and and that expresses the degree to which the boundary of a set is uncertainty U(A) = log2 |A| uncertainty non-sharp. L5: Features L5: Features L6 Features Relates to the nonspeciﬁcity inherent in each set. L6 Features Essential requirements: L7: Distances L7: Distances Generalized Hartley function: 1 f (A) = 0 iﬀ A is a crisp set L8: Set L8: Set operations h(A) operations 2 f (A) attains its maximum iﬀ A(x) = 0.5 for all x ∈ X 1 log2 |αA| dα L9: Fuzzy L9: Fuzzy numbers U(A) = numbers 3 f (A) ≤ f (B) when set A is “undoubtedly” sharper than h(A) 0 L10: Fuzzy L10: Fuzzy set B logic logic L11: Fuzzy Average of the α-cuts of the normalized counterpart of A. L11: Fuzzy a) A(x) ≤ B(x) when B(x) ≤ 0.5 control Fuzzy sets that are equal when normalized have the same control b) A(x) ≥ B(x) when B(x) ≥ 0.5 L12: Fuzzy L12: Fuzzy segmentation nonspeciﬁcity. segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (31/146) Joakim Lindblad, 2010-03-16 (32/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Fuzziness of fuzzy sets Techniques Fuzziness of fuzzy sets Joakim Joakim Lindblad Lindblad Outline Outline A simple and intuitive distance measure is the Hamming L1: Intro L1: Intro L1–3: Basics L1–3: Basics distance. L4: Constr. One way to measure fuzziness of a set A is to measure the L4: Constr. d(A, B) = |A(x) − B(x)| and uncertainty distance between A and the nearest crisp set. and uncertainty The measure of fuzziness as the distance to the complement, L5: Features L5: Features Another way is to view the fuzziness of a set as the lack of then becomes L6 Features L6 Features L7: Distances distinction between the set and its complement. L7: Distances ¯ ¯ f (A) = d(X , X ) − d(A, A) L8: Set L8: Set operations operations Both views require a distance measure. = (1 − |A(x) − (1 − A(x))|) L9: Fuzzy L9: Fuzzy numbers numbers L10: Fuzzy L10: Fuzzy = (1 − |2A(x) − 1|) logic logic L11: Fuzzy L11: Fuzzy control control L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (33/146) Joakim Lindblad, 2010-03-16 (34/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Information gain? Techniques L5: Fuzzy thresholding, Fuzzy Joakim Joakim Lindblad Lindblad c-means clustering Outline Outline L1: Intro L1: Intro L1–3: Basics Fuzziness and nonspeciﬁcity are distinct types of uncertainty L1–3: Basics L4: Constr. and totally independent of each other. L4: Constr. and and uncertainty They are also totally diﬀerent in their connections to uncertainty L5: Features information. When nonspeciﬁcity is reduced, we view this as a L5: Features L6 Features gain in information, regardless of any associated change in L6 Features L7: Distances L7: Distances fuzziness. The opposite, however, is not true. L8: Set L8: Set operations A reduction of fuzziness is reasonable to consider as a gain of operations L9: Fuzzy numbers information only if the nonspeciﬁcity also decreases or remains L9: Fuzzy numbers L10: Fuzzy the same. L10: Fuzzy logic logic L11: Fuzzy L11: Fuzzy control control L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (35/146) Joakim Lindblad, 2010-03-16 (36/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Thresholding Techniques Fuzzy c-means clustering Joakim Joakim Lindblad Lindblad Bezdek Thresholding and fuzzy thresholding of fuzzy sets, based on Outline Outline L1: Intro diﬀerent ways of measuring and minimizing fuzziness. L1: Intro L1–3: Basics L1–3: Basics L4: Constr. L4: Constr. Chapter 13.2 and and uncertainty uncertainty Algorithm L5: Features L5: Features L6 Features L6 Features • make initial guess for cluster means L7: Distances L7: Distances • iteratively L8: Set L8: Set • use the estimated means to assign samples to clusters operations operations • update means L9: Fuzzy Membership distributions assigned using L9: Fuzzy numbers numbers • until there are no changes in means L10: Fuzzy a) Pal and Rosenfeld (1988) L10: Fuzzy logic logic L11: Fuzzy b) Huang and Wang (1995) L11: Fuzzy control control c) Fuzzy c-means (Bezdek 1981) algorithms. L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (37/146) Joakim Lindblad, 2010-03-16 (38/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Fuzzy c-means clustering Techniques L6: Features of fuzzy sets Joakim Joakim Lindblad Lindblad Outline Outline L1: Intro L1: Intro L1–3: Basics • a partition of the observed set is represented by a c × n L1–3: Basics L4: Constr. matrix U = [uik ], where uik corresponds to the L4: Constr. and and uncertainty membership value (anything between 0 and 1!) of the kth uncertainty L5: Features element (out of n), to the ith cluster (out of c) L5: Features L6 Features • boundaries between subgroups are not crisp L6 Features L7: Distances L7: Distances • each element may belong to more than one cluster - its L8: Set L8: Set operations ”overall” membership equals one operations L9: Fuzzy L9: Fuzzy numbers • objective function includes parameter controlling degree of numbers L10: Fuzzy fuzziness L10: Fuzzy logic logic L11: Fuzzy L11: Fuzzy control control L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (39/146) Joakim Lindblad, 2010-03-16 (40/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques L6: Features of fuzzy sets Techniques L6: Features of fuzzy sets Joakim Joakim Lindblad Lindblad Aggregation over α-cuts Outline Outline L1: Intro L1: Intro L1–3: Basics • Spatial fuzzy sets L1–3: Basics Given a function f : P(X ) → R. L4: Constr. • Scalar descriptors of (spatial) fuzzy sets L4: Constr. We can extends this function to f : F(X ) → R, and and uncertainty • Deﬁnitions uncertainty using one of the following equations L5: Features • Inter-relations L5: Features L6 Features • Other shape descriptors L6 Features 1 L7: Distances L7: Distances f (A) = f (αA) dα, (1) • Vector-valued and non-numerical 0 L8: Set L8: Set operations • Feature estimation operations f (A) = sup [αf (αA)] (2) L9: Fuzzy • Area and higher order moments L9: Fuzzy α∈(0,1] numbers numbers • Perimeter L10: Fuzzy L10: Fuzzy logic logic Both these deﬁnitions provide consistency for the crisp case. L11: Fuzzy L11: Fuzzy control control L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (41/146) Joakim Lindblad, 2010-03-16 (42/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques L6: Features of fuzzy sets Techniques L6: Features of fuzzy sets Joakim Joakim Lindblad The area of a fuzzy set A on X ⊆ R is Lindblad Outline Outline Geometric moments: L1: Intro area(A) = A(x) dx L1: Intro The moment mp,q (A) of a fuzzy set A deﬁned on X ⊂ R2 , is L1–3: Basics X L1–3: Basics 1 L4: Constr. L4: Constr. and = area(αA) dα and mp,q (A) = A(x, y ) x p y q dxdy . uncertainty uncertainty 0 L5: Features L5: Features X L6 Features For a discrete fuzzy set, the area is equal to the cardinality of L6 Features for integers p, q ≥ 0. L7: Distances the set L7: Distances Remark: The area of a set is the m0,0 moment. L8: Set operations area(A) = |A| = A(x) L8: Set operations X Remark: The centroid (centre of gravity) of a set is L9: Fuzzy L9: Fuzzy numbers numbers L10: Fuzzy The perimeter of a fuzzy set A L10: Fuzzy m1,0 (A) m0,1 (A) logic logic (xc , yc ) = , 1 m0,0 (A) m0,0 (A) L11: Fuzzy L11: Fuzzy control perim(A) = perim(αA) dα control L12: Fuzzy 0 L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (43/146) Joakim Lindblad, 2010-03-16 (44/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Inter-relations Techniques Estimation of features Joakim Joakim Lindblad Lindblad Outline Outline L1: Intro L1: Intro All the deﬁnitions listed above reduce to the corresponding Features of a continuous spatial shape S, can be estimated L1–3: Basics L1–3: Basics L4: Constr. customary deﬁnitions for crisp sets. However, some L4: Constr. from features of its digitization D(S). and inter-relations which these notions satisfy in the crisp case, do and The precision of such estimates is limited by the spatial uncertainty uncertainty L5: Features not hold for the generalized (fuzziﬁed) deﬁnitions. L5: Features resolution of the digital representation. L6 Features For example: The isoperimetric inequality, L6 Features For object represented by digital spatial fuzzy sets, where the L7: Distances L7: Distances 2 membership of a point indicates to what extent the pixel/voxel L8: Set 4π · area(A) ≤ perim (A), L8: Set operations operations is covered by the imaged object, signiﬁcant improvements in L9: Fuzzy L9: Fuzzy precision of feature estimates can be obtained. Especially for numbers Bogomolny proposed (1987) modiﬁed deﬁnitions. However, numbers small objects/limited resolution. L10: Fuzzy these deﬁnitions are often seen as less intuitive. L10: Fuzzy logic logic L11: Fuzzy L11: Fuzzy control control L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (45/146) Joakim Lindblad, 2010-03-16 (46/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Estimation of features Techniques Estimation of features Joakim Joakim Lindblad Lindblad Pixel coverage digitization If the membership function correponds to pixel/voxel coverage Outline Outline then it is possible to derive very precise estimates. L1: Intro L1: Intro L1–3: Basics L1–3: Basics Trade-oﬀ between spatial and grey-level resolution L4: Constr. Signiﬁcant improvement in the precision of L4: Constr. and and uncertainty feature estimates can be achieved using a fuzzy uncertainty L5: Features L5: Features 2 approach. 10 Absolute value of rel. error in % L6 Features L6 Features Relative error in % 0 L7: Distances L7: Distances L8: Set Exploiting fuzzy can provide an alternative to L8: Set −2 1 operations operations L9: Fuzzy increasing the spatial resolution of the image. L9: Fuzzy −4 n= 1 n= 2 n= 3 n= 1 n= 2 n= 3 numbers numbers n= 5 n= 5 n=10 n=10 n= ∞ n= ∞ L10: Fuzzy L10: Fuzzy −6 0.1 logic logic 0 200 400 600 800 1000 10 100 1000 Grid resolution Grid resolution L11: Fuzzy L11: Fuzzy control control Figure: Relative error of perimeter estimates for diﬀerent membership L12: Fuzzy L12: Fuzzy segmentation segmentation resolutions. L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (47/146) Joakim Lindblad, 2010-03-16 (48/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Estimation of features Techniques L7: Distances on and between Joakim Joakim Lindblad Lindblad fuzzy sets Outline Outline L1: Intro L1: Intro L1–3: Basics L1–3: Basics • Spatial fuzzy sets are of a particular interest in image L4: Constr. L4: Constr. and analysis. and uncertainty uncertainty L5: Features • Features of spatial fuzzy sets → shape descriptors. L5: Features L6 Features • “Horizontal” and “vertical” approach in deﬁnitions. L6 Features L7: Distances L7: Distances • Particular membership functions → High precision L8: Set L8: Set operations estimates. operations L9: Fuzzy numbers • Fuzzy feature values? (Still relatively unexplored) L9: Fuzzy numbers L10: Fuzzy L10: Fuzzy logic logic L11: Fuzzy L11: Fuzzy control control L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (49/146) Joakim Lindblad, 2010-03-16 (50/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques L7: Distances on and between Techniques Set to set distances Joakim Joakim Lindblad fuzzy sets Lindblad Outline Outline L1: Intro • Set to set distances L1: Intro L1–3: Basics L1–3: Basics • (Point to set distances) Distances between fuzzy sets L4: Constr. L4: Constr. and • Point to point distances and uncertainty uncertainty a) Membership focused (vertical) L5: Features A mix of notions L5: Features b) Spatially focused (horizontal) L6 Features L6 Features L7: Distances • The objects that the distance is measured between (start L7: Distances c) Mix of spatial and membership (tolerance) L8: Set and stop) L8: Set d) Feature distances (low or high dimensional representations) operations operations - crisp or fuzzy, point or set L9: Fuzzy L9: Fuzzy e) Morphological (mixed focus) numbers • The space where a path between start and stop is numbers L10: Fuzzy embedded (spatial cost function) L10: Fuzzy logic logic - Unconstrained (Euclidean) L11: Fuzzy L11: Fuzzy control - Constrained (geodesic/cost function) control L12: Fuzzy • Output: Crisp (number) or fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (51/146) Joakim Lindblad, 2010-03-16 (52/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Membership focused Techniques Membership focused Joakim Joakim Lindblad Lp norm Lindblad Lp norm Outline Outline Discrete version: L1: Intro “The functional approach” L1: Intro L1–3: Basics L1–3: Basics L4: Constr. The most common: L4: Constr. 1/p n and Based on the family of Minkowski distances and uncertainty uncertainty dp (A, B) = |µA (xi ) − µB (xi )|p , p ≥ 1, L5: Features L5: Features i=1 L6 Features L6 Features 1/p d∞ (A, B) = max (|µA (xi ) − µB (xi )|) . i=1...n L7: Distances dp (A, B) = |µA (x) − µB (x)|p dx , p ≥ 1, L7: Distances L8: Set X L8: Set operations dEssSup (A, B) = lim dp (A, B) operations dp for p ≥ 1 are all metrics in the discrete case. L9: Fuzzy p→∞ L9: Fuzzy numbers numbers d∞ (A, B) = sup |µA (x) − µB (x)| . L4: Hamming distance. L10: Fuzzy L10: Fuzzy logic x∈X logic L11: Fuzzy L11: Fuzzy d1 (A, B) = |A(x) − B(x)| control control L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (53/146) Joakim Lindblad, 2010-03-16 (54/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Membership focused Techniques Spatially focused Joakim Joakim Lindblad Set operations approach Lindblad Tversky 1977, et al. deﬁnes a measure of similarity based on Outline Outline three components as follows: L1: Intro L1: Intro L1–3: Basics S(a, b) = θf (A ∩ B) − αf (A − B) − βf (B − A) L1–3: Basics • Nearest point L4: Constr. and L4: Constr. and • Mean distance uncertainty uncertainty • Hausdorﬀ L5: Features L5: Features L6 Features L6 Features L7: Distances L7: Distances Three (four) approaches: L8: Set operations L8: Set operations • fuzzy distance L9: Fuzzy L9: Fuzzy • weighting with membership numbers numbers L10: Fuzzy L10: Fuzzy • morphological and integration of alpha-cuts logic logic L11: Fuzzy L11: Fuzzy control Figure: Representation of two objects that each contains its own control L12: Fuzzy unique features and also contains common features. L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (55/146) Joakim Lindblad, 2010-03-16 (56/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Spatially focused Techniques Spatially focused Joakim Joakim Lindblad Hausdorﬀ Lindblad Hausdorﬀ Outline Crisp: Outline L1: Intro L1: Intro Ralescu and Ralescu (1984) L1–3: Basics dH (A, B) = max{sup inf d(x, y ), sup inf d(x, y )} L1–3: Basics L4: Constr. x∈A y ∈B y ∈A x∈B L4: Constr. 1 and uncertainty + = inf{r ∈ R | A ⊆ Dr (B) ∧ B ⊆ Dr (A)} and uncertainty dH1 (A, B) = dH (αA,α B) dα, 0 L5: Features L5: Features dH∞ (A, B) = sup dH (αA,α B), L6 Features where Dr (A) is the dilation of the set A by a ball of radius r L6 Features α>0 L7: Distances L7: Distances Dr (A) = {y ∈ X | ∃x ∈ A : d(x, y ) ≤ r } L8: Set L8: Set where dH is the Hausdorﬀ distance between two crisp sets, operations operations L9: Fuzzy The Hausdorﬀ distance between A and B is the smallest L9: Fuzzy A serious problem is that the distance between two fuzzy sets A numbers numbers L10: Fuzzy amount that A must be expanded to contain B and vice versa. L10: Fuzzy and B is inﬁnite if height(A) = height(B). logic logic Is a metric on the set of nonempty compact sets. L11: Fuzzy L11: Fuzzy No good solution to that problem is found. control Remark: control L12: Fuzzy segmentation Usually extended with: dH (A, ∅) = ∞ and dH (∅, ∅) = 0 L12: Fuzzy segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (57/146) Joakim Lindblad, 2010-03-16 (58/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Feature distances Techniques Point to point distances Joakim Joakim Lindblad “Pattern recognition approach” Lindblad Outline Outline L1: Intro L1: Intro L1–3: Basics L1–3: Basics L4: Constr. L4: Constr. and Use of a feature representation of the observed sets as an and uncertainty uncertainty intermediate step in the distance calculations. Distances between points in a fuzzy set L5: Features L5: Features L6 Features The distance between sets A and B is then given in terms of L6 Features Deﬁning the cost of traveling along a path L7: Distances the distance between their feature vectors. L7: Distances L8: Set operations Often global shape features are used (think shape matching, L8: Set operations L9: Fuzzy image retrieval). L9: Fuzzy numbers numbers L10: Fuzzy L10: Fuzzy logic logic L11: Fuzzy L11: Fuzzy control control L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (59/146) Joakim Lindblad, 2010-03-16 (60/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Cost function Techniques Cost function Joakim Joakim Lindblad “Snow shoveling” Lindblad variations Outline Outline L1: Intro L1: Intro L1–3: Basics Deﬁne the distance along a path πi between points x and y in L1–3: Basics the fuzzy set A Membership as another dimension L4: Constr. L4: Constr. and and integrate the arc-length uncertainty uncertainty L5: Features dA (πi (x, y )) = A(t) dt L5: Features Bloch 1995, Toivanen 1996: s∈π L6 Features L6 Features 2 L7: Distances L7: Distances dA(t) The distance between points x and y in A is dA (π) = 1+ dt L8: Set operations the distance along the shortest path L8: Set operations s∈π dt L9: Fuzzy L9: Fuzzy numbers dA (x, y ) = inf dA (π) numbers Problem: How to relate scale of membership to spatial distance L10: Fuzzy π∈Π(x,y ) L10: Fuzzy in the domain? logic logic L11: Fuzzy L11: Fuzzy control control L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (61/146) Joakim Lindblad, 2010-03-16 (62/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Constrained distance Techniques Connectedness Joakim Joakim Lindblad Lindblad Outline Outline L1: Intro Geodesic distance – shortest path within the set; not allowed to L1: Intro Connectedness, Rosenfeld 1979 L1–3: Basics go out of the set – a path that descends the least in terms of L1–3: Basics Strength of a path – the strength of its weakest link L4: Constr. and membership. L4: Constr. and uncertainty uncertainty Strength of a link between two points deﬁned by the L5: Features ıtre Bloch and Maˆ 1995: L5: Features membership function. L6 Features L6 Features ds The connectedness of two points x and y in A – π L7: Distances d(x, y ) = inf L7: Distances the strength of the strongest path between x and y L8: Set π∈ΠcA (x,y ) cA (x, y ) L8: Set operations operations L9: Fuzzy numbers where cA (x, y ) is the strength of connectedness of points x L9: Fuzzy numbers cA (x, y ) = sup inf A(t) and y , and ΠcA (x, y ) is the set of path contained within the π∈Π(x,y ) t∈π L10: Fuzzy L10: Fuzzy logic α-cut cAA. logic L11: Fuzzy L11: Fuzzy control control L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (63/146) Joakim Lindblad, 2010-03-16 (64/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques L8: Operations on fuzzy sets Techniques Standard fuzzy operations Joakim Joakim Lindblad Lindblad Outline Outline L1: Intro L1: Intro L1–3: Basics L1–3: Basics L4: Constr. L4: Constr. and and uncertainty uncertainty L5: Features L5: Features ¯ A(x) = 1 − A(x) − fuzzy complement L6 Features L6 Features (A ∩ B)(x) = min[A(x), B(x)] − fuzzy intersection L7: Distances L7: Distances (A ∪ B)(x) = max[A(x), B(x)] − fuzzy union L8: Set L8: Set operations operations L9: Fuzzy L9: Fuzzy numbers numbers L10: Fuzzy L10: Fuzzy logic logic L11: Fuzzy L11: Fuzzy control control L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (65/146) Joakim Lindblad, 2010-03-16 (66/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Properties of the standard Techniques Aggregation operators Joakim Joakim Lindblad operations Lindblad Outline Outline L1: Intro L1: Intro L1–3: Basics L1–3: Basics Aggregation operators are used to combine several fuzzy sets in L4: Constr. • They are generalizations of the corresponding (uniquely L4: Constr. order to produce a single fuzzy set. and and uncertainty deﬁned!) classical set operations. uncertainty Associative aggregation operations L5: Features • They satisfy the cutworthy and strong cutworthy L5: Features L6 Features L6 Features • (general) fuzzy intersections - t-norms properties. They are the only ones that do. L7: Distances L7: Distances • (general) fuzzy unions - t-conorms L8: Set • The standard fuzzy intersection of two sets contains (is L8: Set operations bigger than) all other fuzzy intersections of those sets. operations Non-associative aggregation operations L9: Fuzzy L9: Fuzzy numbers • The standard fuzzy union of two sets is contained in (is numbers • averaging operations - idempotent aggregation operations L10: Fuzzy smaller than) all other fuzzy unions of those sets. L10: Fuzzy logic logic L11: Fuzzy L11: Fuzzy control control L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (67/146) Joakim Lindblad, 2010-03-16 (68/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Fuzzy complements Techniques Generators Joakim Joakim Lindblad Axiomatic requirements Lindblad Increasing generators Outline Outline • Increasing generator L1: Intro L1: Intro is a strictly increasing continuous function g : [0, 1] → R, L1–3: Basics L1–3: Basics such that g (0) = 0. L4: Constr. L4: Constr. • A pseudo-inverse of increasing generator g is deﬁned as and and uncertainty Ax c1. c(0) = 1 and c(1) = 0. boundary condition uncertainty 0 for a ∈ (−∞, 0) L5: Features Ax c2. For all a, b ∈ [0, 1], if a ≤ b, then c(a) ≥ c(b). monotonicity L5: Features g (−1) = g −1 (a) for a ∈ [0, g (1)] L6 Features c1 and c2 are called axiomatic skeleton for fuzzy complements L6 Features 1 for a ∈ (g (1), ∞) L7: Distances L7: Distances Ax c3. c is a continuous function. L8: Set L8: Set • An example: operations Ax c4. c is involutive, i.e., c(c(a)) = a, for each a ∈ [0, 1]. operations L9: Fuzzy numbers L9: Fuzzy numbers = ap , p > 0 g (a) L10: Fuzzy L10: Fuzzy 0 for a ∈ (−∞, 0) 1 logic logic g (−1) (a) = a p for a ∈ [0, 1] L11: Fuzzy L11: Fuzzy 1 for a ∈ (1, ∞) control control L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- Joakim Lindblad, 2010-03-16 (69/146) L13: Defuzzi- Similarly for Decreasing generators (70/146) Joakim Lindblad, 2010-03-16 ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Generating fuzzy complements Techniques Fuzzy intersections Joakim Joakim Lindblad Lindblad Axiomatic requirements Outline Theorem Outline L1: Intro (First Characterization Theorem of Fuzzy Complements.) L1: Intro L1–3: Basics Let c be a function from [0, 1] to [0, 1]. Then c is a L1–3: Basics For all a, b, d ∈ [0, 1], L4: Constr. and (involutive) fuzzy complement iﬀ there exists an increasing L4: Constr. and Ax i1. i(a, 1) = a. boundary condition uncertainty generator g such that, for all a ∈ [0, 1] uncertainty Ax i2. b ≤ d implies i(a, b) ≤ i(a, d). monotonicity L5: Features L5: Features Ax i3. i(a, b) = i(b, a). commutativity −1 L6 Features c(a) = g (g (1) − g (a)). L6 Features Ax i4. i(a, i(b, d)) = i(i(a, b), d). associativity L7: Distances L7: Distances L8: Set L8: Set operations Theorem operations Axioms i1 - i4 are called axiomatic skeleton for fuzzy L9: Fuzzy numbers (Second Characterization Theorem of Fuzzy Complements.) L9: Fuzzy numbers intersections. L10: Fuzzy Let c be a function from [0, 1] to [0, 1]. Then c is a L10: Fuzzy If the sets are crisp, i becomes the classical (crisp) logic logic (involutive) fuzzy complement iﬀ there exists an decreasing intersection. L11: Fuzzy L11: Fuzzy control generator f such that, for all a ∈ [0, 1] control L12: Fuzzy L12: Fuzzy segmentation c(a) = f −1 (f (0) − f (a)). segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (71/146) Joakim Lindblad, 2010-03-16 (72/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Fuzzy intersections Techniques Fuzzy intersections Joakim Joakim Lindblad Additional (optional) requirements Lindblad Examples of t-norms frequently used Outline Outline L1: Intro L1: Intro • Drastic intersection L1–3: Basics For all a, b, d ∈ [0, 1], L1–3: Basics 8 < a if b = 1 L4: Constr. L4: Constr. i(a, b) = b if a = 1 and Ax i5. i is a continuous function. continuity and 0 otherwise : uncertainty uncertainty Ax i6. i(a, a) ≤ a. subidempotency L5: Features L5: Features • Bounded diﬀerence Ax i7. a1 < a2 and b1 < b2 implies i(a1 , b1 ) < i(a2 , b2 ). i(a, b) = max[0, a + b − 1] L6 Features L6 Features strict monotonicity L7: Distances L7: Distances • Algebraic product L8: Set L8: Set i(a, b) = ab operations operations Note: • Standard intersection L9: Fuzzy L9: Fuzzy i(a, b) = min[a, b] numbers The standard fuzzy intersection, i(a, b) = min[a, b], is the only numbers L10: Fuzzy idempotent t-norm. L10: Fuzzy logic logic L11: Fuzzy L11: Fuzzy control control • imin (a, b) ≤ max(0, a + b − 1) ≤ ab ≤ min(a, b). L12: Fuzzy L12: Fuzzy segmentation segmentation • For all a, b ∈ [0, 1], imin (a, b) ≤ i(a, b) ≤ min[a, b]. L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (73/146) Joakim Lindblad, 2010-03-16 (74/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Fuzzy intersections Techniques Fuzzy unions Joakim Joakim Lindblad How to generate t-norms Lindblad Axiomatic requirements Outline Outline L1: Intro L1: Intro L1–3: Basics L1–3: Basics For all a, b, d ∈ [0, 1], L4: Constr. L4: Constr. Ax u1. u(a, 0) = a. boundary condition and uncertainty Theorem and uncertainty Ax u2. b ≤ d implies u(a, b) ≤ u(a, d). monotonicity L5: Features (Characterization Theorem of t-norms) Let i be a binary L5: Features Ax u3. u(a, b) = u(b, a). commutativity L6 Features operation on the unit interval. Then, i is an Archimedean L6 Features Ax u4. u(a, u(b, d)) = u(u(a, b), d). associativity L7: Distances t-norm iﬀ there exists a decreasing generator f such that L7: Distances L8: Set L8: Set operations i(a, b) = f (−1) (f (a) + f (b)), for a, b ∈ [0, 1]. operations Axioms u1 - u4 are called axiomatic skeleton for fuzzy unions. L9: Fuzzy L9: Fuzzy They diﬀer from the axiomatic skeleton of fuzzy intersections only numbers numbers in boundary condition. L10: Fuzzy L10: Fuzzy logic logic For crisp sets, u behaves like a classical (crisp) union. L11: Fuzzy L11: Fuzzy control control L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (75/146) Joakim Lindblad, 2010-03-16 (76/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Fuzzy unions Techniques Fuzzy unions Joakim Joakim Lindblad Additional (optional) requirements Lindblad Examples of t-conorms frequently used Outline Outline L1: Intro L1: Intro • Drastic union For all a, b, d ∈ [0, 1], L1–3: Basics L1–3: Basics 8 < a if b = 0 L4: Constr. Ax u5. u is a continuous function. continuity L4: Constr. u(a, b) = b if a = 0 and and 1 otherwise : uncertainty uncertainty Ax u6. u(a, a) ≥ a. superidempotency L5: Features L5: Features • Bounded sum Ax u7. a1 < a2 and b1 < b2 implies u(a1 , b1 ) < u(a2 , b2 ). u(a, b) = min[1, a + b] L6 Features L6 Features strict monotonicity L7: Distances L7: Distances • Algebraic sum L8: Set L8: Set u(a, b) = a + b − ab Note: operations operations • Standard intersection L9: Fuzzy Requirements u5 - u7 are analogous to Axioms i5 - i7. L9: Fuzzy u(a, b) = max[a, b] numbers numbers The standard fuzzy union, u(a, b) = max[a, b], is the only idempotent L10: Fuzzy L10: Fuzzy logic t-conorm. logic L11: Fuzzy control L11: Fuzzy control • max[a, b] ≤ a + b − ab ≤ min(1, a + b) ≤ umax (a, b). L12: Fuzzy L12: Fuzzy • For all a, b ∈ [0, 1], max[a, b] ≤ u(a, b) ≤ umax (a, b). segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (77/146) Joakim Lindblad, 2010-03-16 (78/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Combinations of set operations Techniques Dual triples - Six theorems (1) Joakim Joakim Lindblad De Morgan laws and duality of fuzzy operations Lindblad Theorem Outline Outline The triples min, max, c and imin , umax , c are dual with respect to any L1: Intro L1: Intro fuzzy complement c. De Morgan laws in classical set theory: L1–3: Basics L1–3: Basics L4: Constr. ¯ ¯ ¯ ¯ L4: Constr. Theorem and A∩B =A∪B and A ∪ B = A ∩ B. and uncertainty uncertainty Given a t-norm i and an involutive fuzzy complement c, the binary L5: Features L5: Features operation u on [0, 1], deﬁned for all a, b ∈ [0, 1] by The union and intersection operation are dual with respect to L6 Features L6 Features the complement. u(a, b) = c(i(c(a), c(b))) L7: Distances L7: Distances L8: Set L8: Set is a t-conorm such that i, u, c is a dual triple. operations De Morgan laws for fuzzy sets: operations L9: Fuzzy L9: Fuzzy numbers c(i(A, B)) = u(c(A), c(B)) and c(u(A, B)) = i(c(A), c(B)) numbers Theorem L10: Fuzzy L10: Fuzzy Given a t-conorm u and an involutive fuzzy complement c, the binary logic logic operation i on [0, 1], deﬁned for all a, b ∈ [0, 1] by L11: Fuzzy for a t-norm i, a t-conorm u, and fuzzy complement c. L11: Fuzzy control control i(a, b) = c(u(c(a), c(b))) L12: Fuzzy Notation: i, u, c denotes a dual triple. L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- is a t-norm such that i, u, c is a dual triple. Joakim Lindblad, 2010-03-16 (79/146) Joakim Lindblad, 2010-03-16 (80/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Dual triples - Six theorems (2) Techniques Aggregation operations Joakim Joakim Lindblad Lindblad Deﬁnition Outline Outline L1: Intro Theorem L1: Intro L1–3: Basics Given an involutive fuzzy complement c and an increasing generator g of L1–3: Basics Aggregations on fuzzy sets are operations by which several L4: Constr. c, the t-norm and the t-conorm generated by g are dual with respect to c. L4: Constr. fuzzy sets are combined in a desirable way to produce a single and and uncertainty uncertainty fuzzy set. L5: Features Theorem L5: Features L6 Features Let i, u, c be a dual triple generated by an increasing generator g of the L6 Features Deﬁnition L7: Distances involutive fuzzy complement c. Then the fuzzy operations i, u, c satisfy the L7: Distances Aggregation operation on n fuzzy sets (n ≥ 2) is a function L8: Set law of excluded middle, and the law of contradiction. L8: Set operations operations h : [0, 1]n → [0, 1]. L9: Fuzzy L9: Fuzzy numbers Theorem numbers Applied to fuzzy sets A1 , A2 , . . . , An , function h produces an aggregate L10: Fuzzy L10: Fuzzy logic Let i, u, c be a dual triple that satisﬁes the law of excluded middle and logic fuzzy set A, by operating on membership grades to these sets for each the law of contradiction. Then i, u, c does not satisfy the distributive x ∈ X: L11: Fuzzy L11: Fuzzy control laws. control A(x) = h(A1 (x), A2 (x), . . . , An (x)). L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (81/146) Joakim Lindblad, 2010-03-16 (82/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Axiomatic requirements Techniques Averaging operations Joakim Joakim Lindblad Lindblad Outline Outline L1: Intro L1: Intro Ax h1 h(0, 0, . . . , 0) = 0 and h(1, 1, . . . , 1) = 1. boundary conditions • If an aggregation operator h is monotonic and idempotent (Ax L1–3: Basics L1–3: Basics L4: Constr. Ax h2 For any (a1 , a2 , . . . , an ) and (b1 , b2 , . . . , bn ), such that ai , bi ∈ [0, 1] L4: Constr. h2 and Ax h5), then for all (a1 , a2 , . . . , an ) ∈ [0, 1]n and and ai ≤ bi for i = 1, . . . , n, and uncertainty uncertainty h(a1 , a2 , . . . , an ) ≤ h(b1 , b2 , . . . , bn ). min(a1 , a2 , . . . , an ) ≤ h(a1 , a2 , . . . , an ) ≤ max(a1 , a2 , . . . , an ). L5: Features L5: Features L6 Features h is monotonic increasing in all its arguments. L6 Features • All aggregation operators between the standard fuzzy L7: Distances Ax h3 h is continuous. L7: Distances intersection and the standard fuzzy union are idempotent. L8: Set Ax h4 h is a symmetric function in all its arguments; for any permutation p L8: Set operations on {1, 2, . . . , n} operations • The only idempotent aggregation operators are those between L9: Fuzzy h(a1 , a2 , . . . , an ) = h(ap(1) , ap(2) , . . . , ap(n) ). L9: Fuzzy standard fuzzy intersection and standard fuzzy union. numbers numbers L10: Fuzzy Ax h5 h is an idempotent function; for all a ∈ [0, 1] L10: Fuzzy logic h(a, a, . . . , a) = a. logic Idempotent aggregation operators are called averaging L11: Fuzzy L11: Fuzzy operations. control control L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (83/146) Joakim Lindblad, 2010-03-16 (84/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Averaging operations Techniques Do we need more than standard Joakim Joakim Lindblad Lindblad operations? Outline Generalized means: Outline L1: Intro 1 L1: Intro L1–3: Basics α α α a1 + a2 + · · · + an α L1–3: Basics Intersection: No positive compensation (trade-oﬀ) hα (a1 , a2 , . . . , an ) = , between the memberships of the fuzzy sets observed. L4: Constr. n L4: Constr. and and uncertainty uncertainty Union: Full compensation of lower degrees of membership L5: Features for α ∈ R, and α = 0, and for α < 0 ai = 0. L5: Features by the maximal membership. L6 Features • Geometric mean: For α → 0, L6 Features 1 L7: Distances lim hα (a1 , a2 , . . . , an ) = (a1 · a2 · · · · · an ) n ; α→0 L7: Distances In reality of decision making, rarely either happens. L8: Set L8: Set operations • Harmonic mean: For α = −1, operations L9: Fuzzy h−1 (a1 , a2 , . . . , an ) = n ; L9: Fuzzy (non-verbal) “merging connectives” → (language) connectives 1 1 1 numbers a1 + a2 + ··· + an numbers {’and’, ’or’,...,}. L10: Fuzzy logic • Arithmetic mean: For α = 1, L10: Fuzzy logic 1 L11: Fuzzy h1 (a1 , a2 , . . . , an ) = (a1 + a2 + . . . an ). L11: Fuzzy Aggregation operations called compensatory and are needed control n control L12: Fuzzy L12: Fuzzy to model fuzzy sets representing to, e.g., managerial decisions. segmentation Functions hα satisfy axioms Ax h1 - Ax h5. segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (85/146) Joakim Lindblad, 2010-03-16 (86/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques An Application: Fuzzy Techniques L9: Fuzzy numbers and fuzzy Joakim Joakim Lindblad morphologies Lindblad arithmetics Outline Morphological operations Outline L1: Intro L1: Intro L1–3: Basics • Mathematical morphology is completely based on set L1–3: Basics L4: Constr. L4: Constr. and theory. Fuzziﬁcation started in 1980s. and uncertainty uncertainty • Basic morphological operations are dilation and erosion. L5: Features L5: Features L6 Features Many others can be derived from them. L6 Features L7: Distances • Dilation and erosion are, in crisp case, dual operations L7: Distances L8: Set with respect to the complementation: D(A) = c(E (cA)). L8: Set operations operations L9: Fuzzy • In crisp case, dilation and erosion fulﬁl a certain number of L9: Fuzzy numbers numbers properties. L10: Fuzzy L10: Fuzzy logic logic Main construction principles: L11: Fuzzy L11: Fuzzy control α-cut decomposition; control L12: Fuzzy segmentation fuzziﬁcation of set operations. L12: Fuzzy segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (87/146) Joakim Lindblad, 2010-03-16 (88/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Interval numbers Techniques Interval numbers Joakim Joakim Lindblad Lindblad For closed intervals A = [a1 , a2 ] and B = [b1 , b2 ], the four Outline Outline arithmetic operations are deﬁned as follows (equivalent with L1: Intro An interval number, representing an uncertain real number L1: Intro L1–3: Basics L1–3: Basics deﬁnition on previous slide) L4: Constr. A = [a1 , a2 ] = {x | a1 ≤ x ≤ a2 , x ∈ R} L4: Constr. and and uncertainty uncertainty A+B = [a1 + b1 , a2 + b2 ] L5: Features For intervals A and B, and operator ∗ ∈ {+, −, ·, /} L5: Features L6 Features we deﬁne L6 Features A−B =A+ B− = [a1 − b2 , a2 − b1 ] L7: Distances A ∗ B = {a ∗ b | a ∈ A, b ∈ B} L7: Distances L8: Set L8: Set A·B = [min(a1 b1 , a1 b2 , a2 b1 , a2 b2 ), operations Division, A/B, is not deﬁned when 0 ∈ B. operations L9: Fuzzy L9: Fuzzy max(a1 b1 , a1 b2 , a2 b1 , a2 b2 )] numbers numbers The result of an arithmetic operation on closed intervals is and, if 0 ∈ [b1 , b2 ] / L10: Fuzzy L10: Fuzzy 1 1 logic again a closed interval. logic A/B = A · B −1 = [a1 , a2 ] · [ b2 , b1 ] a a a a a a a a L11: Fuzzy control L11: Fuzzy control = [min( b1 , b1 , b2 , b2 ), max( b1 , b1 , b2 , b2 )]. 1 2 1 2 1 2 1 2 L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (89/146) Joakim Lindblad, 2010-03-16 (90/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Fuzzy numbers and fuzzy intervals Techniques Fuzzy numbers and fuzzy intervals Joakim Joakim Lindblad Lindblad Outline Outline L1: Intro L1: Intro Theorem (4.1) L1–3: Basics A fuzzy number is a fuzzy set on R L1–3: Basics Let A ∈ F(R). Then, A is a fuzzy number iﬀ there exists a L4: Constr. L4: Constr. closed interval [a, b] = ∅ such that and uncertainty A : R → [0, 1] and uncertainty L5: Features L5: Features 1 for x ∈ [a, b] L6 Features such that L6 Features A(x) = l(x) for x ∈ (−∞, a) L7: Distances L7: Distances r (x) for x ∈ (b, ∞) (i) A is normal (height(A) = 1) L8: Set L8: Set operations (ii) αA is a closed interval for all α ∈ (0, 1] operations where l : (−∞, a) → [0, 1] is monotonic non-decreasing, L9: Fuzzy L9: Fuzzy (iii) The support of A, Supp(A) = 0+A, is bounded numbers numbers continuous from the right, and l(x) = 0 for x < ω1 L10: Fuzzy logic L10: Fuzzy logic and r : (b, ∞) → [0, 1] is monotonic non-increasing, continuous L11: Fuzzy L11: Fuzzy from the left, and r (x) = 0 for x > ω2 . control control L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (91/146) Joakim Lindblad, 2010-03-16 (92/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Arithmetics on fuzzy numbers Techniques Arithmetics on fuzzy numbers Joakim Joakim Lindblad Lindblad Outline Moving from interval numbers, we can deﬁne arithmetics on Outline L1: Intro fuzzy numbers based on two principles: L1: Intro Theorem (4.2) L1–3: Basics 1 Cutworthiness (thanks to inclusion monotonicity of L1–3: Basics L4: Constr. intervals) L4: Constr. Let ∗ ∈ {+, −, ·, /}, and let A, B denote continuous fuzzy and and uncertainty α (A ∗ B) =αA ∗αB uncertainty numbers. Then, the fuzzy set A ∗ B deﬁned by the extension L5: Features L5: Features principle (prev. slide) is a continuous fuzzy number. L6 Features in combination with L6 Features L7: Distances A∗B = α (A ∗ B) L7: Distances Lemma L8: Set operations α∈(0,1] L8: Set operations (A ∗ B)(z) = sup min [A(x), B(x)] ⇒ α(A ∗ B) =αA ∗αB z=x∗y L9: Fuzzy L9: Fuzzy numbers 2 or the extension principle numbers So the two deﬁnitions are equivalent for continuous fuzzy L10: Fuzzy logic L10: Fuzzy logic numbers. (The proof is built on continuity.) L11: Fuzzy (A ∗ B)(z) = sup min [A(x), B(x)] L11: Fuzzy control z=x∗y control L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (93/146) Joakim Lindblad, 2010-03-16 (94/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques MIN and MAX operators Techniques MIN and MAX operators Joakim Joakim Lindblad Lindblad Outline MIN(A, B)(z) = sup min [A(x), B(x)] , Outline L1: Intro z=min(x,y ) L1: Intro L1–3: Basics MAX(A, B)(z) = sup min [A(x), B(x)] L1–3: Basics L4: Constr. z=max(x,y ) L4: Constr. and and uncertainty uncertainty Again, for continuous fuzzy numbers, this is equivalent with a L5: Features L5: Features deﬁnition based on cutworthiness. L6 Features L6 Features α α α L7: Distances (MIN(A, B)) = MIN( A, B), L7: Distances α L8: Set operations (MAX(A, B)) = MAX(αA,α B), ∀α ∈ (0, 1]. L8: Set operations L9: Fuzzy L9: Fuzzy numbers numbers L10: Fuzzy logic Where, for intervals [a1 , a2 ], [b1 , b2 ] L10: Fuzzy logic L11: Fuzzy MIN([a1 , a2 ], [b1 , b2 ]) = [min(a1 , b1 ), min(a2 , b2 )], L11: Fuzzy control control MAX([a1 , a2 ], [b1 , b2 ]) = [max(a1 , b1 ), max(a2 , b2 )]. L12: Fuzzy L12: Fuzzy Figure: Comparison of the operators MIN, min, MAX, and max. segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (95/146) Joakim Lindblad, 2010-03-16 (96/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Arithmetics on fuzzy numbers Techniques Linguistic variables Joakim Joakim Lindblad Lindblad When fuzzy numbers are connected to linguistic concepts, such as very small, small, medium, and interpreted in a particular Outline We can deﬁne a partial ordering on the set of fuzzy numbers Outline context, the resulting constructs are usually called linguistic L1: Intro L1: Intro R by variables. L1–3: Basics L1–3: Basics L4: Constr. and A B ⇔ MIN(A, B) = A or, alternatively L4: Constr. and uncertainty uncertainty A B ⇔ MAX(A, B) = B L5: Features L5: Features L6 Features L6 Features L7: Distances Not all fuzzy numbers are comparable (only partial order). L7: Distances L8: Set However, values of linguistic variables are often deﬁned by L8: Set operations fuzzy numbers that are comparable. operations L9: Fuzzy L9: Fuzzy numbers For example: numbers L10: Fuzzy L10: Fuzzy logic logic very small small medium large very large L11: Fuzzy L11: Fuzzy control control L12: Fuzzy L12: Fuzzy Figure: An example of a linguistic variable. segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (97/146) Joakim Lindblad, 2010-03-16 (98/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Interval equations Techniques Fuzzy equations Joakim Joakim Lindblad Equation A + X = B Lindblad The solution to a fuzzy equation can be obtained by solving a Outline Outline set of interval equations, one for each nonzero α in the level set L1: Intro L1: Intro L1–3: Basics L1–3: Basics Λ(A) ∪ Λ(B). L4: Constr. A+X =B L4: Constr. and and The equation A + X = B has a solution iﬀ uncertainty Let X = [x1 , x2 ]. uncertainty L5: Features Then [a1 + x1 , a2 + x2 ] = [b1 , b2 ] follows immediately. L5: Features (i) αb1 −α a1 ≤α b2 −α a2 for every α ∈ (0, 1], and L6 Features Clearly: x1 = b1 − a1 and x2 = b2 − a2 . L6 Features (ii) α ≤ β implies L7: Distances L7: Distances αb −α a ≤β b −β a ≤β b −β a ≤α b −α a . 1 1 1 1 2 2 2 2 L8: Set Since X must be an interval, it is required that x1 ≤ x2 . L8: Set operations That is, the equation has a solution iﬀ b1 − a1 ≤ b2 − a2 . operations If a solution αX exists for every α ∈ (0, 1] (property (i)), L9: Fuzzy L9: Fuzzy numbers numbers and property (ii) is satisﬁed, then the solution X is given by Then X = [b1 − a1 , b2 − a2 ] is the solution. L10: Fuzzy L10: Fuzzy logic logic X = αX L11: Fuzzy L11: Fuzzy control control α∈(0,1] L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (99/146) Joakim Lindblad, 2010-03-16 (100/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Fuzzy equations Techniques L10: Fuzzy logic and approximate Joakim Joakim Lindblad Equation A · X = B Lindblad reasoning Outline Outline L1: Intro Similarly as A + X = B L1: Intro L1–3: Basics L1–3: Basics The equation A · X = B has a solution iﬀ L4: Constr. L4: Constr. and αb /αa and uncertainty (i) 1 1 ≤αb2 /αa2 for every α ∈ (0, 1], and uncertainty L5: Features (ii) α ≤ β implies αb1 /αa1 ≤βb1 /βa1 ≤βb2 /βa2 ≤αb2 /αa2 . L5: Features L6 Features L6 Features L7: Distances If the solution exists, it has the form L7: Distances L8: Set L8: Set operations X = αX operations L9: Fuzzy L9: Fuzzy numbers α∈(0,1] numbers L10: Fuzzy L10: Fuzzy logic where αX = [αb1 /αa1 ,α b2 /αa2 ]. logic L11: Fuzzy L11: Fuzzy control Again, X = B/A is not a solution of the equation. control L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (101/146) Joakim Lindblad, 2010-03-16 (102/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Classical logic: A brief overview Techniques Classical logic: A brief overview Joakim Joakim Lindblad Logic functions Lindblad Logic functions of two variables Outline Outline v2 1 1 0 0 Function Adopted v1 1 0 1 0 name symbol L1: Intro L1: Intro ω1 0 0 0 0 Zero function 0 L1–3: Basics L1–3: Basics ω2 0 0 0 1 NOR function v1 ↓ v2 L4: Constr. L4: Constr. and and ω3 0 0 1 0 Inhibition v1 > v2 uncertainty Logic function assigns a truth value to a combination of truth uncertainty ω4 0 0 1 1 Negation ¯ v2 L5: Features values of its variables: L5: Features ω5 0 1 0 0 Inhibition v1 < v2 L6 Features L6 Features ω6 0 1 0 1 Negation ¯ v1 n L7: Distances f : {true, false} → {true, false} L7: Distances ω7 0 1 1 0 Exclusive OR v1 ⊕ v2 L8: Set L8: Set ω8 0 1 1 1 NAND function v1 |v2 operations n 2n operations 2 choices of n arguments → 2 logic functions of n variables. ω9 1 0 0 0 Conjunction v1 ∧ v2 L9: Fuzzy L9: Fuzzy numbers numbers ω10 1 0 0 1 Equivalence v1 ⇔ v2 L10: Fuzzy L10: Fuzzy ω11 1 0 1 0 Assertion v1 logic logic ω12 1 0 1 1 Implication v1 ⇐ v2 L11: Fuzzy L11: Fuzzy control control ω13 1 1 0 0 Assertion v2 L12: Fuzzy L12: Fuzzy ω14 1 1 0 1 Implication v1 ⇒ v2 segmentation segmentation ω15 1 1 1 0 Disjunction v1 ∨ v2 L13: Defuzzi- ﬁcation Joakim Lindblad, 2010-03-16 (103/146) L13: Defuzzi- ﬁcation ω16 1 Joakim Lindblad, 2010-03-16 1 1 (104/146) 1 One function 1 Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Classical logic: A brief overview Techniques Classical logic: A brief overview Joakim Joakim Lindblad Logic primitives Lindblad Logic formulae Outline Outline L1: Intro L1: Intro L1–3: Basics L1–3: Basics L4: Constr. L4: Constr. and and Deﬁnition uncertainty We can express all the logic functions of n variables by using uncertainty L5: Features only a small number of simple logic functions. Such a set is a L5: Features 1. If v is a logic variable, then v and v are logic formulae; ¯ L6 Features complete set of logic primitives. L6 Features 2. If v1 and v2 are logic formulae, then v1 ∧ v2 and v1 ∨ v2 L7: Distances L7: Distances Examples: are also logic formulae; L8: Set L8: Set operations {negation, conjunction, disjunction}, operations 3. Logic formulae are only those deﬁned (obtained) by the L9: Fuzzy {negation, implication}. L9: Fuzzy numbers numbers two previous rules. L10: Fuzzy L10: Fuzzy logic logic L11: Fuzzy L11: Fuzzy control control L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (105/146) Joakim Lindblad, 2010-03-16 (106/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Classical logic: A brief overview Techniques Fuzzy propositions Joakim Joakim Lindblad Inference rules Lindblad Outline Outline The range of truth values of fuzzy propositions is not only L1: Intro Tautology is (any) logic formula that corresponds to a logic L1: Intro {0, 1}, but [0, 1]. L1–3: Basics function one. L1–3: Basics The truth of a fuzzy proposition is a matter of degree. L4: Constr. and Contradiction is (any) logic formula that corresponds to a L4: Constr. and uncertainty logic function zero. uncertainty Classiﬁcation of fuzzy propositions: L5: Features L5: Features • Unconditional and unqualiﬁed propositions L6 Features Inference rules are tautologies used for making deductive L6 Features “The temperature is high.” L7: Distances L7: Distances inferences. • Unconditional and qualiﬁed propositions L8: Set L8: Set operations Examples: operations “The temperature is high is very true.” L9: Fuzzy L9: Fuzzy numbers • (a ∧ (a ⇒ b)) ⇒ b modus ponens numbers • Conditional and unqualiﬁed propositions L10: Fuzzy ¯ L10: Fuzzy “If the temperature is high, then it is hot.” logic • (b ∧ (a ⇒ b)) ⇒ ¯ a modus tollens logic L11: Fuzzy L11: Fuzzy • Conditional and qualiﬁed propositions control • (a ⇒ b) ∧ (b ⇒ c)) ⇒ (a ⇒ c) hypothetical syllogism control “If the temperature is high, then it is hot is true.” L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (107/146) Joakim Lindblad, 2010-03-16 (108/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Linguistic hedges (modiﬁers) Techniques Modiﬁers Joakim Joakim Lindblad Lindblad Outline Outline Strong modiﬁer reduces the truth value of a proposition. L1: Intro L1: Intro Weak modiﬁer increases the truth value of a proposition (by L1–3: Basics • Linguistic hedges are linguistic terms by which other L1–3: Basics weakening the proposition). L4: Constr. and linguistic terms are modiﬁed. L4: Constr. and uncertainty uncertainty One commonly used class of modiﬁers is “Tina is young is true.” L5: Features L5: Features “Tina is very young is true.” hα (a) = aα , for α ∈ R + and a ∈ [0, 1]. L6 Features L6 Features “Tina is young is very true.” L7: Distances L7: Distances “Tina is very young is very true.” For α < 1, hα is a weak modiﬁer. L8: Set operations L8: Set operations √ • Fuzzy predicates and fuzzy truth values can be modiﬁed. Example: H : fairly ↔ h(a) = a. L9: Fuzzy L9: Fuzzy numbers Crisp predicates cannot be modiﬁed. numbers For α > 1, hα is a strong modiﬁer. L10: Fuzzy • Examples of hedges: very, fairly, extremely. L10: Fuzzy Example: H : very ↔ h(a) = a2 . logic logic L11: Fuzzy L11: Fuzzy control control h1 is the identity modiﬁer. L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (109/146) Joakim Lindblad, 2010-03-16 (110/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Fuzzy quantiﬁers Techniques Fuzzy propositions Joakim Joakim Lindblad Lindblad Unconditional and unqualiﬁed propositions Outline Outline The canonical form L1: Intro L1: Intro L1–3: Basics L1–3: Basics p : ν is F L4: Constr. and • Absolute quantiﬁers: L4: Constr. and ν is a variable on some universal set V uncertainty “about 10”; “much more than 100”, ... uncertainty F is a fuzzy set on V that represents a fuzzy predicate L5: Features • Relative quantiﬁers: L5: Features (e.g., low, tall, young, expensive...) L6 Features L6 Features L7: Distances “almost all”; “about half”, ... L7: Distances The degree of truth of p is L8: Set Examples: L8: Set operations p: “There are about 3 high-ﬂuent students in the group.” operations T (p) = F (v ), for v ∈ ν. L9: Fuzzy L9: Fuzzy numbers q: “Almost all students in the group are high-ﬂuent.” numbers T is a fuzzy set on V . Its membership function is derived form the L10: Fuzzy L10: Fuzzy logic logic membership function of a fuzzy predicate F . L11: Fuzzy L11: Fuzzy The role of a function T is to connect fuzzy sets and fuzzy propositions. control control L12: Fuzzy L12: Fuzzy In case of unconditional and unqualiﬁed propositions, the identity function segmentation segmentation is used. L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (111/146) Joakim Lindblad, 2010-03-16 (112/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Fuzzy propositions Techniques Fuzzy propositions Joakim Joakim Lindblad Unconditional and qualiﬁed propositions Lindblad Conditional and unqualiﬁed propositions Outline Outline L1: Intro L1: Intro L1–3: Basics The canonical form L1–3: Basics The canonical form L4: Constr. L4: Constr. and and uncertainty p : ν is F is S (truth qualiﬁed proposition) uncertainty p : If X is A, then Y is B, L5: Features L5: Features L6 Features where ν is a variable on some universal set V , L6 Features where X , Y are variables on X , Y respectively, L7: Distances F is a fuzzy set on V that represents a fuzzy predicate, L7: Distances and A, B are fuzzy sets on X , Y respectively. L8: Set operations and S is a fuzzy truth qualiﬁer. L8: Set operations Alternative form: L9: Fuzzy To calculate the degree of truth T (p) of the proposition p, we L9: Fuzzy X,Y is R numbers numbers use: L10: Fuzzy L10: Fuzzy where R(x, y ) = J (A(x), B(x)) is a fuzzy set on X × Y logic T (p) = S(F (v )) logic representing a suitable fuzzy implication. L11: Fuzzy L11: Fuzzy control control L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (113/146) Joakim Lindblad, 2010-03-16 (114/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Fuzzy propositions Techniques Fuzzy implications Joakim Joakim Lindblad Conditional and qualiﬁed propositions Lindblad Deﬁnition(s) Outline Outline L1: Intro L1: Intro A fuzzy implication J of two fuzzy propositions p and q is a L1–3: Basics L1–3: Basics function of the form L4: Constr. L4: Constr. and and uncertainty The canonical form uncertainty J : [0, 1] × [0, 1] → [0, 1], L5: Features L5: Features L6 Features p : If X is A, then Y is B is S L6 Features which for any truth values a = T (p) and b = T (q) deﬁnes the L7: Distances L7: Distances truth value J (a, b) of the conditional proposition where X , Y are variables on X , Y respectively, L8: Set L8: Set “if p, then q”. operations A, B are fuzzy sets on X , Y respectively, operations Fuzzy implications as extensions of the classical logic implication: L9: Fuzzy numbers and S is a truth qualiﬁer. L9: Fuzzy numbers Crisp implication a ⇒ b Fuzzy implication J (a, b) L10: Fuzzy L10: Fuzzy (S) ¯∨b a u(c(a), b) logic logic (R) max{x ∈ {0, 1} | a ∧ x ≤ b} sup{x ∈ [0, 1] | i(a, x) ≤ b} L11: Fuzzy L11: Fuzzy (QL) ¯ ∨ (a ∧ b) a u(c(a), i(a, b)) control control (QL) a ¯ (¯ ∧ b) ∨ b u(i(c(a), c(b)), b) L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (115/146) Joakim Lindblad, 2010-03-16 (116/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Fuzzy implications Techniques Binary fuzzy relations Joakim Joakim Lindblad How to select fuzzy implication Lindblad • A crisp binary relation R on sets X , Y is any (crisp) Outline Outline subset of X × Y . L1: Intro L1: Intro L1–3: Basics L1–3: Basics • xRy L4: Constr. Look at Table 11.2 , Table 11.3, and Table 11.4 L4: Constr. ( x ∈ X is in relation R with y ∈ Y ) iﬀ (x, y ) ∈ R and and uncertainty (pp. 315-317). uncertainty • A fuzzy binary relation R on sets X , Y is any fuzzy subset L5: Features One good choice: L5: Features of X × Y . L6 Features 1 a≤b L6 Features • Elements x ∈ X and y ∈ Y are in relation R up to some Js (a, b) = L7: Distances 0 a>b L7: Distances extent. L8: Set L8: Set operations One frequently used implication: Lukasiewicz operations The standard composition of two fuzzy relations, P(X , Y ) and Q(Y , Z ), L9: Fuzzy L9: Fuzzy is a binary relation R(X , Z ) deﬁned by numbers Ja (a, b) = min[1, 1 − a + b] numbers L10: Fuzzy L10: Fuzzy R(x, z) = [P ◦ Q](x, z) = max min[P(x, y ), Q(y , z)] logic logic y ∈Y L11: Fuzzy L11: Fuzzy control control for all x ∈ X and all z ∈ Z . L12: Fuzzy L12: Fuzzy segmentation segmentation This composition is based on standard t-norm, and standard t-conorm. It L13: Defuzzi- Joakim Lindblad, 2010-03-16 (117/146) L13: Defuzzi- is also referred to as max-min composition. Joakim Lindblad, 2010-03-16 (118/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Binary fuzzy relations Techniques Inference rules Joakim Joakim Lindblad Lindblad To represent (fuzzy) binary relations, membership matrices Outline are convenient. Outline Fuzzy inference rules are basis for approximate reasoning. L1: Intro L1: Intro As an example, three classical inference rules L1–3: Basics R = [rxy ], where rxy = R(x, y ). L1–3: Basics (Modus ponens, Modus Tollens, Hypothetical syllogism) L4: Constr. L4: Constr. and An example: and are generalized by using compositional rule of inference uncertainty uncertainty Determine R = P ◦ Q = [rij ] = [pik ] ◦ [qkj ] = [maxk min(pik , qkj )] L5: Features L5: Features For a given fuzzy relation R on X × Y , and a given fuzzy set A′ L6 Features 2 0.3 0.5 0.8 3 2 0.9 0.5 0.7 0.7 3 L6 Features on X , a fuzzy set B ′ on Y can be derived for all y ∈ Y , so that L7: Distances R = P ◦ Q = 4 0.0 0.7 1.0 5 ◦ 4 0.3 0.2 0.0 0.9 5 L7: Distances 0.4 0.6 0.5 1.0 0.0 0.5 0.5 L8: Set L8: Set B ′ (y ) = sup min[A′ (x), R(x, y )]. operations operations 2 3 x∈X L9: Fuzzy 0.8 0.3 0.5 0.5 L9: Fuzzy numbers = 4 1.0 0.2 0.5 0.7 5 . numbers 0.5 0.4 0.5 0.6 In matrix form, compositional rule of inference is L10: Fuzzy L10: Fuzzy logic logic L11: Fuzzy For example L11: Fuzzy B′ = A′ ◦ R control control L12: Fuzzy r23 = max[min(0.0, 0.7), min(0.7, 0.0), min(1.0, 0.5)] L12: Fuzzy segmentation = max[0.0, 0.0, 0.5] = 0.5. segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (119/146) Joakim Lindblad, 2010-03-16 (120/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Inference rules Techniques Multiconditional approximate Joakim Joakim Lindblad Example: Generalized modus ponens Lindblad reasoning Outline Outline L1: Intro L1: Intro L1–3: Basics L1–3: Basics L4: Constr. Rule: If X is A, then Y is B L4: Constr. General schema is of the form: and and uncertainty Fact: X is A′ uncertainty Rule 1: If X is A1 , then Y is B1 L5: Features Conclusion: Y is B ′ L5: Features Rule 2: If X is A2 , then Y is B2 L6 Features L6 Features ... L7: Distances In this case, L7: Distances Rule n: If X is An , then Y is Bn R(x, y ) = J [A(x), B(y )] L8: Set L8: Set Fact: X is A′ operations operations and Conclusion: Y is B ′ L9: Fuzzy numbers B ′ (y ) = sup min[A′ (x), R(x, y )]. L9: Fuzzy numbers x∈X L10: Fuzzy L10: Fuzzy A′ , Aj are fuzzy sets on X , logic logic L11: Fuzzy L11: Fuzzy B ′ , Bj are fuzzy sets on Y , for all j. control control L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (121/146) Joakim Lindblad, 2010-03-16 (122/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Approximate reasoning Techniques Multiconditional approximate Joakim Joakim Lindblad Method of interpolation Lindblad reasoning Most common way to determine B ′ is by using Method of interpolation-Example Outline Outline method of interpolation. L1: Intro L1: Intro L1–3: Basics Step 1. Calculate the degree of consistency between the given fact L1–3: Basics L4: Constr. and the antecedent of each rule. L4: Constr. and Use height of intersection of the associated sets: and uncertainty uncertainty L5: Features rj (A′ ) = h(A′ ∧ Aj ) = sup min[A′ (x), Aj (x)]. L5: Features L6 Features x∈X L6 Features L7: Distances L7: Distances ′ ′ L8: Set Step 2. Truncate each Bj by the value rj (A ) and determine B as L8: Set operations the union of truncated sets: operations L9: Fuzzy L9: Fuzzy numbers B ′ (y ) = sup min[rj (A′ ), Bj (y )], for all y ∈ Y . numbers L10: Fuzzy j∈Nn L10: Fuzzy logic logic L11: Fuzzy A special case of the composition rule of inference, with L11: Fuzzy control control R(x, y ) = sup min[Aj (x), Bj (y )] L12: Fuzzy j∈Nn L12: Fuzzy segmentation segmentation L13: Defuzzi- whereLindblad,B ′ (y ) = supx∈X min[A′ (x), R(x, y )] = (A′ ◦ R)(y ). Joakim then 2010-03-16 (123/146) L13: Defuzzi- Joakim Lindblad, 2010-03-16 (124/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques L11: Fuzzy control Techniques Conventional control system Joakim Joakim Lindblad Lindblad Outline Outline L1: Intro L1: Intro L1–3: Basics L1–3: Basics L4: Constr. L4: Constr. and and uncertainty uncertainty L5: Features L5: Features L6 Features L6 Features L7: Distances L7: Distances L8: Set L8: Set operations operations PID Control (Proportional-Integral-Derivative) L9: Fuzzy L9: Fuzzy numbers numbers The PID controller is the workhorse of the process industries. L10: Fuzzy L10: Fuzzy logic logic t L11: Fuzzy L11: Fuzzy dε control control Output = bias + KP ε + KI ε dt + KD L12: Fuzzy L12: Fuzzy 0 dt segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (125/146) Joakim Lindblad, 2010-03-16 (126/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Fuzzy logic control Techniques L11: Fuzzy control Joakim Lindblad Joakim Lindblad Useful cases Methodology ﬁrst developed by Mamdani in 1975 used to 1 The control processes are too complex to analyze by conventional Outline Outline control a steam plant. Based on work by Zadeh (1973) on quantitative techniques. L1: Intro L1: Intro fuzzy algorithms for complex systems and decision processes. 2 The available sources of information are interpreted qualitatively, L1–3: Basics L1–3: Basics inexactly, or uncertainly. L4: Constr. and In a manner analogous to conventional control systems, L4: Constr. and Advantage of Fuzzy logic control uncertainty inputs of a system are mapped to outputs using fuzzy uncertainty L5: Features logic rather than diﬀerential equations. L5: Features • Flexible L6 Features L6 Features • Universal approximator L7: Distances • Can be used for systems that are diﬃcult or impossible to L7: Distances • Easy to understand L8: Set model mathematically. L8: Set • Powerful – yet simple operations operations L9: Fuzzy • Can also be applied to processes that are too complex or L9: Fuzzy • Linguistic control numbers numbers nonlinear to be controlled with traditional strategies. • linguistic terms – human knowledge L10: Fuzzy L10: Fuzzy logic • Human operators often are capable of managing complex logic • Tolerant of imprecision / Robust control • more than 1 control rules - an error of a rule is not fatal L11: Fuzzy control situations of a plant without knowing anything about L11: Fuzzy control • limited trust in input data L12: Fuzzy diﬀerential equations. L12: Fuzzy • Parallel or distributed control segmentation segmentation • multiple fuzzy rules - complex nonlinear system L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (127/146) Joakim Lindblad, 2010-03-16 (128/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques L11: Fuzzy control Techniques Fuzzy control Joakim Joakim Lindblad Lindblad Outline Outline L1: Intro L1: Intro L1–3: Basics L1–3: Basics L4: Constr. L4: Constr. and Disadvantages and uncertainty uncertainty Four main components L5: Features • More complex than PID L5: Features L6 Features L6 Features 1 The fuzziﬁcation interface : transforms input crisp values • More parameters to tune into fuzzy values L7: Distances L7: Distances L8: Set • Diﬃcult to analyze mathematically (stability?) L8: Set 2 The knowledge base : contains a knowledge of the operations operations application domain and the control goals. L9: Fuzzy L9: Fuzzy numbers numbers 3 The decision-making logic : performs inference for fuzzy L10: Fuzzy L10: Fuzzy logic logic control actions L11: Fuzzy L11: Fuzzy 4 The defuzziﬁcation interface : provides a crisp control control control L12: Fuzzy L12: Fuzzy action out segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (129/146) Joakim Lindblad, 2010-03-16 (130/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Five steps... Techniques An example Joakim Joakim Lindblad Lindblad Outline How to build a fuzzy controller in ﬁve easy steps. . . Outline L1: Intro L1: Intro 1 Partition input and output spaces: L1–3: Basics L1–3: Basics L4: Constr. Select meaningful linguistic states for each variable and L4: Constr. and uncertainty express them as appropriate fuzzy sets. and uncertainty L5: Features 2 Fuzziﬁcation of input: L5: Features L6 Features Introduce a fuzziﬁcation function for each input variable L6 Features L7: Distances to express the associated measurement uncertainty. L7: Distances L8: Set L8: Set operations 3 Formulate a set of inference rules: operations L9: Fuzzy If ǫ = A and dǫ = B, then C . dt L9: Fuzzy numbers numbers L10: Fuzzy 4 Design an inference engine: L10: Fuzzy logic Use method of interpolation (Lecture 10). logic L11: Fuzzy L11: Fuzzy control 5 Select a suitable defuzziﬁcation method (Lecture 13). control L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (131/146) Joakim Lindblad, 2010-03-16 (132/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques L12: Fuzzy segmentation Techniques Fuzzy connectedness Joakim Joakim Lindblad Lindblad Outline Outline L1: Intro L1: Intro L1–3: Basics L1–3: Basics L4: Constr. L4: Constr. Hanging-togetherness natural grouping of voxels constituting and and uncertainty uncertainty an object a human viewer readily sees in a display of the scene L5: Features L5: Features as a Gestalt in spite of intensity heterogeneity. L6 Features L6 Features Basic idea: L7: Distances L7: Distances L8: Set L8: Set Compute global hanging-togetherness from local operations operations hanging-togetherness. L9: Fuzzy L9: Fuzzy numbers numbers L10: Fuzzy L10: Fuzzy logic logic L11: Fuzzy L11: Fuzzy control control L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (133/146) Joakim Lindblad, 2010-03-16 (134/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Fuzzy connectedness Techniques Fuzzy connectedness Joakim Joakim Lindblad Lindblad Outline Outline Fuzzy spel adjacency is a reﬂexive and symmetric fuzzy Strength of a path – the strength of its weakest link relation α in Zn and assigns a value to a pair of spels (c, d) L1: Intro L1: Intro L1–3: Basics A. Rosenfeld 1979 L1–3: Basics based on how close they are spatially. L4: Constr. Strength of a link between two points deﬁned by the L4: Constr. Fuzzy spel aﬃnity is a reﬂexive and symmetric fuzzy relation κ and and uncertainty membership function. uncertainty in Zn and assigns a value to a pair of spels (c, d) based on how L5: Features J. K. Udupa and S. Samarasekera 1996 L5: Features close they are spatially and intensity-based-property-wise (local L6 Features Strength of a link between two points deﬁned by aﬃnity L6 Features hanging-togetherness). L7: Distances L7: Distances L8: Set The connectedness of two points x and y in A – L8: Set operations the strength of the strongest path between x and y operations µκ (c, d) = h(µα (c, d), µ(c), µ(d), c, d) L9: Fuzzy L9: Fuzzy numbers numbers The fuzzy κ-connectedness assigns a value to a pair of spels (c, L10: Fuzzy cA (x, y ) = sup inf A(t) L10: Fuzzy logic π∈Π(x,y ) t∈π logic d) that is the maximum of the strengths of connectedness L11: Fuzzy L11: Fuzzy assigned to all possible paths from c to d (global control control hanging-togetherness). L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (135/146) Joakim Lindblad, 2010-03-16 (136/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Components of fuzzy aﬃnity Techniques Fuzzy connectedness Joakim Joakim Lindblad Lindblad Outline Outline L1: Intro Fuzzy spel adjacency µα (c, d) indicates the degree of spatial L1: Intro Computation – A graph search problem L1–3: Basics adjacency of spels The homogeneity-based component L1–3: Basics L4: Constr. µψ (c, d) indicates the degree of local hanging-togetherness of L4: Constr. Dynamic programming solution (think distance transform or and and uncertainty spels due to their similarities of intensities The uncertainty level sets computation) L5: Features L5: Features object-feature-based component µϕ (c, d) indicates the Practical usage examples: L6 Features L6 Features degree of local hanging-togetherness of spels with respect to L7: Distances L7: Distances • Seed foreground (one or multiple seeds), threshold at some some given object feature L8: Set L8: Set level of fuzzy connectedness. operations Example: operations L9: Fuzzy 1 L9: Fuzzy • Seed diﬀerent regions and let them compete (relative fc, numbers µκ = µα (µψ + µϕ ) numbers iterated relative fc). L10: Fuzzy 2 L10: Fuzzy logic logic L11: Fuzzy L11: Fuzzy control control L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (137/146) Joakim Lindblad, 2010-03-16 (138/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Pixel coverage representations Techniques Pixel coverage representations Joakim Joakim Lindblad Lindblad Coverage representation – constrained fuzziness Outline Outline L1: Intro • Restrict to one speciﬁc meaning of memberships. L1: Intro L1–3: Basics L1–3: Basics • Pixel coverage representations are shown to be superior to • Restrict to crisp imaged objects (no clouds or ﬂames). L4: Constr. L4: Constr. crisp image object representations for many reasons. and and uncertainty uncertainty L5: Features Pixel coverage digitization L5: Features • By suitably utilizing information available in images it is L6 Features ⇒ Let the value of a pixel/voxel L6 Features possible to perform pixel coverage segmentations. L7: Distances be equal to the part of it being L7: Distances • We observe that for reasonable noise levels, the achieved L8: Set covered by the object. L8: Set operations operations pixel coverage representation provides a more accurate L9: Fuzzy • Keep the good sides of fuzzy (robust, precise features, L9: Fuzzy representation of image objects than a perfect, noise free, numbers numbers information rich representation). crisp representation. L10: Fuzzy L10: Fuzzy logic • Enable strong theoretical results. logic L11: Fuzzy L11: Fuzzy control • Clear meaning of feature values – features describe the control L12: Fuzzy continuous crisp object. L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (139/146) Joakim Lindblad, 2010-03-16 (140/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques L13: Defuzziﬁcation Techniques L13: Defuzziﬁcation Joakim Joakim Lindblad Lindblad Outline Outline L1: Intro L1: Intro L1–3: Basics L1–3: Basics Defuzziﬁcation is a process that maps a fuzzy set to a crisp set. L4: Constr. and L4: Constr. and Approaches: uncertainty uncertainty L5: Features L5: Features • Defuzziﬁcation to a point. L6 Features L6 Features • Defuzziﬁcation to a set. L7: Distances L7: Distances L8: Set L8: Set operations operations • Generating a good representative of a fuzzy set. L9: Fuzzy L9: Fuzzy numbers numbers • Recovering a crisp original set. L10: Fuzzy L10: Fuzzy logic logic L11: Fuzzy L11: Fuzzy control control L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (141/146) Joakim Lindblad, 2010-03-16 (142/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Defuzziﬁcation to a point Techniques Defuzziﬁcation to a set Joakim Joakim Lindblad Examples of common methods Lindblad A.k.a. Averaging procedures Outline Outline L1: Intro • Centre of gravity (Set of real numbers) L1: Intro L1–3: Basics Pxmax L1–3: Basics xmin x · A(x) L4: Constr. COG (A) = Pxmax . L4: Constr. • α-cuts and xmin A(x) and uncertainty uncertainty chosen at various levels α (often α = 0.5 or α = 1). L5: Features L5: Features • Mean of maxima (Set of real numbers) L6 Features L6 Features • Average α-cuts based on an integration of set-valued function, P L7: Distances x∈core(A) x L7: Distances MeOM(A) = . L8: Set |core(A)| L8: Set called Kudo-Aumann integration. operations operations • Feature distance minimization L9: Fuzzy • Centre of area (COA) L9: Fuzzy numbers COA(A) is the value that minimizes the expression numbers ﬁnd the crisp set at the minimal feature distance to L10: Fuzzy logic ŕ ŕ L10: Fuzzy logic the given fuzzy set. ŕ COA(A) sup(X ) ŕ L11: Fuzzy ŕ X X ŕ L11: Fuzzy control ŕ ŕ A(x) − A(x)ŕŕ . control ŕ x=inf(X ) x=COA(A) ŕ L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (143/146) Joakim Lindblad, 2010-03-16 (144/146) ﬁcation ﬁcation Fuzzy Sets Fuzzy Sets and Fuzzy and Fuzzy Techniques Defuzziﬁcation to a set Techniques Defuzziﬁcation by feature distance Joakim Joakim Lindblad Average α-cuts Lindblad minimization Outline Let a fuzzy set A be given by a membership function Outline L1: Intro µ : R → [0, 1]. L1: Intro L1–3: Basics • Sets F (w ) are α cuts, Aα of the fuzzy set A, for α ∈ [0, 1]; L1–3: Basics Deﬁnition L4: Constr. and • Selectors are ϕ(α) = inf Aα and ϕ(α) = sup Aα . L4: Constr. and An optimal defuzziﬁcation D(A) of a fuzzy set A on a reference uncertainty uncertainty set X , with respect to the distance d, is L5: Features Then, the average α-cut of A is L5: Features L6 Features L6 Features D(A) ∈ {C ∈ P(X ) | d(A, C ) = min [d(A, B)]} . (3) L7: Distances Aµ = inf Aα dα, sup Aα dα . L7: Distances B∈P(X ) L8: Set [0,1] [0,1] L8: Set operations operations L9: Fuzzy L9: Fuzzy I.e., the crisp set that is at smallest distance to the fuzzy set. numbers numbers L10: Fuzzy L10: Fuzzy Use a feature distance containing both local and global logic logic features. Find a minimum using a suitable optimization L11: Fuzzy L11: Fuzzy control control method. L12: Fuzzy L12: Fuzzy segmentation segmentation L13: Defuzzi- L13: Defuzzi- Joakim Lindblad, 2010-03-16 (145/146) Joakim Lindblad, 2010-03-16 (146/146) ﬁcation ﬁcation

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