1. Fuzzy Logic and Fuzzy Set Theorywith examples from Image Processing By: Rafi Steinberg 4/2/20081
2. Some Fuzzy Background LoftiZadeh has coined the term “Fuzzy Set” in 1965 and opened a new field of research and applications A Fuzzy Set is a class with different degrees of membership. Almost all real world classes are fuzzy! Examples of fuzzy sets include: {‘Tall people’}, {‘Nice day’}, {‘Round object’} … If a person’s height is 1.88 meters is he considered ‘tall’? What if we also know that he is an NBA player? 2
4. Overview L. Zadeh D. Dubois H. Prade J.C. Bezdek R.R. Yager M. Sugeno E.H. Mamdani G.J. Klir J.J. Buckley 4
5. A Crisp Definition of Fuzzy Logic Does not exist, however … - Fuzzifies bivalent Aristotelian (Crisp) logic Is “The sky are blue”True or False? Modus Ponens IF <Antecedent == True> THEN <Do Consequent> IF (X is a prime number) THEN (Send TCP packet) Generalized Modus Ponens IF “a region is green and highly textured” AND “the region is somewhat below a sky region” THEN “the region contains trees with high confidence” 5
7. Fuzzy Vs. Probability Walking in the desert, close to being dehydrated, you find two bottles of water: The first contains deadly poison with a probability of 0.1 The second has a 0.9 membership value in the Fuzzy Set “Safe drinks” Which one will you choose to drink from??? 7
8. Membership Functions (MFs) What is a MF? Linguistic Variable A Normal MF attains ‘1’ and ‘0’ for some input How do we construct MFs? Heuristic Rank ordering Mathematical Models Adaptive (Neural Networks, Genetic Algorithms …) 8
15. Aggregation Operations (3) Yager S-Norm b (=0.8) a (=0.3) Yager S-Norm for varying w Generalized Mean Drastic T-Norm Zadehian min Geometric Zadehian max Bounded Sum Drastic S-Norm Product Harmonic Algebraic (Mean) 14
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17. Law of Excluded Middle and Non-Contradiction hold:Crisp Membership Function Intersection (AND) , Union (OR), and Negation (NOT) are fixed 15
18. Image Processing Binary Gray Level Color (RGB,HSV etc.) Can we give a crisp definition to light blue? 16
20. Fuzziness “As the complexity of a system increases, our ability to make precise and yet significant statements about its behavior diminishes” – L. Zadeh A possible definition of fuzziness of an image: 18
22. Mathematical Morphology Operates on predefined geometrical objects in an image Structured Element (SE) represents the shape of interest Initially developed for binary images; extended to grayscale using aggregation operations from Fuzzy Logic Some Examples: Dilation, Erosion, Open, Close, Hit&Miss, Skeleton 20
24. Some Basic Concepts Universe of Discourse: Power Set of X= P(X)= {Null , {a} , {b} , {c} ,{a , b},{b , c}, {a , c}, {a,b,c}} Singletons of the Power Set of X: { {a} , {b} , {c} } An Event=An Element of the Power Set Basic Probability Assignment (BPA) Consonant Body of Evidence Focal Element m(A)=0.2 22
25. Fuzzy Measures Additive, Sub Additive, Super Additive Measures Examples: {Probability}, {Belief, Plausibility}, {Necessity, Possibility} (1) Boundary Condition: (2) Monotonicity: (3) Uniform Convergence increasing sequence of measurable sets we have uniform convergence: 23
29. Sugeno Measures Sugeno Measure’s Additional Axiom: Compute λ from the normalization rule: Sugeno Inverse: Sugeno Inverse for λ={-0.99, -0.9, -0.5, 0, 1, 10} Optimistic/Pessimistic Aggregation of Evidence 27
30. Finding the Sugeno Measure We need to solve the third order equation: Solutions: {0, -15, 5/3} Since λ=0 is the trivial additive solution and since λ =-15 is out of range (λ>-1) we choose λ=5/3 and obtain: 28
31. Example: Sugeno Integral Calculation -> We cannot aggregate with the Sugeno Union since the segmenting alpha cut values are not part of our initial frame of discernment -> Zadehian Max-Min are ‘good’ default operators 29 h(q) is the alpha cut that entirely includes the measure of q.
33. O.K. So Now What? We have a fuzzy result, however in many cases we need to make a crisp decision (On/Off) Methods of defuzzifying are: Centroid (Center of Mass) Maximum Other methods 31
34. Fuzzy Inference (Expert) Systems Fuzzify: Apply MF on input Generalized Modus Ponens with specified aggregation operations Defuzzify: Method of Centroid, Maximum, ... 32
35. Automatic Speech Recognition (ASR) via Automatic Reading of Speech Spectrograms Phoneme Classes: Vowels Semi-vowels/Diphthongs Nasals Plosives Fricatives Silence Examples of Fuzzy Variables: Distance between formants (Large/Small) Formant location (High/Mid/Low) Formant length (Long/Average/Short) Zero crossings (Many/Few) Formant movement (Descending/Ascending/Fixed) VOT= Voice Onset Time (Long/Short) Phoneme duration (Long/Average/Short) Pitch frequency (High/Low/Undetermined) Blob (F1/F2/F3/F4/None) “Don’t ask me to carry…" 33
37. Suggested Fuzzy Inference System Assign a Fuzzy Value for each Phoneme, Output Highest N Values to a Linguistic model Output Fuzzy MF for each Phoneme 35
45. Heavy-Tail Distributions 38 Bonus Slide Examples: Alpha Stable (Cauchy, Pareto), Weibull, Student-T, Log-Normal … Problem – different samples with very low probability occur very frequently Solution: Smoothing the probability density function; Good or Bad?? Another Solution: Use Possibility (Membership function) and Necessity as envelopes Example: Amazon sells far more books that are ‘very unpopular’ than popular books Another example: Automatic translation – most words in English have a very low frequency of occurrence. However, we often find such rare words in a sentence.
Notas del editor
Linguistic Variables
Following Bezdek
Buckley: Experiment – ask many people if statement A, B, A AND B is true. Then check the prior correlation coefficient. The result shows which method to use. The assumption is that with a large population model, the TRUE/FALSE values converge to the probability that a person would say that the statement is true.
Following the work of Klir
We obtain a negative value for lamda if the fuzzy measure of the singletons that span our set sum to more than unity. We have cancellation of evidence in this case. On the other hand, for (0.2+0.2+0.3) <1 we obtain a positive lamda
