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- 1. Fuzzy Logic and Fuzzy Set Theory
- 2. Some Fuzzy Background Lofti Zadeh 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
- 3. Some Related Fields Fuzzy Logic & Fuzzy Set Theory Evidence Theory Pattern Recognition & Image Processing Control Theory Knowledge Engineering 3
- 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 Membership Functions Linguistic Hedges Aggregation Operations Image Processing Fuzzy Morphology Fuzzy Measures Fuzzy Integrals Fuzzy Expert Systems Speech Spectrogram Reading
- 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
- 6. Fuzzy Inference (Expert) Systems Input_1 Fuzzy IF-THEN Rules OutputInput_2 Input_3 6
- 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 …) 1 2 1 2, 1, 0A Ax x x x 8
- 9. Membership Function Examples TrapezoidalTriangular 1 , , 1 smf a x c f x a c e Sigmoid 2 2 2 ; , x c gmff x c e Gaussian ; , , , max min ,1, ,0 x a d x f x a b c d b a d c ; , , max min , , 0 x a c x f x a b c b a c b 9
- 10. Alpha Cuts AA x X x AA x X x Strong Alpha Cut Alpha Cut 0 0.2 0.5 0.8 1 10
- 11. Linguistic Hedges Operate on the Membership Function (Linguistic Variable) 1. Expansive (“Less”, ”Very Little”) 2. Restrictive (“Very”, “Extremely”) 3. Reinforcing/Weakening (“Really”, “Relatively”) Less x 4Very Little x 2 Very x 4 Extremely x A Ax x c 11
- 12. Aggregation Operations 1 21 21 ,,, n aaa aaah n n 0 0, ,1iand a i i n , min 1 , 0 , 1 , , max h h Harmonic Mean h Geometric Mean h Algebraic Mean h Generalized Mean: 12
- 13. Aggregation Operations (2) • Fixed Norms (Drastic, Product, Min) • Parametric Norms (Yager) T-norms: , 1 , , 1 0 , D b if a T a b a if b otherwise Drastic Product , min ,ZT a b a b ,T a b a b Zadehian ,BSS a b a b a b , 0 , , 0 1 , D b if a S a b a if b otherwise S-Norm Duals: , max ,ZS a b a b Bounded Sum DrasticZadehian 13
- 14. Aggregation Operations (3) Drastic T-Norm Product Zadehian min Generalized Mean Zadehian max Bounded Sum Drastic S-Norm Algebraic (Mean) Geometric Harmonic b (=0.8)a (=0.3) 1 , min 1, 0,w w w u a b a b for w Yager S-Norm Yager S-Norm for varying w 14
- 15. Crisp Vs. Fuzzy Fuzzy Sets • Membership values on [0,1] • Law of Excluded Middle and Non- Contradiction do not necessarily hold: • Fuzzy Membership Function • Flexibility in choosing the Intersection (T-Norm), Union (S- Norm) and Negation operations Crisp Sets • True/False {0,1} • Law of Excluded Middle and Non- Contradiction hold: • Crisp Membership Function • Intersection (AND) , Union (OR), and Negation (NOT) are fixed A A A A A A A A 15
- 16. Binary Gray Level Color (RGB,HSV etc.) Can we give a crisp definition to light blue? 16
- 17. Fuzziness Vs. Vagueness Vagueness=Insufficient Specificity “I will be back sometime” Fuzzy Vague “I will be back in a few minutes” Fuzzy Fuzziness=Unsharp Boundaries 17
- 18. 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: 2 min ,ij ij i j Fuzz M N 18
- 19. Example: Finding an Image Threshold Membership Value Gray Level 1 , , 1 smf a x c f x a c e 19
- 20. Fuzzy Inference (Expert) Systems Service Time Fuzzy IF-THEN Rules Tip Level Food Quality Ambiance Fuzzify: Apply MF on input Generalized Modus Ponens with specified aggregation operations Defuzzify: Method of Centroid, Maximum, ... 20
- 21. 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…" 21
- 22. Applying the Segmentation Algorithm 22
- 23. Suggested Fuzzy Inference System Feature Vector from Spectrogram Identify Phoneme Class using Fuzzy IF-THEN Rules Vowels Find Vowel Fricatives Nasals Output Fuzzy MF for each Phoneme 23 Assign a Fuzzy Value for each Phoneme, Output Highest N Values to a Linguistic model
- 24. Summary 24 • Fuzzy Logic can be useful in solving Human related tasks • Evidence Theory gives tools to handle knowledge • Membership functions and Aggregation methods can be selected according to the problem at hand Some things we didn’t talk about: • Fuzzy C-Means (FCM) clustering algorithm • Dempster-Schafer theory of combining evidence • Fuzzy Relation Equations (FRE) • Compositions • Fuzzy Entropy
- 25. References [1] G. J. Klir ,U. S. Clair, B. Yuan“Fuzzy Set Theory: Foundations and Applications “, Prentice Hall PTR 1997, ISBN: 978-0133410587 [2] H.R. Tizhoosh;“Fast fuzzy edge detection” Fuzzy Information Processing Society, Annual Meeting of the North American, pp. 239 – 242, 27-29 June 2002. [3] A.K. Hocaoglu; P.D. Gader; “ An interpretation of discrete Choquet integrals in morphological image processing Fuzzy Systems “, Fuzzy Systems, FUZZ '03. Vol. 2, 25-28, pp. 1291 – 1295, May 2003. [4] E.R. Daugherty, “An introduction to Morphological Image Processing”, SPlE Optical Engineering Press, Bellingham, Wash., 1992. [5] A. Dumitras, G. Moschytz, “Understanding Fuzzy Logic – An interview with Lofti Zadeh”, IEEE Signal Processing Magazine, May 2007 [6] J.M. Yang; J.H. Kim, ”A multisensor decision fusion strategy using fuzzy measure theory ”, Intelligent Control, Proceedings of the 1995 IEEE International Symposium on, pp. 157 – 162, Aug. 1995 [7] R. Steinberg, D. O’Shaugnessy ,”Segmentation of a Speech Spectrogram using Mathematical Morphology ” ,To be presented at ICASSP 2008. [8] J.C. Bezdek, J. Keller, R. Krisnapuram, N.R. Pal, ” Fuzzy Models and Algorithms for Pattern Recognition and Image Processing ” Springer 2005, ISBN: 0-387-245 15-4 [9] W. Siler, J.J. Buckley,“Fuzzy Expert Systems and Fuzzy Reasoning“, John Wiley & Sons, 2005, Online ISBN: 9780471698500 [10] http://pami.uwaterloo.ca/tizhoosh/fip.htm [11] "Heavy-tailed distribution." Wikipedia, The Free Encyclopedia. 22 Jan 2008, 17:43 UTC. Wikimedia Foundation, Inc. 3 Feb 2008 http://en.wikipedia.org/w/index.php?title=Heavy-tailed_distribution&oldid=186151469 [12] T.J. Ross, “Fuzzy Logic with Engineering Applications”, McGraw-Hill 1997. ISBN: 0070539170 25

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