Fuzzy Logic Ppt


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  • 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
  • Fuzzy Logic Ppt

    1. 1. Fuzzy Logic and Fuzzy Set Theorywith examples from Image Processing<br />By: Rafi Steinberg<br />4/2/20081<br />
    2. 2. Some Fuzzy Background<br />LoftiZadeh has coined the term “Fuzzy Set” in 1965 and opened a new field of research and applications<br />A Fuzzy Set is a class with different degrees of membership. Almost all real world classes are fuzzy!<br />Examples of fuzzy sets include: {‘Tall people’}, {‘Nice day’}, {‘Round object’} …<br />If a person’s height is 1.88 meters is he considered ‘tall’?<br />What if we also know that he is an NBA player? <br />2<br />
    3. 3. Some Related Fields<br />3<br />
    4. 4. Overview<br />L. Zadeh<br />D. Dubois<br />H. Prade<br />J.C. Bezdek<br />R.R. Yager<br />M. Sugeno<br />E.H. Mamdani<br />G.J. Klir<br />J.J. Buckley<br />4<br />
    5. 5. A Crisp Definition of Fuzzy Logic<br />Does not exist, however …<br /> - Fuzzifies bivalent Aristotelian (Crisp) logic<br /> Is “The sky are blue”True or False?<br />Modus Ponens<br />IF &lt;Antecedent == True&gt; THEN &lt;Do Consequent&gt;<br />IF (X is a prime number) THEN (Send TCP packet)<br />Generalized Modus Ponens<br />IF “a region is green and highly textured” <br />AND “the region is somewhat below a sky region”<br />THEN “the region contains trees with high confidence”<br />5<br />
    6. 6. Fuzzy Inference (Expert) Systems<br />6<br />
    7. 7. Fuzzy Vs. Probability<br />Walking in the desert, close to being dehydrated, you find two bottles of water:<br />The first contains deadly poison with a probability of 0.1<br />The second has a 0.9 membership value in the Fuzzy Set “Safe drinks”<br />Which one will you choose to drink from???<br />7<br />
    8. 8. Membership Functions (MFs)<br />What is a MF? <br />Linguistic Variable<br />A Normal MF attains ‘1’ and ‘0’ for some input<br />How do we construct MFs?<br />Heuristic<br />Rank ordering<br />Mathematical Models<br />Adaptive (Neural Networks, Genetic Algorithms …)<br />8<br />
    9. 9. Membership Function Examples<br />Sigmoid<br />Gaussian<br />Trapezoidal<br />Triangular<br />9<br />
    10. 10. Alpha Cuts<br />Alpha Cut<br />Strong Alpha Cut<br />10<br />
    11. 11. Linguistic Hedges<br />Operate on the Membership Function (Linguistic Variable)<br />Expansive (“Less”, ”Very Little”)<br />Restrictive (“Very”, “Extremely”)<br />Reinforcing/Weakening (“Really”, “Relatively”)<br />11<br />
    12. 12. Aggregation Operations<br />Generalized Mean:<br />12<br />
    13. 13. Aggregation Operations (2)<br />T-norms:<br /><ul><li> Fixed Norms (Drastic, Product, Min)
    14. 14. Parametric Norms (Yager)</li></ul>S-Norm Duals:<br />Bounded Sum <br />Drastic<br />Zadehian<br />Drastic<br />Product <br />Zadehian<br />13<br />
    15. 15. Aggregation Operations (3)<br />Yager S-Norm<br />b (=0.8)<br />a (=0.3)<br />Yager S-Norm for varying w<br />Generalized Mean<br />Drastic<br />T-Norm<br />Zadehian min<br />Geometric<br />Zadehian max<br />Bounded<br />Sum<br />Drastic <br />S-Norm<br />Product<br />Harmonic<br />Algebraic (Mean)<br />14<br />
    16. 16. Crisp Vs. Fuzzy<br />Fuzzy Sets <br />Membership values on [0,1]<br />Law of Excluded Middle and Non-Contradiction do not necessarily hold:<br />Fuzzy Membership Function<br />Flexibility in choosing the Intersection (T-Norm), Union (S-Norm) and Negation operations<br />Crisp Sets<br /><ul><li>True/False {0,1}
    17. 17. Law of Excluded Middle and Non-Contradiction hold:</li></ul>Crisp Membership Function<br /> Intersection (AND) , Union (OR), and Negation (NOT) are fixed<br />15<br />
    18. 18. Image Processing<br />Binary<br />Gray Level<br />Color (RGB,HSV etc.)<br />Can we give a crisp definition to light blue?<br />16<br />
    19. 19. Fuzziness Vs. Vagueness<br />Vagueness=Insufficient Specificity<br />Fuzziness=Unsharp Boundaries<br />17<br />
    20. 20. Fuzziness<br />“As the complexity of a system increases, our ability to make precise and yet significant statements about its behavior diminishes” – L. Zadeh<br />A possible definition of fuzziness of an image:<br />18<br />
    21. 21. Example: Finding an Image Threshold<br />Membership Value<br />Gray Level<br />19<br />
    22. 22. Mathematical Morphology<br />Operates on predefined geometrical objects in an image<br />Structured Element (SE) represents the shape of interest<br />Initially developed for binary images; extended to grayscale using aggregation operations from Fuzzy Logic<br />Some Examples: Dilation, Erosion, Open, Close, Hit&Miss, Skeleton<br />20<br />
    23. 23. Fuzzy Mathematical Morphology<br />“Does it fit” “How well does a SE fit”<br />For B=0<br />21<br />
    24. 24. Some Basic Concepts<br />Universe of Discourse:<br />Power Set of X= P(X)= {Null , {a} , {b} , {c} ,{a , b},{b , c}, {a , c}, {a,b,c}}<br />Singletons of the Power Set of X: { {a} , {b} , {c} }<br />An Event=An Element of the Power Set<br />Basic Probability Assignment (BPA)<br />Consonant Body of Evidence<br />Focal Element<br />m(A)=0.2<br />22<br />
    25. 25. Fuzzy Measures<br />Additive, Sub Additive, Super Additive Measures<br />Examples: {Probability}, {Belief, Plausibility}, {Necessity, Possibility}<br />(1) Boundary Condition:<br />(2) Monotonicity:<br />(3) Uniform Convergence<br />increasing sequence of measurable sets we have uniform convergence:<br />23<br />
    26. 26. Example: Fuzzy Measure<br />24<br />
    27. 27. The Choquet Integral<br />Is defined over a Fuzzy Measure<br />Consider a gray level input<br />25<br />
    28. 28. Example: Choquet Integral Calculation<br />26<br />
    29. 29. Sugeno Measures<br />Sugeno Measure’s Additional Axiom:<br />Compute λ from the normalization rule:<br />Sugeno Inverse:<br />Sugeno Inverse for λ={-0.99, -0.9, -0.5, 0, 1, 10}<br />Optimistic/Pessimistic Aggregation of Evidence<br />27<br />
    30. 30. Finding the Sugeno Measure<br /> We need to solve the third order equation:<br /> Solutions: {0, -15, 5/3}<br /> Since λ=0 is the trivial additive solution and since <br />λ =-15 is out of range (λ&gt;-1) we choose λ=5/3 and obtain:<br />28<br />
    31. 31. Example: Sugeno Integral Calculation<br />-&gt; We cannot aggregate with the Sugeno Union since the segmenting alpha cut values are not part of our initial frame of discernment<br />-&gt; Zadehian Max-Min are ‘good’ default operators<br />29<br />h(q) is the alpha cut that entirely includes the measure of q.<br />
    32. 32. Example: Finding Edges<br />30<br />
    33. 33. O.K. So Now What?<br />We have a fuzzy result, however in many cases we need to make a crisp decision (On/Off)<br />Methods of defuzzifying are:<br />Centroid (Center of Mass)<br />Maximum<br />Other methods<br />31<br />
    34. 34. Fuzzy Inference (Expert) Systems<br />Fuzzify: <br />Apply MF on input<br />Generalized Modus Ponens with specified aggregation operations<br />Defuzzify: <br />Method of Centroid, Maximum, ...<br />32<br />
    35. 35. Automatic Speech Recognition (ASR) via Automatic Reading of Speech Spectrograms<br />Phoneme Classes:<br />Vowels<br />Semi-vowels/Diphthongs<br />Nasals<br />Plosives<br />Fricatives<br />Silence<br />Examples of Fuzzy Variables:<br />Distance between formants (Large/Small)<br />Formant location (High/Mid/Low)<br />Formant length (Long/Average/Short)<br />Zero crossings (Many/Few)<br />Formant movement (Descending/Ascending/Fixed)<br />VOT= Voice Onset Time (Long/Short)<br />Phoneme duration (Long/Average/Short)<br />Pitch frequency (High/Low/Undetermined)<br />Blob (F1/F2/F3/F4/None)<br />“Don’t ask me to carry…&quot;<br />33<br />
    36. 36. Applying the Segmentation Algorithm<br />34<br />
    37. 37. Suggested Fuzzy Inference System<br />Assign a Fuzzy Value for each Phoneme, Output <br />Highest N Values to a Linguistic model<br />Output Fuzzy MF<br /> for each Phoneme<br />35<br />
    38. 38. Summary<br />36<br /><ul><li>Fuzzy Logic can be useful in solving Human related tasks
    39. 39. Evidence Theory gives tools to handle knowledge
    40. 40. Membership functions and Aggregation methods can be selected according to the problem at hand</li></ul>Some things we didn’t talk about:<br /><ul><li> Fuzzy C-Means (FCM) clustering algorithm
    41. 41. Dempster-Schafer theory of combining evidence
    42. 42. Fuzzy Relation Equations (FRE)
    43. 43. Compositions
    44. 44. Fuzzy Entropy</li></li></ul><li>References<br />[1] G. J. Klir ,U. S. Clair, B. Yuan“Fuzzy Set Theory: Foundations and Applications “,Prentice Hall PTR 1997, ISBN: 978-0133410587 <br />[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. <br />[3] A.K. Hocaoglu; P.D. Gader; “ An interpretation of discrete Choquet integrals in morphological image processingFuzzy Systems “, Fuzzy Systems, FUZZ &apos;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.<br />[5] A. Dumitras, G. Moschytz, “Understanding Fuzzy Logic – An interview with LoftiZadeh”, IEEE Signal Processing Magazine, May 2007 <br />[6] J.M. Yang; J.H. Kim, ”A multisensordecision fusion strategyusingfuzzymeasuretheory ”, Intelligent Control, Proceedings of the 1995 IEEE International Symposium on, pp. 157 – 162, Aug. 1995 <br />[7] R. Steinberg, D. O’Shaugnessy ,”Segmentation of a Speech Spectrogram using Mathematical Morphology ” ,To be presented at ICASSP 2008.<br />[8] J.C. Bezdek, J. Keller, R. Krisnapuram, N.R. Pal, ” FuzzyModels 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<br />[10] http://pami.uwaterloo.ca/tizhoosh/fip.htm<br />[11] &quot;Heavy-tailed distribution.&quot; 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<br />[12] T.J. Ross, “Fuzzy Logic with Engineering Applications”, McGraw-Hill 1997. ISBN: 0070539170<br />37<br />
    45. 45. Heavy-Tail Distributions<br />38<br />Bonus Slide<br />Examples: Alpha Stable (Cauchy, Pareto), Weibull, Student-T, Log-Normal …<br />Problem – different samples with very low probability occur very frequently<br />Solution: Smoothing the probability density function; Good or Bad??<br />Another Solution: Use Possibility (Membership function) and Necessity as envelopes<br />Example: Amazon sells far more books that are ‘very unpopular’ than popular books<br />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. <br />