Fuzzy logic and fuzzy set theory allow for partial membership in sets rather than binary membership. Some key concepts include fuzzy sets, membership functions, linguistic variables, aggregation operations, and fuzzy inference systems. Fuzzy logic has applications in areas like pattern recognition, control systems, and knowledge engineering by allowing systems to reason with imprecise or uncertain information in a way that resembles human thought processes.

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?
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3. Some Related Fields
Fuzzy
Logic &
Fuzzy Set
Theory
Evidence
Theory
Pattern
Recognition
& Image
Processing
Control
Theory
Knowledge
Engineering
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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
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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”
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6. Fuzzy Inference (Expert) Systems
Input_1 Fuzzy
IF-THEN
Rules
OutputInput_2
Input_3
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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???
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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    
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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 
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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  
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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:
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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
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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
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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
  
  
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16. Binary
Gray Level
Color (RGB,HSV etc.)
Can we give a crisp definition to light blue?
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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
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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
  


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19. Example: Finding an Image Threshold
Membership Value
Gray Level
   
1
, ,
1
smf a x c
f x a c
e 


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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, ...
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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…"
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22. Applying the Segmentation Algorithm
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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
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Assign a Fuzzy Value for
each Phoneme, Output
Highest N Values to a
Linguistic model

24. Summary
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• 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
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