Fuzzy Image ProcessingFuzzy Image ProcessingUniversity of Waterloo  

Fuzzy Sets
What is FIP
Why FIP?
History
Theory
Examples
Literature
Software
Conferences
Related Links
Contact
 

Fuzzy Logic & Probability Theory:
Clarification towards Building a Bridge

Fuzzy logic and probability theory are the most powerful tools to overcome the imperfection (see Fig.1). Fuzzy logic is mainly responsible for representation and processing of vague data (ill-defined, fuzzy). Probability theory is mainly responsible for representation and processing of uncertainty (randomness).


Fig.1. Imperfection and theories to handle it.

Following table clarifies the differences between the two theories.

Probability Measure

Membership Function

Calculates the probability that an
ill-known variable x ranging on U hits the well-known set A

Calculates the membership of a
well-known variable x ranging on U hits the ill-known set A

Before an event happens

After it happened

Measure Theory

Set Theory

Domain is 2U (Boolean Algebra)

Domain is [0,1]U (Cannot be a Boolean Algebra)

A Bridge

Consider the following statements:

  • It is probable that it will rain a lot tomorrow.

  • It is probable that the image will be very dark.

  • It is probable that her new friend is handsome.

In such cases (which are very usual in pattern recognition, for instance), we are interested in probability of an event that cannot be defined exactly. Therefore, the only sophisticated way is to calculate the probability of a fuzzy event represented by a fuzzy set:

Probability space

Membership function

The probability of the fuzzy event F

For more details see following papers/books:

  1. Probability measures of fuzzy events, L.A.Zadeh, Journal Math. Anal. Appl., vol 23, pp. 421-427, 1968

  2. Fuzzy Sets as a basis for a theory of possibility, L.A. Zadeh, Fuzzy Sets and Systems, vol. 1, pp. 3-28, 1978

  3. Possibility Theory, D.Dubois, H. Prade, Plenum Press, 1988

  4. Fuzzy sets and probability : Misunderstandings, bridges and gaps, D.Dubois, H. Prade, Proc. of the Second IEEE Inter. Conf. on Fuzzy Systems, volume 2, pp. 1059-1068, 1993
 
 
 

Content by:
H. R. Tizhoosh

Created by:
Log Web Design

Powered by:
PAMI Lab

Created: June 1997
Updated: Nov 2004