Image Processing Reference

In-Depth Information

The construction of possibility distributions can also be done from probabilistic

learning, followed by a transformation of probability into possibility. Several meth-

ods have been suggested for this purpose. The main advantage in signal and image

processing is that statistical information is often available, particularly the histogram,

which is well suited for applying statistical learning methods. We then get probability

distributions
p
k
. Their transformation into possibility distributions
π
k
(both distribu-

tions are assumed to be discrete and 1

≤

≤

K
) is achieved according to various

criteria [DEV 85, DUB 83, KLI 92], such as not changing the order, normalization

constraints, conserving the uncertainty measured by the entropy [KLI 92], the consis-

tency
p

k

−

π
expressed by [DEL 87]:

p
k
,

which is not very satisfactory (an unlikely class can be possible), or [ZAD 78]:

∀

k, π
k
≤

K

p
k
π
k
=
c

k
=1

where
c
is a constant in [0
,
1], or also a more general relation involving all of the

subsets
A
[DUB 80]:

N
(
A
)

≤

P
(
A
)

≤

Π(
A
)
.

A comparison of these methods can be found in [KLI 92].

Other methods try to directly estimate the membership functions based on the

histogram, in order to optimize entropy criteria [CHE 95] or minimal specificity and

consistency criteria [CIV 86].

In any case, these methods attempt to find a similarity between the histogram and

the membership functions or the possibility distributions and do not take into consid-

eration interpretations that are specific to fuzziness because they invalidate some of

these similarities. For example, the tails of the histogram correspond to classes with

little representation, hence with values that can be very low, even if the points involved

belong to the corresponding classes. The method suggested in [BLO 97] provides a

way of avoiding this problem with the help of a criterion that combines the similari-

ties of membership functions and the histogram where they have meaning, with an
a

priori
form of the functions that correspond to the desired interpretation. The parame-

ters of the membership functions are then estimated in order to optimize this criterion

by using a simulated annealing method.

8.10. Combining and choosing the operators

One of the advantages of fuzzy set and possibility theory, beyond the fact that it

imposes few constraints on modeling, is that it offers a wide variety of combination

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