Image Processing Reference

In-Depth Information

a possible application of this theory consists of choosing
S
=
D
(the class set) and

defining the measure
M
i

by:

M
i
(
x
)=
π
j
C
i
,

[8.101]

i.e. as the possibility degree for the class to which
x
belongs to have the value
C
i
,

according to the source
I
j
. This defines a possibility distribution for each source and

each element
x
. The possibility and the necessity for each class are then written:

Π
j
C
i
=
π
j
C
i
,

N
j
C
i
=inf
1

π
j
C
k
,C
k

=
C
i
.

−

[8.102]

For any subset
A
of
D
, the possibility and the necessity are calculated using for-

mulae [8.24] and [8.26].

This modeling assumes that classes are crisp, whereas the fuzzy model defined by

equation [8.100] assumes fuzzy classes.

Generally speaking, there are three interpretations of fuzziness in other works, in

terms of plausibilities, similarities, preferences [DUB 99]. The same interpretations

are used in signal and image fusion. The interpretation in terms of plausibilities is

used for the membership to a class, in the definition of a fuzzy spatial object (an

object with imprecise limits). The interpretation in terms of similarities is that used

for the definition of a fuzzy class in a characteristic space as a function of the distance

to a prototype, for example, of linguistic variables representing information or knowl-

edge about spatial objects, or also of degrees of satisfaction of a relation, a constraint.

Finally, preferences are used in the expression of choice criteria (for example, for

planning applications in robotics), which are often related to constraints or knowledge

outside the image.

8.9. Defining membership functions or possibility distributions

Constructing membership functions or possibility distributions can be done in sev-

eral ways.

In most applications, this construction is done either by taking ideas directly from

probabilistic learning methods, from heuristics, from neuromimetic methods used for

learning the parameters of particular forms of membership functions, or finally by

minimizing classification criteria [BEZ 81]. Below is a description of the major meth-

ods.

A first method consists of defining a fuzzy class membership function based on

the image's intensity function
I
(the gray levels):

μ
i
(
x
)=
F
i
I
(
x
)
,

[8.103]

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