Fuzzy Sets and Possibility Theory - Information Fusion in Signal and Image Processing - page 158

This only assumes knowing an order of reliability of the sources.

If, additionally, we have access to numerical values of reliability (a much more
stringent constraint than the previous hypothesis), we can then transform the possibil-
ity distributions into distributions with equivalent reliabilities. Let w j be the reliability
coefficient of π j ; if the source is completely reliable, this coefficient is equal to 1 and
it is equal to 0 if the source is not reliable at all. The transformation of π j

works

according to the formula:

max π j , 1 w j

− [8.83]

which amounts to conducting a disjunction between π j and a constant distribution of

value 1 w j . Thus, if the source is completely reliable, the corresponding distribution

is not modified, whereas if it had not been reliable at all, the distribution would have
become constant and equal to 1, which represents absence of knowledge (any element
is completely possible). Once the distributions have been transformed, they can be
combined conjunctively.

−

Other operators of this kind can be found in [DUB 99], but we will not discuss
them in detail here.

These operators can also be used conditionally to the classes, to take into account
the specificities of the sources for each class. Two sources may, for example, be in
conflict over a class but not over the others, a source may be reliable for certain classes
and not others, etc. Although these ideas are still not often applied in fuzzy fusion of
signals and images, the theoretical framework allows it.

8.6. Linguistic variables

It often happens to have numerical representations that are not suited for the de-
scription of a situation. For example, if a variable has a wide variation range, it can be
difficult to assign a precise value to each specific situation and the preferred method
will consist of using more qualitative terms, taken from natural language, to gener-
ally crudely define subsets that are typical of interesting situations. For example, to
describe the size of an object, it can be easier and more appropriate to only use a few
terms with flexible boundaries, such a small, medium, large. This corresponds to a
certain granularity of information. According to [ZAD 96], the concept of “granule”
is the starting point for “computing with words” theories. Zadeh defined a granule
as “a fuzzy set of points having the form of a clump of elements drawn together by
similarity” [ZAD 96]. A word then becomes a label for a granule. In order to per-
form calculations with such representations, specific tools have to be developed. The
field of fuzzy reasoning particularly benefits from these tools, as well as the field of
knowledge fusion or approximate or crude data.

These types of representations are referred to as linguistic variables. They are vari-
able whose values are words, phrases or sentences [ZAD 75]. Their advantage lies