Belief Function Theory - Information Fusion in Signal and Image Processing - page 103

This type of approach is very effective if such information is available, but it
remains supervised and therefore can only be applied to problems with a reasonable
number of elements in D .
7.3.4. Learning composite focal elements
Learning methods for focal elements often rely on prior classifications conducted
in each source separately. Typically, based on confusion matrices, it is possible to
identify classes confused according to a source, whose reunion will constitute a focal
element of the mass function assigned to this source.
In a completely non-supervised way, the intersection between the classes detected
in a source and those detected in another source can define the singletons and the frame
of discernment, with the classes detected in each source then becoming disjunctions
[MAS 97].
Dissonance and consonance measures are given in [MEN 96]. The idea consists
of modifying an initial mass function, involving only the singletons, by discounting
the masses of the singletons depending on their levels of consonance and by creating
masses for disjunctions of two classes depending on the level of dissonance between
these two classes. This method was applied to the fusion of several classifiers. The
consonance of a class is calculated based on the number of elements affected to that
class by all of the classifiers and the dissonance based on the number of elements that
are classified differently.
In the case of elements that are characterized by a measurement in a one-dimen-
sional space (represented typically by a histogram), the masses on the composite
hypotheses can be defined in the areas of overlapping or ambiguity between two neigh-
boring classes. Another method, based on thresholding by hierarchy, is suggested in
[ROM 99] where each peak of the histogram corresponds to a singleton. Then the
histogram is progressively thresholded at decreasing heights and disjunctions are cre-
ated when maxima are grouped together. This method can be compared to component
trees, which are used, for example, in mathematical morphology with the concept of
cup topology [DOK 00], and to confidence intervals and their relations with possibility
distributions [DUB 99, MER 05] (see Chapter 8).
7.3.5. Introducing disjunctions by mathematical morphology
Without being restricted to one-dimensional representation spaces, the method
suggested in [BLO 97a] allows for the calculation of masses for disjunctions, by ero-
sions and dilations of masses first defined for singletons. The properties of these mor-
phological operations (of duality in particular) make it possible to interpret them as
beliefs and plausibilities, from which the masses can then be inferred.