Belief Function Theory - Information Fusion in Signal and Image Processing - page 100

The simplest way imaginable consists of calculating the masses on the singletons
in a source (an image, for example) I j by:
m j C i ( x )= M i ( x ) ,

[7.12]

where M i ( x ) is estimated most often as a probability. The masses on all of the other
subsets D are then equal to zero. Clearly, this model is very simplistic and does not put
to use the interesting characteristics of belief function theory. However, many meth-
ods rely on an initial model like this one, or only use certain composite hypotheses, in
a simplifying and often very heuristic method [CLE 91, LEE 87, RAS 90, ZAH 92].
Recent work addressed the problem of estimating belief functions from sample data.
For instance, belief functions are estimated from realizations of a random variable,
with the constraint that they converge towards the probability distribution of this vari-
able when the sample size goes towards infinity (see [DEN 06]). But other methods
can also be considered. In the following sections, we present a few of the models found
in other works.

7.3.1. Modification of probabilistic models

The simplest and most often used model consists of using the discounting tech-
nique [SHA 76]. The new masses m are calculated based on the initial masses m as
follows (the index j representing the source of information, as well as the element x
we are reasoning on, are omitted here):
m C i = αm C i , [7.13]

m ( D )=1 − α + αm ( D ) , [7.14]

where α [0 , 1] is the discounting coefficient. In the case where the initial masses

are learned from singletons only, for example, based on probabilities, then m ( D )=0
and m ( D )=1

∈

α . This technique is often used to weaken a source depending on its
reliability and makes it possible to assign to D a mass that will be small if the source
is reliable and high if it is not. In extreme cases, the value α =0is used for a source
that is not reliable at all and all of the mass is then assigned to D , which represents the
total absence of knowledge. The value α =1is used for a reliable source in which all
of the mass is assigned to the singletons and there is no ambiguity between classes.

−

This type of model is very simple. Learning the masses of the singletons can bene-
fit from the usual techniques of statistical learning. However, hypothesis disjunctions
are not modeled, which strongly reduces the applicability of this model.

Two models based on the probabilistic approach have been suggested by Appriou
[APP 93], taking into account disjunctions other than D . These models assume that
initial estimations have been conducted of the conditional probabilities p ( f ( x ) C i )

(where f ( x ) refers to the characteristics of x extracted from the source and on which
the fusion is based), which are denoted more simply by p ( x

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C i ). The mass function