Belief Function Theory - Information Fusion in Signal and Image Processing - page 105

and

m 1 ⊕

m l ( m 2 ⊕···⊕

∅

)=0

if the denominator of equation [7.27] is not equal to zero, i.e. if:
k =
B 1 ∩···∩ B l = ∅

m 1 B 1 m 2 B 2 ··· m l B l < 1 .

[7.28]

Therefore, this quantity (which measures the conflict between sources) is directly
taken into account in the combination in the form of normalization factor. It represents
the mass that would be assigned to the empty set if we did not have this normalization
(equation [7.26]). It is important to take into account this value in order to appreciate
the quality of the combination: it may not make much sense in case of a strong conflict
and lead to decisions that could be disputed.

and two mass
functions with only the singletons as their focal elements and the following values:

Let us consider a simple example in which D = { C 1 ,C 2 ,C 3 }

C 1 C 2 C 3

m 1 0.9 0.0 0.1

m 2 0.0 0.9 0.1

Their non-normalized and normalized fusions lead to:

C 1 C 2 C 3 ∅

m 1 ⊕ m 2 non-normalized 0.0 0.0 0.01 0.99

m 1 ⊕ m 2 normalized 0.0 0.0 1.0 0.0

All of the mass is then concentrated in C 3 , which is the only class in which both
sources agree, but only to say that the solution is hardly plausible. The normalization
masked the conflict. The non-normalized form is often preferable in case of conflict.
Here, it allows us to assign the essential part of the mass to the empty set and the
origin of the conflict can be attributed to the open world hypothesis, a low reliability
of at least one of the two sources, or to the fact that one source sees a class C 1 object
whereas the second source see another class C 2 object.