Belief Function Theory - Information Fusion in Signal and Image Processing - page 110

, then both functions cause conflict, since a non-zero
mass is assigned to the empty set. When s 1 s 2

Particularly, if A 1 ∩ A 2 = ∅

=1, the resulting function is referred

to as a separable mass function.

7.4.7. Complexity

In the general case, as shown by formula [7.26], the combination has an exponen-
tial complexity. In practice, it is rare to have to take into account all of the subsets of
D and the complexity is often more reasonable. A linear complexity is obtained if the
masses are modeled according to Barnett's structure [BAR 81], i.e. if the focal ele-
ments of each source are only the singletons and their complements (separable func-
tions). This structure is suited for shape recognition problems in which each source is
a detector that can be used to distinguish one class from all the others. But it is not
general and cannot be applied to sources which require focal elements that can be any
disjunctions.

7.5. Other combination modes

Other combination modes, such as disjunctive or compromise modes, are possible
by replacing the intersection in formula [7.26] with another set operation. For exam-
ple, disjunctive fusion is obtained by taking the union [SME 93]:
m 1 ⊕ ∪ ···⊕ ∪ m l ( A )=
B 1 ∪···∪ B l = A

m 1 B 1 ··· m l B l .

[7.34]

Let us note that this combination cannot lead to conflict. It widens the focal ele-
ments therefore providing less precise information from each of the sources. This
fusion can be useful if we are unable to model beforehand the reliabilities, ambigu-
ities and imprecisions of the sources. For example, if a source is focused in A and
another one in B with A

∩ B = ∅ , one way of not solving the conflict is to conclude

that the truth is in A ∪ B , thus allowing a disjunctive fusion.

However, in most image fusion applications, the goal is to obtain a combined mass
function that is more focused than the initial masses. This is why conjunctive fusion is
the preferred method, since it implies that the imprecisions, reliabilities, ambiguities
of each source are taken into account in the modeling stage. It then constitutes the
most crucial phase and requires the most attention.

7.6. Decision

Once the combined mass functions have been combined, the belief and plausibility
functions are inferred from equations [7.3] and [7.6]. The last step is the decision