Belief Function Theory - Information Fusion in Signal and Image Processing - page 112

Mixed rules have also been suggested, in which plausibility is used for certain
classes and belief for others. This makes it possible to favor the detection of classes
for which plausibility is considered [MIL 01].

The decision can also be made in favor of a disjunction. In this case, it is imprecise
but does allow us to take into account class mixture or ambiguities remaining after the
fusion. This type of decision is interesting, for example, when taking into account the
partial volume effect and the voxels affected by it will thus be classified as mixture
voxels, rather than voxels from pure classes, as our intuition would tell us [BLO 96].
The decision also allows us to indicate the elements for which fusion is not enough to
clear up the ambiguities and therefore to suggest the acquisition of new information,
as well as the use of active fusion [GAN 96, PIN 95].

Finally, decision rules with costs have been suggested [DEN 95]. For any function
f of D in , the lower and upper expectations of f relative to a belief function Bel,in

Dempster's sense, are defined by:
E ∗ ( f )=
A ⊆ D

R

f C i ,

[7.40]

m ( A )min
C i ∈ A

E ∗ ( f )=
A ⊆ D

f C i .

m ( A )max
C i ∈ A

[7.41]

Decision rules with costs are then obtained by choosing for f a function that
expresses the cost of an action when the element to which the decision pertains belongs
to the class C i . This cost function can also be introduced with a traditional proba-
bilistic decision rule with costs, using pignistic probability. Thus, the decision can be
optimistic if the lower expectation is minimized, pessimistic if the upper expectation
is minimized, or intermediate if the pignistic probability is used.

7.7. Application example in medical imaging

The application we have chosen here, to give the reader an idea of the poten-
tial of belief function theory, is the classification of MRI images presenting a pathol-
ogy known as adrenoleukodystrophy (ALD), which are acquired with two echo times
[BLO 96]. For doctors, obtaining significant measurements requires a segmentation
of both the pathological areas and ventricles, which are visible on different images.
The initial images are represented in Figure 7.1. This figure shows a good discrimi-
nation between the brain, the ventricles (V) and the cerebrospinal fluid (CSF) on the
first image, but white matter (WM) cannot be distinguished from gray matter (GM),
or WM from CSF. On the other hand, the ALD area is clearly visible on the second
image (in white). This image presents small differences between WM and GM, but the
ventricles have almost the same gray levels as the GM and their contours are indistinct.