Belief Function Theory - Information Fusion in Signal and Image Processing - page 113

Figure 7.1. Example of an MRI section of the brain acquired with two echo times
(Saint-Vincent de Paul hospital, radiology service, Professor Catherine Adamsbaum).
The pathological area corresponds to the whiter areas in the upper part of the image

These two images constitute a typical example to illustrate the fusion by belief
function theory. We will only present here the results obtained for three classes ( C 1 =
WM + GM, C 2 = V + CSF and C 3 = ALD).

The definition of the focal elements is supervised using a reasoning method that
takes into account the knowledge available and the characteristics of the image with
respect to the classes we are focusing on. For the example described here, the focal
elements of the mass function m 1 assigned to the first image are C 2 , C 1 ∪ C 3 , since C 1

and C 3 are not well discriminated on this image. Zero mass functions are assigned to
the other composite hypotheses, since the corresponding classes cannot be confused.
On the second image, it is on the other hand difficult to separate the brain from the ven-
tricles and therefore the focal elements of m 2 are C 3 and C 1 ∪

C 2 . We will discuss later
the introduction of overall absence of knowledge and of a mass explicitly representing
the partial volume. The mass functions are chosen with a simple trapezoidal shape,
whose parameters are automatically determined on the histograms [BLO 97b]. This is
a crude model, but it has proven to be sufficient for this application. The functions are

then normalized so as to satisfy the normalization constraint A ⊂ D m ( A )=1. With
this model, the classification is performed only based on the gray levels and the fusion
is performed on the pixel level, therefore without spatial information.

Conjunctive combination by Dempster's rule only provides focal elements which
are the singletons C 1 , C 2 , C 3 . The conflict is not equal to zero in this case.

The last phase is the decision making. Always making a decision in favor of a
simple hypothesis forces us in fact to always making a clear decision, which is not
adapted to all of the actual situations in medical imaging, where pixels can belong to
a union of classes but also to none strictly. However, because Bel( A )

≥

Bel( C i ) for