Belief Function Theory - Information Fusion in Signal and Image Processing - page 102

then the masses are normalized. If no probability is significant, the mass is assigned
entirely to D .

7.3.2. Modification of distance models

An approach using shape recognition is suggested in [DEN 95]. If each class C i is
represented by a prototype (also called a center) x i , then a mass function associated
with each class can be defined, in which C i and D are the only focal elements:
m i C i ( x )= αe − γd 2 ( x,x i ) , [7.24]

αe − γd 2 ( x,x i ) .

m i ( D )( x )=1

− [7.25]

The parameters α and γ allow us to modify the amount of absence knowledge and
the types of mass functions. Using the distance d 2 ( x, x i ), the mass can be set so as to
be high when x “is similar” to the class C i . The m i are then combined according to
Dempster's rule (see section 7.4) in order to have a mass that takes into account the
information about all of the classes.

This approach can also be applied to the k closest neighbors. The distance is the
one between x and one of its neighbors, and the mass is assigned, according to the
previous model, to the class to which this neighbor belongs and to D . The functions
calculated for each of the neighbors of x are then combined using Dempster's rule.

7.3.3. A priori information on composite focal elements (disjunctions)

In many applications, it is possible to have a priori information available that
can be used to determine, under some supervision, which focal elements should be
taken into account. These methods were used, for example, in [BLO 96, MIL 00,
MIL 01, TUP 99]. In [BLO 96], images of the brain are combined to detect patholo-
gies (see section 7.7). Mass functions are automatically estimated based on gray levels
[BLO 97b] and the classes that cannot be distinguished in certain images from their
gray levels are grouped together in disjunctions. In [TUP 99], the results from sensors
in several structures are fused in order to interpret a radar image. The abilities of a
sensor to tell the difference or not between various classes of structures is what makes
it possible to define focal elements and the class disjunctions that need to be taken
into account. In [MIL 00, MIL 01], attributes extracted from images provided by dif-
ferent sensors are combined in order to distinguish mines from harmless objects, in a
humanitarian demining program. The measurements to combine can be specific to a
class or to the entire frame of discernment. For example, the depth of the objects can
be used to assign a mass to harmless objects if it is high, but cannot be used to tell the
difference between objects if it is low and the mass is then assigned to D .