Image Processing Reference
In-Depth Information
the phase that consists of modeling and representing the information and knowledge
available.
From now on, we will very often illustrate our discussion with the example of
multi-source classification.
7.2. Modeling
Belief function theory, like possibility theory, allows us, as we will see in the fol-
lowing chapter, to represent both imprecision and uncertainty using mass functions
m
, plausibility functions Pls and belief functions Bel [GUA 91, SHA 76, SME 90a].
Mass functions are defined for all of the subsets in the space
D
, referred to here as the
frame of discernment (containing, for example, the classes we are interested in), and
not simply the singletons such as probabilities which only measure the probability of
belonging to a given class.
where each
C
i
refers to a hypothesis
that supports a decision (typically a class in a multi-source classification problem).
A mass function is defined as a function of 2
D
Let us assume that
D
=
{
C
1
,C
2
,...,C
n
}
(the sets of subsets of
D
)into[0
,
1].
Usually, the condition
m
(
∅
)=0is imposed, as well as a normalization of the form:
m
(
A
)=1
,
[7.1]
A
⊆
D
which guarantees the commensurability between several sets of masses.
)=0corresponds to a closed world hypothesis, in which all of
the possible situations are in fact represented in
D
(which implies that we are capable
of making a list of them). If this constraint is relaxed and if we accept having a mass
that is strictly positive over
The constraint
m
(
∅
∅
, this then corresponds to an open world hypothesis, in
which the solutions outside of
D
can be considered.
A focal element is a subset
A
of
D
such that
m
(
A
)
>
0. The collection of focal
elements is called the core.
A belief function Bel is a totally increasing function defined from 2
D
into [0
,
1]:
2
D
,...A
k
∈
2
D
,
∀
A
1
∈
Bel
∪
i
=1
···
k
A
i
≥
1)
|
I
|
+1
Bel
∩
i
∈
I
A
i
,
[7.2]
(
−
I
⊆{
1
···
k
}
,I
=
∅
where
|
I
|
refers to the number of elements in
I
and such that Bel(
∅
)=0, Bel(
D
)=
1.
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