Image Processing Reference
In-Depth Information
which, this time, is perfectly satisfactory. This particular mass function plays an
important part in the combination rule since it is its identity element, which confirms
its interpretation in terms of complete lack of knowledge, which cannot modify
another mass function.
Smets's second idea is the transferable belief model, which defines conditioning.
The problem is stated as follows: given a new information, which makes it possible
to state that the truth is located in a subset
B
of the frame of discernment
D
,how
can a set of masses
m
be modified to take into account this new information? The
expression suggested by Smets is as follows:
m
(
A
)=
X
⊆
m
(
A
∪
X
)
∀
A
⊆
B
B
[B.4]
=0otherwise
where
m
refers to the new set of masses. These formulae can be modified if needed,
although this is not necessary and can even be detrimental to the extent that it masks
conflict [SME 90], as we saw in Chapter 7. Normalization also poses continuity prob-
lems in the vicinity of the total conflict [DUB 86].
This formula is interpreted as follows. If we decompose a subset
A
into the union
A
1
∪
B
, the mass
m
(
A
1
∪
A
2
) is entirely transferred to
A
1
(hence the model's name). In specific cases where
A
2
=
A
2
with
A
1
⊆
B
and
A
2
⊆
∅
(
A
⊆
B
), the mass of
B
), the mass of
A
becomes zero.
A
is not modified and if
A
1
=
∅
(
A
⊆
In Shafer's theory [SHA 76] and in our presentation of it in Chapter 7, the con-
ditioning formula is inferred from the combination rule, whereas here, it precedes it
and is constructed simply by logical considerations. Note that the conditioning rule on
plausibilities:
B
)=
Pls(
A
B
)
Pls(
B
)
∩
|
Pls(
A
can itself be justified by supposing that:
B
)=
T
Pls(
A
B
)
,
Pls(
B
)
Pls(
A
∩
|
and by applying a method similar to the one that lead us to the first functional equation
(section A.3.2) [DUB 86].
In a third step, Smets defines two axioms that he wishes to see satisfied by the
combination rule, denoted by
⊕
:
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