Image Processing Reference
In-Depth Information
All of the sources have to be “conditioned” by
m
B
, in order to account for the
fact that the truth can only be in
B
. Conditioning is done simply by combining a mass
function
m
with
m
B
:
m
B
(
A
)=
A
=
B
∩
C
∀
A
⊆
D, m
⊕
m
(
C
)
,
[7.31]
which can also be written:
∀
A
⊆
D, A
⊆
B, m
⊕
m
B
(
A
)=0
,
[7.32]
m
B
(
A
)=
X
⊆
B
C
∀
A
⊆
D, A
⊆
B, m
⊕
m
(
A
∪
X
)
.
[7.33]
Conditioning is performed in accordance with the transferable belief model
[SME 90a]: knowledge of
B
leads us to transferring all of the mass on the subsets
included in
B
. Thus, the belief initially assigned to a subset
A
=
A
1
∪
A
2
(with
A
1
⊆
B
C
) represented the fact that the truth could be anywhere in
A
. Knowl-
edge of
B
can now be used to specify the information and to reduce
A
to
A
1
.Ina
way, the diffuse belief in
A
is now concentrated in the only part that is included in
B
.
B
and
A
2
⊆
Conditioning performed according to the conjunctive rule is the equivalent, in the
framework of belief functions, of conditional probabilities, which also corresponds to
a conjunction. This is because we have:
B
)=
P
(
X
B
)
P
(
B
)
∩
P
(
X
|
.
7.4.6.
Separable mass functions
We now consider simple support mass functions. If
m
1
and
m
2
are simple support
functions with the same support
A
, with weights
s
1
and
s
2
, then the combination
yields a function with the same support and a weight
s
1
+
s
2
−
s
1
s
2
. Such functions
are never cause for conflict.
If both functions have different supports
A
1
and
A
2
, then the combination leads
to:
m
1
⊕
m
2
A
1
∩
A
2
=
s
1
s
2
m
1
⊕
m
2
A
1
=
s
1
1
s
2
−
m
1
⊕
m
2
A
2
=
s
2
1
s
1
−
m
1
⊕
m
2
(
D
)=
1
s
1
1
s
2
−
−
m
1
⊕
m
2
(
B
)=0
∀
B, B
=
A
1
,B
=
A
2
,B
=
A
1
∩
A
2
,B
=
D.
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