Image Processing Reference

In-Depth Information

A1:
(Bel
1
⊕

Bel
2
)(
A
) has to be a function of only the functions
m
1
and
m
2
and

of
A
.
2

A2:

⊕

has to be commutative.

A3:

⊕

has to be associative.

A4:

If
m
2
(
B
)=1, then
m
1
⊕

m
2
has to satisfy the conditioning law, meaning

that:

m
2
(
A
)=

X
⊆
B

m
1
⊕

m
1
(
A

∪

X
)

∀

A

⊆

B

[B.5]

=0otherwise
.

A5:
The law has to satisfy an internal symmetry property (invariance under per-

mutation of the simple hypotheses).

A
(auto-

A6:
For
A

=
D
, (
m
1
⊕

m
2
)(
A
) does not depend on
m
1
(
X
) for
X

⊆

functionality property).

A7:
There are at least 3 elements in
D
.

A8:
The law has to satisfy a continuity property:

m
2
(
A
)=1

−

ε, m
2
(
D
)=
ε, m
A
(
A
)=1

ε
→
0
m
1
⊕

m
2
(
X
)=
m
1
⊕

m
A
(
X
)
,

[B.6]

=

⇒∀

X,
lim

with
m
1
being a function of any mass; this property makes it possible to eliminate

degenerate cases.

B.2. Inference of the combination rule

Based on the previous axioms, Smets obtains the only possible combination rule

that satisfies these axioms. To do this, he relies on communality functions defined by:

D, q
(
A
)=

A
⊆
X, X
⊆
D

∀

A

⊆

m
(
X
)
.

[B.7]

2. In [KLA 92], the axioms are for the most part similar to those used by Smets. The major

difference lies in the use of relations between spaces of discernment that are included in one

another rather than of dependence relations as used by Smets.

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