Image Processing Reference
A1: (Bel 1 ⊕
Bel 2 )( A ) has to be a function of only the functions m 1 and m 2 and
of A . 2
has to be commutative.
has to be associative.
If m 2 ( B )=1, then m 1 ⊕
m 2 has to satisfy the conditioning law, meaning
m 2 ( A )=
X ⊆ B
m 1 ⊕
m 1 ( A
A5: The law has to satisfy an internal symmetry property (invariance under per-
mutation of the simple hypotheses).
A6: For A
= D , ( m 1 ⊕
m 2 )( A ) does not depend on m 1 ( X ) for X
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 ) ,
with m 1 being a function of any mass; this property makes it possible to eliminate
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
m ( X ) .
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.