Image Processing Reference
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His demonstration, which we will not give in detail, relies on the properties of
triangular norms and of absolutely monotonic functions and is divided into three parts.
First of all, axioms A1 to A4 lead to the existence of a function f such that the result of
the combination depends only on A and on the communalities of the subsets included
in A :
q 1
q 2 ( A )= f A, q 1 ( X ); X
A .
A, q 2 ( X ); X
[B.8]
Then, by adding axioms A5 and A6 , it is possible to specify the form of f , which
now only depends on A and on q 1 ( A ) and q 2 ( A ):
q 1
q 2 ( A )= f A, q 1 ( A ) ,q 2 ( A ) .
[B.9]
Finally, the set of axioms A1 to A8 allows us to determine the final form of the
combination rule:
q 1
q 2 ( A )= q 1 ( A ) q 2 ( A ) .
[B.10]
As expected, we get the Dempster-Shafer rule on communalities and this leads us
to the combination of mass or credibility functions.
The advantage of Smets's method is that the axioms it relies on have interpretations
that are close to what our intuition tells us. It is also easier to refute or modify them if
they do not correspond to the problem at hand.
B.3. Relation with Cox's postulates
In this section, we will try to establish the links between Cox's postulates (see
Appendix A) and Smet's axioms, in order to show why they lead to different theories.
First of all, we should specify which framework we chose for this comparison. The
works of Cox and those of Smets do not deal with exactly the same problems, since
Cox attempted to justify probabilities and their properties, whereas Smets tried to
justify a combination rule. However, it is interesting to note certain analogies between
the two sets of axioms. Furthermore, Cox's axioms make it possible to infer Bayes'
rule (equation [A.8]), which is used in signal and image processing to fuse information
by using conditional probabilities (see Chapter 6). We chose the data fusion point of
view for this comparison. It would also be interesting to compare Cox's axioms with
those introduced by Smets in order to justify credibility and plausibility functions
[SME 93], but this comparison would only involve the modeling phases of the fusion
process and not the combination phases themselves.
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