Image Processing Reference
In-Depth Information
6.4. Combination in a Bayesian framework
In the Bayesian model, fusion can be achieved in equivalent ways on two levels:
- either on the modeling level and we then calculate probabilities in the form:
p
x
I
1
,...,I
l
,
∈
C
i
|
[6.8]
using Bayes' rule:
I
1
,...,I
l
=
p
I
1
,...I
l
|
C
i
p
x
C
i
p
x
x
∈
∈
p
I
1
,...,I
l
∈
C
i
|
,
[6.9]
where the different terms are estimated by learning;
- or from Bayes' rule itself, where the information provided by a sensor updates
the information regarding
x
, which is estimated according to the previous sensors (this
is the only usable form if the elements of information are available one after the other
and not simultaneously):
p
x
I
1
,...,I
l
=
p
I
1
|
∈
C
i
|
C
i
p
I
2
|
C
i
,I
1
···
p
I
l
|
C
i
,I
1
,...,I
l
−
1
p
x
C
i
x
∈
x
∈
x
∈
∈
p
I
1
p
I
2
|
I
1
···
p
I
l
|
I
1
,...,I
l
−
1
.
Very often, because of the complexity of learning using several sensors and the
difficulty of gathering enough statistics, these equations are simplified under the inde-
pendence hypothesis. Again, criteria have been suggested for verifying the validity of
these hypotheses. The previous formulae then become:
I
1
,...,I
l
=
j
=1
p
I
j
|
C
i
p
x
C
i
p
x
x
∈
∈
p
I
1
,...,I
l
∈
C
i
|
.
[6.10]
This equation clearly shows the combination of information as a product, hence
a conjunctive fusion. It is worth noting that the
a priori
probability plays exactly the
same role in the combination as each of the sources with which it is also combined by
a product.
6.5. Combination as an estimation problem
Another way of seeing probabilistic fusion consists of considering that each source
yields a probability (of belonging to a class, for example) and that fusion consists of
combining these probabilities, in order to find the overall probability of belonging to
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