Image Processing Reference

In-Depth Information

Bayes' rule makes it possible to calculate the
a posteriori
probability of each class

conditionally to the two images. The
a priori
probability term is modeled using a

Markovian hypothesis regarding the image of the classes and acts as a spatial regu-

larization. Therefore, the
a posteriori
probability is expressed as the product of three

terms: two terms expressing the probabilities of the gray levels in each of the images

conditionally to the classes (under the conditional independence hypotheses) and a

term expressing the spatial regularities of the classes. The Markovian framework

allows us to express the problem of the
a posteriori
maximum optimization as the

minimization of an energy that includes:

1) data-based terms, that depend on the gray levels of each image, on coefficients

weighting the importance of each image according to the classes and on prior knowl-

edge of the positions of the ventricles in the brain;

2) a regularization term, in the form of a Potts potential, that takes into account the

neighboring pixels of each point.

Therefore, the energy is written in each point
x
assigned to the class
C
i
in the

current step:

Φ
i
f
1
(
x
)
+Φ
i
f
2
(
x
)
+
λ

y
∈V
(
x
)

ω
C
i
,C
y

[9.4]

where Φ
i
(Φ
i
) represents the data-based potential characterizing the class
C
i
in the

first (second) image, which is a function of the gray level
f
1
(
x
) (
f
2
(
x
)) at the point

x
in this image,

V

(
x
) represents the spatial neighborhood of
x
,
C
y
the class to which

the neighbor
y
is assigned in the current iteration and
ω
(
C
i
,C
y
) represents the regu-

larization constraints between the classes
C
i
and
C
y
. The factor
λ
makes it possible

to weight the influence of the regularization with respect to the data-based term. The

data-based potentials are determined automatically based on histograms of gray levels,

whose significant modes are selected using a multi-scale approach. Here, the regular-

ization is simple: it favors the membership to the same class as the neighboring points

(
ω
(
C
i
,C
i
)=0) and puts at a disadvantage the membership of neighboring points to

different classes (
ω
(
C
i
,C
y
)=1if
C
y

=
C
i
).

The results (see Figure 9.1) show the spatial homogenity of the obtained segmen-

tation. The spatial information used in this example is still relatively local, since it

only involves a small neighborhood around each point.

9.4.2.
The modeling and decision level: fusion of structure detectors using belief

function theory

In this example, developed in [TUP 99], the objective is to interpret a radar image

by fusing the results of several structure detectors (roads, slopes, cities, etc.). The

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