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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