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 ) i f 2 ( x ) + λ
y ∈V ( x )
ω C i ,C y
[9.4]
where Φ i 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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