176
6: Teerasit Kasetkasem
an image model can take any valid forms (having a finite PDF), the ICM algorithm only permits the prior probability of an image to be the multiplication
of local characteristics of all sites, i. e.,
Pr(X) = n Pr (X(s) IX (-8\s)) .
(6.25)
SE~
The above assumption does not fit the conventional MRF models, described
in Sects. 6.2 and 6.3. Nevertheless, it still considers the local interactions among
adjacent pixels through a statistical model while allowing the optimization
process to be completed at a faster rate because it deals with individual local characteristics rather than the Gibbs distribution of an entire image. In
addition, Begas pointed out that an image analysis algorithm based on the
conventional MRF model may not always produce an accurate result because
the conventional MRF model tends to favor single color cases (e. g. an entire
image composed of only one intensity value) of X over the more realistic multiple color cases (multiple land cover class attributes) of X. Single or multiple
color cases of X do not have significant influence on the prior probability under
the Begas' model as long as a spatial dependence is characterized correctly. As
a result, the model in (6.25) may be more suitable for the multiple color cases.
There are several approaches to find the optimum solutions of the MAP
problem under the assumption given in (6.25). The simplest one is to visit
one pixel (e.g. s) at a time, and try to replace the configuration (e.g. x(s)) of
this pixel with the configuration that maximizes the posterior probability. As
mentioned above, computation of the posterior probability can be obtained
easily under (6.25) than the conventional MRF model because, for one update,
we need to determine only the summation over all the possible configurations
of a site rather than the summation of all the possible configurations of the
entire image. Furthermore, the convergence of this method is guaranteed as
long as the posterior probability is well defined, since the posterior probability
always increases as the number of iterations increases, and is bounded by one.
The above procedure is summarized in Fig. 6.5. Another approach is to update
the configuration of a site by only considering its local characteristic, i. e.,
Xnew(s) = arg [ mtx (Pr (y(s) II) Pr (I I Ns,old ))] ,
(6.26)
where the subscripts new and old indicate configurations of the current and
previous iterations, respectively. For this second approach, the convergence
may not always be achieved since a higher posterior probability may not be
attained after one complete update. Moreover, based on the same reason,
the configurations may oscillate between two or more values. However, this
approach allows the entire image to be updated at the same time because the
update at a site has no effect on other sites.
6: Teerasit Kasetkasem
an image model can take any valid forms (having a finite PDF), the ICM algorithm only permits the prior probability of an image to be the multiplication
of local characteristics of all sites, i. e.,
Pr(X) = n Pr (X(s) IX (-8\s)) .
(6.25)
SE~
The above assumption does not fit the conventional MRF models, described
in Sects. 6.2 and 6.3. Nevertheless, it still considers the local interactions among
adjacent pixels through a statistical model while allowing the optimization
process to be completed at a faster rate because it deals with individual local characteristics rather than the Gibbs distribution of an entire image. In
addition, Begas pointed out that an image analysis algorithm based on the
conventional MRF model may not always produce an accurate result because
the conventional MRF model tends to favor single color cases (e. g. an entire
image composed of only one intensity value) of X over the more realistic multiple color cases (multiple land cover class attributes) of X. Single or multiple
color cases of X do not have significant influence on the prior probability under
the Begas' model as long as a spatial dependence is characterized correctly. As
a result, the model in (6.25) may be more suitable for the multiple color cases.
There are several approaches to find the optimum solutions of the MAP
problem under the assumption given in (6.25). The simplest one is to visit
one pixel (e.g. s) at a time, and try to replace the configuration (e.g. x(s)) of
this pixel with the configuration that maximizes the posterior probability. As
mentioned above, computation of the posterior probability can be obtained
easily under (6.25) than the conventional MRF model because, for one update,
we need to determine only the summation over all the possible configurations
of a site rather than the summation of all the possible configurations of the
entire image. Furthermore, the convergence of this method is guaranteed as
long as the posterior probability is well defined, since the posterior probability
always increases as the number of iterations increases, and is bounded by one.
The above procedure is summarized in Fig. 6.5. Another approach is to update
the configuration of a site by only considering its local characteristic, i. e.,
Xnew(s) = arg [ mtx (Pr (y(s) II) Pr (I I Ns,old ))] ,
(6.26)
where the subscripts new and old indicate configurations of the current and
previous iterations, respectively. For this second approach, the convergence
may not always be achieved since a higher posterior probability may not be
attained after one complete update. Moreover, based on the same reason,
the configurations may oscillate between two or more values. However, this
approach allows the entire image to be updated at the same time because the
update at a site has no effect on other sites.