ISPRS Commission III, Vol.34, Part 3A „Photogrammetric Computer Vision‘, Graz, 2002
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1
(22);
where 2i is the translation applied and for a given integer, i,
w(y) € [2i-1, 2i+11.
3.2 Boundary Probability
The aim here is to provide a single gradient image given as
input a multi-dimensional set of images. In order to integrate
the contrast information contained in the various channels into
one meaningful result, Drewniok, 1994 extended in a formal
way a gray-level edge detector to multidimensional image data.
The integrating approach combines the contrast information
coming from the different spectral channels in a well-founded
way. The resulting gradient image gives a suitable description
of area boundaries well adapted to the purpose of our
framework. Assuming that the i^ channel produces an image
I;(n,m):
1;(n,m)= R;(n,m)+ N;(n,m) (10)
where Ri(n,m) is the ground truth and N;(n,m) is an additive
Gausssian noise. The produced gradient image, G(n,m), is
corrupted by false-edges due to noise. The plausibility of false
edges follows a Rayleigh distribution (Voorhees and Poggio,
1987). In order to distinguish real edges from false ones,
Voorhees proposed to estimate statistically a threshold that
separate these two populations. The threshold is calculated
according to the estimation of the peak of the Rayleigh
distribution. The threshold, x, required to remove noise with a
confidence value of 99% is defined as follows:
ves (11)
1
where u is the mean of the Rayleigh distribution. We assume
the plausibility of the true edges being described by a single
distribution. Let ug be its expected mean value. As for the
Mahalanobis distance discussed in Section (3.1.2), one need to
normalize the gradient values in order to keep both measures
(gradient and statistics) in the same numerical range.
Therefore, we define the normalized gradient at a given
position y; as follows:
à, )- $03)
JS Ig
(12)
The segmentation field, x, has an isotropic nature and its
distribution is strictly defined in a local neighborhood.
Thereafter, we use an MRF to model it (Bouman and Sauer,
1993). Using the Hammersley-Clifford theorem, the density of
x is given by a Gibbs density on the form:
«e|
e (13)
me
Here, Z is a normalizing constant and the summation is over all
cliques C. A clique is a set of points that are neighbors of each
A- 184
other. The clique potentials Vc depend only on the pixels that
belong to clique C. They are inversely proportional to the
homogeneity of the contour plausibility in the immediate
neighborhood of the considered pixel, x,, (Bouman and Sauer,
1993).
Since the gradient defines a measure of non-homogeneity and
is evaluated in the immediate neighborhood of yy, its response
could be handled as being a transformation that maps the gray-
level of y, to the potential function V-(x;) . This relation is
carried out in a proportional manner as follows:
Ve (xs) G(v,) (14)
In view of this, the MRF field will henceforth be written in the
following, simplified form (where the constant of
proportionality in (14) is dropped for the sake of computational
ease):
ps, )- 7e 60) (15)
4. COMBINATION
As mentioned earlier, the segmentation purpose can be
described as being a the estimation of the suitable values for
the control nodes coordinates that minimize the global energy
of the snake. The estimation of these parameters to find the
boundary is posed as an optimization process, where a MAP-
based objective function measures the strength of the boundary
given the set of control nodes. The maximization of p(x|y)
given the control nodes could be written as follows:
X = arg max [120s / ys) (16)
y^ Uses
By applying the Bayesian formulation, the combined
probability p(x[y) in equation (16) can be expressed as follows:
ply) yo (17)
In the following, we ignore the term p(y) because it is
supposed to be a prior knowledge and it does not modify the
MAP estimation. Thus:
X = arg „a Tob, | (18)
y* ses
By replacing p(x;) and p(y;|x;) by their respective expressions,
X could be evaluated in terms of the snake curvilinear abscise,
r, as follows:
X = arg max Il Le-606)),-v" (vr) (19)
ve v(r)eS
It is clear that equation (19) combines the statistical and the
gradient-based measures in order to find the optimal
segmentation. The main question is to know if the formulation
given below allows a constructive integration of both provided
measures. In other words, we intend to investigate the behavior