ISPRS Commission III, Vol.34, Part 3A „Photogrammetric Computer Vision‘, Graz, 2002 
  
pio) er-(v6)-1°) | 
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