ISPRS Commission II, Vol.34, Part 3A „Photogrammetric Computer Vision“, Graz, 2002 
  
of the MAP and determine if the two parts acts in such a 
manner to preserve the same objective everywhere in the 
image. This could be carried out by a qualitative analysis of the 
MAP. Indeed, when inside a textured region but far from the 
area boundaries, the statistical part may indicate that a pixel 
under investigation have a high confidence to belong to the 
region of interest. However, the gradient value can indicate 
that the pixel belongs to the area boundaries. This means that 
the gradient do not distinguish between the right area edges 
and those generated by the texture or the noise. In such 
situation, the gradient and the statistics are in conflict. Since, 
the MAP estimation is obtained by the product of these two 
parts, the decision made will be altered. Otherwise, when the 
pixel under investigation belongs to the area boundaries, its 
membership given by both gradient and statistics will be low. 
The two parts behave simultaneously to exclude this pixel from 
the segmented area. Knowing that the statistics suffers from a 
poor edges localization, the joint decision made could be 
affected. 
To resolve this conflict, we propose the use of a weight to set 
the contribution of each part into the MAP estimation. The 
weight will act as a smooth switch, which gives priority to 
statistics when the pixel is far from the area boundaries. Also, 
when the pixel is closer to the edge, the priority will be 
transferred to the gradient. The modified MAP is given as 
follows: 
x = arg max II 4 o6), -0- ry? (v(r)) (20) 
ye "v(r)eSs Z 
where 7 is a weight parameter resolving the conflict as 
described above. In the experimental results section, we 
propose a formulation to set this parameter. 
The MAP estimation will be maximized during the evolution 
of the snake. This is performed by setting the external force 
proportionally to the combined probability p(xs|ys). The final 
expression of the external forces will be: 
F 
ext 
(v(r)) Ze 66) @1) 
Finally, the partial derivative equation that govern the snake 
motion given by equation (5) can be rewritten as: 
ov(r.r) = ov. (r,t)+ pv (r, t) = 
ot 
a9 f T e 1096), - 7t y^ ver.) (22) 
ov Z 
This is equivalent to say that the evolution of the snake is 
equivalent to the estimation of the control nodes that maximize 
the criterion given by equation (20). This estimation is 
constrained by the snake topology described by the terms in the 
left side of equation (22) 
5. ALGORITHM 
Assuming the evaluation of the statistical mixtures done, the 
localization step defines the new position of every existing area 
of the database on the image. The operations performed over 
each area vector can be summarized as follow: 
e Set the initial snake parameters. 
e Until the snake energy is not minimum: 
e Set the external force of the snake from equation (21); 
e Compute the new snake location using equation (22). 
While the minimization process is running, the snake a and f 
parameters are decreased in parallel in order to enable the 
snake to fit accurately to the high curved parts of the area 
borders. This decrease of the snake parameters is done by 
multiplying them with a constant factor, A, smaller than one 
(0.75 for instance). Also, updating the control nodes of the 
snakes optimizes the iterative energy minimization process. 
Indeed, two principal operations are performed: addition of 
new control points as the snake expands and deletion of control 
points whenever parts of the snake shrink at a point where 
control points overlap. Finally, the external energy factor is 
moderate considering the changes of direction for a node. The 
idea is to slow down when oscillations occur until it reaches its 
stable point. 
6. EXPERIMENTAL RESULTS 
From the experience of existing works, it has been established 
that it is difficult to set the rigidity and the elasticity 
parameters of the internal energy of the snake correctly 
(Horritt, 1999). In this work, we calculate the rigidity 
parameter f from the average curvature of the initial database 
vectors as proposed in (Bentabet, 2001): 
1 
ß= Bmax 1: (eurv)y (23) 
where Bn: is the maximum value assigned to the rigidity 
parameter, curv is a measure of the average curvature of the 
snake initialization using the segment of the database. The 
parameter q defines the degree of relationship between the 
curvature and the rigidity parameter. The elasticity parameter 
Q is set to a value near zero in order to allow the snake to 
stretch according to the external energy only. 
We propose a formulation for the 7 parameter based on the 
idea of giving priority to statistics when the snake evolves far 
from the boundaries and transfer this priority to the gradient 
when near the area boundaries. In this context, , the c 
parameter is set by an estimation of the proximity to the area 
boundaries that could be deduced from the normalized 
Mahalanobis, y, as follows: 
I, v(v(r))e [0.95.1.05] 24 
= w(v(r))e [0.95,1.05] 29 
The proposed approach was first tested on synthetic images 
presented in figure 1. The first result presents a region defined 
by 2 textures on 3 different layers. The initial vector that 
served to initialize the snake is shown in white on image (a). 
An intermediate image of the iteration process shows the snake 
on images (b) and the final result is presented in image (c). 
Specific parameters values for this result were 0.85 as the 
decreasing factor, A, the Mahalanobis threshold value was 
based on a confidence value of 0.0005, internal parameters a at 
0.05 and f at 30. 
A - 185