ISPRS Commission III, Vol.34, Part 3A ,,Photogrammetric Computer Vision“, Graz, 2002 
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Vol pon) [rir s (6) 
where / is the NxN identity matrix. K is an NxN matrix which 
depends on a, f and the shape of the interpolating curves. Ve 
gives the location of the element nodes at iteration n. 
3. EXTERNAL FORCES CALCULATION 
The calculated external force is defined as a function of the 
region and the boundary information. We assume that the 
image can be modelled as a collection of homogeneous regions 
of uniform or slowly varying intensities. Let the image data be 
described by a random field Y defined on a rectangular grid, S, 
of N points, and the value of Y at a point s € S will be written 
as Y,. When necessary the points in S will be explicitly written 
as integer pairs (n,m). X will denote the random field of a 
given segmentation of Y into uniform regions. Lowercase x and 
y will be realizations of the X and Y respectively. The 
probability P(Z=z) is written as p(z). 
3.1 Statistical Measure 
By definition, topographic features could be described by more 
than one texture. Indeed, for example, both hardwood and 
coniferous trees are included in a vegetation region definition. 
Also, these two types of forest can appear separately or in 
mixed areas. Thus, the statistical properties of the targeted 
area are computed assuming the region of interest is described 
by a number of textures being normally distributed. We 
propose a mixture estimation process independent but 
necessary for the region localization. Indeed, the definition of 
the mixture parameters for different type of topographic feature 
in an image can be recorded in a database and used as a prior 
knowledge. This information is essential to the snake process 
but can be defined in an earlier step. 
The statistical information is described trough a number k of 
Gaussian distributions with the mixture proportion, pj, the 
expected value of the argument, u;, and the covariance matrix, 
X, with j € [1, k]. The dimension of x; and 2; are related to the 
number of bands available in the multi-spectral imagery. 
Indeed, the definition of these parameters through n bands 
provides 4; vectors of dimension n and 2; matrix of size nxn. 
3.1.1 Statistical prior knowledge 
The goal is to define the ideal number of textures contained in 
a sample region witch describes the data. Recent studies 
(Olivier et al., 1999) propose an iterative algorithm for normal 
mixture definition. This operation is a two steps iterative 
process. It is done using the Expectation — Maximization (EM) 
algorithm assuming an adequate initialization of the 
parameters values for a given number of distributions. The 
objective of the EM algorithm is to maximize the log- 
likelihood in an iterative manner. The second step consists of 
evaluating the balance between the accuracy of the model and 
the number of components. Evaluating the Minimum 
Description Length (MDL) of the model provides this measure 
as described by Olivier et al., 1999. 
The mixture model that minimizes the MDL criterion 
corresponds to the appropriate mixture model for the region of 
interest. 
Since the statistical characteristics of the area are estimated, a 
goodness function based on these statistical features can be 
derived. Indeed, this function will define the region probability 
measure part of the external energy. 
3.1.2 Region Probability Measure 
Considering a realization of the field Y noted, y, the aim here is 
to evaluate p(y|x). The statistical goodness measure is based on 
the calculation of a distance, y(y), that evaluates the proximity 
of a pixel to the statistical properties of the region. This value 
can be defined using the Mahalanobis distance. 
Having k textures for a given region, we associate to the 
sample y the smallest distance from all k distributions defined 
within the region. In order to consider this value in a generic 
way, one must take into consideration the number of bands 
from the multi-spectral image involved. Since we assume that y 
follows a Gaussian distribution, we know that the Mahalanobis 
distance follows a x? distribution (Saporta, 1990) with n 
degrees of freedom (n being the number of bands for the multi- 
spectral image). Thus, we can set a threshold value based on 
some confidence value (0.00596 for instance). This threshold 
value will serve to normalize the calculated distance for tested 
samples and also to identify outliers. Thus, the goodness 
function, w(y), can be defined as follow: 
min, ne -u) Y. 3 (y —H; ) 
v(v)= 
V, 
where y, is a threshold given by the x? table value according 
  
  
(7) 
to the dimension of the data and the desired confidence value. 
From Baysian inference, the joint probability distribution, 
p(y,0), comprises two parts, a prior distribution p(0) and a 
likelihood p(y|0). In this framework, we do not consider the 
prior distribution of 0, p(0) in the evaluation of the closest 
distribution nor for the goodness function. The idea is to 
consider the closest texture according to the Mahalanobis 
measure for a candidate as the unique potential. 
By observing the distance defined by equation (7), we find that 
when it is greater than 1, odds are that this pixel is outside the 
region. On the opposite, as the value decreases near zero, the 
indication of being inside the region increases. This 
information allows the definition of the direction of growth. 
Finally, the likelihood p(y|x) when the snake evolves inside the 
considered region (i.e., Y(y)<l), is given by a Gaussian 
distribution as follows: 
px) Sule od (8) 
(2x)2V31 
For cases, where the snake is outside the considered region 
(i.e. w(y)>1), the likelihood is described by a translated form 
of the Gaussian given in equation (8). The general case is 
defined as follows: 
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