ISPRS Commission III, Vol.34, Part 3A „Photogrammetric Computer Vision“, Graz, 2002 
  
theoretic method. The game theory as a concept has its roots in 
decision making under a conflicting and often hostile 
environment. This method processes by maintaining the 
modularity of the system involved and by allowing the modules 
(i.e., region-based and boundary-based modules) to interact by 
a decentralized mode of decision making. The contribution of 
each module is determined by achieving the Nash equilibrium 
(Chakraborty and Duncan, 1999). This method allows high 
performance when applied to noisy images, especially when 
applied to deformable models. However, the existence of the 
Nash equilibrium depends on weighting parameters of the goal 
function. While for simple problems it might be simple to 
mathematically choose right values for the weights, often, for 
complicated problems, it is almost impossible. 
Our goal here is to develop a fully automated formalism for 
integrating boundary finding and region-based methods. This 
formalism will be used to model the external force acting on 
the deformable model. The region-based modeling is achieved 
at a global level by a statistical characterization. Thus, the 
cluster of interest could be considered as being a mixture of 
distributions. The boundary finding part is handled by the 
gradient information. Since the gradient defines a measure of 
non-homogeneity in the pixel neighborhood, its response is 
modeled as a potential function, that generates a Gibbs 
distribution of a Markov random field (MRF). The 
combination relies on an approximate maximum a posteriori 
(MAP) estimate that gives the likely segmentation according to 
the observed data. In order, to resolve the conflicting situation 
that could appear, each part of the MAP is weighted by a 
measure that ensures the selection of the suitable MAP 
configuration. The paper is organized as follows. Section 2 
gives a brief overview of the deformable model approach as 
introduced by Kass et al., 1988 and the snake discretization 
using finite element used in this framework. Section 3 details 
the methods for both region-based and multi-spectral boundary 
finding formulation respectively. In section 4, we introduce the 
MAP combination and conflict resolving formulation for the 
external forces calculation purpose. In section 5, we resume the 
final algorithm. Finally, in the rest of the paper, we show and 
discuss the results obtained on both synthetic and LANDSAT7 
images. A conclusion is given with the possible extensions of 
our work. 
2. DEFORMABLE MODELS APPROACH 
Active contours models or snakes were introduced by Kass et 
al., 1988 as a novel solution to the low-level imaging task of 
finding step edges. A snake is defined, in the image plane 
(n,m), as a parametric curve of the curvilinear abscise, r, by 
v(n,m)=v(n(r,t),m(r,t)). The snake is allowed to deform from 
some arbitrary initial location within an image towards the 
desired final location. Thus, the use of snakes involves a two 
steps process being an initialization and the iterative 
minimizing process. To initialize the snake, we first perform 
the discretization of the database vectors. The final snake 
location is obtained through minimization process acting upon 
the global energy of the snake defined as follows (Kass et al., 
1988): 
Ejot (E) * Ej, (t) - Es, (t) (1) 
where Eex(t) is the external energy, E;n(f) the internal energy. 
As described in classical snake modeling approach (Kass et al., 
A- 182 
1988), the internal energy acts as a stabilizer to the external 
data irregularities. The standard internal formulation is given 
as follow: 
vea] * 
Eunl)= 
where a and ß are Tikhonov stabilizers which controls 
respectively the elasticity and rigidity of the snake. In active 
contours framework, the external energy, E..(?), is used to 
derive the external forces, F4;(f), which act on the snake to 
deform its shape and location. The relationship between 
external energy and forces is given by the following equation: 
  
  
vea Jos (2) 
E ext (r) = § Fext (v(r.t))as (3) 
As mentioned earlier, the initial location of the snake is 
provided by the database vectors. In the next step, the initial 
vectors are approximated by parametric curves. This task is 
undergone by the use of the finite elements method (FEM). 
The FEM enables accurate discretization of derivatives and 
smooth shape representation for the snake. The interested 
reader will find all details in (Bentabet et al, 2001). To 
summarize, the discretizing of the parametric curve w(r) by 
FEM leads to an expression for each element given by: 
Xy YF 
v 6)e(NC). N20)... NO) X2 | @ 
Ag uyg 
where, N;(r) define a vector of interpolation polynomials. In 
our case, the interpolation is carried out using a Hermite basis 
function. The two-columns matrix V4 — (x € Y* | contains the 
coordinates of the control nodes which are inserted at regular 
intervals of the initial curve issued from the database. 
Consequently, both snake topology and location will be entirely 
defined by the knowledge of the control nodes. Therefore, the 
segmentation purpose can be described as being a process of 
estimating the suitable values for the control nodes coordinates 
that minimize the global energy of the snake energy defined in 
equation (1). 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 snake evolution is governed 
by a partial derivative equation of motion obtained by resolving 
the Euler Lagrange equation (VE;,,—0) which can be expressed 
as follows (Bentabet, 2001): 
  
y wrt) d*v.t), B 9^ v(r,t) __ OFext (v(r,£)) (5) 
Bil A dy 
where y controls the speed of evolution of the snake. A 
discretized form of Equation (5) can be derived using the finite 
elements formalism, which yields the following iterative 
process: