International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B7. Istanbul 2004 
  
supervised classification that doesn't require any knowledge on 
the classes. 
Many classification models can be found in the field of 
stochastic approaches (discrete models) with the use of Markov 
Random Field (MRF) theory (Pony and al., 2000). Structural 
approaches as splitting, merging and region growing have also 
been developed. Works on the classification by variational 
models (continuous models) have been conducted lately, mainly 
because the notion of classes has a discrete nature. 
The variational approaches are always associated with 
resolution of partial differential equations (PDEs), and have for 
interest, that they allow to get in many cases the results of 
existence and uniqueness of the solution. They can be 
implemented by powerful numeric methods (Deriche, R. and 
Faugeras, O., 1995). 
It often happens that acquired images have less than desirable 
quality due to various imperfections and/or physical limitations 
in the image formation and transmission processes. The 
acquired image may look blurry due to the motion of camera 
for example or atmospheric turbulence. Noise may be 
introduced owing to measurement errors, quantization, etc. The 
aim of restoration is to find the original image from the 
observed one. This problem can be identified by inverse 
problem. 
In this paper, we present a model proposed by C.Samson that 
combine in the same process, image classification and image 
restoration (Aubert, G. and Kornprobst, P., 2002). This 
deterministic model is based on variational calculus and 
resolution of partial differential equations (PDEs). It is inspired 
from Van Der Walls-Cahn-Hilliard works on phase transition in 
mechanic, and uses Gamma-convergence theory. The 
classification-restoration is achieved by minimization of a 
sequence of functional that contains at least one term for 
classification and other one for restoration. 
We suppose that discriminant feature between classes is the 
spatial distribution of intensity. Of course, other discriminant 
features like the texture can be used. We also assume that the 
distribution of intensity is Gaussian for each class. Under these 
assumptions, classes can be characterized by their means and its 
standard deviations. 
2. IMAGE RESTORATION 
Generally, image degradation can be modeled by a linear and 
translation invariant blur and additive noise. The equation 
relating observed image / and original one / can be written as: 
I-Kf*n (1) 
  
fx) —R| kG9) s 16) 
n(x) 
Figure 1. Linear image degradation model 
  
  
  
Where K is a convolution operator with the impulse response of 
the system K (K1- k * I), and n is an additive white noise. In 
practice, the noise can be considered as Gaussian. The 
restoration consists of recovering the original image f from the 
observed one I. One simple method consists in minimizing the 
half quadratic error given by equation 2: 
2 
Gn) [(roa-100) à 2) 
Q 
Many restoration methods are performed under the condition 
that the blur operator is known. Unfortunately, the true image 
must be identified directly from the degraded image by using 
partial or no information about the blurring process and the true 
image. Such estimation problem is called blind deconvolution, 
and consists of finding alternately an estimate to the original 
image and the impulse response. 
The problem of recovering an image that has been blurred and 
corrupted with additive noise is an inverse problem and is 
always ill-posed in the sense of Hadamard. The existence and 
uniqueness of the solution are not guaranteed. It is therefore 
necessary to introduce an a priori constraint on the solution. 
This operation is the regularization. We can distinguish two 
types of regularization: the linear one and the non-linear. The 
regularized solution is computed by minimizing the functional: 
J (f.I) = fHfix) =) y dx oT ren (3) 
Q 
Where À is a real parameter. 
The most important linear regularization is the Tikhonov one, 
(Aubert, G. and Kornprobst, P., 2002) described by equation 4: 
gu EB (4) 
This regularization leads to a solution without edge preserving. 
To overcome this problem, the non linear regularization is used. 
On the homogeneous regions that correspond to weak gradient, 
an important smoothing is done. On the contours that 
correspond to strong gradient smoothing is very weak. So the 
noise in the image can be minimized while preserving the 
contours of the objects. Among the non linear methods that 
have been proposed, the most successful ones are the total 
variation (TV) restoration (Bertalmio, M., and al, 2003) 
(Rudin, L., and Osher, S., 1994), (Vogel, C.R., and Oman, M. 
E., 1996) and the regularization with a ¢ function (Samson, C., 
and al, 2000). For our implementation, we have used the ¢ 
function regularization, and we have assumed that the image is 
not blurred. In this case, the equation 3 can be written as: 
1 69 füreo- 16) 2 Jolvrps 6) 
Q 
Q 
In the table below, we present some ¢ functions and their 
property in relation to the convexity: 
  
ext) e(t)! 2t Convexity 
Total n 1/2] t] if t20 . Yes 
Variation 
  
  
  
  
  
  
  
1210