— 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B7. Istanbul 2004 
  
Vx, —»x  liminf f, (x)? f(x) (9) 
x0 
and 
3x, —»x  limsup f;(x) < f(x) (10) 
op 
are fulfilled for every x € X. The I'-limit, if it exits, is unique. 
The T-convergence is stable under continuous perturbations, 
that is, (f;*v) F-converge to (f*v) if /; l-converge to fand v is 
continuous (Aubert, G., and al, 2002). The most important 
property of [-convergence is the following: 
if {x.}¢is asymptotically minimizing, i.e: 
li (x) - inf f.) 20 
Tim (f, 5) = fn Je) (11) 
and if {xen}n converge to x for some sequence £y —J 0, then x 
minimizes f. 
6. EXPRESSION OF THE FUNCTIONAL AND 
IMPLEMENTATION 
By applying the properties of Gamma-convergence, the solution 
of the equation 8 is obtained by minimizing the functional Je 
when & is approaching the zero value. 
Jr [| ovr) + Face + [Creo — I(x)) de 
: (12) 
7 = im, {ar min py) 
E— op 
We can note that this functional is composed by three terms: 
regularization term, classification term and data fidelity term. 
The first term is weighted by a parameter proportional to £, and 
the classification term is weighted by a parameter proportional 
to 1/e. The convergence of the criterion given by equation 12 is 
reached for little values of e, so that the regularization and the 
classification are not achieved simultaneously. For high values 
of £, the regularization is privileged, and progressively with & 
decreasing, the process changes its behavior, and becomes 
classification process. 
The power of the regularization by ¢ functions lies in its 
nonlinearity. This later criterion leads to difficulties for 
optimization calculation. If ¢ is quadratic, the function to be 
minimized is quadratic, therefore the minimum is single and 
easy to calculate. To bring back itself to a quadratic model, the 
semi quadratic theorem is used, and consists of introducing an 
auxiliary variable b. 
Vt,..p(t) = inflor? + v(^) (13) 
L<b<M 
ks el ) (14) 
where: lim 2 UY L5. dim 90). M 
[0 Df (50r, 21 
  
  
w(b)- g((g'(b)-b(g'(b) and gH) = (VO 
The equation 12 can be rewritten as: 
Jf) [ure - Hy dx 4 e$ [lst + (Bic + 
Q Q (15) 
7 fm f )dx 
T 
Q 
For minimizing the sequence of functional 15, we use Euler 
Lagrange equation and the minimization problem is 
transformed to a problem of resolution of partial differential 
equation (PDE), given by: 
2 
[re)- 109] T-'re)-er devo — a9 
Where div is the divergence operator. 
7. EXPERIMENTAL RESULTS 
To validate the approach of classification suggested, we 
initially tested it on a synthetic image before applying it to the 
real satellite image. The synthetic image contains 4 classes 
detailed in table 2. 
  
  
  
  
  
  
  
  
  
classe Hu, o, 
1 22.46 4.66 
2 63.62 5.07 
3 107.13 4.58 
4 232.02 4.73 
  
Table 5. Characteristics of synthetic image classes 
The figure 6 illustrates the image to be classified and the figure 
7 the graph of the potential W, with 4 wells. In figure 8 we can 
see the localization of training areas, and in figure 9 we show 
the classified image. 
in W(potential) 
F (gray level) 
  
Figure 6. Original image Figure 7. The potential W 
1212