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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B7. Istanbul 2004 
  
2 
  
  
  
  
  
Tikhonov | / ] Yes 
Geman & | 1/(1+t%) (120)? No 
McClure : 
Green log(cosh(t)) tang(ty2t iftz0 | Yes 
Hebert & | Log (1+t%) 1/(1+t) No 
Leahy 
rs AE Mena | v4Ass yes 
Perona & | 1 _ a o No 
Malik 
  
  
  
  
  
  
Table 2. Some q functions 
a 
a 
t3 x 
  
  
  
  
041 2 3°4 5 6 7 6 9-19 11 12.13 14 15 18 17 15 19 20 1 
Figure 3. Graphs of ¢ functions 
1 e()- à 
2. o(t)=1 
3 do) lodi?) 
4. o(r) - 2414? -2 
5. lt) = exp(—+”) 
3. VAN DER WALLS-CAHN-HILLIARD THEORY 
Van der waals-cahn-Hilliard theory on phase transitions has 
been studied extensively in mechanics to describe the steady 
states of the physical systems constituted of unsteady phases 
(Aubert, G. and Kornprobst, P., 2002). Let's consider a physical 
system constituted of a fluid of which the energy of Gibbs by 
unit of volume (potential) is a function W depending on the 
distribution of density of the fluid. If the fluid is constituted of 
two different phases described by the levels p(x)=a and 
u(x)=b, then the potential W is double well potential with two 
minima. Ww 
  
Density of fluid 
Figure 4. Double well potential 
At stability, the fluid will take two values p (x) =a or p(x)- b. 
The approach consists in characterizing the stability state of the 
system by minimising: 
1211 
p, inf. | Wux))dx 
Q 
Under the constraint ax) =m (6) 
Q 
The mass of the fluid m is constant and € is positive real. 
The regularized solution of this problem is obtained by 
minimizing E, where : 
E,(u)- J AZ +} 
Under the constraint [ucodx =m 
Q 
4. ANALOGY WITH CLASSIFICATION AND 
RESTORATION 
The stability of a mixture of fluids is reached when each of the 
fluids forms a homogeneous entity separated of the other by 
interfaces of minimal lengths. Mathematically, this state is 
gotten by the minimisation of the energy E;. 
We can note the similarity that exists between image 
classification and the stability of fluids in mechanics. Indeed, 
the classification consists in partitioning an image into 
homogeneous regions, of minimal interfaces. 
Energy is then defined on the image, so that its minimum 
corresponds to a classified image. This configuration of the 
image is equivalent to the steady state of the fluids for which 
the criterion is minimal. 
The potential W defined on image is K wells, where K is the 
number of classes (Samson, C., and al, 2000). 
By analogy, the problem of classification and restoration can be 
deduced directly from equation 7 and can be written as: 
min, (f) 
f 
Il 
2 
Be Je [locates nl (8) 
> € 
under constraint [(/(x)~1(x)) dx& o? 
Q 
Where c, is the standard deviation of noise, and n is a real 
parameter. 
n?/s W(f) is a classification term, that attract gray level of pixels 
to the K means of classes. 
5. GAMMA CONVERGENCE THEORY 
Let X be a metric space, and let f;: X — e [0, +oo[ be a family 
of functions indexed by £70. 
We say that /. F-converge as € —P 0° to /: X —> [0, +oo] if 
the following two conditions