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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B7. Istanbul 2004 
  
into one of M classes; that is, x, = Uu. 2... M where M is the 
number of classes assumed to be known in supervised 
classification process. This optimisation is executed from the view 
point of the maximum a posterior (MAP) estimation as follows: 
Am = argmax P(X]Y)| © 
NeQ 
Where Q is labelled configurations set. Following Bayes theorem, 
equation (3) becomes: 
JP(Y/X )P(X) (4) 
= argmax | P(Y) 
XeQ 
X 
MAP 
The modelling of both class conditional distribution P(Y/X) and 
prior distribution P(X) becomes an essential task (Khedam ef al., 
2001). P(Y) is the probability distribution of the observed data and 
doesn't depend on the labelling X. Note that the estimate (4) 
becomes the pixel-wise non—contextual (punctual) classifier if the 
prior probability doesn't have any consequence in formulating (4). 
P(Y/X) is the conditional probability distribution of the observation 
Y given the labelling X. A commonly used model for P(Y/X) is that 
the feature vector observed Y, is drawn from a "gaussian 
distribution". For a Markov Random Field (MRF) of field 
labelled X and so, according to the Hammerslay-Clifford theory, 
P(X) can be expressed as a Gibbs distribution with “Potts model” 
as energy function model. The global MAP estimate given by 
equation (3) is equivalent to the minimisation of the followed a 
posterior global energy function (khedam ef al., 2002): 
Xuw=argmin{ U (X/Y) } (5) 
Xe U 
Where: 
  
UV) Si E. dd = 
se. 
I 
+38 3 (1-8 (x30) © 
q€Q reVs 
Where x and S are statistic parameters of class xs 
ph. 
estimated during training step process and B is parameter 
regularisation and is frequently user specified. @ is Kroeneker 
function calculated on the neighbourhood Vs of site s on all clique 
q from set clique Q (figure 1). Neighbourhood system Vs can be 
4-connexivity given by equation 7 or 8-connexivity given by 
equation 8. 
Viel «pe s. o«i-Xy«G-Ysi j à 
2 
t D e S, o«i- by G-Iys2 ] € 
4-Connexivi Order ! 9 i) 
Oo 
Pe: Order 2 1 
  
  
2e 
  
  
  
  
  
  
  
o (k, 1) 
8-Connexivity ! 
o © 0 Order 3 c ... 
O @ o 
Order 4 LL) ... 
OQ Oo o 
  
  
Figure 1. Neighbourhood and cliques 
Once MAP classification problem is formulated as an energy 
minimisation problem, it can be solved by an optimisation 
algorithm. Among the most effective algorithms for optimisation in 
the framework of image MRF modelling are Simulated Annealing 
(SA) (Geman et al., 1984) whose the computational demands are 
well known and Iterated Conditional Modes (ICM) (Besag, 1986) 
which is a computationally feasible alternative of the SA with a 
local minimum convergence of the energy function. To use ICM 
algorithm, global minimisation energy function of equation (6) 
must be transformed on the followed local minimisation energy 
function under the assumption of the independence of conditional 
probabilities: 
U (x.J/v» is do -i Xs y. S ; (y.-i Xs ) nd Is 
  
ij 
+3 3 (7-3 (x«,x)) 
re Vs 
The first term on the right hand of (9) called data attach term, can 
be consider as the potential function of one-order clique and is 
often used to provide an initial configuration for the contextual 
classification process. The second term called regularisation term, 
defines pair-cliques potential function which explicitly describes 
local spatial interactions in neighbourhood Vs . ICM flow chart is 
presented on figure 2.It can be resumed on five steps as follows: 
Step 1: Estimate statistic parameters set ( 4« , 22s ) from the 
training samples of each class from classes set A. 
Step 2: Based on ( xs , Zs ), estimate an initial classification using 
the non-contextual pixe-wise maximum likelihood decision rule. 
We use first term of equation (9). 
Step 3: Choose an appropriate value of D, an appropriate shape 
and size of neighbourhood system Vs and an appropriate 
convergence criterion. 
Step 4: Perform the local minimisation defined by equation (9) at 
each pixel in specified order: update y, by the class x, that 
minimises equation (9). 
Step 5: Repeat step (3) until convergence. Iterative algorithms 
often pose convergence problem. Convergence criterion which 
we have adopted in this study is a zero number of pixels changing 
classes between two consecutive iterations. This number of pixels 
is calculated on the whole image and thus for all classes. We