Wy = arg max{p(w| X)} = arg max{p(X |w)p(w)} (0 
^ 
where W, = MAP (Maximum a Posteriori) estimation of 
"Map 
the field of class labels which maximizes the posterior cost 
function (1). 
p(w)= prior probability distribution 
p(X]w)- class-conditional distribution 
Therefore, the modeling of both the p(w) and p(X|w) becomes 
an essential task. 
2.1. Prior Distribution Model-MRF 
The introduction of MRF can be found in many texts 
(Chellappa, 1983; 1985). The image function w(s) can be taken 
a two-dimensional random, and expressed by Markov random 
field as: 
p(s) [WS —5)} = p{(w(s) | w(0s)} (2) 
where S = image lattice 
Os = neighborhood system 
So for a given point in a two-dimensional random, its class 
label is only dependent on its neighbors and unrelated with 
other pixels of image. 
For a given neighborhood system, a Gibbs distribution is 
defined as any distribution p(w) that can be expressed in (Julian, 
1986) as: 
1 1 c 
p(w) = at V (w)] (3) 
where  J(w)- arbitrary function of w on the clique c, 
C - the set of all cliques 
z = normalizing constant called a partition coefficient 7 
= analogous to temperature. 
The prior distribution based on the first order neighborhood 
system as: 
plu) = i | exp[-— Y. V:Qwyj--l-expl-BY.r(Q] — (5 
ceC z ceC 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XXXIX-B7, 2012 
XXII ISPRS Congress, 25 August — 01 September 2012, Melbourne, Australia 
   
where pf = weight emphasizing the significance of interactions 
among adjacent pixels inside the clique, 
f(w) = V'(w) mathematically. 
So (1) can be further written as: 
Wig = AIG min 5 (- In p(X |w(s)* B» .tt()] 6 
ses ceC 
where w(s) = class label at s € S. 
22. Modeling the Conditional Probability Density 
Function 
As the impact of speckle noise of SAR image in synthesizing 
classification of the optical and microwave images, it is difficult 
to obtain the conditional probability density function of the 
multisource remote sensing data, maximum likelihood classifier 
with modified M-estimates of mean and covariance (MMLM) 
can be used to classify the multisource images and get the 
initial class labels and the conditional probability density 
function of each class. From the Reference (Yonhong, 1996), 
we see that MLMM can obtain a good precision of 
classification and proper conditional probability density 
function, also restrain the speckle noise of SAR images. 
2.3. Classification by Iterated Conditional Modes (ICM) 
The ICM is computationally feasible since it updates the class 
assignments iteratively (Julian, 1986), the objective is to 
estimate the class label of a pixel given the estimates of class 
labels for all other pixels inside the rectangular lattice. Then the 
optimization problem of (5) becomes: 
W(s) = arg max[ p(w(s) | X, W(S — s))] (6) 
w(s) 
Applying the Bayes' rule and considering the Markov 
property (Julian, 1986), the argument of (6) becomes: 
W(s) = armar (s) | W(s)} p{w(s) | W(@s)}] (7) 
From the Hammersley-Clifford theory (Geman, 1984) we 
know: 
piw(s)| w(05)5 — Zexp t- BU [w(s), W(8s)]) (8) 
[w(s), M(@s)1= 3 [178,5 55] (9) 
ceC