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