Image Change Detection and Fusion Using MRF Models
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configurations of the same site in a pair of images must be equal. This equal
configuration value is denoted by x~CHk. For notational convenience, let us
denote X/8 CHk ) and Xi(.8 NCHk ) by XfHk and Xf CHk , respectively. The joint
PDF of the pair is given by
L1
!Xf Hk = XfHk,xfcHk = x~CHk'l
P(Xi,Xj IHd =P CH
CH
NCH
'IvCH
x. k = x. k,X. k = x.. k
}
J
J
I}
=P {XC;Hk = XC;Hk IXNCHk = x~CHk }
I
I
I
I}
X P {X~Hk = x~Hk IX NCHk = x~CHk }
}
J
J
I}
X P {X NCHk = X~CHk = x~CHk }
(12.6)
I
}
I}
•
Since Xi and Xj correspond to the same scene, we assume that they are statistically identical. By summing over all possible configurations of the changed
sites, the joint PDF can be expressed as,
P(Xi, Xj IH k) = P {XfHk = XfHk Ixf cHk = x~CHk }
X P {X~Hk = X CHk IX NCHk = x~CHk }
J
}
}
IJ
X
' " ' P(X CHk = A X NCHk = x~CHk)
~
I
' I
I } ·
(12.7)
AEA"CH
Substituting the Gibbs potential V C and using the properties of conditional
probability, (12.7) can be written as
exp [- L Vc(Xi) - L Vc(Xj)]
1
Cc-8
Cc-8
P(Xi,Xj IH k) = -
(12.8)
Zx
[
(NCH)]
L, exp - L Vc A,X ij k
AEA"CHk
Cc-8
By using the notation L to denote the sum over the sets C that contain site 5,
C35
the term L Vc(Xi) can be decomposed as
Cc-8
L Vc(Xi) = L Vc (xfHk,ax~Hk) + L Vc (x~CHk) ,
Cc-8
C3-8 CHk
CC-8NCHk
(12.9)
where the boundaries of the changed region, denoted by ax~H are contained
within the changed region. Substituting (12.9) into (12.8), we obtain
exp [-Ek (Xi, Xj)]
P(Xi,Xj IH k) = ,
(12.lO)
Zx