Image Change Detection and Fusion Using MRF Models
281
statistical models for the given images as well as for the change image is crucial. Here, we assume that the given images are obtained by the summation
of noiseless images (NIM) and noises in the image. Both the NIMs and the
change image are assumed to have MRF properties. Furthermore, we assume
that configurations (intensity values) of the NIMs are the same for the unchanged sites. In addition, configurations of the changed sites from one NIM
are independent of configurations of the changed sites from the other NIM
when the configurations of the unchanged sites are given. Based on the above
assumptions, the a posteriori probability is determined and the MAP criterion
is used to select the optimum change image.
Due to the size of the search space, the solution of the MAP detection
problem cannot be obtained directly. As a result, a stochastic search method
such as the simulated annealing (SA) algorithm (see Sect. 6.4.1 of Chap. 6
and Geman and Geman 1984) is employed. Here, the SA algorithm generates
a random sequence of change images in which a new configuration depends
only on the previous change image and observed images by using the Gibbs
sampling procedure (Geman and Geman 1984; Winkler 1995; Bremaud 1999;
Begas 1986). The randomness of the new change image gradually decreases as
the number of iterations increases. Eventually, this sequence of change images
converges to the solution of the MAP detector.
12.2.1
Image Change Detection (lCD) Algorithm
The ICD algorithm uses the MAP detector whose structure is based on statistical knowledge. Therefore, statistical models for the given images as well as
for the change image (CI) are required.
Noiseless Image and Change Image Models
Let -8 be asetof sites 5, andit = {O, 1, ... , L - I} bethe phase space. Furthermore,
we write {Xi( J)} i>O E its as a sequence of image (configuration) vectors taken
at time ti, i E {O,l, ... ,N -I} where to < tl < ... < ti < ... < tN-I. Here the
value of N is 2. We assume that Xi( J) satisfies MRF properties with a Gibbs
potential V dXi). Note that Xi( -8) are noiseless image models (NIM) of the
scene. These will be used extensively in our discussion. Due to noise, we
cannot observe Xi directly. Instead, we observe noisy images (NI) Yi( -8) E e S
given by
Yi(-8) = Xi(J) + Wi(J); i = 0, 1, .. . ,N ,
(12.1)
where e = {qO' ql, . . . qf-I } is the phase space corresponding to noisy observed
images and Wi( -8) is a vector of additive Gaussian noises with mean zero and
covariance matrix 0 2 1. Here, I is an identity matrix of size M x M, where
M = 1-81.
Obviously, there are a total of K = 2M possible change images (CIs) that
may occur between any pair of NIMs (including the no-change event). Let
281
statistical models for the given images as well as for the change image is crucial. Here, we assume that the given images are obtained by the summation
of noiseless images (NIM) and noises in the image. Both the NIMs and the
change image are assumed to have MRF properties. Furthermore, we assume
that configurations (intensity values) of the NIMs are the same for the unchanged sites. In addition, configurations of the changed sites from one NIM
are independent of configurations of the changed sites from the other NIM
when the configurations of the unchanged sites are given. Based on the above
assumptions, the a posteriori probability is determined and the MAP criterion
is used to select the optimum change image.
Due to the size of the search space, the solution of the MAP detection
problem cannot be obtained directly. As a result, a stochastic search method
such as the simulated annealing (SA) algorithm (see Sect. 6.4.1 of Chap. 6
and Geman and Geman 1984) is employed. Here, the SA algorithm generates
a random sequence of change images in which a new configuration depends
only on the previous change image and observed images by using the Gibbs
sampling procedure (Geman and Geman 1984; Winkler 1995; Bremaud 1999;
Begas 1986). The randomness of the new change image gradually decreases as
the number of iterations increases. Eventually, this sequence of change images
converges to the solution of the MAP detector.
12.2.1
Image Change Detection (lCD) Algorithm
The ICD algorithm uses the MAP detector whose structure is based on statistical knowledge. Therefore, statistical models for the given images as well as
for the change image (CI) are required.
Noiseless Image and Change Image Models
Let -8 be asetof sites 5, andit = {O, 1, ... , L - I} bethe phase space. Furthermore,
we write {Xi( J)} i>O E its as a sequence of image (configuration) vectors taken
at time ti, i E {O,l, ... ,N -I} where to < tl < ... < ti < ... < tN-I. Here the
value of N is 2. We assume that Xi( J) satisfies MRF properties with a Gibbs
potential V dXi). Note that Xi( -8) are noiseless image models (NIM) of the
scene. These will be used extensively in our discussion. Due to noise, we
cannot observe Xi directly. Instead, we observe noisy images (NI) Yi( -8) E e S
given by
Yi(-8) = Xi(J) + Wi(J); i = 0, 1, .. . ,N ,
(12.1)
where e = {qO' ql, . . . qf-I } is the phase space corresponding to noisy observed
images and Wi( -8) is a vector of additive Gaussian noises with mean zero and
covariance matrix 0 2 1. Here, I is an identity matrix of size M x M, where
M = 1-81.
Obviously, there are a total of K = 2M possible change images (CIs) that
may occur between any pair of NIMs (including the no-change event). Let