298
12: Teerasit Kasetkasem, Pramod K. Varshney
restated here,
1) lim T(n) = 0 ,
n----+oo
L1r
2) T(n) 2: log(n) ,
(12.38)
where L1 is the maximum value of the absolute energy change when a pixel is
updated and r is a constant defined in Chap. 6.
Theoretically speaking, regardless of the initial HI, the induced Markov
chain eventually converges to the solution of (12.36) if necessary conditions
given in (12.38) are satisfied. However, this convergence may require an infinite
number of iterations, which are not feasible in practice. To actually implement
this algorithm, the Metropolis algorithm must be terminated after a certain
number of iterations. As a result, we must carefully select the initial HI that
is relatively close to the desired solution so that the algorithm terminates in
a reasonable time. Here, we pick the initial HI to be the average of intensity
values of pixels in Y 1 and the maximum likelihood estimate (MLE) of HI based
on Y2, i.e.,
1
Xinitiai = - (Yl + MLE(Y2)) ,
2
(12.39)
where MLE(·) denotes the MLE of HI based on observation Y 2. The main
reason for employing this procedure is the fact that the MLE of HI for a given
Y 2 usually has high resolution, but the important information cannot be clearly
seen whereas Y 1 has high contrast and the vital information is clearly seen,
but its resolution is very poor. By using the above procedure, we expect the
initial HI to have high contrast making the vital information more visible while
the background and texture information of the high-resolution image is also
present. To use the MLE, the conditional probability of the HI given Y2 must
be estimated. Here, we employ the joint histogram of Y2 and Y 1• Hence, we
have
(
) number of pixels that have Y2 = a and Y 1 = b
PY21X alb = ----"---------:----------number of pixels that have Y 1 = b
(12.40)
Furthermore, even with a reasonably good initial estimate given in (12.40),
the total number of iterations required for the induced Markov chain to converge to a single value is still very large since we are dealing with gray-scaled
images under the GMRF model. The Gibbs energy function under the GMRF
model is in a quadratic form under which small changes (increase or decrease
of one or two intensity values) of a configuration of a site has little effect
on the Gibbs energy function unless the temperature is very low. This fact
creates a noise-like effect on the fused image even with a fairly large number of iterations (500 in our examples.) To minimize this effect and to allow
our algorithm to terminate in a reasonable time, we average the resulting HIs
generated during the last few iterations. Obviously, this method violates the
12: Teerasit Kasetkasem, Pramod K. Varshney
restated here,
1) lim T(n) = 0 ,
n----+oo
L1r
2) T(n) 2: log(n) ,
(12.38)
where L1 is the maximum value of the absolute energy change when a pixel is
updated and r is a constant defined in Chap. 6.
Theoretically speaking, regardless of the initial HI, the induced Markov
chain eventually converges to the solution of (12.36) if necessary conditions
given in (12.38) are satisfied. However, this convergence may require an infinite
number of iterations, which are not feasible in practice. To actually implement
this algorithm, the Metropolis algorithm must be terminated after a certain
number of iterations. As a result, we must carefully select the initial HI that
is relatively close to the desired solution so that the algorithm terminates in
a reasonable time. Here, we pick the initial HI to be the average of intensity
values of pixels in Y 1 and the maximum likelihood estimate (MLE) of HI based
on Y2, i.e.,
1
Xinitiai = - (Yl + MLE(Y2)) ,
2
(12.39)
where MLE(·) denotes the MLE of HI based on observation Y 2. The main
reason for employing this procedure is the fact that the MLE of HI for a given
Y 2 usually has high resolution, but the important information cannot be clearly
seen whereas Y 1 has high contrast and the vital information is clearly seen,
but its resolution is very poor. By using the above procedure, we expect the
initial HI to have high contrast making the vital information more visible while
the background and texture information of the high-resolution image is also
present. To use the MLE, the conditional probability of the HI given Y2 must
be estimated. Here, we employ the joint histogram of Y2 and Y 1• Hence, we
have
(
) number of pixels that have Y2 = a and Y 1 = b
PY21X alb = ----"---------:----------number of pixels that have Y 1 = b
(12.40)
Furthermore, even with a reasonably good initial estimate given in (12.40),
the total number of iterations required for the induced Markov chain to converge to a single value is still very large since we are dealing with gray-scaled
images under the GMRF model. The Gibbs energy function under the GMRF
model is in a quadratic form under which small changes (increase or decrease
of one or two intensity values) of a configuration of a site has little effect
on the Gibbs energy function unless the temperature is very low. This fact
creates a noise-like effect on the fused image even with a fairly large number of iterations (500 in our examples.) To minimize this effect and to allow
our algorithm to terminate in a reasonable time, we average the resulting HIs
generated during the last few iterations. Obviously, this method violates the