Image Change Detection and Fusion Using MRF Models
299
necessary conditions for the convergence of the Metropolis algorithm, but after
a large number of iterations, the temperature change is very small due to the
fact that the inverse of the log function (for example, 1/log(400) = 0.167 and
l/log(SOO) = 0.161) is used to determine temperature in (12.34). As a result,
over a short period of time, an induced Markov chain under the Metropolis
algorithm (when parameters are fixed) after a large number of iterations has
similar properties as homogeneous Markov chains. Hence, the average of the
resulting HIs should provide a reasonable estimate of the most likely HI for
the given observations.
In addition, (12.36) depends on several unknown parameters such as the
noise variance and the low-pass filter F as described earlier. Therefore, we
couple parameter estimation with the Metropolis algorithm so that parameter estimation and the search for a solution of (12.36) can be accomplished
simultaneously. Here also, the maximum likelihood estimate (MLE) is chosen
because of its simplicity in implementation. The use of the MLE can sometimes
severely affect the convergence of the induced Markov chain in (12.37) since, in
many cases, the difference between parameters before and after estimation is
too large and may cause the induced Markov chain to diverge. To deal with this
problem, we limit the difference to the rate of n- 1 • With this rate, parameters
are permitted to change significantly in the early stages of the search whereas,
in the later stages, these can only be changed by small amounts to allow the
induced Markov chain to converge. More details of this algorithm have been
presented in Chap. 6.
12.S
Illustrative Examples of Image Fusion
We test our algorithm on two sets of remote sensing data: multispectral and
hyperspectral in the two examples respectively. Visual inspection is used for
performance evaluation of the fused products in both the examples.
12.S.1
Example 1: Multispectral Image Fusion
The IKONOS data set for the Syracuse University campus is used to investigate
the effectiveness of our image fusion algorithm for resolution merging, i. e.,
fused image resulting from images of two different resolutions. Four multispectral images of red, green, blue, and near infrared (NIR) color spectra and
one panchronometric (PAN) image are used in this example. Figure 12.9 and
Fig. 12.10 display false color composites of NIR, red and green spectra of the
multispectral image, and the PAN image, respectively. Clearly, the PAN image
has sharper feature boundaries than the multispectral image, specifically for
road features.
We first apply principal component analysis (PCA) to the 4-band multispectral image to transform the multispectral image into four uncorrelated images.
Then, the image corresponding to the highest eigenvalue (highest power) is
299
necessary conditions for the convergence of the Metropolis algorithm, but after
a large number of iterations, the temperature change is very small due to the
fact that the inverse of the log function (for example, 1/log(400) = 0.167 and
l/log(SOO) = 0.161) is used to determine temperature in (12.34). As a result,
over a short period of time, an induced Markov chain under the Metropolis
algorithm (when parameters are fixed) after a large number of iterations has
similar properties as homogeneous Markov chains. Hence, the average of the
resulting HIs should provide a reasonable estimate of the most likely HI for
the given observations.
In addition, (12.36) depends on several unknown parameters such as the
noise variance and the low-pass filter F as described earlier. Therefore, we
couple parameter estimation with the Metropolis algorithm so that parameter estimation and the search for a solution of (12.36) can be accomplished
simultaneously. Here also, the maximum likelihood estimate (MLE) is chosen
because of its simplicity in implementation. The use of the MLE can sometimes
severely affect the convergence of the induced Markov chain in (12.37) since, in
many cases, the difference between parameters before and after estimation is
too large and may cause the induced Markov chain to diverge. To deal with this
problem, we limit the difference to the rate of n- 1 • With this rate, parameters
are permitted to change significantly in the early stages of the search whereas,
in the later stages, these can only be changed by small amounts to allow the
induced Markov chain to converge. More details of this algorithm have been
presented in Chap. 6.
12.S
Illustrative Examples of Image Fusion
We test our algorithm on two sets of remote sensing data: multispectral and
hyperspectral in the two examples respectively. Visual inspection is used for
performance evaluation of the fused products in both the examples.
12.S.1
Example 1: Multispectral Image Fusion
The IKONOS data set for the Syracuse University campus is used to investigate
the effectiveness of our image fusion algorithm for resolution merging, i. e.,
fused image resulting from images of two different resolutions. Four multispectral images of red, green, blue, and near infrared (NIR) color spectra and
one panchronometric (PAN) image are used in this example. Figure 12.9 and
Fig. 12.10 display false color composites of NIR, red and green spectra of the
multispectral image, and the PAN image, respectively. Clearly, the PAN image
has sharper feature boundaries than the multispectral image, specifically for
road features.
We first apply principal component analysis (PCA) to the 4-band multispectral image to transform the multispectral image into four uncorrelated images.
Then, the image corresponding to the highest eigenvalue (highest power) is