Optimization-Based Fusion - Hyperspectral Image Fusion - page 111

the desired qualities, and the rate of convergence of the solution. The values of these
parameters, thus, play an important role in the process of fusion. The selection of
regularization parameters is known to be a difficult problemwhich is typically solved
using a cross-validation technique [64]. However, fortunately, the final result of fusion
is not very sensitive to the exact value of λ v , but depends on the order of the value, i.e.,

. We have used the values in the range of 10 2 which have
been found to provide a good balance among the competing terms in the objective
function.
The value of

λ v = 1 , 10 , 100 , 1000 , ···

λ s serves as the relative weightage
given to the smoothness term of the minimization functional. It should be noted that
the smoothness penalty should not be very strong, as it may produce almost similar
values of fusion weights for neighboring pixels. Thus, a high value of this term may
reduce contrast in the fused image which would lead to spectral averaging of the
image bands. We have selected this value to be 5-10% of the regularization weight
assigned to the variance term.
Several strategies to stop the iterative minimization process may be employed.
We have followed the commonly used relative cost based criteria to conclude the
iterative procedure as the one employed in the previous chapters of this monograph
as well. During this procedure, the total cost of the functional J ( m ) is computed after
every iteration

λ s should be less than λ v ,as

. The change in the value of the functional over the successive
iterations is calculated, i.e.,

( m )

J ( m ) = J ( m ) − J ( m − 1 ) , 1. The stopping rule is

defined in terms of this relative difference of the cost functional, i.e.,

∇ m ≥

J ( m )
J ( m )

∇

.Itwas

seen that typically the fusion process took 8-10 iterations to converge.

7.5 Experimental Results

In order to maintain the uniformity across chapters, we have used the same two
datasets for the demonstration of this fusion technique. More results are provided in
Chap. 10 .
The urban hyperspectral data consist of 242 bands with dimensions (512

256)
each. To generate a single grayscale fused image, we have processed the entire data
cube at once, while to produce an RGB output, we have partitioned the data into 3
subsets. These subsets undergo fusion independently to generate three images which
are then assigned to the red, green, and blue channels to provide a fused RGB image.
The assignment of colors is not directly related to the actual wavelengths of these
primary colors, and hence, several pseudo-color schemes may be used to present the
result in an enhanced manner. Figure 7.1 a shows the result of combining all bands
using the optimization-based solution. This result represents fusion over the spectral
bandwidth of nearly 2200nm. An RGB version of the fused image is shown in
Fig. 7.1 b which is a result of assignment of pseudo colors to the resultants of fusion
of nearly one-third data each. The results of fusing the moffett 2 dataset have been
provided in Fig. 7.2 . While the figure on the left provides a grayscale version of the

×