Image Processing Reference
InDepth Information
For image processing and vision problems, we mostly deal with functionals and
EulerLagrange equations of two variables. The usefulness of these equations will
be evident in the next section where these have been employed to obtain the resultant
fused image.
6.3 Variational Solution
We would like to develop an algorithm that will iteratively produce the resultant
image that preserves most of the salient features of the set of input hyperspectral
bands. We want to design the fusion weights
w
k
(

mattes for every band in the data such that the said goal is achieved. The pixels with
a high local contrast provide a good amount of visual information to the observer.
Also, features such as edges and boundaries possess the characteristic of high local
contrast due to significant differences in gray values from their local neighborhood.
Therefore, the choice of local contrast as the weighing parameter is quite justified.
The contrast being a dissimilarity measure, these pixels might be quite dissimilar
in gray values. However, we want the merging process to be smooth, providing a
natural appearance to the resultant fused image.
One of the easy ways to generate the
x
,
y
)
which are essentially the
α
matte is to select just one pixel from
all available bands at every spatial location using certain quality criterion, and the
contribution from the rest of the bands is zero. The fusion weight
w
k
(
α
,
)
x
y
takes
{
,
}
(
,
)
one of the values in
in any hyperspectral band
I
k
.This
approach has two major shortcomings. First, this approach essentially selects only
one pixel per spectral array, and therefore, a large amount of data is discarded. In order
to make the solution efficient, one has to select the quality measure carefully. Often,
a set of multiple quality measures is needed as a single measure may not be able to
efficiently capture the saliency of the pixel data. Secondly, when the choice of matte
selection depends on certain quality criterion of the fused image, the fusion procedure
leads to a combinatorial search problem which is computationally demanding. We
can, however, as explained in previous two chapters, relax the condition on fusion
weights
w
k
(
0
1
at any location
x
y
allowing it to take real values in the range [0, 1]. The fusion weights
have to satisfy the usual properties, i.e., nonnegativity, and unity sum. As the fusion
weights are restricted to the aforementioned range, these are essentially nonnegative
by formulation. The normalization condition refers to the summation of all weights
along the spectral dimension for every spatial location
x
,
y
)
(
x
,
y
)
being equal to one. We
have to impose the normalization constraint explicitly, i.e.,
K
w
k
(
x
,
y
)
=
1
∀
(
x
,
y
).
k
=
1
Subject to this constraint, we can employ the variational framework to generate
a fused image that combines input hyperspectral bands where fusion weights have