Image Processing Reference
InDepth Information
(e.g., edges, boundaries), get preserved in the output which is formed through a linear
combination of the input hyperspectral bands. Thus, if the input bands have a good
contrast, it does not get deteriorated during fusion. The smoothness objective for the
α
matte can be written as,
d
x
d
y
K
2
2
ε
3
(α)
=
α
k
x
(
,
)
+
α
k
y
(
,
)
,
x
y
x
y
(7.4)
k
=
1
x
y
where
α
k
x
and
α
k
y
denote derivatives of the fusion weight
α
k
in the
x
 and
y

directions, respectively.
Now, let us formulate the overall objective function for a constrainedminimization
by combining Eqs. (
7.2
)(
7.4
)asshowninEq.(
7.5
).
J
(α)
=−
ε
1
(α)
−
λ
v
ε
2
(α)
+
λ
s
ε
3
(α)
(7.5)
K
subject to,
α
k
(
x
,
y
)
≥
0
∀
k
,
and
1
α
k
(
x
,
y
)
=
1
∀
(
x
,
y
),
(7.6)
k
=
where
λ
s
are the regularization parameters that define the weightage given
to the variance term, and the smoothness term as compared to the entropy term,
respectively. From Eq. (
7.5
), we can infer that the fused image can be obtained by
solving the problem of calculus of variation.
λ
v
and
7.3.2 Variational Solution
We have discussed two constraints on the fusion weights, viz., unity norm and non
negativity. The first constraint refers to the sum of all the weights across all spectral
observations at a given spatial location. This value should be unity which indicates
the relative contribution of all the observations toward the final combination. The
normalization constraint can easily be incorporated into the cost function with the
help of a Lagrangian multiplier. An explicit term enforcing the sum of the fusion
weights to be one can be easily augmented with the existing cost function, as it will
be shown later.
An addition of the nonnegativity constraint on the fusion weights, however, con
verts the problem into a computationally demanding constrained optimization prob
lem. We need a computationally simple solution without sacrificing on the positivity
of the fusion weights. We show how to accomplish this task with the use of an aux
iliary variable. Here we introduce a set of auxiliary variables
w
, and define it as the
positive square root of the matte
α
. Thus the original weights
α
can be replaced as,
w
k
(
α
k
(
x
,
y
)
x
,
y
)
∀
(
x
,
y
).
(7.7)