Image Processing Reference
In-Depth Information
s
2
w
(
m
)
1
log
s
T
w
(
m
)
2
1
δ
w
(
m
+
1
)
=
w
(
m
)
−
¯
◦
+
2
4
λ
s
+
μδ
1
4
λ
s
s
T
w
(
m
)
2
x
y
s
T
w
(
m
)
2
XY
−
log
(
0
.
5e
)
−
2
λ
v
−
s
T
w
(
m
)
−
x
y
s
T
w
(
m
)
2
XY
2
λ
v
XY
w
(
m
)
+
s
◦
,
(7.15)
x
y
where
(
m
)
is the index of iteration. The scalar
μ
appears only as a part of a positive
scaling factor in Eq. (
7.15
). Also, the purpose of
is only to enforce the unit length
of the weight vector. If we want to avoid this scaling factor, we have to explicitly nor-
malize the weight vector
w
(
m
+
1
)
at each iteration to satisfy the necessary constraint
given in Eq. (
7.8
) [76]. Here we introduce an intermediate variable
z
to represent
un-normalized weights,
w
. The final solution is thus given by Eq. (
7.16
).
μ
s
1
2
log
s
T
w
(
m
)
2
−
δ
z
(
m
+
1
)
= ¯
w
(
m
)
−
w
(
m
)
◦
+
log
(
0
.
5e
)
4
λ
s
s
T
w
(
m
)
2
x
y
s
T
w
(
m
)
2
XY
−
2
λ
v
−
s
T
w
(
m
)
2
x
y
s
T
w
(
m
)
2
XY
2
λ
v
XY
w
(
m
)
+
−
s
◦
(7.16)
x
y
z
(
m
+
1
)
◦
z
(
m
+
1
)
w
(
m
+
1
)
=+
T
z
(
m
+
1
)
.
(7.17)
z
(
m
+
1
)
The above equation provides a solution of the unconstrained optimization problem
to solve fusion problem. As explained earlier, the resultant fused image is formed
by linear weighted combination of the input hyperspectral bands, while the fusion
weights have been computed using the aforementioned unconstrained optimization
process. The basic process of fusion is, thus, provided by Eq. (
7.1
) with
α
k
(
x
,
y
)
=
w
k
(
. We expect the fused image to be centered around the mean radiometric
value of the data cube
I
, and to have a high contrast. The estimated
x
,
y
)
α
-matte is locally
smooth, but not necessarily the fused image.
7.4 Implementation
The solution presented in this chapter requires two regularization parameters that
define the relative weightage for each of the corresponding objectives. These weights
essentially determine the nature of the fused image in terms of the relative strength of
