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