Image Processing Reference

In-Depth Information

Here, we need tomodify the cost function and the constraints inEq. (
7.5
) appropriately

in terms of
w
as the following. The expressions for entropy and variance terms are

modified by replacing

by
w
2
. In case of the smoothness term, however, the smooth-

ness in
w
k
which is a positive square root of
w
k
also implies smoothness in the actual

weights
w
k
. Therefore, we can as well impose the smoothness in
w
k
as an explicit

constraint to ensure the smoothness in the actual weights
w
k
. It can be easily observed

that the actual weights

α

α
k
(i.e.,
w
k
) are always non-negative, irrespective of the sign

of
w
. The constraint for unity norm, however, should be explicitly added to ensure

that for every pixel

, the sum of the weights should equal 1. After modifying

the weights to the square term, the unity norm constraint can be written as,

(

x

,

y

)

K

w
k
(

x

,

y

)
=

1

.

(7.8)

k

=

1

Consider the constraint given in Eq. (
7.5
) which specifies the condition on the

fusion weights

α
k
. The summation constraint indicates that the fusion weights for

a given observation lie on the hyperplane
α
k

1 in the first quadrant. The

constraint given by the auxiliary variables specifies that the fusion weights in the

form of auxiliary variables should lie on a unit hyper-sphere. It will be seen later

that such a unity norm constraint can be easily enforced externally while solving

Eq. (
7.5
) as an unconstrained optimization problem.

Now we provide the solution to the multi-objective optimization problem using

the Euler-Lagrange equation that we have already discussed in the previous chapter.

Since we are mostly dealing with the 1-D spectral array at a given location in a

2-D spatial plane, instead of working with 2-D bands
I
k
,

=

K
,itis

more convenient to work with the spectral array notation as explained earlier in

k

=

1

,

2

,
···
,

(

x

,

y

)

has been referred to as

K
.The
k
-th element of this vector is denoted by
s
k
(

s

.It

should be, however, noted that the change of notation is purely meant for the ease

of understanding, and to avoid any possible confusion when the same data is being

referred from different dimensions. The entire hyperspectral image as a cube can be

denoted by
I
. We shall denote the weight vector (in
w
) at the spatial location

(

x

,

y

)

, where
s

∈ R

x

,

y

)

(

x

,

y

)

by
w
2

, where
w
2

K
. This vector represents an element-wise product

(

x

,

y

)

(

x

,

y

)
∈ R

K
with itself. This is also known as the
Hadamard product
.

Mathematically, we can write this expression by Eq. (
7.9
).

of the vector
w

(

x

,

y

)
∈ R

w
2

(

,

)
={

w
k
(

,

)

w
k
(

,

)
∀

}=

(

,

)
◦

(

,

)

x

y

x

y

x

y

k

w

x

y

w

x

y

(7.9)

where

represents an element-wise product operator. Using the vector notation, the

resultant fused image
F

◦

(

x

,

y

)

can be represented in the form of a dot product of the

input data vector
s

(

x

,

y

)

at the corresponding pixel location

(

x

,

y

)

with the weight

vector
w
2

(

x

,

y

)

at the same spatial location, i.e.,

w
2

s
T

w
2

F

(

x

,

y

)
=

s

(

x

,

y

).

(

x

,

y

)
=

(

x

,

y

)

(

x

,

y

).

(7.10)