Image Processing Reference

In-Depth Information

equal to the measure of local contrast. A similar weighting function based on the

intensity difference has been proposed in [102, 145]. It is also interesting to note that

if
K
hyperspectral bands are considered to be different noisy (assumed Gaussian)

observations as discussed in the previous chapter, then this solution would lead to

an optimal noise smoothing filter. The fusion weights can now be computed using

Eq. (
6.7
).

)
=
(σ

2
.

2

C
)(

w
k
(

x

,

y

k
(

x

,

y

)
+

F

(

x

,

y

)
−

I
k
(

x

,

y

))

(6.7)

It may be noted that one may employ various other quality measures such as spatial

gradient to evaluate the saliency of the pixel. These fusion weights, by definition,

are guaranteed to be non-negative. However, we should explicitly normalize them

for every pixel. The normalized fusion weights are given by Eq. (
6.8
)as,

(σ

2

2

C
)(

k
(

x

,

y

)
+

F

(

x

,

y

)
−

I
k
(

x

,

y

))

w
k
(

x

,

y

)
=

2
,

(6.8)

k
=
1
(σ

2

C
)(

k
(

x

,

y

)
+

F

(

x

,

y

)
−

I
k
(

x

,

y

))

which is merely dividing by the sum of all weights at a given pixel over all input

images to ensure that the weights sum to unity for a given pixel

.

We have transformed the fusion problem into an optimization problem where we

seek to obtain the resultant image that iteratively refines itself in order to merge

input hyperspectral bands smoothly. We provide the solution to this problem of

calculus of variations using the corresponding Euler-Lagrange equation described in

the previous section. The basic variational formulation is given in Eq. (
6.6
), while the

fusion weights
w
k
(

(

x

,

y

)

are computed using Eq. (
6.8
). For the purpose of brevity, we

omit the arguments
x
and
y
, and restate the same for a quick reference in Eq. (
6.9
).

x

,

y

)

⎧

⎨

⎫

⎬

F

2

K

+
λ
var

2

2

F

=

argmin

F

−

w
k
I
k

||

F
x
||

+||

F
y
||

dx dy

,

(6.9)

⎩

⎭

x

y

k

=

1

where
F
x
and
F
y
refer to the partial derivatives of the fused image
F
with respect to

x
and
y
, respectively. At every iteration, the solution generates the output image
F

which progressively minimizes the above expression. The newly formed fused image

is subjected to further refinement during the next iteration till a satisfactory result is

obtained. The discretized version of the solution of the Euler-Lagrange equation is

given by Eq. (
6.10
).

F
(
m
)
−

w
k
I
k
1

K

K

A
(
m
)
D
(
m
)

k

−
E
(
m
)
B
(
m
)

k

1

λ
var

F
(
m
+
1
)
=
F
(
m
)
−

−

I
k

A
(
m
)
2

k

=

1

k

=

1

(6.10)