Image Processing Reference

In-Depth Information

pixels with small values of the gradient in their local neighborhood, this term in

the expression for sharpness measure (
Q
2
) is quite close to the constant
C
, resulting

in comparatively smaller values of the corresponding measure
Q
2
. It should be noted

that the range of
Q
2
depends on the choice of the constant. We need to explicitly

scale
Q
2
appropriately over
k
for each spatial location

(

x

,

y

)

, and confine it to the

range [0, 1].

The constant
C
serves two purposes—first, it provides a non-zero value to the

quality measure
Q
2
in the uniform or homogeneous regions. This prevents the possi-

bility of the sensor selectivity factor

attaining zero values solely due to the uniform

nature of the neighborhood region. Secondly, the constant
C
brings the flexibility to

the fusion system. Very few pixels from a particular band possess a significant value

of
Q
2
when
C
is small. That is, only those pixels which have a very high value of

sharpness in terms of their spatial gradient will get selected. With increase in the

value of
C
, more and more pixels get selected for the actual process of fusion. The

constant acts as the fixed bias or the offset for the selection of good pixels. Thus, with

the help of constant
C
, the user can tune the fusion algorithm as per the requirements

of the output.

We can obtain the sensor selectivity factor

β

for every pixel of the data by

substituting Eqs. (
5.6
) and (
5.7
)inEq.(
5.5
). These values, however, have been

calculated independently on a per pixel basis. The adjacent pixels in the hyperspec-

tral image bands usually have the same or similar material composition, and thus,

their reflectance responses are also highly similar. The degree of spatial correlation

among the intra-band pixels is quite high. As the sensor selectivity factor

β

β

indicates

visual quality of the pixel, one may naturally expect this factor also to possess some

degree of spatial closeness.We do not expect sharply discontinuous nature of

within

a local region in a given hyperspectral band. It should, however, be noted that the

β

β

values within a small neighborhood should not be exactly the same as each pixel is

unique. We redefine the problem of computation of

in a regularization framework

to address this requirement of spatial smoothness. We incorporate a regularizing

term that enforces the spatial smoothness constraint on the sensor selectivity factor

β

β

, from

smoothness, and prevents sharp discontinuities within the given band. The compu-

tation of

. This additional term penalizes departure of the sensor selectivity factor

β

β
k
(

x

,

y

)

proceeds as the minimization of the following expression given in

Eq. (
5.8
).

2

Q
j
I
k
(

)
−
β
k
(

)
2

β
k
(

x

,

y

)
=

argmin

β
k

x

,

y

x

,

y

(5.8)

x

y

j

=

1

)
dx dy

+
λ
β
β

2

2

k
x
(

x

,

y

)
+
β

k
y
(

x

,

y

,

where

indicates the relative importance of the smoothness constraint, also known

as the regularization parameter.

λ
β

β
k
x
and

β
k
y

indicate the spatial derivatives along the