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
 
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