Image Processing Reference
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spatially smooth resultant. The MRFbased model represents local behavior among
the pixels in the form of the Gibbs distribution, where the value of the fused pixel is
a function of pixels in the spatial neighborhood of the corresponding location. The
use of the MRFbased prior typically produces spatially smooth images by favoring
the radiometrically similar pixels, and penalizing sharp discontinuities. However, the
GMRF priors tend to blur the edges and boundaries by oversmoothing them, unless
the edge fields have been separately modeled. The corresponding output cannot be
a good approximation of the image to be estimated if it contains edges [28]. Remote
sensing images typically contain a large number of small objects, which can be
identified quickly only if the corresponding edges and boundaries are sharp and
clear. Therefore, the use of any prior that leads to blurring or smearing of edges,
should be avoided. We would like to use the prior that preserves and enhances the
edge information in the data. Rudin et al. proposed the use of the total variation (TV)
norm based on the
L
1
norm of the derivatives of image [155]. Initially proposed for
image denoising, the TVnorm has also proved to be very effective in several image
processing and image restoration applications [27, 51, 57, 122]. The TV normbased
penalty functions can be approximately modeled as a Laplacian pdf prior, and thus,
one may use them as a prior within the MAP formulation also. The TV norm based
priors preserve the discontinuities in the image, and thus, retain sharp boundaries and
edges. The fused image preserves small objects and features in the scene, whichwould
get washed away or smeared during the process of fusion otherwise. When employed
in the deterministic framework, it is also claimed to provide a better quality as
compared to the one obtained by the use of the
L
2
normbased penalty functions under
exactly the same conditions [155]. The TV normbased penalty or prior functions
also outperform the
L
2
normbased penalty functions in terms of noise removal.
In [95], the TV norm has been incorporated as it provides a fused image which is
less noisy and has pronounced edges as compared to the one obtained using the
L
2
norm based penalty function. The formal expression of the TV norm for image
F
is
given by Eq. (
5.13
).
F
x
F
y
dx dy
P(
F
)
≡
TV
(
F
)
=
y
∇
F

dx dy
=
+
,
(5.13)
x
x
y
where
F
x
and
F
y
denote the spatial derivatives of the image in the
x
and
y
directions,
respectively.
We may combine the data term from Eq. (
5.12
), and the prior term from Eq. (
5.13
)
into the basic formulation from Eq. (
5.10
) to estimate the fused image
F
. Some of
the terms that do not affect the estimation process can be eliminated. We rewrite
the Bayesian solution by appropriately introducing the regularization parameter
λ
F
,
and eliminating the terms that do not affect the process of estimation to formulate
Eq. (
5.14
).
dx dy
F
2
=
argmax
F
−
s
−
β
F

−
λ
F
∇
F

.
(5.14)
x
y