Image Processing Reference
InDepth Information
such as spatial smoothness, and a natural appearance. One of the common strategies
for improvements in the ML estimate is to incorporate some prior information on
the image to be estimated. This process transforms the problem into the maximum
a posteriori
(MAP) estimation.
The problemof MAPbased estimation of the fused image
F
can now be described
F
that maximizes the following joint probability density
as obtaining the function
function
F
(
x
,
y
)
=
argma
F
P(
s
(
x
,
y
)

F
(
x
,
y
))P(
F
(
x
,
y
)).
(5.10)
The MAP formulation, as explained, is composed of two terms. The first term on the
RHS of Eq. (
5.10
) represents the likelihood function of
F
, i.e., the conditional pdf
of the hyperspectral observations
s
. The second
term represents the pdf of
F
known as the prior on
F
. The prior favors certain desired
outputs, and thus, reduce the uncertainty in the solution.
First we consider the image formation model given by Eq. (
5.4
). Having com
puted the sensor selectivity factor
(
x
,
y
)
given the fused image
F
(
x
,
y
)
, we want to estimate the fused image
F
from the noisy observations. The noise term
β
k
(
x
,
y
)
η
k
(
,
)
x
y
represents the departure of
the corresponding pixel
s
k
(
from the image formation model. The noise term
for each observation at any given location
x
,
y
)
is assumed to be an independent
and identically distributed (i.i.d.) Gaussian process in [95, 164, 192, 193]. We also
model the noise by a zero mean Gaussian i.i.d. process with the variance value of
σ
(
x
,
y
)
2
. We can write the conditional probability density function (pdf) of the observed
spectral array
s
k
given the sensor selectivity factor
β
k
, and the fused image
F
using
Eq. (
5.11
).
exp
2
1
−

s
(
x
,
y
)
−
β(
x
,
y
)
F
(
x
,
y
)

P
(
s
(
x
,
y
)

F
(
x
,
y
))
=
√
2
∀
(
x
,
y
).
2
σ
2
(
πσ
2
)
K
(5.11)
We omit the function arguments
x
and
y
for the purpose of brevity (i.e.,
β
≡
β(
x
,
y
)
,
etc.), and write the loglikelihood function as the following.
log
exp
2
1
−

s
−
β
F

log
P(
s

F
)
=
√
2
∀
(
x
,
y
)
2
2
σ
2
K
(
πσ
)
2
K
2
log
)
−

s
−
β
F

2
log
P(
s

F
)
=−
(
2
πσ
∀
(
x
,
y
).
(5.12)
2
σ
2
Now, since we have provided the basic MAP formulation, let us consider the
second term in Eq. (
5.10
) which deals with the
apriori
information on the fused
image
F
. One can derive the prior term from the available information about the
function to be estimated. The prior information can also be obtained from some of the
physical properties that the estimated quantity is expected to have. A GaussMarkov
random field (GMRF) is one of the popular prior in image and vision problems.
In [192, 193], the fused image has been modeled as an MRF in order to obtain a