Digital Signal Processing Reference
In-Depth Information
Selecting an appropriate prior model for the image data is critically impor-
tant for the accuracy of Bayesian estimation. Typically, a Gaussian distribution
is also assumed for the image, such that
Z
exp
1
1
2
(
T
−
1
p
(
x
)
=
−
x
−
µ
x
)
(
x
−
µ
x
)
.
(9.21)
Here,
denotes the
image covariance matrix, and
Z
is a normalizing constant. A sparse covariance
matrix is often assumed, with a narrow band of nonzero elements along the
diagonal to model an image with strong correlations among neighboring
pixels but zero correlation among distant pixels. A Gaussian image prior is
an accurate model for globally smooth image data, but that is usually not a
good assumption for real-world digital imagery.
For Gaussian noise and image models, the Bayesian MAP estimate becomes
µ
x
represents the image mean (possibly spatially varying),
1
2
,
1
2
(
2
−
1
T
x
=
arg min
x
2
y
−
Hx
+
x
−
µ
x
)
(
x
−
µ
x
)
(9.22)
σ
resulting in the linear estimate
−
1
H
T
H
2
−
1
2
−
1
H
T
y
x
=
(
+
σ
)
(σ
µ
+
).
(9.23)
x
Note the similarity between this linear Bayesian estimate and the least-squares
estimate in Equation 9.11. Both of these solutions generally possess smooth
object edges and discontinuities that appear somewhat blurred.
The advantage of utilizing the Bayesian framework for inverse problems
in image processing is that it is possible to stochastically model the image
data more accurately. A Gaussian image density is often assumed for math-
ematical and computational convenience. Non-Gaussian densities can more
closely approximate the true underlying image distribution, and the non-
linear estimates that result are often more accurate than linear solutions. To
incorporate a non-Gaussian image prior into the Bayesian MAP estimation
problem formulation, a two-dimensional Markov random field (MRF) must
be used to represent the image pixels. The Gibbs distribution is the probability
distribution over an MRF, with the corresponding density
Z
exp
1
1
2
p
(
x
)
=
−
V
c
(
x
)
.
(9.24)
β
c
∈C
Equation 9.24 is known as the Gibbs prior model.
20
In this expression,
β
is the
“temperature” parameter of the distribution,
V
c
(
x
)
is a potential function of
a local group of points
c
known as cliques, and
C
denotes the set of all image
cliques.
Digitized images of interest are assumed to be piecewise smooth. In other
words, only small variations exist between neighboring sample values, with
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