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ECE Technical Reports Electrical and Computer Engineering
1-1-1992
A Generalized Gaussian Image Model for Edge-
Preserving MAP Estimation
Charles Bouman
Purdue University School of Electrical Engineering
Ken Sauer
University of Notre Dame, Department of Electrical Engineering
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Bouman, Charles and Sauer, Ken, "A Generalized Gaussian Image Model for Edge-Preserving MAP Estimation" (1992). ECE Technical
Reports. Paper 277.
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A Generalized Gaussian Image
Model for Edge
-
Preserving
MAP Estimation
Charles Bouman
Ken Sauer
TR
-
EE 92
-
1
January
1992
A
Generalized Gaussian Image Model for
Edge-Preserving MAP Estimation
Charles Bournan
School of Electrical Engineering
Purdue University
West Lafayette, IN 47907
-
0501
(317) 494
-
0340
Ken Sauer
Department of Electrical Engineering
University of
Notre Dame
Notre Dame, IN 46556
(219) 239
-
6999
March 19, 1991
Abstract
We present a Markov random field model intended to allow realistic edges in maximum
a
poste-
riori
(MAP)
image estimates, while providing stable solutions. Similar to the generalized Gaussian
distribution used in robust detection and estimation, we proposed the generalized Gaussian Markov
random field (GGMRF). This model satisfies several desirable analytical and computational prop
-
erties for
MAP
estimation, including continuous dependence of the estimate on the data, invariance
of the character of solutions to scaling of data, and a solution which lies at the unique local min
-
imum of the
a
posteriori
log likelihood function. The GGMRF is demonstrated to be useful for
image reconstruction in low dosage transmission tomography.
1
Introduction
Many important problems in image processing and computer vision require the estimation of
an image or other 2D field,
X,
from noisy data
Y.
For example, tomographic reconstruction
and 2D depth estimation are two seemingly dissimilar problems which
fit
into this structure.
When the data is of good quality and sufficient quantity, these problems may be solved well
by straightforward deterministic inverse formulae. However, when data is sparse or noisy,
direct inversion is usually excessively sensitive to noise. If the data is sufficiently sparse,
the inverse problem will be underdetermined or ill
-
posed. In such cases, the result can be
significantly improved by exploiting prior information about X's behavior.
Bayesian estimation is a statistical approach for incorporating prior information through
the choice of an
a
priori
distribution for the random field
X.
While many Bayesian estimation
techniques exist, a common choice for image estimation problems is the maximum
a
posteriori
(MAP) estimator. The MAP estimate has the appealing attribute that it yields the most
likely image given the observed data. In addition,
it
results in an optimization problem
which may be approached using a variety of established techniques.
The specific choice of prior distribution for
X
is, of course, a critical component in MAP
estimation. The Markov random field (MRF) has been applied widely during the recent
~ast[l, 2,
3,
41,
due to its power to usefully represent many image sources, and the local
nature of the resulting estimation operations.
A
variety of distinct models exist within the
class of
MRFs, depending on the choice of the
potential functions.
Each potential function
characterizes the interactions among a local group of pixels by assigning a larger cost to
configurations of pixels which are less likely to occur. In particular, we will restrict our
attention to potential functions
p(xi
-
xj),
which act on pairs of pixels. The shape of p(A),
where
A
is the difference between pixel values, then indicates the attributes of our model
for
X.
One of the more troublesome elements of applying MRFs to image estimation is coping
with edges. Because most potential functions penalize large differences in neighboring pixels,
sharp edges are often discouraged. This is especially true for the Gaussian MRF, which
penalizes the square of local pixel differences. Many approaches to ameliorate this effect
have been introduced.
Geman and Geman[2], incorporated a
"
line process
"
into their MRF
to describe sharp discontinuities. Others limited the penalty of any local difference at some
prescribed
threshold[5, 61, or created other potential functions which become flat at large
magnitudes of their
arguments[7,
8,
91.
Since such functions are non
-
convex, the global
optimization required in MAP estimation can not be exactly computed, and an approximate
MAP estimate must be used. In addition, we show that there is a second equally important
liability to using MRFs with non
-
convex potential functions: the MAP estimate may not be
a continuous function of the input data. This means that the position of the
2
with globally
minimal cost may undergo
a
large shift due to
a
small perturbation in
E'.
Therefore, the
MAP estimator is an unstable and ill
-
posed inverse operation.
Several researchers have proposed the use of convex potential functions. Stevenson and
Delp[lO] used the convex Huber function[ll], which is quadratic for small values of A, and
linear for large values. The point of transition between the quadratic and linear regions
of the function is a predetermined threshold,
T.
Green[l2] and Lange[l3] included the
strict convexity criterion, also for the sake of computational tractability. Green's choice
of
logcosh(A) has
a
shape similar to that of the Huber function, but with the transition
point from approximately quadratic to approximately linear at
T
=
1. Lange also derived
several other potential functions in
[13], each satisfying convexity and several other desired
properties.
The restriction to convex potential functions makes the computation of the exact MAP
estimate feasible, but the approaches listed above still exhibit a limitation: their effect in
MAP estimation is dependent on the
scaling
of
X
and
Y.
The transition threshold for the
Huber function, for example, should be related to the magnitude of edges expected in
X.
If
