4.1.5
The convexity of 
leads to a convex energy
(See also [Li et al. 1995]). Convex models have
several advantages. The convexity guarantees the stability with respect
to the input [Bouman and sauer 1993]. It makes the solution less sensitive to
changes in the parameters. Parameter graduation or annealing is not
necessary in convex minimization and this reduces the complexity.
Besides Shulman and Herve's convex (not strictly convex, though) APF
(3.34), other convex models also exist.
Hebert and Leahy (1989) examine the following three potential functions
.
in Bayesian reconstruction from emission tomography data and find that
the quality of the reconstruction could benefit from the third function
which is a compromise between the first and the second priors. Obviously
the first is exactly the quadratic potential function. The second
potential function has been used by Geman and McClure in
[Geman and McClure 1985]. In fact, both the second and the third
potential functions have the property 
and therefore are non-convex.
Green (1990) suggests the use of
as the potential function for the same problem. It is approximately
quadratic for small 
, and linear for large values, similar to the
Huber function. It is convex and satisfies all the properties of the
convex DA model.
Lange (1990) proposes seven properties for the suggested
potential functions, two of which guarantees robust penalties (bounded
smoothing) and convexity. But they require the functions to be twice
differentiable and strictly convex. These requirements have been
alleviated in our model. In that model, a positive, integrable function
is specified on 
; the corresponding 
can be recovered via
In the DA model, an APF is simply recovered by integrating the APF once ( cf. (3.28)).
Bouman (1993) emphasizes the importance of the convexity in Gibbs distribution and constructs a scale-invariant Gaussian MRF model by using the potential function
where 
. When p=0 it becomes the quadratic
(standard) regularizer which estimates smooth parameter fields. For
p=1 the corresponding estimator is the sample median and will allow
discontinuities. This class of potential functions are strictly convex
but unbounded, though the smoothing over discontinuities is fairly
limited when p is small enough. This model controls the degree of
discontinuities allowed in solutions through the choice of p. But, as
pointed out by [Stevenson et al. 1994], this choice for MAP estimation
will not allow consistent adjustment of the degree that discontinuities
will be allowed. When the data is sparse the model seems to be very
sensitive to the selection of p.
Stevenson et al.
(1994) presents a systematic Al
study on both convex and nonconvex regularization and summarizes four
desirable properties for Gibbs models to have good behaviors: convexity,
symmetry, restricted smoothing and the adjustability of the degree that
discontinuities are allowed in solutions. The restricted smoothing here
requires that 
for large 
, which is more relaxed
than the requirement of bounded smoothing in the DA model. In that work,
a class of convex potential functions is defined as
where 
, and generally 
. Actually when 
and 
, it is the case of Bouman's model
[Bouman and Sauer 1993] and when 
and 
, that is the Huber
function. There are variations on the choice of p and q and the
suitable choice will depend on the specific application and the a
priori information that is known. In general, when p is chosen to be
near 
and q chosen to be around 
, an appropriate threshold
T will lead to a quite satisfactory solution.