3.2.2
The first three instantiated models behave in a similar way to the
quadratic prior model when 
or 
,
as noticed by [Li 1990b]. This can be understood by looking at
the power series expansion of 
, 
where 
are constants with
(the expansion can also involve a trivial additive constant
). When 
,
Thus in this situation of sufficiently small 
, the
adaptive model inherits the convexity of the quadratic model.
The interaction function h also well explains the differences between various regularizers. For the quadratic regularizer, the interaction is constant everywhere
and the smoothing strength is proportional to 
. This is why the
quadratic regularizer leads to oversmoothing at discontinuities where
is infinite. In the LP model, the interaction is piecewise
constant
Obviously, it inhibits oversmoothing by switching off smoothing when
exceeds 
in a binary manner.
In the LP approximations using the Hopfield approach [Koch et al. 1986 ; Yuille 1987] and mean field theory [Geiger and Girosi 1991], the line process variables are approximated by (3.13). This approximation effectively results in the following interaction function
As the temperature 
decreases toward zero, the above approaches
, that is,
.
Obviously, 
with nonzero 
is a member of the AIF
family, i.e.
and therefore the
approximated LP models are instances of the DA model.
It is interesting to note an observation made by Geiger and Girosi:
``sometimes a finite 
solution may be more desirable or robust"
([Geiger and Girosi 1991], pages 406-407) where 
.
They further suggest that there is ``an optimal (finite) temperature
(
)'' for the solution. An algorithm is presented in
[Herbert and Leahy 1992] for estimating an optimal 
. The LP approximation
with finite 
or nonzero T>0 is more an instance of the
DA than the LP model which they aimed to approximate. It will be shown
in Section 3.2.4 that the DA model is indeed more stable
than the LP model.
Anisotropic diffusion [Perona and Malik 1990] is a scale-space method for
edge-preserving smoothing. Unlike fixed coefficients in the traditional
isotropic scale-space filtering [Witkin 1983], anisotropic diffusion
coefficients are spatially varying according to the gradient
information. A so-called biased anisotropic diffusion
[Nordstrom 1990] model is obtained if anisotropic diffusion is
combined with a closeness term. Two choices of APFs are used in those
anisotropic diffusion models: 
and 
.
Shulman and Herve (1989) propose to use the following Huber's robust error penalty function [Huber 1981] as the adaptive potential
where 
plays a similar role to 
in 
. The
above is a convex function and has the first derivative as:
for 
, and
for other 
. Comparing 
with (3.24), we
find that the corresponding AIF is 
for 
and 
for other 
. This
function allows bounded but nonzero smoothing at discontinuities. The
same function has also been applied by [Stevenson and Delp 1990] to curve
fitting. A comparative study on the DA model and robust statistics is
presented in Chapter 4 (see also [Li 1995a]).
The approximation (3.18) of Leclerc's minimal length
model [Leclerc 1989] is in effect the same as the DA with APF 1.
Eq.(3.17) may be one of the best cost functions for
the piecewise constant restoration; for more general piecewise
continuous restoration, one needs to use (3.18) with a
nonzero 
, which is a DA instance. Regarding the continuity
property of domains, Mumford and Shah's model [Mumford and Shah 1985] and
Terzopoulos' continuity-controlled regularization model
[Terzopoulos 1986b] can also be defined on continuous domains
as the DA model.
