4.1.3
Resemblances between M estimation with outliers and adaptive smoothing
with discontinuities have been noted by several authors
[Besl et al. 1988 ; Shulman and Herve 1989 ; Black and Anandan 1993 ; Black and Rangarajan 1994 ; Li 1995a]. We can
compare the M estimator with the DA model studied in
Chapter 3. The influence of the datum 
on the
estimate f is proportional to 
. This compares to the
smoothing strength 
given after equ.(). A very
large 
value, due to 
being far from f, are likely an
outlier. This is similar to saying that very large 
value is
likely due to a step (discontinuity) in the signal there. The
resemblance suggests that the definition of the DA model can also be
used to define M estimators [Li 1995e].
We replace the scale estimate in the M estimator by a parameter
and choose to use the adaptive interaction function
and the adaptive
potential function 
for
the M estimation. However, 
needs only be 
continuous for
the location estimation from discrete data. Theoretically, the
definitions give an infinite number of suitable choices of the M
estimators. Table 3.2.1 and Fig.3.1 showed
four such possibilities. With
and 
, we can define the energy under 
as
and thereby the minimum energy estimate
This defines a class of M estimators which are able to deal with outliers as the traditional M Estimators are. Its performance in the solution quality is significantly enhanced by the use of an annealing procedure.