Smoothers for Discontinuous Signals

First we explain the interplay between robust loss functions,
nonlinear filters and Bayes smoothers for edge-preserving image
reconstruction. Then we prove the surprising fact that maximum
posterior smoothers are nonlinear filters. A (generalized) Potts
prior for segmentation and piecewise smoothing of noisy signals and
images is adopted. For one-dimensional signals, an exact solution
for the maximum posterior mode - based on dynamic programming - is
derived. After some results on the performance of nonlinear filters
on jumps and ramps we finally introduce a cascade of nonlinear
filters with varying scale parameters and discuss the choice of
parameters for segmentation and piecewise smoothing.