Winkler, Gerhard and Liebscher, V. and Aurich, Volker
Smoothers for Discontinuous Signals.
Collaborative Research Center 386, Discussion Paper 146
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.