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