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We discuss the interplay between local M -smoothers, Bayes smoothers and some nonlinear filters for edge-preserving signal reconstruction. We prove that all smoothers in question are nonlinear filters in a precise sense and characterize their fixed points. Then a Potts model is adopted for segmentation. For 1-d signals, an exact algorithm for the computation of maximum posterior modes is derived and applied to a phantom and to 1-d fMRI-data.

We discuss a nonindependent (beam) splitting for which the related thinning leaves the class of equilibrium states for a one mode electromagnetic field invariant. The thinning affects only the parameters of the state, showing a nonlinear loss of energy. After the splitting, the energy values of both split parts are independent. This independence is a characteristic property of the geometric distribution, the distribution of energy values in the equilibrium state. Also, we observe that the class of states where the full states of the split parts are independent is formed by the so-called phase states.