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Removing noise from piecewise constant (PWC) signals, is a challenging signalprocessing problem arising in many practical contexts. For example, inexploration geosciences, noisy drill hole records need separating intostratigraphic zones, and in biophysics, jumps between molecular dwell statesneed extracting from noisy fluorescence microscopy signals. Many PWC denoisingmethods exist, including total variation regularization, mean shift clustering,stepwise jump placement, running medians, convex clustering shrinkage andbilateral filtering; conventional linear signal processing methods arefundamentally unsuited however. This paper shows that most of these methods areassociated with a special case of a generalized functional, minimized toachieve PWC denoising. The minimizer can be obtained by diverse solveralgorithms, including stepwise jump placement, convex programming, finitedifferences, iterated running medians, least angle regression, regularizationpath following, and coordinate descent. We introduce novel PWC denoisingmethods, which, for example, combine global mean shift clustering with localtotal variation smoothing. Head-to-head comparisons between these methods areperformed on synthetic data, revealing that our new methods have a useful roleto play. Finally, overlaps between the methods of this paper and others such aswavelet shrinkage, hidden Markov models, and piecewise smooth filtering aretouched on.

Generalized Methods and Solvers for Noise Removal from Piecewise Constant Signals