Generalized Methods and Solvers for Noise Removal from Piecewise Constant Signals

TL;DR

This paper introduces a unified framework for denoising piecewise constant signals, compares various existing and new methods, and demonstrates their effectiveness through synthetic data experiments.

Contribution

It presents a generalized functional for PWC denoising, introduces novel combined methods, and compares their performance with existing techniques.

Findings

New methods outperform some traditional approaches on synthetic data

Unified framework links various PWC denoising techniques

Head-to-head comparisons highlight strengths of proposed methods

Abstract

Removing noise from piecewise constant (PWC) signals, is a challenging signal processing problem arising in many practical contexts. For example, in exploration geosciences, noisy drill hole records need separating into stratigraphic zones, and in biophysics, jumps between molecular dwell states need extracting from noisy fluorescence microscopy signals. Many PWC denoising methods exist, including total variation regularization, mean shift clustering, stepwise jump placement, running medians, convex clustering shrinkage and bilateral filtering; conventional linear signal processing methods are fundamentally unsuited however. This paper shows that most of these methods are associated with a special case of a generalized functional, minimized to achieve PWC denoising. The minimizer can be obtained by diverse solver algorithms, including stepwise jump placement, convex programming, finite differences, iterated running medians, least angle regression, regularization path following, and coordinate descent. We introduce novel PWC denoising methods, which, for example, combine global mean shift clustering with local total variation smoothing. Head-to-head comparisons between these methods are performed on synthetic data, revealing that our new methods have a useful role to play. Finally, overlaps between the methods of this paper and others such as wavelet shrinkage, hidden Markov models, and piecewise smooth filtering are touched on.