Generalized Methods and Solvers for Noise Removal from Piecewise Constant Signals

Removing noise from signals which are piecewise constant (PWC) is a challenging signal processing problem that arises in many practical scientific and engineering contexts. For example, in exploration geosciences, noisy drill hole records must be separated into constant stratigraphic zones, and in biophysics, the jumps between states and dwells of a molecular structure need to be determined from noisy fluorescence microscopy signals. This problem is one for which conventional linear signal processing methods are fundamentally unsuited. A wide range of PWC denoising methods exists, including total variation regularization, mean shift clustering, stepwise jump placement, running median filtering, convex clustering shrinkage, bilateral filtering, wavelet shrinkage and hidden Markov models. This paper builds on results from the image processing community to show that the majority of these algorithms, and more proposed in the wider literature, are each associated with a special case of a generalized functional, that, when minimized, solves the PWC denoising problem. We show how the minimizer can be obtained by a range of computational solver algorithms, including stepwise jump placement, quadratic or linear programming, finite differences with and without adaptive step size, iterated running medians, least angle regression, piecewise-linear regularization path following, or coordinate descent. Using this generalized functional, we introduce several novel PWC denoising methods, which, for example, combine the global behaviour of mean shift clustering with the local smoothing of total variation diffusion, and show example solver algorithms for these new methods. 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 generalized methods of this paper and others such as wavelet shrinkage, hidden Markov models, and piecewise smooth filtering are touched on.