  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.
