Title: PACO: Signal Restoration via PAtch COnsensus,
Abstract: Many signal processing algorithms operate by breaking the target signal into
possibly overlapping segments (typically called windows or patches), processing
them separately, and then stitching them back into place to produce a unified
output. In most cases where pach overlapping occurs, the final value of those
samples that are estimated by more than one patch is resolved by averaging
those estimates; this includes many recent image processing algorithms. In
other cases, typically frequency-based restoration methods, the average is
implicitly weighted by some window function such as Hanning, Blackman, etc.
which is applied prior to the Fourier/DCT transform in order to avoid Gibbs
oscillations in the processed patches. Such averaging may incidentally help in
covering up artifacts in the restoration process, but more often will simply
degrade the overall result, posing an upper limit to the size of the patches
that can be used. In order to avoid such drawbacks, we propose a new
methodology where the different estimates of any given sample are forced to be
identical. We show that, together, these consensus constraints constitute a
non-empty convex feasible set, provide a general formulation of the resulting
constrained optimization problem which can be applied to a wide variety of
signal restoration tasks, and propose an efficient algorithm for finding the
corresponding solutions. Finally, we describe in detail the application of the
proposed methodology to three different signal processing problems, in some
cases surpassing the state of the art by a significant margin.