The L1-Potts functional for robust jump-sparse reconstruction

TL;DR

This paper studies the $L^1$-Potts functional for reconstructing jump-sparse signals, proving convergence from discrete to continuous models, developing an efficient algorithm, and demonstrating its effectiveness in blind deconvolution and deblurring tasks.

Contribution

It establishes $ ext{Gamma}$-convergence of discrete to continuous $L^1$-Potts functionals, introduces an $O(n^2)$ algorithm for exact minimization, and explores blind deconvolution capabilities.

Findings

Discrete $L^1$-Potts functionals converge to continuous models.

An efficient $O(n^2)$ algorithm computes exact minimizers.

The method effectively reconstructs mildly blurred jump-sparse signals.

Abstract

We investigate the non-smooth and non-convex $L^1$-Potts functional in discrete and continuous time. We show $\Gamma$-convergence of discrete $L^1$-Potts functionals towards their continuous counterpart and obtain a convergence statement for the corresponding minimizers as the discretization gets finer. For the discrete $L^1$-Potts problem, we introduce an $O(n^2)$ time and $O(n)$ space algorithm to compute an exact minimizer. We apply $L^1$-Potts minimization to the problem of recovering piecewise constant signals from noisy measurements $f.$ It turns out that the $L^1$-Potts functional has a quite interesting blind deconvolution property. In fact, we show that mildly blurred jump-sparse signals are reconstructed by minimizing the $L^1$-Potts functional. Furthermore, for strongly blurred signals and known blurring operator, we derive an iterative reconstruction algorithm.