The L1-Potts functional for robust jump-sparse reconstruction

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.

References in corpus (3)

• Consistencies and rates of convergence of jump-penalized least squares estimators
• Jump-sparse and sparse recovery using Potts functionals
• Scale space consistency of piecewise constant least squares estimators -- another look at the regressogram

Cited by in corpus (7)

• Mumford-Shah and Potts Regularization for Manifold-Valued Data with Applications to DTI and Q-Ball Imaging
• Exact algorithms for $L^1$-TV regularization of real-valued or circle-valued signals
• Smoothing for signals with discontinuities using higher order Mumford-Shah models
• Variational Model-Based Reconstruction Techniques for Multi-Patch Data in Magnetic Particle Imaging
• Smoothing splines for discontinuous signals
• Reconstruction Formulae for 3D Field-Free Line Magnetic Particle Imaging
• Degrees-of-freedom penalized piecewise regression