Signal recovery via TV-type energies

TL;DR

This paper investigates one-dimensional total variation denoising, providing new regularity results for minimizers and analyzing the associated boundary value problem to deepen understanding of signal recovery methods.

Contribution

It offers novel regularity insights for 1D TV denoising minimizers and analyzes the Euler-Lagrange equation using ODE techniques, extending previous multidimensional work.

Findings

Strong regularity results for standard signal examples

Analysis of the Euler-Lagrange equation as a boundary value problem

Enhanced understanding of signal recovery via TV energies

Abstract

We consider one-dimensional variants of the classical first order total variation denoising model introduced by Rudin, Osher and Fatemi. This study is based on our previous work on various denoising and inpainting problems in image analysis, where we applied variational methods in arbitrary dimensions. More than being just a special case, the one-dimensional setting allows us to study regularity properties of minimizers by more subtle methods that do not have correspondences in higher dimensions. In particular, we obtain quite strong regularity results for a class of data functions that contains many of the standard examples from signal processing such as rectangle- or triangle signals as a special case. An analysis of the related Euler-Lagrange equation, which turns out to be a second order two-point boundary value problem with Neumann conditions, by ODE methods completes the picture of our investigation.