Boundedness and unboundedness in total variation regularization

TL;DR

This paper investigates the boundedness of minimizers in total variation regularization for linear inverse problems, providing conditions for boundedness, explicit unbounded examples, and extensions to higher-order regularizations.

Contribution

It offers a simple proof of boundedness under certain conditions, constructs explicit unbounded minimizers, and explores extensions to higher-order regularization functionals.

Findings

Boundedness of minimizers under fixed regularization parameter

Existence of explicit unbounded minimizers in certain cases

Extension of results to infimal convolution of first and second order total variation

Abstract

We consider whether minimizers for total variation regularization of linear inverse problems belong to $L^\infty$ even if the measured data does not. We present a simple proof of boundedness of the minimizer for fixed regularization parameter, and derive the existence of uniform bounds for sufficiently small noise under a source condition and adequate a priori parameter choices. To show that such a result cannot be expected for every fidelity term and dimension we compute an explicit radial unbounded minimizer, which is accomplished by proving the equivalence of weighted one-dimensional denoising with a generalized taut string problem. Finally, we discuss the possibility of extending such results to related higher-order regularization functionals, obtaining a positive answer for the infimal convolution of first and second order total variation.