Analytical aspects of spatially adapted total variation regularisation

TL;DR

This paper analyzes the structure of solutions in one-dimensional weighted total variation regularisation, revealing how the weight function influences discontinuities and establishing well-posedness even with vanishing weights.

Contribution

It provides a detailed analysis of the relationship between weight functions and solution discontinuities, and proves well-posedness of the weighted total variation problem despite lack of coercivity.

Findings

Weighted total variation solutions are well-posed even with zero weights.

The total variation of the solution is bounded by that of the data.

Weighted total variation and fidelity problems can yield different solutions.

Abstract

In this paper we study the structure of solutions of the one dimensional weighted total variation regularisation problem, motivated by its application in signal recovery tasks. We study in depth the relationship between the weight function and the creation of new discontinuities in the solution. A partial semigroup property relating the weight function and the solution is shown and analytic solutions for simply data functions are computed. We prove that the weighted total variation minimisation problem is well-posed even in the case of vanishing weight function, despite the lack of coercivity. This is based on the fact that the total variation of the solution is bounded by the total variation of the data, a result that it also shown here. Finally the relationship to the corresponding weighted fidelity problem is explored, showing that the two problems can produce completely different solutions even for very simple data functions.