# A total variation based regularizer promoting piecewise-Lipschitz reconstructions

## TL;DR

This paper introduces a novel total variation-based regularizer that enforces piecewise-Lipschitz properties in reconstructions, offering an intuitive parameter and spatial adaptivity, with proven regularization properties and competitive performance.

## Contribution

A new regularizer in the total variation family promoting spatially adaptive Lipschitz constraints, with theoretical analysis and practical advantages over existing methods.

## Key findings

- Achieves similar performance to total generalized variation.
- Provides an intuitive interpretation of the regularizer parameter.
- Enables spatially adaptive regularization.

## Abstract

We introduce a new regularizer in the total variation family that promotes reconstructions with a given Lipschitz constant (which can also vary spatially). We prove regularizing properties of this functional and investigate its connections to total variation and infimal convolution type regularizers TVLp and, in particular, establish topological equivalence. Our numerical experiments show that the proposed regularizer can achieve similar performance as total generalized variation while having the advantage of a very intuitive interpretation of its free parameter, which is just a local estimate of the norm of the gradient. It also provides a natural approach to spatially adaptive regularization.