Total Variation Regularisation with Spatially Variable Lipschitz Constraints

TL;DR

This paper introduces a novel first-order Total Variation regulariser with spatially variable Lipschitz constraints, enabling adaptive control of variation in reconstructions, with demonstrated efficiency and comparable quality to second-order methods.

Contribution

It proposes a new Total Variation regulariser with spatially adaptive Lipschitz bounds, connecting it to existing methods and providing an efficient primal dual optimisation scheme.

Findings

Achieves reconstruction quality similar to Total Generalised Variation.

Requires significantly less computational time.

Demonstrates effective spatially adaptive regularisation.

Abstract

We introduce a first order Total Variation type regulariser that decomposes a function into a part with a given Lipschitz constant (which is also allowed to vary spatially) and a jump part. The kernel of this regulariser contains all functions whose Lipschitz constant does not exceed a given value, hence by locally adjusting this value one can determine how much variation is the reconstruction allowed to have. We prove regularising properties of this functional, study its connections to other Total Variation type regularisers and propose a primal dual optimisation scheme. Our numerical experiments demonstrate that the proposed first order regulariser can achieve reconstruction quality similar to that of second order regularisers such as Total Generalised Variation, while requiring significantly less computational time.