Unified Models for Second-Order TV-Type Regularisation in Imaging - A New Perspective Based on Vector Operators

TL;DR

This paper introduces a unified regulariser based on vector calculus operators that generalizes and interpolates between various TV-type regularisation methods, providing insights into their nullspaces and invariance properties.

Contribution

It presents a novel regulariser using vector operators that unifies multiple TV-type methods and analyzes their nullspaces and invariance properties.

Findings

The regulariser generalizes TV, ICTV, and TGV$^2$ methods.

Including shear in the penalty is crucial for high-quality results.

Proper discretisation ensures physically meaningful reconstructions.

Abstract

We introduce a novel regulariser based on the natural vector field operations gradient, divergence, curl and shear. For suitable choices of the weighting parameters contained in our model it generalises well-known first- and second-order TV-type regularisation methods including TV, ICTV and TGV$^2$ and enables interpolation between them. To better understand the influence of each parameter, we characterise the nullspaces of the respective regularisation functionals. Analysing the continuous model, we conclude that it is not sufficient to combine penalisation of the divergence and the curl to achieve high-quality results, but interestingly it seems crucial that the penalty functional includes at least one component of the shear or suitable boundary conditions. We investigate which requirements regarding the choice of weighting parameters yield a rotational invariant approach. To guarantee physically meaningful reconstructions, implying that conservation laws for vectorial differential operators remain valid, we need a careful discretisation that we therefore discuss in detail.