Compact Representation of n-th order TGV

TL;DR

This paper introduces two simple, implementable representations for n-th order Total Generalized Variation (TGV), addressing the lack of general algorithms for higher-order TGV regularization in image reconstruction.

Contribution

It provides the first straightforward, practical representations for higher-order TGV, enabling broader application and computational feasibility.

Findings

Two simple representations of n-th order TGV are proposed

The new methods facilitate implementation of higher-order TGV

Potential for improved image reconstruction with higher-order regularization

Abstract

Although regularization methods based on derivatives are favored for their robustness and computational simplicity, research exploring higher-order derivatives remains limited. This scarcity can possibly be attributed to the appearance of oscillations in reconstructions when directly generalizing TV-1 to higher orders (3 or more). Addressing this, Bredies et. al introduced a notable approach for generalizing total variation, known as Total Generalized Variation (TGV). This technique introduces a regularization that generates estimates embodying piece-wise polynomial behavior of varying degrees across distinct regions of an image.Importantly, to our current understanding, no sufficiently general algorithm exists for solving TGV regularization for orders beyond 2. This is likely because of two problems: firstly, the problem is complex as TGV regularization is defined as a minimization problem with non-trivial constraints, and secondly, TGV is represented in terms of tensor-fields which is difficult to implement. In this work we tackle the first challenge by giving two simple and implementable representations of n th order TGV