Higher-order total variation approaches and generalisations

TL;DR

This paper reviews higher-order total variation methods for inverse problems, discussing theoretical foundations, algorithms, and applications in imaging, highlighting their advantages over traditional TV in various imaging tasks.

Contribution

It provides a unified framework for higher-order TV approaches, including well-posedness, convergence, and practical algorithms, with extensive applications in imaging.

Findings

Higher-order TV methods improve image reconstruction quality.

Numerical algorithms for all models are developed and analyzed.

Applications demonstrate benefits in medical imaging and other inverse problems.

Abstract

Over the last decades, the total variation (TV) evolved to one of the most broadly-used regularisation functionals for inverse problems, in particular for imaging applications. When first introduced as a regulariser, higher-order generalisations of TV were soon proposed and studied with increasing interest, which led to a variety of different approaches being available today. We review several of these approaches, discussing aspects ranging from functional-analytic foundations to regularisation theory for linear inverse problems in Banach space, and provide a unified framework concerning well-posedness and convergence for vanishing noise level for respective Tikhonov regularisation. This includes general higher orders of TV, additive and infimal-convolution multi-order total variation, total generalised variation (TGV), and beyond. Further, numerical optimisation algorithms are developed and discussed that are suitable for solving the Tikhonov minimisation problem for all presented models. Focus is laid in particular on covering the whole pipeline starting at the discretisation of the problem and ending at concrete, implementable iterative procedures. A major part of this review is finally concerned with presenting examples and applications where higher-order TV approaches turned out to be beneficial. These applications range from classical inverse problems in imaging such as denoising, deconvolution, compressed sensing, optical-flow estimation and decompression, to image reconstruction in medical imaging and beyond, including magnetic resonance imaging (MRI), computed tomography (CT), magnetic-resonance positron emission tomography (MR-PET), and electron tomography.