# Higher-order total variation approaches and generalisations

arXiv: https://arxiv.org/abs/1912.01587 · HTML: https://mlsanity.com/paper/1912.01587/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Dec 2019.

## Built on by (most cited first, 14 of 14)

- [Variation Spaces for Multi-Output Neural Networks: Insights on Multi-Task Learning and Network Compression](https://mlsanity.com/paper/2305.16534/) (2023)
- [Solving Inverse Problems with FLAIR](https://mlsanity.com/paper/2506.02680/) (2025)
- [Approximation of Lipschitz Functions using Deep Spline Neural Networks](https://mlsanity.com/paper/2204.06233/) (2022)
- [Posterior-Variance-Based Error Quantification for Inverse Problems in Imaging](https://mlsanity.com/paper/2212.12499/) (2022)
- [On Optimal Regularization Parameters via Bilevel Learning](https://mlsanity.com/paper/2305.18394/) (2023)
- [Unsupervised Knowledge-Transfer for Learned Image Reconstruction](https://mlsanity.com/paper/2107.02572/) (2021)
- [Measuring Complexity of Learning Schemes Using Hessian-Schatten Total Variation](https://mlsanity.com/paper/2112.06209/) (2021)
- [Deep unrolling for learning optimal spatially varying regularisation parameters for Total Generalised Variation](https://mlsanity.com/paper/2502.16532/) (2025)
- [Unsupervised energy disaggregation via convolutional sparse coding](https://mlsanity.com/paper/2207.09785/) (2022)
- [Outlier-robust neural network training: variation regularization meets trimmed loss to prevent functional breakdown](https://mlsanity.com/paper/2308.02293/) (2023)
- [Semi-sparsity Priors for Image Structure Analysis and Extraction](https://mlsanity.com/paper/2308.09141/) (2023)
- [Why do we regularise in every iteration for imaging inverse problems?](https://mlsanity.com/paper/2411.00688/) (2024)
- [OASIS: A Deep Learning Framework for Universal Spectroscopic Analysis Driven by Novel Loss Functions](https://mlsanity.com/paper/2509.11499/) (2025)
- [Learning spatially varying regularisation parameters of low regularity for image reconstruction](https://mlsanity.com/paper/2608.25127/) (2026)

## Abstract (arXiv)

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.