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We consider inverse problems with large null spaces, which arise in important applications such as in inverse ECG and EEG procedures. Standard regularization methods typically produce solutions in or near the orthogonal complement of the…

Numerical Analysis · Mathematics 2025-12-05 Martin Burger , Ole Løseth Elvetun , Bjørn Fredrik Nielsen

Total variation regularization has proven to be a valuable tool in the context of optimal control of differential equations. This is particularly attributed to the observation that TV-penalties often favor piecewise constant minimizers with…

Optimization and Control · Mathematics 2025-10-03 Giacomo Cristinelli , José A. Iglesias , Daniel Walter

A computational method is introduced for choosing the regularization parameter for total variation (TV) regularization. The approach is based on computing reconstructions at a few different resolutions and various values of regularization…

A class of mixed-order \emph{PDE}-constraint regularizer for image processing problem is proposed, generalizing the standard first order total variation $(TV)$. A semi-supervised (bilevel) training scheme, which provides a simultaneous…

Analysis of PDEs · Mathematics 2019-03-19 Pan Liu

The objectives of this chapter are: (i) to introduce a concise overview of regularization; (ii) to define and to explain the role of a particular type of regularization called total variation norm (TV-norm) in computer vision tasks; (iii)…

Computer Vision and Pattern Recognition · Computer Science 2016-04-01 Vania V. Estrela , Hermes Aguiar Magalhaes , Osamu Saotome

We consider the image denoising problem using total variation (TV) regularization. This problem can be computationally challenging to solve due to the non-differentiability and non-linearity of the regularization term. We propose an…

Optimization and Control · Mathematics 2014-08-26 Zhiwei Qin , Donald Goldfarb , Shiqian Ma

Total Variation (TV) based regularization has been widely applied in restoration problems due to its simple derivative filters based formulation and robust performance. While first order TV suffers from staircase effect, second order TV…

Signal Processing · Electrical Eng. & Systems 2019-04-08 Sanjay Viswanath , Muthuvel Arigovindan

Regularization approaches have demonstrated their effectiveness for solving ill-posed problems. However, in the context of variational restoration methods, a challenging question remains, which is how to find a good regularizer. While total…

Optimization and Control · Mathematics 2011-10-25 Nelly Pustelnik , Caroline Chaux , Jean-Christophe Pesquet

This work is about the total variation (TV) minimization which is used for recovering gradient-sparse signals from compressed measurements. Recent studies indicate that TV minimization exhibits a phase transition behavior from failure to…

Information Theory · Computer Science 2019-09-17 Sajad Daei , Farzan Haddadi , Arash Amini

We present a parallel version of the cut-pursuit algorithm for minimizing functionals involving the graph total variation. We show that the decomposition of the iterate into constant connected components, which is at the center of this…

Data Structures and Algorithms · Computer Science 2019-05-08 Hugo Raguet , Loic Landrieu

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…

Optimization and Control · Mathematics 2020-12-30 Kristian Bredies , Martin Holler

We explore the use of the recently proposed "total nuclear variation" (TNV) as a regularizer for reconstructing multi-channel, spectral CT images. This convex penalty is a natural extension of the total variation (TV) to vector-valued…

Medical Physics · Physics 2015-06-23 David Rigie , Patrick La Riviere

The total variation (TV) regularization method is an effective method for image deblurring in preserving edges. However, the TV based solutions usually have some staircase effects. In this paper, in order to alleviate the staircase effect,…

Numerical Analysis · Mathematics 2015-05-20 Gang Liu , Ting-Zhu Huang , Jun Liu , Xiao-Guang Lv

This paper addresses the solution of inverse problems in imaging given an additional reference image. We combine a modification of the discrete geodesic path model for image metamorphosis with a variational model,actually the $L^2$-$TV$…

Numerical Analysis · Mathematics 2019-05-22 Sebastian Neumayer , Johannes Persch , Gabriele Steidl

Based on transformed $\ell_1$ regularization, transformed total variation (TTV) has robust image recovery that is competitive with other nonconvex total variation (TV) regularizers, such as TV$^p$, $0<p<1$. Inspired by its performance, we…

Computer Vision and Pattern Recognition · Computer Science 2024-06-05 Elisha Dayag , Kevin Bui , Fredrick Park , Jack Xin

In this paper, we consider the efficient numerical minimization of Tikhonov functionals resulting from total-variation (TV) regularization of linear inverse problems. Since the TV penalty is non-smooth, this is typically done either via…

Numerical Analysis · Mathematics 2026-05-13 Helmut Gfrerer , Simon Hubmer , Stefan Kindermann , Jaakko Kultima , Ronny Ramlau , Tanja Tarvainen

Total variation (TV) regularization is a popular reconstruction method for ill-posed imaging problems, and particularly useful for applications with piecewise constant targets. However, using TV for medical cone-beam computed X-ray…

Medical Physics · Physics 2024-12-11 Alexander Meaney , Mikael A. K. Brix , Miika T. Nieminen , Samuli Siltanen

Various problems in computer vision and medical imaging can be cast as inverse problems. A frequent method for solving inverse problems is the variational approach, which amounts to minimizing an energy composed of a data fidelity term and…

Computer Vision and Pattern Recognition · Computer Science 2020-06-17 Erich Kobler , Alexander Effland , Karl Kunisch , Thomas Pock

Total variation (TV) is a powerful regularization method that has been widely applied in different imaging applications, but is difficult to apply to diffuse optical tomography (DOT) image reconstruction (inverse problem) due to complex and…

Computer Vision and Pattern Recognition · Computer Science 2019-04-08 Wenqi Lu , Jinming Duan , David Orive-Miguel , Lionel Herve , Iain B Styles

In this paper, we propose an adaptive finite difference scheme in order to numerically solve total variation type problems for image processing tasks. The automatic generation of the grid relies on indicators derived from a local estimation…

Numerical Analysis · Mathematics 2024-10-18 Thomas Jacumin , Andreas Langer