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In the context of image processing, given a $k$-th order, homogeneous and linear differential operator with constant coefficients, we study a class of variational problems whose regularizing terms depend on the operator. Precisely, the…

Numerical Analysis · Mathematics 2022-11-15 Valerio Pagliari , Kostas Papafitsoros , Bogdan Raiţă , Andreas Vikelis

This work proposes a novel convex-non-convex formulation of the image segmentation and the image completion problems. The proposed approach is based on the minimization of a functional involving two distinct regularization terms: one…

Numerical Analysis · Mathematics 2025-09-01 Mohamed El Guide , Anas El Hachimi , Khalide Jbilou , Lothar Reichel

This study presents the development of a spatially adaptive weighting strategy for Total Variation regularization, aimed at addressing under-determined linear inverse problems. The method leverages the rapid computation of an accurate…

Numerical Analysis · Mathematics 2025-01-20 Elena Morotti , Davide Evangelista , Andrea Sebastiani , Elena Loli Piccolomini

The parameter selection is crucial to regularization based image restoration methods. Generally speaking, a spatially fixed parameter for regularization item in the whole image does not perform well for both edge and smooth areas. A larger…

Image and Video Processing · Electrical Eng. & Systems 2021-04-07 Tingting Zhang , Jie Chen , Caiying Wu , Zhifei He , Tieyong Zeng , Qiyu Jin

Depth estimation is an essential component in understanding the 3D geometry of a scene, with numerous applications in urban and indoor settings. These scenes are characterized by a prevalence of human made structures, which in most of the…

Computer Vision and Pattern Recognition · Computer Science 2020-09-03 Mattia Rossi , Mireille El Gheche , Andreas Kuhn , Pascal Frossard

An adaptive regularization algorithm for unconstrained nonconvex optimization is proposed that is capable of handling inexact objective-function and derivative values, and also of providing approximate minimizer of arbitrary order. In…

Optimization and Control · Mathematics 2021-11-30 N. I. M. Gould , Ph. L. Toint

Adaptive cubic regularization methods have emerged as a credible alternative to linesearch and trust-region for smooth nonconvex optimization, with optimal complexity amongst second-order methods. Here we consider a general/new class of…

Optimization and Control · Mathematics 2018-11-20 Coralia Cartis , Nicholas I. M. Gould , Philippe L. Toint

The importance of regularization has been well established in image reconstruction -- which is the computational inversion of imaging forward model -- with applications including deconvolution for microscopy, tomographic reconstruction,…

Image and Video Processing · Electrical Eng. & Systems 2021-06-29 Sanjay Viswanath , Manu Ghulyani , Muthuvel Arigovindan

Total Variation (TV) and related extensions have been popular in image restoration due to their robust performance and wide applicability. While the original formulation is still relevant after two decades of extensive research, its…

Image and Video Processing · Electrical Eng. & Systems 2021-06-02 Sanjay Viswanath , Simon de Beco , Maxime Dahan , Muthuvel Arigovindan

Ill-posed linear inverse problems (ILIP), such as restoration and reconstruction, are a core topic of signal/image processing. A standard approach to deal with ILIP uses a constrained optimization problem, where a regularization function is…

Optimization and Control · Mathematics 2016-11-15 Manya V. Afonso , Jose M. Bioucas-Dias , Mario A. T. Figueiredo

In multiband fusion, an image with a high spatial and low spectral resolution is combined with an image with a low spatial but high spectral resolution to produce a single multiband image having high spatial and spectral resolutions. This…

Image and Video Processing · Electrical Eng. & Systems 2022-10-11 Unni V. S. , Pravin Nair , Kunal N. Chaudhury

While variational methods have been among the most powerful tools for solving linear inverse problems in imaging, deep (convolutional) neural networks have recently taken the lead in many challenging benchmarks. A remaining drawback of deep…

Computer Vision and Pattern Recognition · Computer Science 2019-08-20 Tim Meinhardt , Michael Moeller , Caner Hazirbas , Daniel Cremers

In this work, we propose Regularization-by-Equivariance (REV), a novel structure-adaptive regularization scheme for solving imaging inverse problems under incomplete measurements. This regularization scheme utilizes the equivariant…

Optimization and Control · Mathematics 2022-02-15 Junqi Tang

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

We propose a new fast algorithm for solving one of the standard formulations of image restoration and reconstruction which consists of an unconstrained optimization problem where the objective includes an $\ell_2$ data-fidelity term and a…

Optimization and Control · Mathematics 2015-05-14 Manya V. Afonso , José M. Bioucas-Dias , Mário A. T. Figueiredo

Dynamic inverse problems are challenging to solve due to the need to identify and incorporate appropriate regularization in both space and time. Moreover, the very large scale nature of such problems in practice presents an enormous…

Numerical Analysis · Mathematics 2025-01-23 Toluwani Okunola , Mirjeta Pasha , Misha Kilmer , Melina Freitag

In this paper, we use composite optimization algorithms to solve sigmoid networks. We equivalently transfer the sigmoid networks to a convex composite optimization and propose the composite optimization algorithms based on the linearized…

Optimization and Control · Mathematics 2023-07-10 Huixiong Chen , Qi Ye

Many inverse problems and signal processing problems involve low-rank regularizers based on the nuclear norm. Commonly, proximal gradient methods (PGM) are adopted to solve this type of non-smooth problems as they can offer fast and…

Signal Processing · Electrical Eng. & Systems 2025-11-25 Rodrigo A. Lobos , Javier Salazar Cavazos , Raj Rao Nadakuditi , Jeffrey A. Fessler

Image smoothing is a fundamental procedure in applications of both computer vision and graphics. The required smoothing properties can be different or even contradictive among different tasks. Nevertheless, the inherent smoothing nature of…

Graphics · Computer Science 2019-11-28 Wei Liu , Pingping Zhang , Yinjie Lei , Xiaolin Huang , Jie Yang , Ian Reid

Owing to the edge preserving ability and low computational cost of the total variation (TV), variational models with the TV regularization have been widely investigated in the field of multiplicative noise removal. The key points of the…

Computer Vision and Pattern Recognition · Computer Science 2015-03-18 Dai-Qiang Chen , Li-Zhi Cheng