research-article

Second-order Cone Programming Methods for Total Variation-Based Image Restoration

Authors:

Donald Goldfarb,

Wotao YinAuthors Info & Claims

SIAM Journal on Scientific Computing, Volume 27, Issue 2

Pages 622 - 645

https://doi.org/10.1137/040608982

Published: 01 January 2005 Publication History

Abstract

In this paper we present optimization algorithms for image restoration based on the total variation (TV) minimization framework of Rudin, Osher, and Fatemi (ROF). Our approach formulates TV minimization as a second-order cone program which is then solved by interior-point algorithms that are efficient both in practice (using nested dissection and domain decomposition) and in theory (i.e., they obtain solutions in polynomial time). In addition to the original ROF minimization model, we show how to apply our approach to other TV models, including ones that are not solvable by PDE-based methods. Numerical results on a varied set of images are presented to illustrate the effectiveness of our approach.

References

[1]

F. Alizadeh, D. Goldfarb, Second‐order cone programming, Math. Program., 95 (2003), 3–51, ISMP 2000, Part 3 (Atlanta, GA)

[2]

F. Alizadeh and S. Schmieta, Optimization with Semidefinite, Quadratic, and Linear Constraints, Technical report, RUTCOR, Rutgers University, Piscataway, NJ, 1997.

[3]

Knud Andersen, Edmund Christiansen, Andrew Conn, Michael Overton, An efficient primal‐dual interior‐point method for minimizing a sum of Euclidean norms, SIAM J. Sci. Comput., 22 (2000), 243–262

Digital Library

[4]

Gilles Aubert, Pierre Kornprobst, Mathematical problems in image processing, Applied Mathematical Sciences, Vol. 147, Springer‐Verlag, 2002xxvi+286, Partial differential equations and the calculus of variations; With a foreword by Olivier Faugeras

[5]

Antonin Chambolle, An algorithm for total variation minimization and applications, J. Math. Imaging Vision, 20 (2004), 89–97, Special issue on mathematics and image analysis

Digital Library

[6]

Antonin Chambolle, Pierre‐Louis Lions, Image recovery via total variation minimization and related problems, Numer. Math., 76 (1997), 167–188

[7]

R. Chan, T. Chan, and H.‐M. Zhou, Continuation Method for Total Variation Denoising Problems, CAM Report 95‐18, UCLA, 1995.

[8]

T. Chan and S. Esedoglu, Aspects of Total Variation Regularized $L^{1}$ Function Approximation, CAM Report 04‐07, UCLA, 2004.

[9]

Tony Chan, Gene Golub, Pep Mulet, A nonlinear primal‐dual method for total variation‐based image restoration, SIAM J. Sci. Comput., 20 (1999), 1964–1977

Digital Library

[10]

A. George, Nested dissection of a regular finite element mesh, SIAM J. Numer. Anal., 10 (1973), pp. 345–363.

Digital Library

[11]

D. Goldfarb and K. Scheinberg, Numerically Stable Product‐Form Cholesky $({LDL}^{T})$ Factorization Based Implementations of Interior Point Methods for Second‐Order Cone Programming, CORC Report TR‐2003‐05, IEOR Department, Columbia University, New York, 2003.

[12]

M. Hintermüller, K. Kunisch, Total bounded variation regularization as a bilaterally constrained optimization problem, SIAM J. Appl. Math., 64 (2004), 1311–1333

Digital Library

[13]

M. Hintermüller and G. Stadler, An Infeasible Primal‐Dual Algorithm for $TV$ ‐Based inf‐Convolution‐type Image Restoration, Technical report TR04‐15, CAAM Department, Rice University, Houston, TX, 2004.

[14]

M. Lysaker, A. Lundervold, and X.‐C. Taj, Noise removal using fourth‐order partial differential equations with applications to medical magnetic resonance images in space and time, IEEE Trans. Image Process., 12 (2003), pp. 1579–1590.

Digital Library

[15]

Yves Meyer, Oscillating patterns in image processing and nonlinear evolution equations, University Lecture Series, Vol. 22, American Mathematical Society, 2001x+122, The fifteenth Dean Jacqueline B. Lewis memorial lectures

Digital Library

[16]

Stanley Osher, Andrés Solé, Luminita Vese, Image decomposition and restoration using total variation minimization and the

H^{- 1}

norm, Multiscale Model. Simul., 1 (2003), 349–370

[17]

Ilya Pollak, Alan Willsky, Yan Huang, Nonlinear evolution equations as fast and exact solvers of estimation problems, IEEE Trans. Signal Process., 53 (2005), 484–498

Digital Library

[18]

Donald Rose, Gregory Whitten, A recursive analysis of dissection strategies, Academic Press, New York, 1976, 59–83

[19]

L. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), pp. 259–268.

Digital Library

[20]

David Strong, Tony Chan, Edge‐preserving and scale‐dependent properties of total variation regularization, Inverse Problems, 19 (2003), 0–0S165–S187, Special section on imaging

[21]

L. Vese, A study in the BV space of a denoising‐deblurring variational problem, Appl. Math. Optim., 44 (2001), 131–161

Digital Library

[22]

Luminita Vese, Stanley Osher, Modeling textures with total variation minimization and oscillating patterns in image processing, J. Sci. Comput., 19 (2003), 553–572, Special issue in honor of the sixtieth birthday of Stanley Osher

Digital Library

[23]

C. Vogel, M. Oman, Iterative methods for total variation denoising, SIAM J. Sci. Comput., 17 (1996), 227–238, Special issue on iterative methods in numerical linear algebra (Breckenridge, CO, 1994)

Digital Library

[24]

W. Yin, D. Goldfarb, and S. Osher, Total Variation‐Based Image Cartoon‐Texture Decomposition, CORC Report, TR‐2005‐01, IEOR Department, Columbia University, New York, 2005.

Cited By

Parzer FKirisits CScherzer O(2024)Uncertainty Quantification for Scale-Space Blob DetectionJournal of Mathematical Imaging and Vision10.1007/s10851-024-01194-x66:4(697-717)Online publication date: 1-Aug-2024
https://dl.acm.org/doi/10.1007/s10851-024-01194-x
Jensen TJørgensen JHansen PJensen S(2022)Implementation of an optimal first-order method for strongly convex total variation regularizationBIT10.1007/s10543-011-0359-852:2(329-356)Online publication date: 11-Mar-2022
https://dl.acm.org/doi/10.1007/s10543-011-0359-8
Yuan GGhanem B(2019)ℓ0 TVIEEE Transactions on Pattern Analysis and Machine Intelligence10.1109/TPAMI.2017.278393641:2(352-364)Online publication date: 1-Feb-2019
https://dl.acm.org/doi/10.1109/TPAMI.2017.2783936
Show More Cited By

Index Terms

Second-order Cone Programming Methods for Total Variation-Based Image Restoration

Index terms have been assigned to the content through auto-classification.

Recommendations

A Nonlinear Primal-Dual Method for Total Variation-Based Image Restoration

We present a new method for solving total variation (TV) minimization problems in image restoration. The main idea is to remove some of the singularity caused by the nondifferentiability of the quantity $|\nabla u|$ in the definition of the TV-norm ...
A Hybrid Proximal Extragradient Self-Concordant Primal Barrier Method for Monotone Variational Inequalities

This paper presents a hybrid proximal extragradient (HPE) self-concordant primal barrier method for solving a monotone variational inequality over a closed convex set endowed with a self-concordant barrier and with an underlying map that has Lipschitz ...
Hessian Barrier Algorithms for Linearly Constrained Optimization Problems

In this paper, we propose an interior-point method for linearly constrained---and possibly nonconvex---optimization problems. The method---which we call the Hessian barrier algorithm (HBA)---combines a forward Euler discretization of Hessian--Riemannian ...

Comments

Information & Contributors

Information

Published In

cover image SIAM Journal on Scientific Computing

SIAM Journal on Scientific Computing Volume 27, Issue 2

2005

371 pages

ISSN:1064-8275

Issue’s Table of Contents

Copyright © 2005 Society for Industrial and Applied Mathematics.

Publisher

Society for Industrial and Applied Mathematics

United States

Publication History

Published: 01 January 2005

Author Tags

Author Tags

Qualifiers

Research-article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

31
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 10 Nov 2024

Other Metrics

View Author Metrics

Citations

Cited By

Parzer FKirisits CScherzer O(2024)Uncertainty Quantification for Scale-Space Blob DetectionJournal of Mathematical Imaging and Vision10.1007/s10851-024-01194-x66:4(697-717)Online publication date: 1-Aug-2024
https://dl.acm.org/doi/10.1007/s10851-024-01194-x
Jensen TJørgensen JHansen PJensen S(2022)Implementation of an optimal first-order method for strongly convex total variation regularizationBIT10.1007/s10543-011-0359-852:2(329-356)Online publication date: 11-Mar-2022
https://dl.acm.org/doi/10.1007/s10543-011-0359-8
Yuan GGhanem B(2019)ℓ0 TVIEEE Transactions on Pattern Analysis and Machine Intelligence10.1109/TPAMI.2017.278393641:2(352-364)Online publication date: 1-Feb-2019
https://dl.acm.org/doi/10.1109/TPAMI.2017.2783936
Hanasusanto GRoitch VKuhn DWiesemann W(2017)Ambiguous Joint Chance Constraints Under Mean and Dispersion InformationOperations Research10.1287/opre.2016.158365:3(751-767)Online publication date: 1-Jun-2017
https://dl.acm.org/doi/10.1287/opre.2016.1583
Liao FShao S(2017)An Image Denoising Fast Algorithm for Weighted Total VariationProceedings of the 2nd International Conference on Intelligent Information Processing10.1145/3144789.3144809(1-7)Online publication date: 17-Jul-2017
https://dl.acm.org/doi/10.1145/3144789.3144809
Gong MJiang XLi H(2017)Optimization methods for regularization-based ill-posed problemsFrontiers of Computer Science: Selected Publications from Chinese Universities10.1007/s11704-016-5552-011:3(362-391)Online publication date: 1-Jun-2017
https://dl.acm.org/doi/10.1007/s11704-016-5552-0
Li MHan CWang RGuo T(2017)Shrinking gradient descent algorithms for total variation regularized image denoisingComputational Optimization and Applications10.1007/s10589-017-9931-868:3(643-660)Online publication date: 1-Dec-2017
https://dl.acm.org/doi/10.1007/s10589-017-9931-8
Yokota TZhao QCichocki A(2016)Smooth PARAFAC Decomposition for Tensor CompletionIEEE Transactions on Signal Processing10.1109/TSP.2016.258675964:20(5423-5436)Online publication date: 15-Oct-2016
https://dl.acm.org/doi/10.1109/TSP.2016.2586759
Su KChen JWang WSu L(2016)Reconstruction algorithm for block-based compressed sensing based on mixed variational inequalityMultimedia Tools and Applications10.1007/s11042-015-2975-975:23(16417-16438)Online publication date: 1-Dec-2016
https://dl.acm.org/doi/10.1007/s11042-015-2975-9
Dang CDai KLan G(2014)A linearly convergent first-order algorithm for total variation minimisation in image processingInternational Journal of Bioinformatics Research and Applications10.1504/IJBRA.2014.05877510:1(4-26)Online publication date: 1-Jan-2014
https://dl.acm.org/doi/10.1504/IJBRA.2014.058775
Show More Cited By

View Options

View options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

Media

Figures

Other

Tables

View Issue’s Table of Contents