Abstract
Since their introduction in a classic paper by Rudin, Osher, and Fatemi [51],total variation minimizing models have become one of the most popular andsuccessful methodologies for image restoration. New developments continue toexpand the capability of the basic method in various aspects. Many fasternumerical algorithms and more sophisticated applications have been proposed.This chapter reviews some of these recent developments.
Keywords
- Total Variation Minimization
- Bregman Iteration
- Split Bregman Method
- Total Variation Denoising
- Split Bregman Iteration
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References and Further Reading
Acar A, Vogel C (1994) Analysis of bounded variation penalty methods for ill-posed problems. Inverse Probl 10(6):1217–1229
Adams R, Fournier J (2003) Sobolev spaces, vol 140 of Pure and applied mathematics, 2nd edn. Academic, New York
Aujol J-F (2009) Some first-order algorithms for total variation based image restoration. J Math Imaging Vis 34(3):307–327
Aujol J-F, Gilboa G, Chan T, Osher S (2006) Structure-texture image decomposition – modeling, algorithms, and parameter selection. Int J Comput Vis 67(1):111–136
Bect J, Blanc-Féraud L, Aubert G, Chambolle A (2004) A l 1-unified variational framework for image restoration. In Proceedings of ECCV, vol 3024 of Lecture notes in computer sciences, pp 1–13
Bioucas-Dias J, Figueiredo M, Nowak R (2006) Total variation-based image deconvolution: a majorization-minimization approach. In Proceedings of IEEE international conference on acoustics, speech and signal processing ICASSP 2006, vol 2, pp 14–19
Blomgren P, Chan T (1998) Color TV: total variation methods for restoration of vector-valued images. IEEE Trans Image Process 7:304–309
Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, Cambridge
Bresson X, Chan T (2008) Non-local unsupervised variational image segmentation models. UCLA CAM Report, 08–67
Bresson X, Chan T (2008) Fast dual minimization of the vectorial total variation norm and applications to color image processing. Inverse Probl Imaging 2(4):455–484
Buades A, Coll B, Morel J (2005) A review of image denoising algorithms, with a new one. Multiscale Model Simulat 4(2):490–530
Burger M, Frick K, Osher S, Scherzer O (2007) Inverse total variation flow. Multiscale Model Simulat 6(2):366–395
Carter J (2001) Dual methods for total variation-based image restoration. Ph.D. thesis, UCLA, Los Angeles, CA, USA
Chambolle A (2004) An algorithm for total variation minimization and applications. J Math Imaging Vis 20:89–97
Chambolle A, Darbon J (1997) On total variation minimization and surface evolution using parametric maximum flows. Int J Comput Vis 84(3):288–307
Chambolle A, Lions P (1997) Image recovery via total variation minimization and related problems. Numer Math 76:167–188
Chan R, Chan T, Wong C (1999) Cosine transform based preconditioners for total variation deblurring. IEEE Trans Image Process 8:1472–1478
Chan R, Wen Y, Yip A (2009) A fast optimization transfer algorithm for image inpainting in wavelet domains. IEEE Trans Image Process 18(7):1467–1476
Chan T, Vese L (2001) Active contours without edges. IEEE Trans Image Process 10(2):266–277
Chan T, Golub G, Mulet P (1999) A nonlinear primal-dual method for total variation-based image restoration. SIAM J Sci Comp 20:1964–1977
Chan T, Esedoḡlu S, Park F, Yip A (2005) Recent developments in total variation image restoration. In: Paragios N, Chen Y, Faugeras O (eds) Handbook of mathematical models in computer vision. Springer, Berlin, pp 17–32
Chan T, Esedoḡlu S, Nikolova M (2006a) Algorithms for finding global minimizers of image segmentation and denoising models. SIAM J Appl Math 66(5):1632–1648
Chan T, Shen J, Zhou H (2006b) Total variation wavelet inpainting. J Math Imaging Vis 25(1): 107–125
Chan T, Ng M, Yau C, Yip A (2007) Superresolution image reconstruction using fast inpainting algorithms. Appl Comput Harmon Anal 23(1): 3–24
Christiansen O, Lee T, Lie J, Sinha U, Chan T (2007) Total variation regularization of matrix-valued images. Int J Biomed Imaging 2007:27432
Combettes P, Wajs V (2004) Signal recovery by proximal forward-backward splitting. Multiscale Model Simulat 4(4):1168–1200
Darbon J, Sigelle M (2006) Image restoration with discrete constrained total variation part I: Fast and exact optimization. J Math Imaging Vis 26:261–276
Efros A, Leung T (1999) Texture synthesis by non-parametric sampling. In: Proceedings of the IEEE international conference on computer vision, vol 2, Corfu, Greece, pp 1033–1038
Esser E, Zhang X, Chan T (2009) A general framework for a class of first order primal-dual algorithms for TV minimization. UCLA CAM Report, 09–67
Fu H, Ng M, Nikolova M, Barlow J (2006) Efficient minimization methods of mixed l2-l1 and l1-l1 norms for image restoration. SIAM J Sci Comput 27(6):1881–1902
Gilboa G, Osher S (2008) Nonlocal operators with applications to image processing. Multiscale Model Simulat 7(3):1005–1028
Giusti E (1984) Minimal surfaces and functions of bounded variation. Birkhäuser, Boston
Glowinki R, Le Tallec P (1989) Augmented Lagrangians and operator-splitting methods in nonlinear mechanics. SIAM, Philadelphia
Goldfarb D, Yin W (2005) Second-order cone programming methods for total variation based image restoration. SIAM J Sci Comput 27(2): 622–645
Goldstein T, Osher S (2009) The split Bregman method for l 1-regularization problems. SIAM J Imaging Sci 2(2):323–343
Hintermüller M, Kunisch K (2004) Total bounded variation regularization as a bilaterally constrained optimisation problem. SIAM J Appl Math 64:1311–1333
Hintermüller M, Stadler G (2006) A primal-dual algorithm for TV-based inf-convolution-type image restoration. SIAM J Sci Comput 28: 1–23
Hintermüller M, Ito K, Kunisch K (2003) The primal-dual active set strategy as a semismooth Newton’s method. SIAM J Optim 13(3):865–888
Huang Y, Ng M, Wen Y (2008) A fast total variation minimization method for image restoration. Multiscale Model Simulat 7(2):774–795
Kanwal RP (2004) Generalized functions: theory and applications. Birkhäuser, Boston
Krishnan D, Lin P, Yip A (2007) A primal-dual active-set method for non-negativity constrained total variation deblurring problems. IEEE Trans Image Process 16(11):2766–2777
Krishnan D, Pham Q, Yip A (2009) A primal dual active set algorithm for bilaterally constrained total variation deblurring and piecewise constant Mumford-Shah segmentation problems. Adv Comput Math 31(1–3):237–266
Lange K (2004) Optimization. Springer, New York
Lange K, Carson R (1984) EM reconstruction algorithms for emission and transmission tomography. J Comput Assist Tomogr 8:306–316
Law Y, Lee H, Yip A (2008) A multi-resolution stochastic level set method for Mumford-Shah image segmentation. IEEE Trans Image Process 17(12):2289–2300
LeVeque R (2005) Numerical methods for conservation laws, 2nd edn. Birkhäuser, Basel
Mumford D, Shah J (1989) Optimal approximation by piecewise smooth functions and associated variational problems. Commun Pure Appl Math 42:577–685
Ng M, Qi L, Tang Y, Huang Y (2007) On semismooth Newton’s methods for total variation minimization. J Math Imaging Vis 27(3):265–276
Osher S, Burger M, Goldfarb D, Xu J, Yin W (2005) An iterative regularization method for total variation based image restoration. Multiscale Model Simulat 4:460–489
Royden H (1988) Real analysis, 3rd edn. Prentice-Hall, Englewood Cliffs
Rudin L, Osher S, Fatemi E (1992) Nonlinear total variation based noise removal algorithms. Physica D 60:259–268
Sapiro G, Ringach D (1996) Anisotropic diffusion of multivalued images with applications to color filtering. IEEE Trans Image Process 5:1582–1586
Setzer S (2009) Split Bregman algorithm, Douglas-Rachford splitting and frame shrinkage. In: Proceedings of scale-space 2009, pp 464–476
Setzer S, Steidl G, Popilka B, Burgeth B (2009) Variational methods for denoising matrix fields. In: Laidlaw D, Weickert J (eds) Visualization and processing of tensor fields: advances and perspectives, mathematics and visualization. Springer, Berlin, pp 341–360
Shen J, Kang S (2007) Quantum TV and application in image processing. Inverse Probl Imaging 1(3):557–575
Strang G (1983) Maximal flow through a domain. Math Program 26(2):123–143
Tschumperlé D, Deriche R (2001) Diffusion tensor regularization with constraints preservation. In: Proceedings of 2001 IEEE computer society conference on computer vision and pattern recognition, vol 1, Kauai, Hawaii, pp 948–953. IEEE Computer Science Press
Vogel C, Oman M (1996) Iteration methods for total variation denoising. SIAM J Sci Comp 17:227–238
Wang, Y, Yang J, Yin W, Zhang Y (2008) A new alternating minimization algorithm for total variation image reconstruction. SIAM J Imaging Sci 1(3):248–272
Wang Z, Vemuri B, Chen Y, Mareci T (2004) A constrained variational principle for direct estimation and smoothing of the diffusion tensor field from complex DWI. IEEE Trans Med Imaging 23(8):930–939
Weiss P, Aubert G, Blanc-Fèraud L (2009) Efficient schemes for total variation minimization under constraints in image processing. SIAM J Sci Comput 31(3):2047–2080
Wu C, Tai XC (2009) Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models. UCLA CAM Report, 09–76
Yin W, Osher S, Goldfarb D, Darbon J (2008) Bregman iterative algorithms for l 1-minimization with applications to compressed sensing. SIAM J Imaging Sci 1(1):143–168
Zhu M, Chan T (2008) An efficient primal-dual hybrid gradient algorithm for total variation image restoration. UCLA CAM Report, 08–34
Zhu M, Wright SJ, Chan TF (to appear) Duality-based algorithms for total-variation-regularized image restoration. Comput Optim Appl
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer Science+Business Media, LLC
About this entry
Cite this entry
Chan, R., Chan, T., Yip, A. (2011). Numerical Methods and Applications in Total Variation Image Restoration. In: Scherzer, O. (eds) Handbook of Mathematical Methods in Imaging. Springer, New York, NY. https://doi.org/10.1007/978-0-387-92920-0_24
Download citation
DOI: https://doi.org/10.1007/978-0-387-92920-0_24
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-92919-4
Online ISBN: 978-0-387-92920-0
eBook Packages: Mathematics and StatisticsReference Module Computer Science and Engineering