Numerical Methods and Applications in Total Variation Image Restoration

  • Raymond Chan
  • Tony Chan
  • Andy Yip


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.


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.

References and Further Reading

  1. 1.
    Acar A, Vogel C (1994) Analysis of bounded variation penalty methods for ill-posed problems. Inverse Probl 10(6):1217–1229MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Adams R, Fournier J (2003) Sobolev spaces, vol 140 of Pure and applied mathematics, 2nd edn. Academic, New YorkMATHGoogle Scholar
  3. 3.
    Aujol J-F (2009) Some first-order algorithms for total variation based image restoration. J Math Imaging Vis 34(3):307–327MathSciNetCrossRefGoogle Scholar
  4. 4.
    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):111–136CrossRefGoogle Scholar
  5. 5.
    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 1–13Google Scholar
  6. 6.
    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 14–19Google Scholar
  7. 7.
    Blomgren P, Chan T (1998) Color TV: total variation methods for restoration of vector-valued images. IEEE Trans Image Process 7:304–309CrossRefGoogle Scholar
  8. 8.
    Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, CambridgeMATHGoogle Scholar
  9. 9.
    Bresson X, Chan T (2008) Non-local unsupervised variational image segmentation models. UCLA CAM Report, 08–67Google Scholar
  10. 10.
    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):455–484MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Buades A, Coll B, Morel J (2005) A review of image denoising algorithms, with a new one. Multiscale Model Simulat 4(2):490–530MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Burger M, Frick K, Osher S, Scherzer O (2007) Inverse total variation flow. Multiscale Model Simulat 6(2):366–395MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Carter J (2001) Dual methods for total variation-based image restoration. Ph.D. thesis, UCLA, Los Angeles, CA, USAGoogle Scholar
  14. 14.
    Chambolle A (2004) An algorithm for total variation minimization and applications. J Math Imaging Vis 20:89–97MathSciNetCrossRefGoogle Scholar
  15. 15.
    Chambolle A, Darbon J (1997) On total variation minimization and surface evolution using parametric maximum flows. Int J Comput Vis 84(3):288–307CrossRefGoogle Scholar
  16. 16.
    Chambolle A, Lions P (1997) Image recovery via total variation minimization and related problems. Numer Math 76:167–188MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Chan R, Chan T, Wong C (1999) Cosine transform based preconditioners for total variation deblurring. IEEE Trans Image Process 8:1472–1478CrossRefGoogle Scholar
  18. 18.
    Chan R, Wen Y, Yip A (2009) A fast optimization transfer algorithm for image inpainting in wavelet domains. IEEE Trans Image Process 18(7):1467–1476MathSciNetCrossRefGoogle Scholar
  19. 19.
    Chan T, Vese L (2001) Active contours without edges. IEEE Trans Image Process 10(2):266–277MATHCrossRefGoogle Scholar
  20. 20.
    Chan T, Golub G, Mulet P (1999) A nonlinear primal-dual method for total variation-based image restoration. SIAM J Sci Comp 20:1964–1977MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    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 17–32Google Scholar
  22. 22.
    Chan T, Esedoḡlu S, Nikolova M (2006a) Algorithms for finding global minimizers of image segmentation and denoising models. SIAM J Appl Math 66(5):1632–1648MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Chan T, Shen J, Zhou H (2006b) Total variation wavelet inpainting. J Math Imaging Vis 25(1): 107–125MathSciNetCrossRefGoogle Scholar
  24. 24.
    Chan T, Ng M, Yau C, Yip A (2007) Superresolution image reconstruction using fast inpainting algorithms. Appl Comput Harmon Anal 23(1): 3–24MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Christiansen O, Lee T, Lie J, Sinha U, Chan T (2007) Total variation regularization of matrix-valued images. Int J Biomed Imaging 2007:27432CrossRefGoogle Scholar
  26. 26.
    Combettes P, Wajs V (2004) Signal recovery by proximal forward-backward splitting. Multiscale Model Simulat 4(4):1168–1200MathSciNetCrossRefGoogle Scholar
  27. 27.
    Darbon J, Sigelle M (2006) Image restoration with discrete constrained total variation part I: Fast and exact optimization. J Math Imaging Vis 26:261–276MathSciNetCrossRefGoogle Scholar
  28. 28.
    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 1033–1038Google Scholar
  29. 29.
    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, 09–67Google Scholar
  30. 30.
    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):1881–1902MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Gilboa G, Osher S (2008) Nonlocal operators with applications to image processing. Multiscale Model Simulat 7(3):1005–1028MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Giusti E (1984) Minimal surfaces and functions of bounded variation. Birkhäuser, BostonMATHGoogle Scholar
  33. 33.
    Glowinki R, Le Tallec P (1989) Augmented Lagrangians and operator-splitting methods in nonlinear mechanics. SIAM, PhiladelphiaCrossRefGoogle Scholar
  34. 34.
    Goldfarb D, Yin W (2005) Second-order cone programming methods for total variation based image restoration. SIAM J Sci Comput 27(2): 622–645MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Goldstein T, Osher S (2009) The split Bregman method for l 1-regularization problems. SIAM J Imaging Sci 2(2):323–343MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Hintermüller M, Kunisch K (2004) Total bounded variation regularization as a bilaterally constrained optimisation problem. SIAM J Appl Math 64:1311–1333MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Hintermüller M, Stadler G (2006) A primal-dual algorithm for TV-based inf-convolution-type image restoration. SIAM J Sci Comput 28: 1–23MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Hintermüller M, Ito K, Kunisch K (2003) The primal-dual active set strategy as a semismooth Newton’s method. SIAM J Optim 13(3):865–888MATHCrossRefGoogle Scholar
  39. 39.
    Huang Y, Ng M, Wen Y (2008) A fast total variation minimization method for image restoration. Multiscale Model Simulat 7(2):774–795MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Kanwal RP (2004) Generalized functions: theory and applications. Birkhäuser, BostonMATHGoogle Scholar
  41. 41.
    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):2766–2777MathSciNetCrossRefGoogle Scholar
  42. 42.
    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):237–266MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    Lange K (2004) Optimization. Springer, New YorkMATHGoogle Scholar
  44. 44.
    Lange K, Carson R (1984) EM reconstruction algorithms for emission and transmission tomography. J Comput Assist Tomogr 8:306–316Google Scholar
  45. 45.
    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):2289–2300MathSciNetCrossRefGoogle Scholar
  46. 46.
    LeVeque R (2005) Numerical methods for conservation laws, 2nd edn. Birkhäuser, BaselGoogle Scholar
  47. 47.
    Mumford D, Shah J (1989) Optimal approximation by piecewise smooth functions and associated variational problems. Commun Pure Appl Math 42:577–685MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    Ng M, Qi L, Tang Y, Huang Y (2007) On semismooth Newton’s methods for total variation minimization. J Math Imaging Vis 27(3):265–276CrossRefGoogle Scholar
  49. 49.
    Osher S, Burger M, Goldfarb D, Xu J, Yin W (2005) An iterative regularization method for total variation based image restoration. Multiscale Model Simulat 4:460–489MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    Royden H (1988) Real analysis, 3rd edn. Prentice-Hall, Englewood CliffsMATHGoogle Scholar
  51. 51.
    Rudin L, Osher S, Fatemi E (1992) Nonlinear total variation based noise removal algorithms. Physica D 60:259–268MATHCrossRefGoogle Scholar
  52. 52.
    Sapiro G, Ringach D (1996) Anisotropic diffusion of multivalued images with applications to color filtering. IEEE Trans Image Process 5:1582–1586CrossRefGoogle Scholar
  53. 53.
    Setzer S (2009) Split Bregman algorithm, Douglas-Rachford splitting and frame shrinkage. In: Proceedings of scale-space 2009, pp 464–476Google Scholar
  54. 54.
    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 341–360CrossRefGoogle Scholar
  55. 55.
    Shen J, Kang S (2007) Quantum TV and application in image processing. Inverse Probl Imaging 1(3):557–575MathSciNetMATHCrossRefGoogle Scholar
  56. 56.
    Strang G (1983) Maximal flow through a domain. Math Program 26(2):123–143MathSciNetMATHCrossRefGoogle Scholar
  57. 57.
    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 PressGoogle Scholar
  58. 58.
    Vogel C, Oman M (1996) Iteration methods for total variation denoising. SIAM J Sci Comp 17:227–238MathSciNetMATHCrossRefGoogle Scholar
  59. 59.
    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):248–272MathSciNetMATHCrossRefGoogle Scholar
  60. 60.
    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):930–939CrossRefGoogle Scholar
  61. 61.
    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):2047–2080MathSciNetMATHCrossRefGoogle Scholar
  62. 62.
    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, 09–76Google Scholar
  63. 63.
    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):143–168MathSciNetMATHCrossRefGoogle Scholar
  64. 64.
    Zhu M, Chan T (2008) An efficient primal-dual hybrid gradient algorithm for total variation image restoration. UCLA CAM Report, 08–34Google Scholar
  65. 65.
    Zhu M, Wright SJ, Chan TF (to appear) Duality-based algorithms for total-variation-regularized image restoration. Comput Optim ApplGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Raymond Chan
    • 1
  • Tony Chan
    • 2
  • Andy Yip
    • 3
  1. 1.The Chinese University of Hong KongShatinHong Kong
  2. 2.University of California Los AngelesLos Angeles, CAUSA
  3. 3.National University of SingaporeSingaporeSingapore

Personalised recommendations