Skip to main content
SpringerLink
Log in

Nonlinear Multilevel Schemes for Solving the Total Variation Image Minimization Problem

  • Conference paper
Image Processing Based on Partial Differential Equations

Part of the book series: Mathematics and Visualization ((MATHVISUAL))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. R. Acar and C. R. Vogel. Analysis of total variation penalty method for ill-posed problems, Inverse Probs., 10:1217–1229, 1994.

    Article  MathSciNet  Google Scholar 

  2. S. T. Acton. Multigrid anisotropic diffusion, IEEE Trans. Imag. Proc., 3 (3):280-291,1998.

    Article  MathSciNet  Google Scholar 

  3. I. Albarreal, M. C. Calzada, J. L. Cruz, E. Fernández-Cara, J. Galo, and M. Marin. Convergence analysis and error estimates for a parallel algorithm for solving the Navier-Stokes equations. Numer. Math., 93(2):201–221, 2002.

    Article  MathSciNet  Google Scholar 

  4. E. Arian and S. Ta’asan. Multigrid one-shot methods for optimal control problems, ICASE technical report No. 94-52, USA, 1994.

    Google Scholar 

  5. P. Blomgren, T. F. Chan and P. Mulet, Extensions to Total Variation Denoising, Proc. SPIE 97, San Diego, USA, 1997.

    Google Scholar 

  6. P. Blomgren, T. F. Chan, P. Mulet, L. Vese, and W. L. Wan. Variational PDE models and methods for image processing, in Research Notes in Mathematics, 420:43-67. Chapman & Hall/CRC, 2000.

    Google Scholar 

  7. A. Borzi and K. Kunisch. A globalization strategy for the multigrid solution of elliptic optimal control problems. Optim. Methods Softw. 21 (2006), no. 3, 445–459.

    Article  MathSciNet  Google Scholar 

  8. A. Bruhn, J. Weickert, T. Kohlberger and C. Schnörr, A multigrid platform for real-time motion computation with discontinuity-preserving variational methods, Technical Report No. 136, Department of Mathematics, Saarland University, Saarbrücken, Germany, May 2005.

    Google Scholar 

  9. M. Burger, S. Osher, J. Xu and G. Gilboa. Nonlinear inverse scale space methods for image restoration, Comm. Math. Sci., 4 (1), pp.179–212, 2006. (See also UCLA CAM report 05-34, 2005).

    MathSciNet  Google Scholar 

  10. A. Brandt, Multilevel adaptive solutions to boundary value problems, Math. Comp., pp.333-190, 1977.

    Google Scholar 

  11. A. Brandt, Multigrid solvers and multilevel optimization strategies, In J. Cong and J. R. Shinnerl, editors, Multiscale Optimization and VLSI/CAD, pp.1–68. Kluwer Academic (Boston), 2000.

    Google Scholar 

  12. J. L. Carter. Dual method for total variation-based image restoration, CAM report 02-13, PhD thesis, University of California at LA, USA; see http://www.math.ucla.edu/applied/cam/index.html.

  13. E. Casas, K. Kunisch and C. Pola, Regularization of functions of bounded variation and applications to image enhancement, Appl. Math. Optim., 40:229–257, 1999.

    Article  MathSciNet  Google Scholar 

  14. A. Chambolle. An algorithm for total variation minimization and applications, J. Math. Imag. Vis., 20:89–97, 2004.

    Article  MathSciNet  Google Scholar 

  15. A. Chambolle and P.L. Lions. Image recovery via total variation minimization and related problems, Numer. Math., 76 (2):167–188, 1997.

    Article  MathSciNet  Google Scholar 

  16. R. H. Chan, T. F. Chan, and W. L. Wan. Multigrid for differential convolution problems arising from image processing, in R. Chan, T. F. Chan, and G. H. Golub, editors, Proc. Sci. Comput. Workshop. Springer-Verlag, see also CAM report 97-20, UCLA, USA, 1997.

    Google Scholar 

  17. R. H. Chan, Q. S. Chang, and H. W. Sun., Multigrid method for ill-conditioned symmetric Toeplitz systems, SIAM J. Sci. Comput., 19:516–529, 1998.

    Article  MathSciNet  Google Scholar 

  18. R. H. Chan, C. W. Ho, and M. Nikolova. Salt-and-pepper noise removal by median-type noise detectors and detail-preserving regularization, IEEE Trans. Image Proc., to appear, 2005.

    Google Scholar 

  19. R. H. Chan and C. K. Wong. Sine transform based preconditioners for elliptic problems, Numer. Linear Algebra Applic., 4:351–368, 1997.

    Article  MathSciNet  Google Scholar 

  20. T. F. Chan and K. Chen. On a nonlinear multigrid algorithm with primal relaxation for the image total variation minimisation, Numer. Alg., 41:387–411, 2006.

    Article  MathSciNet  Google Scholar 

  21. T. F. Chan and K. Chen. An optimization-based multilevel algorithm for total variation image denoising, SIAM J. Multiscale Mod. Sim., 5(2):615–645, 2006.

    Article  MathSciNet  Google Scholar 

  22. T. F. Chan and S. Esedoglu. Aspects of total variation regularized L1 function approximation, UCLA CAM report 04-07, 2004.

    Google Scholar 

  23. T. Chan, S. Esedoglu, F. Park and A. Yip, Recent Developments in Total Variation Image Restoration, in The Handbook of Math. Models in Computer Vision, eds. N. Paragios, Y. M. Chen and O. Faugeras, Springer-Verlag, pp.17–32, 2005. (See also CAM report 05-01, UCLA, USA.)

    Google Scholar 

  24. T. F. Chan, G. H. Golub, and P. Mulet. A nonlinear primal dual method for total variation based image restoration, SIAM J. Sci. Comput., 20 (6):1964–1977, 1999.

    Article  MathSciNet  Google Scholar 

  25. T. F. Chan and T. P. Mathew. Domain decomposition algorithms, in: Acta Numerica, ed. A. Iserles, pp.61-143, 1994.

    Google Scholar 

  26. T. F. Chan and P. Mulet. Iterative methods for total variation restoration, CAM report 96-38, UCLA, USA, 1996; see http://www.math.ucla.edu/applied/cam/index.html.

    Google Scholar 

  27. Q. S. Chang and I. L. Chern. Acceleration methods for total variation-based image denoising, SIAM J. Sci. Comput., 25:982–994, 2003.

    Article  MathSciNet  Google Scholar 

  28. K. Chen. Matrix Preconditioning Techniques and Applications. Cambridge Monographs on Applied and Computational Mathematics (No. 19). Cambridge University Press, UK, 2005.

    Google Scholar 

  29. K. Chen and X.-C. Tai. A nonlinear multigrid method for total variation minimization from image restoration, see UCLA CAM report 05-26, USA, 2005.

    Google Scholar 

  30. C. Frohn-Schauf, S. Henn, and K. Witsch. Nonlinear multigrid methods for total variation image denoising, Comput Visual Sci., 7:199–206, 2004.

    MathSciNet  Google Scholar 

  31. J. R. Galo, I. Albarreal, M. C. Calzada, J. L. Cruz, E. Fernández-Cara, and M. Marin. Stability and convergence of a parallel fractional step method for the solution of linear parabolic problems. AMRX Appl. Math. Res. Express, (4):117-142, 2005.

    Google Scholar 

  32. J. R. Galo, Isidoro I. Albarreal, M. C. Calzada, J. L. Cruz, E. Fernández-Cara, and M. Marin. A simultaneous directions parallel algorithm for the NavierStokes equations. C. R. Math. Acad. Sci. Paris, 339(3):235–240, 2004.

    MathSciNet  Google Scholar 

  33. J. R. Galo, I. I. Albarreal, M. C. Calzada, J. L. Cruz, E. Fernández-Cara, and M. Marín. Simultaneous directions parallel methods for elliptic and parabolic systems. C. R. Math. Acad. Sci. Paris, 339(2):145–150, 2004.

    MathSciNet  Google Scholar 

  34. D. Goldfarb and W. T. Yin, Second-order cone programming methods for total variation-based image restoration, SIAM J. Sci. Comput., 27 (2):622–645, 2005.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  36. M. Hintermüller and G. Stadler, An infeasible primal-dual algorithm for TV-based infconvolution- type image restoration, Technical Report TR04-15, CAAM Dept. Rice University, USA, 2004.

    Google Scholar 

  37. W. Hinterberger, M. Hintermüller, K. Kunisch, M. von Oehsen and O. Scherzer, Tube Methods for BV Regularization, J. Math. Imaging Vis., 19:219–235, 2003.

    Article  Google Scholar 

  38. K. Ito and K. Kunisch, An active set strategy based on the augmented Lagrangian formulation for image restoration, Math. Mod. Numer. Anal. (M2AN), 33 (1):1–21,1999.

    Article  MathSciNet  Google Scholar 

  39. T. Kärkkäinen and K. Majava, Nonmonotone and monotone active set methods for image restoration II. numerical results, J. Optim. Theory Appl., 106:81–105, 2000.

    Article  MathSciNet  Google Scholar 

  40. T. Kärkkäinen, K. Majava and M. M. Mäkelä, Comparison of formulations and solution methods for image restoration problems, Series B Report No. B 14/2000, Department of Mathematical Information Technology, University of Jyväskylä, Finland, 2000.

    Google Scholar 

  41. C. T. Kelley. Iterative Methods for Solving Linear and Nonlinear Equations. SIAM publications, USA, 1995.

    Google Scholar 

  42. R. Kimmel and I. Yavneh. An algebraic multigrid approach for image analysis, SIAM J. Sci. Comput., 24(4):1218–1231, 2003.

    Article  MathSciNet  Google Scholar 

  43. S. H. Lee and J. K. Seo, Noise removal with Gauss curvature driven diffusion, IEEE Trans. Image Proc., 14 (7):904–909, 2005.

    Article  MathSciNet  Google Scholar 

  44. Y. Y. Li and F. Santosa. A computational algorithm for minimizing total variation in image restoration, IEEE Trans. Image Proc., 5 (6):987–995, 1996.

    Article  Google Scholar 

  45. T. Lu, P. Neittaanmäki, and X.-C. Tai. A parallel splitting up method and its application to Navier-Stokes equations. Appl. Math. Lett., 4(2):25–29, 1991.

    Article  MathSciNet  Google Scholar 

  46. T. Lu, P. Neittaanmäki, and X.-C. Tai. A parallel splitting-up method for partial differential equations and its applications to Navier-Stokes equations. RAIRO Modél. Math. Anal. Numér., 26(6):673–708, 1992.

    Google Scholar 

  47. M. Lysaker, A. Lundervold and X.-C. Tai. Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time, IEEE Trans. Imag. Proc., 12 (12):1579–1590, 2003.

    Article  Google Scholar 

  48. F. Malgouyres. Minimizing the total variation under a general convex constraint for image restoration, IEEE Trans. Imag. Proc., 11 (12):1450–1456, 2002.

    Article  MathSciNet  Google Scholar 

  49. A. Marquina and S. Osher. Explicit Algorithms for a new time dependant model based on level set motion for nonlinear deblurring and noise removal, SIAM J. Sci. Comput., 22(2):387–405, 2000.

    Article  MathSciNet  Google Scholar 

  50. S. Nash. A multigrid approach to discretized optimisation problems, J. Opt. Methods Softw., 14:99–116, 2000.

    Article  MathSciNet  Google Scholar 

  51. M. K. Ng, L. Q. Qi, Y. F. Yang and Y. M. Huang, On semismooth Newton’s methods for total variation minimization, Technical Rep. 413, Dept of Math., Honk Kong Baptist Univ., China, 2005.

    Google Scholar 

  52. M. V. Oehsen, Multiscale Methods for Variational Image Denoising, Logos Verlag, Berlin, 2002.

    Google Scholar 

  53. S. Osher and R. Fedkiw. Level Set Methods and Dynamic Implicit Surfaces. Springer, 2003.

    Google Scholar 

  54. P. Perona and J. Malik, Scale space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intelligence, 12:629–639, 1990.

    Article  Google Scholar 

  55. E. Radmoser, O. Scherzer and J. Schöberl, A cascadic algorithm for bounded variation regularization, SFB-Report No. 00-23, Johannes Kepler University of Linz, Austria, 2000.

    Google Scholar 

  56. L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60:259–268, 1992.

    Article  Google Scholar 

  57. J. Savage and K. Chen, An improved and accelerated nonlinear multigrid method for total-variation denoising, Int. J. Comput. Math., 82 (8):1001–1015, 2005.

    Article  MathSciNet  Google Scholar 

  58. J. Savage and K. Chen. On multigrids for solving a class of improved total variation based PDE models, in this proceeding, 2006.

    Google Scholar 

  59. O. Scherzer. Taut-String Algorithm and Regularization Programs with G-Norm Data Fit, J. Math. Imaging and Vision, 23 (2):135–143, 2005

    Article  MathSciNet  Google Scholar 

  60. K. Stuben. An introduction to algebraic multigrid, in Appendix A of [67]. Also appeared as GMD report 70 from http://www.gmd.de and http://publica.fhg.de/english/index.htm, 2000.

  61. S. Ta’asan. Lecture note 4 of Von-Karman Institute Lectures, Belgium, http://www.math.cmu.edu/~shlomo/VKI-Lectures/lecture4, 1997.

  62. X.-C. Tai. Rate of convergence for some constraint decomposition methods for nonlinear variational inequalities. Numer. Math., 93:755–786, 2000.

    Article  MathSciNet  Google Scholar 

  63. X.-C. Tai and M. Espedal. Rate of convergence of some space decomposition methods for linear and nonlinear problems, SIAM. J. Numer. Anal., 35:1558–1570,1998.

    Article  MathSciNet  Google Scholar 

  64. Xue-Cheng Tai, Bjornove Heimsund and Jin Chao Xu. Rate of convergence for parallel subspace correction methods for nonlinear variational inequalities. In Thirteenth international domain decomposition conference, pages127–138. CIMNE, Barcelona, Spain,2002.Available online at: http://www.mi.uib.no/7%Etai/.

    Google Scholar 

  65. X.-C. Tai and P. Tseng. Convergence rate analysis of an asynchronous space decompostion method for convex minimization, Math. Comp., 71:1105–1135, 2001.

    Article  MathSciNet  Google Scholar 

  66. X.-C. Tai and J. C. Xu. Global and uniform convergence of subspace correction methods for some convex optimization problems, Math. Comp., 71:105–124, 2001.

    Article  MathSciNet  Google Scholar 

  67. U. Trottenberg, C. W. Oosterlee and A. Schuller. Multigrid, Academic Press, London, UK, 2000.

    Google Scholar 

  68. P. Tseng, Convergence of a block coordinate descent method for nondifferentiable minimization, J. Optim. Theory and Applics., 109 (3):475–494, 2001.

    Article  MathSciNet  Google Scholar 

  69. P. S. Vassilevski and J. G. Wade, A comparison of multilevel methods for total variation regularization, Elec. Trans. Numer. Anal., 6:255–270, 1997.

    MathSciNet  Google Scholar 

  70. C. R. Vogel. A multigrid method for TV-based image denoising, in Computation and Control IV, 20, Progress in Systems and Control Theory, eds. K. Bowers and J. Lund, Birkhauser, 1995.

    Google Scholar 

  71. C. R. Vogel. Negative results for multilevel preconditioners in image deblurring, in Scale-Space Theories In Computer Vision, eds. M. Nielson et al, pp.292-304. Springer, 1999.

    Google Scholar 

  72. C. R. Vogel. Computational methods for inverse problems. SIAM publications, USA, 2002.

    Google Scholar 

  73. C. R. Vogel and M. E. Oman. Iterative methods for total variation denoising, SIAM J. Sci. Statist. Comput., 17:227–238, 1996.

    Article  MathSciNet  Google Scholar 

  74. C. R. Vogel and M. E. Oman. Fast, robust total variation-based reconstruction of noisy, blurred images, IEEE Trans. Image Proc., 7:813–824, 1998.

    Article  MathSciNet  Google Scholar 

  75. J. Weickert, B. M. ter Haar Romeny and M. A. Viergever, Efficient and reliable schemes for nonlinear diffusion filtering, IEEE Trans. Image Proc., 7:398–410, 1998.

    Article  Google Scholar 

  76. J. C. Xu. Iteration methods by space decomposition and subspace correction, SIAM Rev., 4:581–613, 1992.

    Article  Google Scholar 

  77. W. T. Yin, D. Goldfarb and S. Osher, Image cartoon-texture decomposition and feature selection using the total variation regularized L1 functional, CAM report CAM05-47, 2005, UCLA, USA.

    Google Scholar 

  78. A. M. Yip and F. Park. Solution dynamics, causality, and critical behavior of the regularization parameter in total variation denoising problems, CAM report 03-59, UCLA, USA, 2003.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, CA 90095-1555, Los Angeles, USA

    Tony F. Chan

  2. Department of Mathematical Sciences, University of Liverpool, Peach Street, L69 7ZL, Liverpool, UK

    Dr. Ke Chen

  3. Department of Mathematics, University of Bergen, Johannes Brunsgate 12, N-5008, Bergen, Norway

    Xue-Cheng Tai

Authors
  1. Tony F. Chan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dr. Ke Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xue-Cheng Tai
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Bergen, Johannes Brunsgate 12, N-5008, Bergen, Norway

    Xue-Cheng Tai

  2. Dept. Applied Math., SINTEF ICT, Blindern, 124, N-0314, Oslo, Norway

    Knut-Andreas Lie

  3. Assistant Director for Math & Physical Sciences Directorate, The National Science Foundation, 4201, Arlington, Virginia, USA

    Tony F. Chan

  4. Department of Mathematics, University of California at Los Angeles, Portola Plaza, 520, CA 90095, Los Angeles, USA

    Stanley Osher

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Chan, T.F., Chen, K., Tai, XC. (2007). Nonlinear Multilevel Schemes for Solving the Total Variation Image Minimization Problem. In: Tai, XC., Lie, KA., Chan, T.F., Osher, S. (eds) Image Processing Based on Partial Differential Equations. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-33267-1_15

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-33267-1_15

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-33266-4

  • Online ISBN: 978-3-540-33267-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation