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On Multigrids for Solving a Class of Improved Total Variation Based Staircasing Reduction Models

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Image Processing Based on Partial Differential Equations

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

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References

  1. P. Blomgren. Total Variation Methods for Restoration of Vector Valued Images. PhD thesis, UCLA, 1998.

    Google Scholar 

  2. P. Blomgren, T. F. Chan, and P. Mulet. Extensions to total variation denoising. In SPIE 97, San Diego, 1997.

    Google Scholar 

  3. W. Briggs. A Multigrid Tutorial. SIAM, Philadelphia, 1987.

    Google Scholar 

  4. M. Burger, S. Osher, J. Xu, and G. Gilboa. Nonlinear inverse scale space methods for image restoration. Technical Report 05-34, UCLA, 2005.

    Google Scholar 

  5. J. Carter. Dual Methods for Total Variation Based Image Restoration. PhD thesis, UCLA, 2001.

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. R. Chan, T. F. Chan, and W. L. Wan. Multigrid for differential-convolution problems arising from image processing. In Proceedings of the Workshop on Scientific Computing, 1997.

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  10. T. F. Chan and K. Chen. An optimization-based multilevel algorithm for total variation image denoising. Multiscale Model. Simul., 5(2):615–645, 2006.

    Article  MathSciNet  Google Scholar 

  11. T. F. Chan and S. Esedoglu. Aspects of total variation regularized l1 function approximation. SIAM J. Appl. Math., 65:1817–1837, 2005 (see also CAM04-07).

    Article  MathSciNet  Google Scholar 

  12. T. F. Chan, S. Esedoglu, and F. Park. Image decomposition combining staircase reduction and texture extraction. Technical Report 05-18, UCLA, 2005.

    Google Scholar 

  13. T. F. Chan, S. Esedoglu, F. Park, and A. Yip. Recent developments in total variation image restoration. In Mathematical Models in Computer Vision: The Handbook. 2004.

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  15. T. F. Chan, A. Marquina, and P. Mulet. Second order differential functionals in total variation-based image restoration. Technical Report 98-35, UCLA, 1998.

    Google Scholar 

  16. T. F. Chan, H. M. Zhou, and R. Chan. Continuation method for total variation denoising problems. Technical Report 95-28, UCLA, 1995.

    Google Scholar 

  17. Q. 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 

  18. K. Chen. Matrix Preconditioning Techniques and Applications, volume 19 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, UK, 2005.

    Google Scholar 

  19. K. Chen and X.-C. Tai. A nonlinear multigrid method for curvature equations related to total variation minimization. Technical Report 05-26, UCLA, 2005.

    Google Scholar 

  20. Y. Chen, S. Levine, and M. Rao. Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math., 66(4):1383–1406, 2006.

    Article  MathSciNet  Google Scholar 

  21. Y. Chen, S. Levine, and J. Stanich. Image restoration via nonstandard diffusion. Technical Report 04-01, Duquesne Univ. Dept. of Math and Comp. Sci., 2004.

    Google Scholar 

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

    MathSciNet  Google Scholar 

  23. V.-E. Henson. Multigrid methods for nonlinear problems: An overview. Technical report, Center for Applied Scientific Computing Lawrence Livermore Laboratory.

    Google Scholar 

  24. K. Ito and K. Kunisch. An active-set strategy based on the augmented lagrangian formulation for image restoration. M2AN Math. Model. Numer. Anal., 33:1–21, 1999.

    Article  MathSciNet  Google Scholar 

  25. K. Ito and K. Kunisch. BV-type regularization methods for convoluted objects with edge flat and grey scales. Inverse Probl., 16:909–928, 2000.

    Article  MathSciNet  Google Scholar 

  26. K. Joo and S. Kim. Pde-based image restoration, i: Anti-staircasing and anti-diffusion. Technical report, University of Kentucky, 2003.

    Google Scholar 

  27. T. Karkkainen and K. Majava. Nonmonotone and monotone active-set methods for image restoration. J. Optim. Theory Appl., 106(1):61–105, 2000.

    Article  MathSciNet  Google Scholar 

  28. T. Karkkainen and K. Majava. Semi-adaptive optimization methodology for image denoising. IEE Proc. Vis. Image Signal Process., 152(1):553–560, 2005.

    Article  Google Scholar 

  29. T. Karkkainen, K. Majava, and M. Makela. Comparisons of formulations and solution methods for image restoration problems. Technical Report B 14/2000, Department of Mathematical Information Technology University of Jyvaskyla, 2000.

    Google Scholar 

  30. S-H. Lee and J. K. Seo. Noise removal with gauss curvature driven diffusion. IEEE Trans. Img. Process., 2005.

    Google Scholar 

  31. S. Levine, M. Ramsey, T. Misner, and S. Schwab. An adaptive model for image decomposition. Technical Report 05-01, Duquesne Univ. Dept. of Math and Comp. Sci., 2005.

    Google Scholar 

  32. 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. Image Process., 12, 2003.

    Google Scholar 

  33. M. Lysaker and X.-C. Tai. Interactive image restoration combining total variation minimization and a second order functional. Int. J. Comp Vis.

    Google Scholar 

  34. 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., 22:387–405, 2000.

    Article  MathSciNet  Google Scholar 

  35. S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin. An iterative regularization method for total variation based image restoration. Multiscale Model. and Simul., 4:460–489, 2005.

    Article  MathSciNet  Google Scholar 

  36. S. Osher, A. Sole, and L. Vese. Image decomposition and restoration using total variation minimization and the h−1 norm. Multiscale Model. and Simul., 1:349–370, 2003.

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  39. U. Trottenberg, C. Oosterlee, and A. Schuller. Multigrid. Academic Press, London, 2001.

    Google Scholar 

  40. L. Vese and S. Osher. Modelling textures and with total variation minimization and oscillating patterns in image processing. Technical Report 02-19, UCLA, 2002.

    Google Scholar 

  41. C. Vogel. A multigrid method for total variation based image denoising. In Computation and Control IV. Birkhauser, 1995.

    Google Scholar 

  42. C. Vogel. Computational Methods for Inverse Problems. SIAM, Philadelphia, 2002.

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  44. T. Washio and C. Oosterlee. Krylov subspace acceleration for nonlinear multi-grid schemes. Electron. Trans. Numer. Anal., 6:271–290, 1997.

    MathSciNet  Google Scholar 

  45. P. Wesseling. An Introduction to Multigrid Methods. Wiley, Chichester, 1992.

    Google Scholar 

  46. W. Yin, D. Goldfarb, and S. Osher. Image cartoon-texture decomposition and feature selection using the total variation regularized l1 functional. Technical Report 05-47, UCLA, 2005.

    Google Scholar 

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Author information

Authors and Affiliations

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

    Joseph Savage

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

    Dr. Ke Chen

Authors
  1. Joseph Savage
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  2. Dr. Ke Chen
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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

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© 2007 Springer-Verlag Berlin Heidelberg

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Savage, J., Chen, K. (2007). On Multigrids for Solving a Class of Improved Total Variation Based Staircasing Reduction Models. 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_5

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  • DOI: https://doi.org/10.1007/978-3-540-33267-1_5

  • 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)

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