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Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Editor: Stephen A. Burns
  • Vol. 26, Iss. 11 — Nov. 1, 2009
  • pp: 2311–2320

Selection of regularization parameter in total variation image restoration

Haiyong Liao, Fang Li, and Michael K. Ng  »View Author Affiliations


JOSA A, Vol. 26, Issue 11, pp. 2311-2320 (2009)
http://dx.doi.org/10.1364/JOSAA.26.002311


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Abstract

We consider and study total variation (TV) image restoration. In the literature there are several regularization parameter selection methods for Tikhonov regularization problems (e.g., the discrepancy principle and the generalized cross-validation method). However, to our knowledge, these selection methods have not been applied to TV regularization problems. The main aim of this paper is to develop a fast TV image restoration method with an automatic selection of the regularization parameter scheme to restore blurred and noisy images. The method exploits the generalized cross-validation (GCV) technique to determine inexpensively how much regularization to use in each restoration step. By updating the regularization parameter in each iteration, the restored image can be obtained. Our experimental results for testing different kinds of noise show that the visual quality and SNRs of images restored by the proposed method is promising. We also demonstrate that the method is efficient, as it can restore images of size 256 × 256 in 20 s in the MATLAB computing environment.

© 2009 Optical Society of America

OCIS Codes
(100.1830) Image processing : Deconvolution
(100.2000) Image processing : Digital image processing
(100.3020) Image processing : Image reconstruction-restoration

ToC Category:
Image Processing

History
Original Manuscript: April 8, 2009
Revised Manuscript: August 13, 2009
Manuscript Accepted: September 8, 2009
Published: October 9, 2009

Citation
Haiyong Liao, Fang Li, and Michael K. Ng, "Selection of regularization parameter in total variation image restoration," J. Opt. Soc. Am. A 26, 2311-2320 (2009)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-26-11-2311


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References

  1. J. M. Morel and S. Solimini, Variational Methods for Image Segmentation (Birkhauser, 1995).
  2. L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259-268 (1992). [CrossRef]
  3. C. Vogel and M. Oman, “Iterative method for total variation denoising,” SIAM J. Sci. Comput. (USA) 17, 227-238 (1996). [CrossRef]
  4. G. Bellettini, V. Caselles, and M. Novaga, “The total variation flow in RN,” J. Differ. Equations 184, 475-525 (2002). [CrossRef]
  5. A. Chambolle and P.-L. Lions, “Image recovery via total variation minimization and related problems,” Numer. Math. 76, 167-188 (1997). [CrossRef]
  6. D. Dobson and F. Santosa, “Reovery of blocky images from noisy and blurred data,” SIAM J. Appl. Math. 56, 1181-1198 (1996). [CrossRef]
  7. D. Strong and T. Chan, “Edge-preserving and scale-dependent properties of total variation regularization,” Inverse Probl. 19, 165-187 (2003). [CrossRef]
  8. P. Blomgren and T. Chan, “Color TV: total variation methods for restoration of vector-valued images,” IEEE Trans. Image Process. 7, 304-309 (1998). [CrossRef]
  9. T. Chan, M. Ng, A. Yau, and A. Yip, “Superresolution image reconstruction using fast inpainting algorithms,” Appl. Comput. Harmonic Anal. 23, 3-24 (2007). [CrossRef]
  10. T. Chan and C. Wong, “Total variation blind deconvolution,” IEEE Trans. Image Process. 7, 370-375 (1998). [CrossRef]
  11. T. Chan and J. Shen, “Variational restoration of nonflat image features: models and algorithms,” SIAM J. Appl. Math. 61, 1338-1361 (2001). [CrossRef]
  12. N. Paragios, Y. Chen, and O. Faugeras, Handbook of Mathematical Models in Computer Vision (Springer, 2006). [CrossRef]
  13. D. Krishnan, P. Lin, and X. Tai, “An efficient operator splitting method for noise removal in images,” Commun. Comput. Phys. 1, 847-858 (2006).
  14. M. Lysaker and X. Tai, “Noise removal using smoothed normals and surface fitting,” IEEE Trans. Image Process. 13, 1345-1357 (2004). [CrossRef] [PubMed]
  15. M. Hintermüller and K. Kunisch, “Total bounded variation regularization as a bilaterally constrained optimization problem,” SIAM J. Appl. Math. 64, 1311-1333 (2004). [CrossRef]
  16. M. Hintermüller and G. Stadler, “An infesible primal-dual algorithm for total bounded variation-based inf-convolution-type image restoration,” SIAM J. Sci. Comput. (USA) 28, 1-23 (2006). [CrossRef]
  17. T. Chan, G. Golub, and P. Mulet, “A nonlinear primal-dual method for total variation-based image restoration,” SIAM J. Sci. Comput. (USA) 20, 1964-1977 (1999). [CrossRef]
  18. M. Hintermüller, K. Ito, and K. Kunisch, “The primal-dual active set strategy as a semismooth Newton's method,” SIAM J. Optim. 13, 865-888 (2003). [CrossRef]
  19. Y. Li and F. Santosa, “A computational algorithm for minimizing total variation in image reconstruction,” IEEE Trans. Image Process. 5, 987-995 (1996). [CrossRef] [PubMed]
  20. H. Fu, M. Ng, M. Nikolova, and J. Barlow, “Efficient minimization methods of mixed l1-l1 and l2-l1 norms for image restoration,” SIAM J. Sci. Comput. (USA) 27, 1881-1902 (2006). [CrossRef]
  21. Y. Huang, M. Ng, and Y. Wen, “A fast total variation minimization method for image restoration,” SIAM J. Multiscale Model. Simul. 7, 774-795 (2008). [CrossRef]
  22. X. Guo, F. Li, and M. Ng, “A fast l1-TV algorithm for image restoration,” SIAM J. Sci. Comput. (USA) (to be published).
  23. Y. Wang, J. Yang, W. Yin, and Y. Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” SIAM J. Imaging Sci. 1, 248-272 (2008). [CrossRef]
  24. J. Yang, W. Yin, Y. Zhang, and Y. Wang, “A fast algorithm for edge-preserving variational multichannel image restoration,” SIAM J. Imaging Sci. 2, 569-592 (2008). [CrossRef]
  25. R. Gonzalez and R. Woods, Digital Image Processing (Addison Wesley, 1992).
  26. H. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, 1996). [CrossRef]
  27. P. Hansen, Rank-Deficient and Discrete Ill-Posed Problems (SIAM, 1998). [CrossRef]
  28. G. Golub, M. Heath, and G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215-223 (1979). [CrossRef]
  29. P. Hansen, J. Nagy, and D. O'leary, Deblurring Images: Matrices, Spectra, and Filtering (SIAM, 2006).
  30. P. Hansen and D. O'Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. (USA) 14, 1487-1503 (1993). [CrossRef]
  31. C. Vogel, Computational Methods for Inverse Problems (SIAM, 1998).
  32. Y. Lin and B. Wohlberg, “Application of the UPRE method to optimal parameter selection for large scale regularization problems,” in Proceedings of the IEEE Southwest Symposium on Image Analysis and Interpretation (IEEE, 2008), pp. 89-92. [CrossRef]
  33. S. D. Babacan, R. Molina, and A. K. Katsaggelos, “Parameter estimation in TV image restoration using variation distribution approximation,” IEEE Trans. Image Process. 17, 326-339 (2008). [CrossRef] [PubMed]
  34. R. Molina, J. Mateos, and A. Katsaggelos, “Blind deconvolution using a variational approach to parameter, image and blur estimation,” IEEE Trans. Image Process. 15, 3715-3727 (2006). [CrossRef] [PubMed]
  35. M. Ng, Iterative Methods for Toeplitz Systems (Oxford Univ. Press, 2004).
  36. A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision 20, 89-97 (2004). [CrossRef]
  37. M. Ng, L. Qi, Y. Yang, and Y. Huang, “On semismooth Newton's methods for total variation minimization,” J. Math. Imaging Vision 27, 265-276 (2007). [CrossRef]
  38. G. Golub and U. Matt, “Generalized cross-validation for large scale problems,” J. Comput. Graph. Stat. 6, 1-34 (1997). [CrossRef]
  39. M. Ng, R. Chan, and W. Tang, “A fast algorithm for deblurring models with Neumann boundary conditions,” SIAM J. Sci. Comput. 21, 851-866 (2000). [CrossRef]
  40. G. Golub and G. Meurant, “Matrices, moments and quadrature,” in Numerical Analysis, D.Griffiths and G.Watson, eds. (Longman, 1994), pp. 105-156.
  41. L. Reichel and H. Sadok, “A new L-curve for ill-posed problems,” J. Comput. Appl. Math. 219, 493-508 (2008). [CrossRef]
  42. L. Reichel, H. Sadok, and A. Shyshkov, “Greedy Tikhonov regularization for large linear ill-posed problems,” Int. J. Comput. Math. 84, 1151-1166 (2007). [CrossRef]

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