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 20s in the MATLAB computing environment.

© 2009 Optical Society of America

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  7. D. Strong and T. Chan, “Edge-preserving and scale-dependent properties of total variation regularization,” Inverse Probl. 19, 165-187 (2003).
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  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).
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  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).
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  10. T. Chan and C. Wong, “Total variation blind deconvolution,” IEEE Trans. Image Process. 7, 370-375 (1998).
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  11. T. Chan and J. Shen, “Variational restoration of nonflat image features: models and algorithms,” SIAM J. Appl. Math. 61, 1338-1361 (2001).
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  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).
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    [CrossRef]
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  27. P. Hansen, Rank-Deficient and Discrete Ill-Posed Problems (SIAM, 1998).
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  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]

2008 (5)

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]

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]

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]

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]

L. Reichel and H. Sadok, “A new L-curve for ill-posed problems,” J. Comput. Appl. Math. 219, 493-508 (2008).
[CrossRef]

2007 (3)

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]

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]

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]

2006 (4)

D. Krishnan, P. Lin, and X. Tai, “An efficient operator splitting method for noise removal in images,” Commun. Comput. Phys. 1, 847-858 (2006).

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]

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]

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]

2004 (3)

A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision 20, 89-97 (2004).
[CrossRef]

M. Lysaker and X. Tai, “Noise removal using smoothed normals and surface fitting,” IEEE Trans. Image Process. 13, 1345-1357 (2004).
[CrossRef] [PubMed]

M. Hintermüller and K. Kunisch, “Total bounded variation regularization as a bilaterally constrained optimization problem,” SIAM J. Appl. Math. 64, 1311-1333 (2004).
[CrossRef]

2003 (2)

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]

D. Strong and T. Chan, “Edge-preserving and scale-dependent properties of total variation regularization,” Inverse Probl. 19, 165-187 (2003).
[CrossRef]

2002 (1)

G. Bellettini, V. Caselles, and M. Novaga, “The total variation flow in RN,” J. Differ. Equations 184, 475-525 (2002).
[CrossRef]

2001 (1)

T. Chan and J. Shen, “Variational restoration of nonflat image features: models and algorithms,” SIAM J. Appl. Math. 61, 1338-1361 (2001).
[CrossRef]

2000 (1)

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]

1999 (1)

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]

1998 (2)

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]

T. Chan and C. Wong, “Total variation blind deconvolution,” IEEE Trans. Image Process. 7, 370-375 (1998).
[CrossRef]

1997 (2)

A. Chambolle and P.-L. Lions, “Image recovery via total variation minimization and related problems,” Numer. Math. 76, 167-188 (1997).
[CrossRef]

G. Golub and U. Matt, “Generalized cross-validation for large scale problems,” J. Comput. Graph. Stat. 6, 1-34 (1997).
[CrossRef]

1996 (3)

D. Dobson and F. Santosa, “Reovery of blocky images from noisy and blurred data,” SIAM J. Appl. Math. 56, 1181-1198 (1996).
[CrossRef]

C. Vogel and M. Oman, “Iterative method for total variation denoising,” SIAM J. Sci. Comput. (USA) 17, 227-238 (1996).
[CrossRef]

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]

1993 (1)

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]

1992 (1)

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259-268 (1992).
[CrossRef]

1979 (1)

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]

Babacan, S. D.

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]

Barlow, J.

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]

Bellettini, G.

G. Bellettini, V. Caselles, and M. Novaga, “The total variation flow in RN,” J. Differ. Equations 184, 475-525 (2002).
[CrossRef]

Blomgren, P.

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]

Caselles, V.

G. Bellettini, V. Caselles, and M. Novaga, “The total variation flow in RN,” J. Differ. Equations 184, 475-525 (2002).
[CrossRef]

Chambolle, A.

A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision 20, 89-97 (2004).
[CrossRef]

A. Chambolle and P.-L. Lions, “Image recovery via total variation minimization and related problems,” Numer. Math. 76, 167-188 (1997).
[CrossRef]

Chan, R.

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]

Chan, T.

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]

D. Strong and T. Chan, “Edge-preserving and scale-dependent properties of total variation regularization,” Inverse Probl. 19, 165-187 (2003).
[CrossRef]

T. Chan and J. Shen, “Variational restoration of nonflat image features: models and algorithms,” SIAM J. Appl. Math. 61, 1338-1361 (2001).
[CrossRef]

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]

T. Chan and C. Wong, “Total variation blind deconvolution,” IEEE Trans. Image Process. 7, 370-375 (1998).
[CrossRef]

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]

Chen, Y.

N. Paragios, Y. Chen, and O. Faugeras, Handbook of Mathematical Models in Computer Vision (Springer, 2006).
[CrossRef]

Dobson, D.

D. Dobson and F. Santosa, “Reovery of blocky images from noisy and blurred data,” SIAM J. Appl. Math. 56, 1181-1198 (1996).
[CrossRef]

Engl, H.

H. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, 1996).
[CrossRef]

Fatemi, E.

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259-268 (1992).
[CrossRef]

Faugeras, O.

N. Paragios, Y. Chen, and O. Faugeras, Handbook of Mathematical Models in Computer Vision (Springer, 2006).
[CrossRef]

Fu, H.

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]

Golub, G.

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]

G. Golub and U. Matt, “Generalized cross-validation for large scale problems,” J. Comput. Graph. Stat. 6, 1-34 (1997).
[CrossRef]

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]

G. Golub and G. Meurant, “Matrices, moments and quadrature,” in Numerical Analysis, D.Griffiths and G.Watson, eds. (Longman, 1994), pp. 105-156.

Gonzalez, R.

R. Gonzalez and R. Woods, Digital Image Processing (Addison Wesley, 1992).

Guo, X.

X. Guo, F. Li, and M. Ng, “A fast l1-TV algorithm for image restoration,” SIAM J. Sci. Comput. (USA) (to be published).

Hanke, M.

H. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, 1996).
[CrossRef]

Hansen, P.

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]

P. Hansen, J. Nagy, and D. O'leary, Deblurring Images: Matrices, Spectra, and Filtering (SIAM, 2006).

P. Hansen, Rank-Deficient and Discrete Ill-Posed Problems (SIAM, 1998).
[CrossRef]

Heath, M.

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]

Hintermüller, M.

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]

M. Hintermüller and K. Kunisch, “Total bounded variation regularization as a bilaterally constrained optimization problem,” SIAM J. Appl. Math. 64, 1311-1333 (2004).
[CrossRef]

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]

Huang, Y.

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]

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]

Ito, K.

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]

Katsaggelos, A.

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]

Katsaggelos, A. K.

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]

Krishnan, D.

D. Krishnan, P. Lin, and X. Tai, “An efficient operator splitting method for noise removal in images,” Commun. Comput. Phys. 1, 847-858 (2006).

Kunisch, K.

M. Hintermüller and K. Kunisch, “Total bounded variation regularization as a bilaterally constrained optimization problem,” SIAM J. Appl. Math. 64, 1311-1333 (2004).
[CrossRef]

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]

Li, F.

X. Guo, F. Li, and M. Ng, “A fast l1-TV algorithm for image restoration,” SIAM J. Sci. Comput. (USA) (to be published).

Li, Y.

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]

Lin, P.

D. Krishnan, P. Lin, and X. Tai, “An efficient operator splitting method for noise removal in images,” Commun. Comput. Phys. 1, 847-858 (2006).

Lin, Y.

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]

Lions, P.-L.

A. Chambolle and P.-L. Lions, “Image recovery via total variation minimization and related problems,” Numer. Math. 76, 167-188 (1997).
[CrossRef]

Lysaker, M.

M. Lysaker and X. Tai, “Noise removal using smoothed normals and surface fitting,” IEEE Trans. Image Process. 13, 1345-1357 (2004).
[CrossRef] [PubMed]

Mateos, J.

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]

Matt, U.

G. Golub and U. Matt, “Generalized cross-validation for large scale problems,” J. Comput. Graph. Stat. 6, 1-34 (1997).
[CrossRef]

Meurant, G.

G. Golub and G. Meurant, “Matrices, moments and quadrature,” in Numerical Analysis, D.Griffiths and G.Watson, eds. (Longman, 1994), pp. 105-156.

Molina, R.

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]

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]

Morel, J. M.

J. M. Morel and S. Solimini, Variational Methods for Image Segmentation (Birkhauser, 1995).

Mulet, P.

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]

Nagy, J.

P. Hansen, J. Nagy, and D. O'leary, Deblurring Images: Matrices, Spectra, and Filtering (SIAM, 2006).

Neubauer, A.

H. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, 1996).
[CrossRef]

Ng, M.

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]

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]

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]

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]

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]

M. Ng, Iterative Methods for Toeplitz Systems (Oxford Univ. Press, 2004).

X. Guo, F. Li, and M. Ng, “A fast l1-TV algorithm for image restoration,” SIAM J. Sci. Comput. (USA) (to be published).

Nikolova, M.

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]

Novaga, M.

G. Bellettini, V. Caselles, and M. Novaga, “The total variation flow in RN,” J. Differ. Equations 184, 475-525 (2002).
[CrossRef]

O'Leary, D.

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]

P. Hansen, J. Nagy, and D. O'leary, Deblurring Images: Matrices, Spectra, and Filtering (SIAM, 2006).

Oman, M.

C. Vogel and M. Oman, “Iterative method for total variation denoising,” SIAM J. Sci. Comput. (USA) 17, 227-238 (1996).
[CrossRef]

Osher, S.

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259-268 (1992).
[CrossRef]

Paragios, N.

N. Paragios, Y. Chen, and O. Faugeras, Handbook of Mathematical Models in Computer Vision (Springer, 2006).
[CrossRef]

Qi, L.

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]

Reichel, L.

L. Reichel and H. Sadok, “A new L-curve for ill-posed problems,” J. Comput. Appl. Math. 219, 493-508 (2008).
[CrossRef]

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]

Rudin, L.

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259-268 (1992).
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L. Reichel and H. Sadok, “A new L-curve for ill-posed problems,” J. Comput. Appl. Math. 219, 493-508 (2008).
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L. Reichel, H. Sadok, and A. Shyshkov, “Greedy Tikhonov regularization for large linear ill-posed problems,” Int. J. Comput. Math. 84, 1151-1166 (2007).
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Santosa, F.

D. Dobson and F. Santosa, “Reovery of blocky images from noisy and blurred data,” SIAM J. Appl. Math. 56, 1181-1198 (1996).
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Y. Li and F. Santosa, “A computational algorithm for minimizing total variation in image reconstruction,” IEEE Trans. Image Process. 5, 987-995 (1996).
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Shen, J.

T. Chan and J. Shen, “Variational restoration of nonflat image features: models and algorithms,” SIAM J. Appl. Math. 61, 1338-1361 (2001).
[CrossRef]

Shyshkov, A.

L. Reichel, H. Sadok, and A. Shyshkov, “Greedy Tikhonov regularization for large linear ill-posed problems,” Int. J. Comput. Math. 84, 1151-1166 (2007).
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Solimini, S.

J. M. Morel and S. Solimini, Variational Methods for Image Segmentation (Birkhauser, 1995).

Stadler, G.

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).
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D. Strong and T. Chan, “Edge-preserving and scale-dependent properties of total variation regularization,” Inverse Probl. 19, 165-187 (2003).
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D. Krishnan, P. Lin, and X. Tai, “An efficient operator splitting method for noise removal in images,” Commun. Comput. Phys. 1, 847-858 (2006).

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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).
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C. Vogel and M. Oman, “Iterative method for total variation denoising,” SIAM J. Sci. Comput. (USA) 17, 227-238 (1996).
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C. Vogel, Computational Methods for Inverse Problems (SIAM, 1998).

Wahba, G.

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]

Wang, Y.

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).
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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]

Wen, Y.

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]

Wohlberg, B.

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]

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T. Chan and C. Wong, “Total variation blind deconvolution,” IEEE Trans. Image Process. 7, 370-375 (1998).
[CrossRef]

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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).
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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]

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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]

Yau, A.

T. Chan, M. Ng, A. Yau, and A. Yip, “Superresolution image reconstruction using fast inpainting algorithms,” Appl. Comput. Harmonic Anal. 23, 3-24 (2007).
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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]

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]

Yip, A.

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]

Zhang, Y.

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]

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).
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Appl. Comput. Harmonic Anal. (1)

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]

Commun. Comput. Phys. (1)

D. Krishnan, P. Lin, and X. Tai, “An efficient operator splitting method for noise removal in images,” Commun. Comput. Phys. 1, 847-858 (2006).

IEEE Trans. Image Process. (6)

M. Lysaker and X. Tai, “Noise removal using smoothed normals and surface fitting,” IEEE Trans. Image Process. 13, 1345-1357 (2004).
[CrossRef] [PubMed]

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]

T. Chan and C. Wong, “Total variation blind deconvolution,” IEEE Trans. Image Process. 7, 370-375 (1998).
[CrossRef]

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]

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]

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]

Int. J. Comput. Math. (1)

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]

Inverse Probl. (1)

D. Strong and T. Chan, “Edge-preserving and scale-dependent properties of total variation regularization,” Inverse Probl. 19, 165-187 (2003).
[CrossRef]

J. Comput. Appl. Math. (1)

L. Reichel and H. Sadok, “A new L-curve for ill-posed problems,” J. Comput. Appl. Math. 219, 493-508 (2008).
[CrossRef]

J. Comput. Graph. Stat. (1)

G. Golub and U. Matt, “Generalized cross-validation for large scale problems,” J. Comput. Graph. Stat. 6, 1-34 (1997).
[CrossRef]

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G. Bellettini, V. Caselles, and M. Novaga, “The total variation flow in RN,” J. Differ. Equations 184, 475-525 (2002).
[CrossRef]

J. Math. Imaging Vision (2)

A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision 20, 89-97 (2004).
[CrossRef]

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]

Numer. Math. (1)

A. Chambolle and P.-L. Lions, “Image recovery via total variation minimization and related problems,” Numer. Math. 76, 167-188 (1997).
[CrossRef]

Physica D (1)

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259-268 (1992).
[CrossRef]

SIAM J. Appl. Math. (3)

D. Dobson and F. Santosa, “Reovery of blocky images from noisy and blurred data,” SIAM J. Appl. Math. 56, 1181-1198 (1996).
[CrossRef]

T. Chan and J. Shen, “Variational restoration of nonflat image features: models and algorithms,” SIAM J. Appl. Math. 61, 1338-1361 (2001).
[CrossRef]

M. Hintermüller and K. Kunisch, “Total bounded variation regularization as a bilaterally constrained optimization problem,” SIAM J. Appl. Math. 64, 1311-1333 (2004).
[CrossRef]

SIAM J. Imaging Sci. (2)

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]

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]

SIAM J. Multiscale Model. Simul. (1)

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]

SIAM J. Optim. (1)

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]

SIAM J. Sci. Comput. (1)

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]

SIAM J. Sci. Comput. (USA) (5)

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]

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]

C. Vogel and M. Oman, “Iterative method for total variation denoising,” SIAM J. Sci. Comput. (USA) 17, 227-238 (1996).
[CrossRef]

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]

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]

Technometrics (1)

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]

Other (11)

P. Hansen, J. Nagy, and D. O'leary, Deblurring Images: Matrices, Spectra, and Filtering (SIAM, 2006).

C. Vogel, Computational Methods for Inverse Problems (SIAM, 1998).

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]

M. Ng, Iterative Methods for Toeplitz Systems (Oxford Univ. Press, 2004).

X. Guo, F. Li, and M. Ng, “A fast l1-TV algorithm for image restoration,” SIAM J. Sci. Comput. (USA) (to be published).

R. Gonzalez and R. Woods, Digital Image Processing (Addison Wesley, 1992).

H. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, 1996).
[CrossRef]

P. Hansen, Rank-Deficient and Discrete Ill-Posed Problems (SIAM, 1998).
[CrossRef]

N. Paragios, Y. Chen, and O. Faugeras, Handbook of Mathematical Models in Computer Vision (Springer, 2006).
[CrossRef]

J. M. Morel and S. Solimini, Variational Methods for Image Segmentation (Birkhauser, 1995).

G. Golub and G. Meurant, “Matrices, moments and quadrature,” in Numerical Analysis, D.Griffiths and G.Watson, eds. (Longman, 1994), pp. 105-156.

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Figures (9)

Fig. 1
Fig. 1

Original images. (a) Cameraman. (b) Barbara.

Fig. 2
Fig. 2

Restored Cameraman images. (a) Gaussian PSF of ( hsize = 7 , σ = 5 ), BSNR = 40 dB . (b) Gaussian PSF of ( hsize = 14 , σ = 5 ), BSNR = 30 dB . (c) Image restored by Algorithm 1 for 2a. (d) Image restored by Algorithm 1 for 2b. (e) Image restored by the method in [26] for 2a. (f) Image restored by the method in [26] for 2b.

Fig. 3
Fig. 3

Restored Barbara images. (a) Out-of-focus PSF of radius = 3 , BSNR = 40 dB . (b) Out-of-focus PSF of radius = 7 , BSNR = 30 dB . (c) Image restored by Algorithm 1 to 3a. (d) Image restored by Algorithm 1 to 3b. (e) Image restored by the method in [26] to 3a. (f) Image restored by the method in [26] to 3b.

Fig. 4
Fig. 4

Restored CarNo images. (a) Motion PSF of length 9, BSNR = 40 dB . (b) Motion PSF of length 15, BSNR = 30 dB . (c) Image restored by Algorithm 1 to 4a. (d) Image restored by Algorithm 1 to 4b. (e) Image restored by the method in [21] to 4a. (f) Image restored by the method in [21] to 4b.

Fig. 5
Fig. 5

Restored images of different levels of salt-and-pepper noise. (a) 10%. (b) Image restored to 5a. (c) 30%. (d) Image restored to 5c. (e) 50%. (f) Image restored to 5e.

Fig. 6
Fig. 6

Images restored for other kinds of noise. (a) Uniform noise. (b) Laplace noise. (c) Partial Gaussian noise (50%). (d) Random-valued noise (20%).

Fig. 7
Fig. 7

Test of convergence of Algorithms 1 and 2. (a) Test of Algorithm 1 on the Cameraman image using the Gaussian PSF of ( hsize = 7 , σ = 5 , BSNR = 40 dB ). (b) Test of Algorithm 2 on the Barbara image using the Gaussian PSF of ( hsize = 7 , σ = 5 ), 40% of random-valued impulse noise.

Fig. 8
Fig. 8

Test of Algorithm 1 on the Gaussian PSF of ( hsize = 7 , σ = 5 ), BSNR = 40 dB . (a) SNRs of the restored images with respect to values of θ, initial β = 1 . (b) SNRs of the restored images with respect to initial values of β, θ = 2 .

Fig. 9
Fig. 9

Test of Algorithm 2 on the out-of-focus PSF of radius = 7 , BSNR = 30 dB . (a) SNRs of the restored images with respect to values of θ, initial β = 1 . SNRs of the restored images with respect to initial values of β, θ = 2 .

Tables (6)

Tables Icon

Table 1 Restoration Results Using the Restoration Method in [21]

Tables Icon

Table 2 Restoration Results Using Algorithm 1

Tables Icon

Table 3 Restoration Results Using Algorithm 1 with the Hutchinson Estimator

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Table 4 Parameters by Different Methods

Tables Icon

Table 5 Comparison between Algorithm 1 and ALG1 in [33]

Tables Icon

Table 6 Restored Images for Different Kinds of Noise Using Algorithm 2

Equations (31)

Equations on this page are rendered with MathJax. Learn more.

g = H f + n .
( f x 2 + f y 2 ) d x d y ,
f x 2 + f y 2 d x d y
min f 1 α H f g + f TV ,
( f ) j , k = ( ( f ) j , k x , ( f ) j , k y )
( f ) j , k x = { f j + 1 , k f j , k , if j < n 0 , if j = n } ,
( f ) j , k y = { f j , k + 1 f j , k , if k < n 0 , if k = n } ,
f TV 1 j , k n | ( f ) j , k | 2 = 1 j , k n | ( f ) j , k x | 2 + | ( f ) j , k y | 2 .
min f , u 1 α H f g 2 2 + β 2 u f 2 2 + u TV .
min f 1 α H f g 2 2 + f TV .
min f , u , v 1 α ( β 2 v ( H f g ) 2 2 + v 1 ) + ( β 2 u f 2 2 + u TV ) .
min f 1 α H f g 1 + f TV .
min f H f g 2 2 + γ f u 2 2 ,
( H T H + γ I ) f = H T g + γ u .
min u β ̂ 2 u f 2 2 + u TV .
min v β ̂ 2 v ( H f g ) 2 2 + v 1 .
v = sign ( H f g ) max { | H f g | 1 β ̂ , 0 } ,
min u β ̂ 2 u f 2 2 + u TV .
min f v ( H f g ) 2 2 + γ u f 2 2 ,
( H T H + γ I ) f = H T ( g + v ) + γ u .
min f H f z 2 2 + γ f u 2 2 ,
j = 1 , j k n 2 ( [ z H u ] j [ H f ] j ) 2 ,
1 n 2 k = 1 n 2 ( [ z H u ] k [ H f γ ( k ) ] k ) 2 .
τ ( γ ) 1 n 2 k = 1 n 2 ( [ z H u ] k [ H f γ ( k ) ] k ) 2 [ 1 m k , k ( γ ) 1 1 n 2 j = 1 n 2 m j , j ( γ ) ] ,
M ( γ ) = H ( H T H + γ I ) 1 H T .
τ ( γ ) = n 2 ( I M ( γ ) ) ( z H u ) 2 2 trace ( I M ( γ ) ) 2 .
τ ( γ ) = n 2 j = 1 n 2 { diag ( γ λ j 2 + γ ) Φ [ z H u ] } j 2 ( j = 1 n 2 γ λ j 2 + γ ) 2 ,
τ ̃ ( γ ) n 2 ( I M ( γ ) ( z H u ) ) 2 2 γ 2 ( x T ( H T H + γ I ) 1 x ) 2 .
SNR = 20 log 10 ( f mean ( f ) 2 u f 2 ) ,
BSNR = 20 log 10 ( g 2 n 2 ) ,
ISNR = 20 log 10 ( f g 2 f u 2 ) ,

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