Abstract

The deconvolution of blurred and noisy satellite images is an ill-posed inverse problem, which can be regularized under the Bayesian framework by introducing an appropriate image prior. In this paper, we derive a new image prior based on the state-of-the-art nonlocal means (NLM) denoising approach under Markov random field theory. Inheriting from the NLM, the prior exploits the intrinsic high redundancy of satellite images and is able to encode the image’s nonsmooth information. Using this prior, we propose an inhomogeneous deconvolution technique for satellite images, termed nonlocal means-based deconvolution (NLM-D). Moreover, in order to make our NLM-D unsupervised, we apply the L-curve approach to estimate the optimal regularization parameter. Experimentally, NLM-D demonstrates its capacity to preserve the image’s nonsmooth structures (such as edges and textures) and outperforms the existing total variation-based and wavelet-based deconvolution methods in terms of both visual quality and signal-to-noise ratio performance.

© 2010 Optical Society of America

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  1. J. Idier, “Convex half-quadratic criteria and interacting auxiliary variables for image restoration,” IEEE Trans. Image Process. 10, 1001–1009 (2001).
    [CrossRef]
  2. S. Teboul, L. Blanc-Feraud, G. Aubert, and M. Barlaud, “Variational approach for edge-preserving regularization using coupled PDE’s,” IEEE Trans. Image Process. 7, 387–397 (1998).
    [CrossRef]
  3. L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D (Amsterdam) 60, 259–268 (1992).
    [CrossRef]
  4. J. P. Oliveira, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Total variation-based image deconvolution: A majorization-minimization approach,” Signal Process. 89, 1683–1693(2009).
    [CrossRef]
  5. A. Buades, B. Coll, and J. M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Simul. 4, 490–530 (2005).
    [CrossRef]
  6. G. Gilboa and S. Osher, “Nonlocal linear image regularization and supervised segmentation,” Multiscale Model. Simul. 6, 595–630 (2007).
    [CrossRef]
  7. M. Protter, M. Elad, H. Takeda, and P. Milanfar, “Generalizing the non-local-means to super-resolution reconstruction,” IEEE Trans. Image Process. 18, 36–51 (2009).
    [CrossRef]
  8. A. Buades, B. Coll, and J. Morel, Image Enhancement by Non-Local Reverse Heat Equation (CMLA, 2006).
  9. Y. Lou, X. Zhang, S. Osher, and A. Bertozzi, “Image recovery via nonlocal operators,” J. Sci. Comput. 42, 185–197 (2009).
    [CrossRef]
  10. W. Zhang, M. Zhao, and Z. Wang, “Adaptive wavelet-based deconvolution method for remote sensing imaging,” Appl. Opt. 48, 4785–4793 (2009).
    [CrossRef] [PubMed]
  11. M. Mignotte, “A non-local regularization strategy for image deconvolution,” Pattern Recogn. Lett. 29, 2206–2212(2008).
    [CrossRef]
  12. X. Zhang, M. Burger, X. Bresson, and S. Osher, “Bregmanized nonlocal regularization for deconvolution and sparse reconstruction,” SIAM J. Imaging Sci 3, 253–276 (2010).
    [CrossRef]
  13. R. Pan and S. J. Reeves, “Efficient Huber-Markov edge-preserving image restoration,” IEEE Trans. Image Process. 15, 3728–3735 (2006).
    [CrossRef] [PubMed]
  14. S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 721–741 (1984).
    [CrossRef]
  15. C. Bouman and K. Sauer, “A generalized Gaussian image model for edge-preserving MAP estimation,” IEEE Trans. Image Process. 2, 296–310 (1993).
    [CrossRef] [PubMed]
  16. G. H. Golub, M. Heath, and G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
    [CrossRef]
  17. G. Golub and U. Matt, “Generalized cross-validation for large scale problems,” J. Comput. Graph. Stat. 6, 1–34 (1997).
    [CrossRef]
  18. H. Liao, F. Li, and M. K. Ng, “Selection of regularization parameter in total variation image restoration,” J. Opt. Soc. Am. A 26, 2311–2320 (2009).
    [CrossRef]
  19. A. Jalobeanu, L. Blanc-FeHraud, and J. Zerubia, “Hyperparameter estimation for satellite image restoration using a MCMC maximum-likelihood method,” Pattern Recogn. 35, 341–352 (2002).
    [CrossRef]
  20. A. Jalobeanu, L. Blanc-Féraud, and J. Zerubia, “An adaptive Gaussian model for satellite image deblurring,” IEEE Trans. Image Process. 13, 613–621 (2004).
    [CrossRef] [PubMed]
  21. R. Molina, A. K. Katsaggelos, and J. Mateos, “Bayesian and regularization methods for hyperparameter estimation in image restoration,” IEEE Trans. Image Process. 8, 231–246(1999).
    [CrossRef]
  22. S. D. Babacan, R. Molina, and A. K. Katsaggelos, “Parameter estimation in TV image restoration using variational distribution approximation,” IEEE Trans. Image Process. 17, 326–339 (2008).
    [CrossRef] [PubMed]
  23. P. C. Hansen and D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1503 (1993).
    [CrossRef]
  24. G. Rodriguez and D. Theis, “An algorithm for estimating the optimal regularization parameter by the L-curve,” Rendiconti di matematica 25, 69–84 (2005).
  25. P. C. Hansen, T. K. Jensen, and G. Rodriguez, “An adaptive pruning algorithm for the discrete L-curve criterion,” J. Comput. Appl. Math. 198, 483–492 (2007).
    [CrossRef]

2010 (1)

X. Zhang, M. Burger, X. Bresson, and S. Osher, “Bregmanized nonlocal regularization for deconvolution and sparse reconstruction,” SIAM J. Imaging Sci 3, 253–276 (2010).
[CrossRef]

2009 (5)

H. Liao, F. Li, and M. K. Ng, “Selection of regularization parameter in total variation image restoration,” J. Opt. Soc. Am. A 26, 2311–2320 (2009).
[CrossRef]

J. P. Oliveira, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Total variation-based image deconvolution: A majorization-minimization approach,” Signal Process. 89, 1683–1693(2009).
[CrossRef]

M. Protter, M. Elad, H. Takeda, and P. Milanfar, “Generalizing the non-local-means to super-resolution reconstruction,” IEEE Trans. Image Process. 18, 36–51 (2009).
[CrossRef]

Y. Lou, X. Zhang, S. Osher, and A. Bertozzi, “Image recovery via nonlocal operators,” J. Sci. Comput. 42, 185–197 (2009).
[CrossRef]

W. Zhang, M. Zhao, and Z. Wang, “Adaptive wavelet-based deconvolution method for remote sensing imaging,” Appl. Opt. 48, 4785–4793 (2009).
[CrossRef] [PubMed]

2008 (2)

M. Mignotte, “A non-local regularization strategy for image deconvolution,” Pattern Recogn. Lett. 29, 2206–2212(2008).
[CrossRef]

S. D. Babacan, R. Molina, and A. K. Katsaggelos, “Parameter estimation in TV image restoration using variational distribution approximation,” IEEE Trans. Image Process. 17, 326–339 (2008).
[CrossRef] [PubMed]

2007 (2)

P. C. Hansen, T. K. Jensen, and G. Rodriguez, “An adaptive pruning algorithm for the discrete L-curve criterion,” J. Comput. Appl. Math. 198, 483–492 (2007).
[CrossRef]

G. Gilboa and S. Osher, “Nonlocal linear image regularization and supervised segmentation,” Multiscale Model. Simul. 6, 595–630 (2007).
[CrossRef]

2006 (1)

R. Pan and S. J. Reeves, “Efficient Huber-Markov edge-preserving image restoration,” IEEE Trans. Image Process. 15, 3728–3735 (2006).
[CrossRef] [PubMed]

2005 (2)

A. Buades, B. Coll, and J. M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Simul. 4, 490–530 (2005).
[CrossRef]

G. Rodriguez and D. Theis, “An algorithm for estimating the optimal regularization parameter by the L-curve,” Rendiconti di matematica 25, 69–84 (2005).

2004 (1)

A. Jalobeanu, L. Blanc-Féraud, and J. Zerubia, “An adaptive Gaussian model for satellite image deblurring,” IEEE Trans. Image Process. 13, 613–621 (2004).
[CrossRef] [PubMed]

2002 (1)

A. Jalobeanu, L. Blanc-FeHraud, and J. Zerubia, “Hyperparameter estimation for satellite image restoration using a MCMC maximum-likelihood method,” Pattern Recogn. 35, 341–352 (2002).
[CrossRef]

2001 (1)

J. Idier, “Convex half-quadratic criteria and interacting auxiliary variables for image restoration,” IEEE Trans. Image Process. 10, 1001–1009 (2001).
[CrossRef]

1999 (1)

R. Molina, A. K. Katsaggelos, and J. Mateos, “Bayesian and regularization methods for hyperparameter estimation in image restoration,” IEEE Trans. Image Process. 8, 231–246(1999).
[CrossRef]

1998 (1)

S. Teboul, L. Blanc-Feraud, G. Aubert, and M. Barlaud, “Variational approach for edge-preserving regularization using coupled PDE’s,” IEEE Trans. Image Process. 7, 387–397 (1998).
[CrossRef]

1997 (1)

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

1993 (2)

C. Bouman and K. Sauer, “A generalized Gaussian image model for edge-preserving MAP estimation,” IEEE Trans. Image Process. 2, 296–310 (1993).
[CrossRef] [PubMed]

P. C. Hansen and D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1503 (1993).
[CrossRef]

1992 (1)

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

1984 (1)

S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 721–741 (1984).
[CrossRef]

1979 (1)

G. H. Golub, M. Heath, and G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[CrossRef]

Aubert, G.

S. Teboul, L. Blanc-Feraud, G. Aubert, and M. Barlaud, “Variational approach for edge-preserving regularization using coupled PDE’s,” IEEE Trans. Image Process. 7, 387–397 (1998).
[CrossRef]

Babacan, S. D.

S. D. Babacan, R. Molina, and A. K. Katsaggelos, “Parameter estimation in TV image restoration using variational distribution approximation,” IEEE Trans. Image Process. 17, 326–339 (2008).
[CrossRef] [PubMed]

Barlaud, M.

S. Teboul, L. Blanc-Feraud, G. Aubert, and M. Barlaud, “Variational approach for edge-preserving regularization using coupled PDE’s,” IEEE Trans. Image Process. 7, 387–397 (1998).
[CrossRef]

Bertozzi, A.

Y. Lou, X. Zhang, S. Osher, and A. Bertozzi, “Image recovery via nonlocal operators,” J. Sci. Comput. 42, 185–197 (2009).
[CrossRef]

Bioucas-Dias, J. M.

J. P. Oliveira, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Total variation-based image deconvolution: A majorization-minimization approach,” Signal Process. 89, 1683–1693(2009).
[CrossRef]

Blanc-FeHraud, L.

A. Jalobeanu, L. Blanc-FeHraud, and J. Zerubia, “Hyperparameter estimation for satellite image restoration using a MCMC maximum-likelihood method,” Pattern Recogn. 35, 341–352 (2002).
[CrossRef]

Blanc-Feraud, L.

S. Teboul, L. Blanc-Feraud, G. Aubert, and M. Barlaud, “Variational approach for edge-preserving regularization using coupled PDE’s,” IEEE Trans. Image Process. 7, 387–397 (1998).
[CrossRef]

Blanc-Féraud, L.

A. Jalobeanu, L. Blanc-Féraud, and J. Zerubia, “An adaptive Gaussian model for satellite image deblurring,” IEEE Trans. Image Process. 13, 613–621 (2004).
[CrossRef] [PubMed]

Bouman, C.

C. Bouman and K. Sauer, “A generalized Gaussian image model for edge-preserving MAP estimation,” IEEE Trans. Image Process. 2, 296–310 (1993).
[CrossRef] [PubMed]

Bresson, X.

X. Zhang, M. Burger, X. Bresson, and S. Osher, “Bregmanized nonlocal regularization for deconvolution and sparse reconstruction,” SIAM J. Imaging Sci 3, 253–276 (2010).
[CrossRef]

Buades, A.

A. Buades, B. Coll, and J. M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Simul. 4, 490–530 (2005).
[CrossRef]

A. Buades, B. Coll, and J. Morel, Image Enhancement by Non-Local Reverse Heat Equation (CMLA, 2006).

Burger, M.

X. Zhang, M. Burger, X. Bresson, and S. Osher, “Bregmanized nonlocal regularization for deconvolution and sparse reconstruction,” SIAM J. Imaging Sci 3, 253–276 (2010).
[CrossRef]

Coll, B.

A. Buades, B. Coll, and J. M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Simul. 4, 490–530 (2005).
[CrossRef]

A. Buades, B. Coll, and J. Morel, Image Enhancement by Non-Local Reverse Heat Equation (CMLA, 2006).

Elad, M.

M. Protter, M. Elad, H. Takeda, and P. Milanfar, “Generalizing the non-local-means to super-resolution reconstruction,” IEEE Trans. Image Process. 18, 36–51 (2009).
[CrossRef]

Fatemi, E.

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

Figueiredo, M. A. T.

J. P. Oliveira, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Total variation-based image deconvolution: A majorization-minimization approach,” Signal Process. 89, 1683–1693(2009).
[CrossRef]

Geman, D.

S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 721–741 (1984).
[CrossRef]

Geman, S.

S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 721–741 (1984).
[CrossRef]

Gilboa, G.

G. Gilboa and S. Osher, “Nonlocal linear image regularization and supervised segmentation,” Multiscale Model. Simul. 6, 595–630 (2007).
[CrossRef]

Golub, G.

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

Golub, G. H.

G. H. Golub, M. Heath, and G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[CrossRef]

Hansen, P. C.

P. C. Hansen, T. K. Jensen, and G. Rodriguez, “An adaptive pruning algorithm for the discrete L-curve criterion,” J. Comput. Appl. Math. 198, 483–492 (2007).
[CrossRef]

P. C. Hansen and D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1503 (1993).
[CrossRef]

Heath, M.

G. H. Golub, M. Heath, and G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[CrossRef]

Idier, J.

J. Idier, “Convex half-quadratic criteria and interacting auxiliary variables for image restoration,” IEEE Trans. Image Process. 10, 1001–1009 (2001).
[CrossRef]

Jalobeanu, A.

A. Jalobeanu, L. Blanc-Féraud, and J. Zerubia, “An adaptive Gaussian model for satellite image deblurring,” IEEE Trans. Image Process. 13, 613–621 (2004).
[CrossRef] [PubMed]

A. Jalobeanu, L. Blanc-FeHraud, and J. Zerubia, “Hyperparameter estimation for satellite image restoration using a MCMC maximum-likelihood method,” Pattern Recogn. 35, 341–352 (2002).
[CrossRef]

Jensen, T. K.

P. C. Hansen, T. K. Jensen, and G. Rodriguez, “An adaptive pruning algorithm for the discrete L-curve criterion,” J. Comput. Appl. Math. 198, 483–492 (2007).
[CrossRef]

Katsaggelos, A. K.

S. D. Babacan, R. Molina, and A. K. Katsaggelos, “Parameter estimation in TV image restoration using variational distribution approximation,” IEEE Trans. Image Process. 17, 326–339 (2008).
[CrossRef] [PubMed]

R. Molina, A. K. Katsaggelos, and J. Mateos, “Bayesian and regularization methods for hyperparameter estimation in image restoration,” IEEE Trans. Image Process. 8, 231–246(1999).
[CrossRef]

Li, F.

Liao, H.

Lou, Y.

Y. Lou, X. Zhang, S. Osher, and A. Bertozzi, “Image recovery via nonlocal operators,” J. Sci. Comput. 42, 185–197 (2009).
[CrossRef]

Mateos, J.

R. Molina, A. K. Katsaggelos, and J. Mateos, “Bayesian and regularization methods for hyperparameter estimation in image restoration,” IEEE Trans. Image Process. 8, 231–246(1999).
[CrossRef]

Matt, U.

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

Mignotte, M.

M. Mignotte, “A non-local regularization strategy for image deconvolution,” Pattern Recogn. Lett. 29, 2206–2212(2008).
[CrossRef]

Milanfar, P.

M. Protter, M. Elad, H. Takeda, and P. Milanfar, “Generalizing the non-local-means to super-resolution reconstruction,” IEEE Trans. Image Process. 18, 36–51 (2009).
[CrossRef]

Molina, R.

S. D. Babacan, R. Molina, and A. K. Katsaggelos, “Parameter estimation in TV image restoration using variational distribution approximation,” IEEE Trans. Image Process. 17, 326–339 (2008).
[CrossRef] [PubMed]

R. Molina, A. K. Katsaggelos, and J. Mateos, “Bayesian and regularization methods for hyperparameter estimation in image restoration,” IEEE Trans. Image Process. 8, 231–246(1999).
[CrossRef]

Morel, J.

A. Buades, B. Coll, and J. Morel, Image Enhancement by Non-Local Reverse Heat Equation (CMLA, 2006).

Morel, J. M.

A. Buades, B. Coll, and J. M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Simul. 4, 490–530 (2005).
[CrossRef]

Ng, M. K.

O’Leary, D. P.

P. C. Hansen and D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1503 (1993).
[CrossRef]

Oliveira, J. P.

J. P. Oliveira, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Total variation-based image deconvolution: A majorization-minimization approach,” Signal Process. 89, 1683–1693(2009).
[CrossRef]

Osher, S.

X. Zhang, M. Burger, X. Bresson, and S. Osher, “Bregmanized nonlocal regularization for deconvolution and sparse reconstruction,” SIAM J. Imaging Sci 3, 253–276 (2010).
[CrossRef]

Y. Lou, X. Zhang, S. Osher, and A. Bertozzi, “Image recovery via nonlocal operators,” J. Sci. Comput. 42, 185–197 (2009).
[CrossRef]

G. Gilboa and S. Osher, “Nonlocal linear image regularization and supervised segmentation,” Multiscale Model. Simul. 6, 595–630 (2007).
[CrossRef]

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

Pan, R.

R. Pan and S. J. Reeves, “Efficient Huber-Markov edge-preserving image restoration,” IEEE Trans. Image Process. 15, 3728–3735 (2006).
[CrossRef] [PubMed]

Protter, M.

M. Protter, M. Elad, H. Takeda, and P. Milanfar, “Generalizing the non-local-means to super-resolution reconstruction,” IEEE Trans. Image Process. 18, 36–51 (2009).
[CrossRef]

Reeves, S. J.

R. Pan and S. J. Reeves, “Efficient Huber-Markov edge-preserving image restoration,” IEEE Trans. Image Process. 15, 3728–3735 (2006).
[CrossRef] [PubMed]

Rodriguez, G.

P. C. Hansen, T. K. Jensen, and G. Rodriguez, “An adaptive pruning algorithm for the discrete L-curve criterion,” J. Comput. Appl. Math. 198, 483–492 (2007).
[CrossRef]

G. Rodriguez and D. Theis, “An algorithm for estimating the optimal regularization parameter by the L-curve,” Rendiconti di matematica 25, 69–84 (2005).

Rudin, L. I.

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

Sauer, K.

C. Bouman and K. Sauer, “A generalized Gaussian image model for edge-preserving MAP estimation,” IEEE Trans. Image Process. 2, 296–310 (1993).
[CrossRef] [PubMed]

Takeda, H.

M. Protter, M. Elad, H. Takeda, and P. Milanfar, “Generalizing the non-local-means to super-resolution reconstruction,” IEEE Trans. Image Process. 18, 36–51 (2009).
[CrossRef]

Teboul, S.

S. Teboul, L. Blanc-Feraud, G. Aubert, and M. Barlaud, “Variational approach for edge-preserving regularization using coupled PDE’s,” IEEE Trans. Image Process. 7, 387–397 (1998).
[CrossRef]

Theis, D.

G. Rodriguez and D. Theis, “An algorithm for estimating the optimal regularization parameter by the L-curve,” Rendiconti di matematica 25, 69–84 (2005).

Wahba, G.

G. H. 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, Z.

Zerubia, J.

A. Jalobeanu, L. Blanc-Féraud, and J. Zerubia, “An adaptive Gaussian model for satellite image deblurring,” IEEE Trans. Image Process. 13, 613–621 (2004).
[CrossRef] [PubMed]

A. Jalobeanu, L. Blanc-FeHraud, and J. Zerubia, “Hyperparameter estimation for satellite image restoration using a MCMC maximum-likelihood method,” Pattern Recogn. 35, 341–352 (2002).
[CrossRef]

Zhang, W.

Zhang, X.

X. Zhang, M. Burger, X. Bresson, and S. Osher, “Bregmanized nonlocal regularization for deconvolution and sparse reconstruction,” SIAM J. Imaging Sci 3, 253–276 (2010).
[CrossRef]

Y. Lou, X. Zhang, S. Osher, and A. Bertozzi, “Image recovery via nonlocal operators,” J. Sci. Comput. 42, 185–197 (2009).
[CrossRef]

Zhao, M.

Appl. Opt. (1)

IEEE Trans. Image Process. (8)

A. Jalobeanu, L. Blanc-Féraud, and J. Zerubia, “An adaptive Gaussian model for satellite image deblurring,” IEEE Trans. Image Process. 13, 613–621 (2004).
[CrossRef] [PubMed]

R. Molina, A. K. Katsaggelos, and J. Mateos, “Bayesian and regularization methods for hyperparameter estimation in image restoration,” IEEE Trans. Image Process. 8, 231–246(1999).
[CrossRef]

S. D. Babacan, R. Molina, and A. K. Katsaggelos, “Parameter estimation in TV image restoration using variational distribution approximation,” IEEE Trans. Image Process. 17, 326–339 (2008).
[CrossRef] [PubMed]

J. Idier, “Convex half-quadratic criteria and interacting auxiliary variables for image restoration,” IEEE Trans. Image Process. 10, 1001–1009 (2001).
[CrossRef]

S. Teboul, L. Blanc-Feraud, G. Aubert, and M. Barlaud, “Variational approach for edge-preserving regularization using coupled PDE’s,” IEEE Trans. Image Process. 7, 387–397 (1998).
[CrossRef]

M. Protter, M. Elad, H. Takeda, and P. Milanfar, “Generalizing the non-local-means to super-resolution reconstruction,” IEEE Trans. Image Process. 18, 36–51 (2009).
[CrossRef]

R. Pan and S. J. Reeves, “Efficient Huber-Markov edge-preserving image restoration,” IEEE Trans. Image Process. 15, 3728–3735 (2006).
[CrossRef] [PubMed]

C. Bouman and K. Sauer, “A generalized Gaussian image model for edge-preserving MAP estimation,” IEEE Trans. Image Process. 2, 296–310 (1993).
[CrossRef] [PubMed]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 721–741 (1984).
[CrossRef]

J. Comput. Appl. Math. (1)

P. C. Hansen, T. K. Jensen, and G. Rodriguez, “An adaptive pruning algorithm for the discrete L-curve criterion,” J. Comput. Appl. Math. 198, 483–492 (2007).
[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]

J. Opt. Soc. Am. A (1)

J. Sci. Comput. (1)

Y. Lou, X. Zhang, S. Osher, and A. Bertozzi, “Image recovery via nonlocal operators,” J. Sci. Comput. 42, 185–197 (2009).
[CrossRef]

Multiscale Model. Simul. (2)

A. Buades, B. Coll, and J. M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Simul. 4, 490–530 (2005).
[CrossRef]

G. Gilboa and S. Osher, “Nonlocal linear image regularization and supervised segmentation,” Multiscale Model. Simul. 6, 595–630 (2007).
[CrossRef]

Pattern Recogn. (1)

A. Jalobeanu, L. Blanc-FeHraud, and J. Zerubia, “Hyperparameter estimation for satellite image restoration using a MCMC maximum-likelihood method,” Pattern Recogn. 35, 341–352 (2002).
[CrossRef]

Pattern Recogn. Lett. (1)

M. Mignotte, “A non-local regularization strategy for image deconvolution,” Pattern Recogn. Lett. 29, 2206–2212(2008).
[CrossRef]

Physica D (Amsterdam) (1)

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

Rendiconti di matematica (1)

G. Rodriguez and D. Theis, “An algorithm for estimating the optimal regularization parameter by the L-curve,” Rendiconti di matematica 25, 69–84 (2005).

SIAM J. Imaging Sci (1)

X. Zhang, M. Burger, X. Bresson, and S. Osher, “Bregmanized nonlocal regularization for deconvolution and sparse reconstruction,” SIAM J. Imaging Sci 3, 253–276 (2010).
[CrossRef]

SIAM J. Sci. Comput. (1)

P. C. Hansen and D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1503 (1993).
[CrossRef]

Signal Process. (1)

J. P. Oliveira, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Total variation-based image deconvolution: A majorization-minimization approach,” Signal Process. 89, 1683–1693(2009).
[CrossRef]

Technometrics (1)

G. H. 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 (1)

A. Buades, B. Coll, and J. Morel, Image Enhancement by Non-Local Reverse Heat Equation (CMLA, 2006).

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Figures (8)

Fig. 1
Fig. 1

Weight ω ( i , i ) between two feature patches centered around point i and i within the search window set Ω.

Fig. 2
Fig. 2

Block diagram of the NLM-D.

Fig. 3
Fig. 3

L-curves obtained by our NLM-D for the experimental setups in Subsection 5B. The solid points in each curve denote the optimal regularization parameters of the L-curves.

Fig. 4
Fig. 4

Deconvolution experiment on the remote sensing image: (a) original image; (b) degraded image, the PSF is a Gaussian blur ( FWHM = 3.3 ) and BSNR = 30 dB ; (c) image restored by TV-MM; (d) image restored by WDALRD; and (e) image restored by NLM-D.

Fig. 5
Fig. 5

Deconvolution experiment on the remote sensing image: (a) original image; (b) degraded image, PSF is a 9 × 9 uniform boxcar, BSNR = 30 dB ; (c) image restored by TV-MM; (d) image restored by WDALRD; and (e) image restored by NLM-D.

Fig. 6
Fig. 6

Zoom on a 131 × 176 area extracted from the image of Fig. 4: (a) original, (b) TV-MM, (c) WDALRD, and (d) NLM-D.

Fig. 7
Fig. 7

Deconvolution experiment on the Cameraman image: (a) original image; (b) degraded image, PSF is a 9 × 9 uniform boxcar, BSNR = 25 dB ; (c) image restored by TV-MM; (d) image restored by WDALRD; and (e) image restored by NLM-D.

Fig. 8
Fig. 8

Zoom on a 74 × 85 area extracted from the image of Fig. 6: (a) original, (b) TV-MM, (c) WDALRD, and (d) NLM-D.

Tables (4)

Tables Icon

Table 1 Restoration Results of Experiment 1

Tables Icon

Table 2 Restoration Results of Experiment 2

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Table 3 Restoration Results of Experiment 3

Tables Icon

Table 4 Restoration Results of NLM-D with the Best Regularization Parameters

Equations (22)

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y = ( h x ) + n             = Hx + n ,
x ^ = arg max { log p y | x ( y | x ) + log p x ( x ) } ,
NL [ x ( i ) ] = k Ω ω ( i ; i + k ) y ( i + k ) ,
ω ( i ; i ) = 1 Z ω ( i ) exp ( u S [ x ( i + u ) x ( i + u ) ] 2 2 h 2 ) ,
p x ( x ) = 1 Z MRF exp ( 1 T U ( x ) ) ,
U ( x ) = c C V c ( x )
U ( x ) = { i } C 1 V 1 ( i ; x ( i ) ) + { i , i } C 2 V 2 ( i , i ; x ( i ) , x ( i ) ) .
V 1 ( i ; x ( i ) ) = 0 , V 2 ( i , i ; x ( i ) , x ( i ) ) = G ( i ; i ) ρ [ x ( i ) x ( i ) ] ,
U ( x ) = i M i N i G ( i ; i ) | x ( i ) x ( i ) | = i M k Ω G ( i ; i + k ) | x ( i ) x ( i + k ) | = k Ω i M G ( i ; i + k ) | x ( i ) x ( i + k ) | = k Ω W k D k x 1 ,
J ( x ) = y Hx 2 + α U ( x ) = y Hx 2 + α k Ω W k D k x 1
x ^ n + 1 = x ^ n λ J ( x ^ ) = x ^ n λ [ H T ( H x ^ n y ) + α U ( x ) x | x = x ^ n ] ,
U ( x ) x u 1 ( x ) = k Ω D k T W k sign ( D k x ) .
x ^ n + 1 = x ^ n λ [ H T ( H x ^ n y ) + α k Ω D k T W k sign ( D k x ^ n ) ] .
x ^ n + 1 ( t ) x ^ n ( t ) x ^ n ( t ) < δ s d ,
x ^ N ( t + 1 ) x ^ N ( t ) x ^ N ( t ) < δ nlmd ,
L ( α ) = log ( y H x ^ α 2 ) ,
φ ( α ) = log U ( x ^ α ) .
SNR = 10 log ( x 2 x ^ α x 2 ) .
U ( x ) x ( m ) = { Σ i Σ k G ( i ; i + k ) | x ( i ) x ( i + k ) | } x ( m ) = Σ i Σ k G ( i ; i + k ) | x ( i ) x ( i + k ) | x ( m ) u 1 [ x ( m ) ] + Σ i Σ k G ( i ; i + k ) x ( m ) | x ( i ) x ( i + k ) | u 2 [ x ( m ) ] ,
u 1 [ x ( m ) ] = i M k Ω G ( i ; i + k ) | x ( i ) x ( i + k ) | x ( m ) k Ω [ G ( m ; m + k ) sign ( x ( m ) x ( m + k ) ) G ( m k ; m ) sign ( x ( m ) x ( m k ) ) ] ,
u 2 [ x ( m ) ] = i M k Ω G ( i ; i + k ) x ( m ) | x ( i ) x ( i + k ) | = k Ω u S [ G ( t ; t + k ) [ x ( m ) x ( m + k ) h 2 ] | x ( t ) x ( t + k ) | G ( t k ; t ) [ x ( m ) x ( m k ) h 2 ] | x ( t ) x ( t k ) | ] ,
u 1 ( x ) = k Ω D k T W k sign ( D k x ) .

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