Abstract

In this paper, we consider the nonnegatively constrained multichannel image deblurring problem and propose regularizing active set methods for numerical restoration. For image deblurring problems, it is reasonable to solve a regularizing model with nonnegativity constraints because of the physical meaning of the image. We consider a general regularizing lplq model with nonnegativity constraints. For p and q equaling 2, the model is in a convex quadratic form, therefore, the active set method is proposed since the nonnegativity constraints are imposed naturally. For p and q not equaling 2, we present an active set method with a feasible Newton-conjugate gradient solution technique. Numerical experiments are presented for ill-posed three-channel blurred image restoration problems.

© 2009 Optical Society of America

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2009 (1)

Y. F. Wang, S. Q. Ma, H. Yang, J. D. Wang, and X. W. Li, “On the effective inversion by imposing a priori information for retrieval of land surface parameters,” Sci. China (Ser. D) 52, 1-10 (2009).
[CrossRef]

2008 (1)

2007 (2)

Y. F. Wang, S. F. Fan, and X. Feng, “Retrieval of the aerosol particle size distribution function by incorporating a priori information,” J. Aerosol Sci. 38885-901 (2007).
[CrossRef]

Y. F. Wang and S. Q. Ma, “Projected Barzilai-Borwein method for large scale nonnegative image restoration,” Inverse Probl. Sci. Eng. 15, 559-583 (2007).
[CrossRef]

2006 (2)

D. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289-1306 (2006).
[CrossRef]

A. Beck, A. Ben, and Y. C. Eldar, “Robust mean-squared error estimation of multiple signals in linear systems affected by model and noise uncertainties,” Math. Program. 107, 155-187(2006).
[CrossRef]

2005 (4)

Z. W. Wen and Y. F. Wang, “ A new trust region algorithm for image restoration,” Sci. China Ser. A Math., Phys., Astron. 48, 169-184 (2005).
[CrossRef]

Y. F. Wang and Y. Yuan, “Convergence and regularity of trust region methods for nonlinear ill-posed inverse problems,” Inverse Probl. 21, 821-838 (2005).
[CrossRef]

Y. C. Eldar, A. Ben-Tal, and A. Nemirovski, “Robust mean-squared error estimation in the presence of model uncertainties,” IEEE Trans. Signal Process. 53, 168-181 (ISPRS Working Groups WG VII/1, 2005).
[CrossRef]

M.-C. Hong, T. Stathaki, and A. K. Katsasggelos, “Iterative regularized mixed norm multichannel image restoration,” J. Electron. Imaging 14, 013004 (2005).
[CrossRef]

2004 (1)

J. Bardsley and C. R. Vogel, “A nonnegatively constrained convex programming method for image reconstruction,” SIAM J. Sci. Comput. 25, 1326-1343 (2004).
[CrossRef]

2003 (1)

Y. F. Wang, “On the regularity of trust region-CG algorithm: with application to image deconvolution problem,” Sci. China Ser. A Math., Phys., Astron. 46, 312-325 (2003).

2002 (1)

M.-C. Hong, T. Stathaki, and A. K. Katsasggelos, “Iterative regularized least-mean mixed-norm image restoration,” Opt. Eng. 41, 2525-2524 (2002).
[CrossRef]

2001 (1)

T. W. S. Chow, X. D. Li, and K.-T. Ng, “Double-regularization approach for blind restoration of multichannel imagery,” IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 48, 1075-1085 (2001).
[CrossRef]

2000 (2)

M. Hanke, J. Nagy, and C. R. Vogel, “Quasi-Newton approach to nonnegative image restorations,” Numer. Linear Algebra Appl. 316, 223-236 (2000).
[CrossRef]

J. Nagy and Z. Strakos, “Enforcing nonnegativity in image reconstruction algorithms,” Proc. SPIE 4121, 182-190(2000).

1998 (1)

C. R. Vogel and M. E. Oman, “A fast, robust algorithm for total variation based reconstruction of noisy, blurred images,” IEEE Trans. Image Process. 7, 813-824 (1998).
[CrossRef]

1996 (1)

M. Hanke and J. Nagy, “Restoration of atmospherically blurred images by symmetric indefinite conjugate gradient techniques,” Inverse Probl. 12, 157-173 (1996).
[CrossRef]

1995 (2)

R. R. Schultz and R. L. Stevenson, “Stochastic modeling and estimation of multispectral image data,” IEEE Trans. Image Process. 4, 1109-1119 (1995).
[CrossRef]

W. W. Zhu, Nikolas P. Galatsanos, and A. K. Katsaggelos, “Regularized multichannel image restoration using cross-validation,” Graphical Models 57, 345-356 (1995).

1994 (1)

G. Angelopoulos and I. Pitas, “Multichannel Wiener filters in color image restoration,” IEEE Trans. Circuits Syst. Video Technol. 4, 83-87 (1994).
[CrossRef]

1992 (1)

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

1991 (3)

N. P. Galatsanos and R. T. Chin, “Restoration of color images by multichannel Kalman filtering,” IEEE Trans. Signal Process. 39, 2237-2252 (1991).
[CrossRef]

N. P. Galatsanos, A. K. Katsaggelos, R. T. Chin, and A. D. Hillery, “Least squares restoration of multichannel images,” IEEE Trans. Signal Process. 39, 2222-2236 (1991).
[CrossRef]

J. J. More and G. Toraldo, “On the solution of large quadratic programming problems with bound constraints,” SIAM J. Control Optim. 1, 93-113 (1991).
[CrossRef]

1989 (2)

N. P. Galatsanos and R. T. Chin, “Digital restoration of multichannel images,” IEEE Trans. Acoust., Speech Signal Process. 37, 415-421 (1989).
[CrossRef]

M. A. Sezan and H. J. Trussell, “Use of a priori knowledge in multispectral image restoration,” in IEEE Trans. Acoust., Speech Signal Process. 3, 1429-1432 (1989).
[CrossRef]

1988 (3)

1987 (1)

D. L. Angwin and H. Haufman, “Effects of modeling domains on recursive color image restoration,” in IEEE Trans. Acoust., Speech, Signal Process. 12, 1229-1231 (1987).

1984 (1)

B. R. Hunt and O. Kubler, “Karhunen-Loeve multispectral image restoration, part I: theory,” IEEE Trans. Acoust., Speech Signal Process. ASSP-32, 592-600 (1984).
[CrossRef]

1982 (1)

D. P. Bertsekas, “Projected Newton methods for optimization problems with simple constraints,” SIAM J. Control Optim. 20, 221-246 (1982).
[CrossRef]

1980 (1)

D. P. O'Leary, “A generalized conjugate gradient algorithm for solving a class of quadratic programming problems,” Numer. Linear Algebra Appl. 34, 371-399 (1980).
[CrossRef]

Altamirano, J. H.

Angelopoulos, G.

G. Angelopoulos and I. Pitas, “Multichannel Wiener filters in color image restoration,” IEEE Trans. Circuits Syst. Video Technol. 4, 83-87 (1994).
[CrossRef]

Angwin, D. L.

D. L. Angwin and H. Haufman, “Effects of modeling domains on recursive color image restoration,” in IEEE Trans. Acoust., Speech, Signal Process. 12, 1229-1231 (1987).

Badique, E.

Bakushinsky, A.

A. Bakushinsky and A. Goncharsky, Ill-Posed Problems: Theory and Applications (Kluwer Academic, 1994).
[CrossRef]

Bardsley, J.

J. Bardsley and C. R. Vogel, “A nonnegatively constrained convex programming method for image reconstruction,” SIAM J. Sci. Comput. 25, 1326-1343 (2004).
[CrossRef]

Barrett, R.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and Vorst H. Van der, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Society for Applied and Industrial Mathematics, 1994).
[CrossRef]

Beck, A.

A. Beck, A. Ben, and Y. C. Eldar, “Robust mean-squared error estimation of multiple signals in linear systems affected by model and noise uncertainties,” Math. Program. 107, 155-187(2006).
[CrossRef]

Ben, A.

A. Beck, A. Ben, and Y. C. Eldar, “Robust mean-squared error estimation of multiple signals in linear systems affected by model and noise uncertainties,” Math. Program. 107, 155-187(2006).
[CrossRef]

Ben-Tal, A.

Y. C. Eldar, A. Ben-Tal, and A. Nemirovski, “Robust mean-squared error estimation in the presence of model uncertainties,” IEEE Trans. Signal Process. 53, 168-181 (ISPRS Working Groups WG VII/1, 2005).
[CrossRef]

Berry, M.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and Vorst H. Van der, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Society for Applied and Industrial Mathematics, 1994).
[CrossRef]

Bertero, M.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, 1998).
[CrossRef]

Bertsekas, D. P.

D. P. Bertsekas, “Projected Newton methods for optimization problems with simple constraints,” SIAM J. Control Optim. 20, 221-246 (1982).
[CrossRef]

Bescos, J.

Boccacci, P.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, 1998).
[CrossRef]

Chan, T. F.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and Vorst H. Van der, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Society for Applied and Industrial Mathematics, 1994).
[CrossRef]

Chin, R. T.

N. P. Galatsanos and R. T. Chin, “Restoration of color images by multichannel Kalman filtering,” IEEE Trans. Signal Process. 39, 2237-2252 (1991).
[CrossRef]

N. P. Galatsanos, A. K. Katsaggelos, R. T. Chin, and A. D. Hillery, “Least squares restoration of multichannel images,” IEEE Trans. Signal Process. 39, 2222-2236 (1991).
[CrossRef]

N. P. Galatsanos and R. T. Chin, “Digital restoration of multichannel images,” IEEE Trans. Acoust., Speech Signal Process. 37, 415-421 (1989).
[CrossRef]

Chow, T. W. S.

T. W. S. Chow, X. D. Li, and K.-T. Ng, “Double-regularization approach for blind restoration of multichannel imagery,” IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 48, 1075-1085 (2001).
[CrossRef]

Demmel, J.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and Vorst H. Van der, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Society for Applied and Industrial Mathematics, 1994).
[CrossRef]

Donato, J.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and Vorst H. Van der, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Society for Applied and Industrial Mathematics, 1994).
[CrossRef]

Dongarra, J.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and Vorst H. Van der, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Society for Applied and Industrial Mathematics, 1994).
[CrossRef]

Donoho, D.

D. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289-1306 (2006).
[CrossRef]

Eijkhout, V.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and Vorst H. Van der, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Society for Applied and Industrial Mathematics, 1994).
[CrossRef]

Eldar, Y. C.

A. Beck, A. Ben, and Y. C. Eldar, “Robust mean-squared error estimation of multiple signals in linear systems affected by model and noise uncertainties,” Math. Program. 107, 155-187(2006).
[CrossRef]

Y. C. Eldar, A. Ben-Tal, and A. Nemirovski, “Robust mean-squared error estimation in the presence of model uncertainties,” IEEE Trans. Signal Process. 53, 168-181 (ISPRS Working Groups WG VII/1, 2005).
[CrossRef]

Engl, H. W.

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

Fan, S. F.

Y. F. Wang, S. F. Fan, and X. Feng, “Retrieval of the aerosol particle size distribution function by incorporating a priori information,” J. Aerosol Sci. 38885-901 (2007).
[CrossRef]

Fatemi, E.

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

Feng, X.

Y. F. Wang, S. F. Fan, and X. Feng, “Retrieval of the aerosol particle size distribution function by incorporating a priori information,” J. Aerosol Sci. 38885-901 (2007).
[CrossRef]

Fletcher, R.

R. Fletcher, Practical Methods of Optimization, 2nd ed.(Wiley,, 1987).

Galatsanos, N. P.

N. P. Galatsanos, A. K. Katsaggelos, R. T. Chin, and A. D. Hillery, “Least squares restoration of multichannel images,” IEEE Trans. Signal Process. 39, 2222-2236 (1991).
[CrossRef]

N. P. Galatsanos and R. T. Chin, “Restoration of color images by multichannel Kalman filtering,” IEEE Trans. Signal Process. 39, 2237-2252 (1991).
[CrossRef]

N. P. Galatsanos and R. T. Chin, “Digital restoration of multichannel images,” IEEE Trans. Acoust., Speech Signal Process. 37, 415-421 (1989).
[CrossRef]

Galatsanos, Nikolas P.

W. W. Zhu, Nikolas P. Galatsanos, and A. K. Katsaggelos, “Regularized multichannel image restoration using cross-validation,” Graphical Models 57, 345-356 (1995).

Goncharsky, A.

A. Bakushinsky and A. Goncharsky, Ill-Posed Problems: Theory and Applications (Kluwer Academic, 1994).
[CrossRef]

Guan, Y. N.

Y. F. Wang, X. W. Li, S. Q. Ma, H. Yang, Z. Nashed, and Y. N. Guan, “BRDF model inversion of multiangular remote sensing: ill-posedness and interior point solution method,” in Proceedings of the 9th International Symposium on Physical Measurements and Signature in Remote Sensing (ISPMSRS) (ISPRS Working Groups WG VII/1, 2005), Vol. XXXVI, pp. 328-330.

Hanke, M.

M. Hanke, J. Nagy, and C. R. Vogel, “Quasi-Newton approach to nonnegative image restorations,” Numer. Linear Algebra Appl. 316, 223-236 (2000).
[CrossRef]

M. Hanke and J. Nagy, “Restoration of atmospherically blurred images by symmetric indefinite conjugate gradient techniques,” Inverse Probl. 12, 157-173 (1996).
[CrossRef]

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

Hansen, P. C.

P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion (Society for Industrial and Applied Mathematics, 1998).
[CrossRef]

Haufman, H.

D. L. Angwin and H. Haufman, “Effects of modeling domains on recursive color image restoration,” in IEEE Trans. Acoust., Speech, Signal Process. 12, 1229-1231 (1987).

Hillery, A. D.

N. P. Galatsanos, A. K. Katsaggelos, R. T. Chin, and A. D. Hillery, “Least squares restoration of multichannel images,” IEEE Trans. Signal Process. 39, 2222-2236 (1991).
[CrossRef]

Honda, T.

Hong, M.-C.

M.-C. Hong, T. Stathaki, and A. K. Katsasggelos, “Iterative regularized mixed norm multichannel image restoration,” J. Electron. Imaging 14, 013004 (2005).
[CrossRef]

M.-C. Hong, T. Stathaki, and A. K. Katsasggelos, “Iterative regularized least-mean mixed-norm image restoration,” Opt. Eng. 41, 2525-2524 (2002).
[CrossRef]

Hunt, B. R.

B. R. Hunt and O. Kubler, “Karhunen-Loeve multispectral image restoration, part I: theory,” IEEE Trans. Acoust., Speech Signal Process. ASSP-32, 592-600 (1984).
[CrossRef]

Katsaggelos, A. K.

W. W. Zhu, Nikolas P. Galatsanos, and A. K. Katsaggelos, “Regularized multichannel image restoration using cross-validation,” Graphical Models 57, 345-356 (1995).

N. P. Galatsanos, A. K. Katsaggelos, R. T. Chin, and A. D. Hillery, “Least squares restoration of multichannel images,” IEEE Trans. Signal Process. 39, 2222-2236 (1991).
[CrossRef]

A. K. Katsaggelos and J. K. Paik, “Iterative color image restoration algorithms,” IEEE Trans. Acoust., Speech Signal Process. 2, 1028-1031 (1988).

Katsasggelos, A. K.

M.-C. Hong, T. Stathaki, and A. K. Katsasggelos, “Iterative regularized mixed norm multichannel image restoration,” J. Electron. Imaging 14, 013004 (2005).
[CrossRef]

M.-C. Hong, T. Stathaki, and A. K. Katsasggelos, “Iterative regularized least-mean mixed-norm image restoration,” Opt. Eng. 41, 2525-2524 (2002).
[CrossRef]

Kubler, O.

B. R. Hunt and O. Kubler, “Karhunen-Loeve multispectral image restoration, part I: theory,” IEEE Trans. Acoust., Speech Signal Process. ASSP-32, 592-600 (1984).
[CrossRef]

Li, X. D.

T. W. S. Chow, X. D. Li, and K.-T. Ng, “Double-regularization approach for blind restoration of multichannel imagery,” IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 48, 1075-1085 (2001).
[CrossRef]

Li, X. W.

Y. F. Wang, S. Q. Ma, H. Yang, J. D. Wang, and X. W. Li, “On the effective inversion by imposing a priori information for retrieval of land surface parameters,” Sci. China (Ser. D) 52, 1-10 (2009).
[CrossRef]

Y. F. Wang, X. W. Li, S. Q. Ma, H. Yang, Z. Nashed, and Y. N. Guan, “BRDF model inversion of multiangular remote sensing: ill-posedness and interior point solution method,” in Proceedings of the 9th International Symposium on Physical Measurements and Signature in Remote Sensing (ISPMSRS) (ISPRS Working Groups WG VII/1, 2005), Vol. XXXVI, pp. 328-330.

Ma, S. Q.

Y. F. Wang, S. Q. Ma, H. Yang, J. D. Wang, and X. W. Li, “On the effective inversion by imposing a priori information for retrieval of land surface parameters,” Sci. China (Ser. D) 52, 1-10 (2009).
[CrossRef]

Y. F. Wang and S. Q. Ma, “Projected Barzilai-Borwein method for large scale nonnegative image restoration,” Inverse Probl. Sci. Eng. 15, 559-583 (2007).
[CrossRef]

Y. F. Wang, X. W. Li, S. Q. Ma, H. Yang, Z. Nashed, and Y. N. Guan, “BRDF model inversion of multiangular remote sensing: ill-posedness and interior point solution method,” in Proceedings of the 9th International Symposium on Physical Measurements and Signature in Remote Sensing (ISPMSRS) (ISPRS Working Groups WG VII/1, 2005), Vol. XXXVI, pp. 328-330.

More, J. J.

J. J. More and G. Toraldo, “On the solution of large quadratic programming problems with bound constraints,” SIAM J. Control Optim. 1, 93-113 (1991).
[CrossRef]

Nagy, J.

J. Nagy and Z. Strakos, “Enforcing nonnegativity in image reconstruction algorithms,” Proc. SPIE 4121, 182-190(2000).

M. Hanke, J. Nagy, and C. R. Vogel, “Quasi-Newton approach to nonnegative image restorations,” Numer. Linear Algebra Appl. 316, 223-236 (2000).
[CrossRef]

M. Hanke and J. Nagy, “Restoration of atmospherically blurred images by symmetric indefinite conjugate gradient techniques,” Inverse Probl. 12, 157-173 (1996).
[CrossRef]

Nashed, Z.

Y. F. Wang, X. W. Li, S. Q. Ma, H. Yang, Z. Nashed, and Y. N. Guan, “BRDF model inversion of multiangular remote sensing: ill-posedness and interior point solution method,” in Proceedings of the 9th International Symposium on Physical Measurements and Signature in Remote Sensing (ISPMSRS) (ISPRS Working Groups WG VII/1, 2005), Vol. XXXVI, pp. 328-330.

Nemirovski, A.

Y. C. Eldar, A. Ben-Tal, and A. Nemirovski, “Robust mean-squared error estimation in the presence of model uncertainties,” IEEE Trans. Signal Process. 53, 168-181 (ISPRS Working Groups WG VII/1, 2005).
[CrossRef]

Neubauer, A.

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

Ng, K.-T.

T. W. S. Chow, X. D. Li, and K.-T. Ng, “Double-regularization approach for blind restoration of multichannel imagery,” IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 48, 1075-1085 (2001).
[CrossRef]

Nocedal, J.

J. Nocedal and S. J. Wright, Numerical Optimization (Springer-Verlag, 1999).
[CrossRef]

Ohyama, N.

O'Leary, D. P.

D. P. O'Leary, “A generalized conjugate gradient algorithm for solving a class of quadratic programming problems,” Numer. Linear Algebra Appl. 34, 371-399 (1980).
[CrossRef]

Oman, M. E.

C. R. Vogel and M. E. Oman, “A fast, robust algorithm for total variation based reconstruction of noisy, blurred images,” IEEE Trans. Image Process. 7, 813-824 (1998).
[CrossRef]

Osher, S.

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

Paik, J. K.

A. K. Katsaggelos and J. K. Paik, “Iterative color image restoration algorithms,” IEEE Trans. Acoust., Speech Signal Process. 2, 1028-1031 (1988).

Pavlovic, G.

A. M. Tekalp and G. Pavlovic, “Space-variant and color image restoration using Kalman filtering,” in Multidimensional Signal Processing Workshop, Vol. 6 (IEEE, 1989), pp. 186-187
[CrossRef]

Pitas, I.

G. Angelopoulos and I. Pitas, “Multichannel Wiener filters in color image restoration,” IEEE Trans. Circuits Syst. Video Technol. 4, 83-87 (1994).
[CrossRef]

Pozo, R.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and Vorst H. Van der, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Society for Applied and Industrial Mathematics, 1994).
[CrossRef]

Rojas, M.

M. Rojas and T. Steihaug, “An interior-point trust-region-based method for large-scale nonnegative regularization,” CERFACS Tech. Rep. TR/PA/01/11 (Centre Européen de Recherche et de Formation Avancée en Calcul Scientifique, 2002).

Romine, C.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and Vorst H. Van der, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Society for Applied and Industrial Mathematics, 1994).
[CrossRef]

Rudin, L.

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

Santamaria, J.

Santisteban, A.

Schultz, R. R.

R. R. Schultz and R. L. Stevenson, “Stochastic modeling and estimation of multispectral image data,” IEEE Trans. Image Process. 4, 1109-1119 (1995).
[CrossRef]

Sezan, M. A.

M. A. Sezan and H. J. Trussell, “Use of a priori knowledge in multispectral image restoration,” in IEEE Trans. Acoust., Speech Signal Process. 3, 1429-1432 (1989).
[CrossRef]

Stathaki, T.

M.-C. Hong, T. Stathaki, and A. K. Katsasggelos, “Iterative regularized mixed norm multichannel image restoration,” J. Electron. Imaging 14, 013004 (2005).
[CrossRef]

M.-C. Hong, T. Stathaki, and A. K. Katsasggelos, “Iterative regularized least-mean mixed-norm image restoration,” Opt. Eng. 41, 2525-2524 (2002).
[CrossRef]

Steihaug, T.

M. Rojas and T. Steihaug, “An interior-point trust-region-based method for large-scale nonnegative regularization,” CERFACS Tech. Rep. TR/PA/01/11 (Centre Européen de Recherche et de Formation Avancée en Calcul Scientifique, 2002).

Stevenson, R. L.

R. R. Schultz and R. L. Stevenson, “Stochastic modeling and estimation of multispectral image data,” IEEE Trans. Image Process. 4, 1109-1119 (1995).
[CrossRef]

Strakos, Z.

J. Nagy and Z. Strakos, “Enforcing nonnegativity in image reconstruction algorithms,” Proc. SPIE 4121, 182-190(2000).

Sun, W. Y.

Y. X. Yuan and W. Y. Sun, Theory and Methods for Optimization (Science Press, 1997).

Tekalp, A. M.

A. M. Tekalp and G. Pavlovic, “Space-variant and color image restoration using Kalman filtering,” in Multidimensional Signal Processing Workshop, Vol. 6 (IEEE, 1989), pp. 186-187
[CrossRef]

Toraldo, G.

J. J. More and G. Toraldo, “On the solution of large quadratic programming problems with bound constraints,” SIAM J. Control Optim. 1, 93-113 (1991).
[CrossRef]

Trussell, H. J.

M. A. Sezan and H. J. Trussell, “Use of a priori knowledge in multispectral image restoration,” in IEEE Trans. Acoust., Speech Signal Process. 3, 1429-1432 (1989).
[CrossRef]

Tsujiuchi, J.

Van der, Vorst H.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and Vorst H. Van der, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Society for Applied and Industrial Mathematics, 1994).
[CrossRef]

Vogel, C. R.

J. Bardsley and C. R. Vogel, “A nonnegatively constrained convex programming method for image reconstruction,” SIAM J. Sci. Comput. 25, 1326-1343 (2004).
[CrossRef]

M. Hanke, J. Nagy, and C. R. Vogel, “Quasi-Newton approach to nonnegative image restorations,” Numer. Linear Algebra Appl. 316, 223-236 (2000).
[CrossRef]

C. R. Vogel and M. E. Oman, “A fast, robust algorithm for total variation based reconstruction of noisy, blurred images,” IEEE Trans. Image Process. 7, 813-824 (1998).
[CrossRef]

C. R. Vogel, Computational Methods for Inverse Problems (Society for Applied and Industrial Mathematics, 2002).
[CrossRef]

Wang, J. D.

Y. F. Wang, S. Q. Ma, H. Yang, J. D. Wang, and X. W. Li, “On the effective inversion by imposing a priori information for retrieval of land surface parameters,” Sci. China (Ser. D) 52, 1-10 (2009).
[CrossRef]

Wang, Y. F.

Y. F. Wang, S. Q. Ma, H. Yang, J. D. Wang, and X. W. Li, “On the effective inversion by imposing a priori information for retrieval of land surface parameters,” Sci. China (Ser. D) 52, 1-10 (2009).
[CrossRef]

Y. F. Wang and C. C. Yang, “A regularizing active set method for retrieval of atmospheric aerosol particle size distribution function,” J. Opt. Soc. Am. A 25, 348-356 (2008).
[CrossRef]

Y. F. Wang and S. Q. Ma, “Projected Barzilai-Borwein method for large scale nonnegative image restoration,” Inverse Probl. Sci. Eng. 15, 559-583 (2007).
[CrossRef]

Y. F. Wang, S. F. Fan, and X. Feng, “Retrieval of the aerosol particle size distribution function by incorporating a priori information,” J. Aerosol Sci. 38885-901 (2007).
[CrossRef]

Z. W. Wen and Y. F. Wang, “ A new trust region algorithm for image restoration,” Sci. China Ser. A Math., Phys., Astron. 48, 169-184 (2005).
[CrossRef]

Y. F. Wang and Y. Yuan, “Convergence and regularity of trust region methods for nonlinear ill-posed inverse problems,” Inverse Probl. 21, 821-838 (2005).
[CrossRef]

Y. F. Wang, “On the regularity of trust region-CG algorithm: with application to image deconvolution problem,” Sci. China Ser. A Math., Phys., Astron. 46, 312-325 (2003).

T. Y. Xiao, Sh. G. Yu, and Y. F. Wang, Numerical Methods for the Solution of Inverse Problems (Beijing: Science Press, 2003).

Y. F. Wang, X. W. Li, S. Q. Ma, H. Yang, Z. Nashed, and Y. N. Guan, “BRDF model inversion of multiangular remote sensing: ill-posedness and interior point solution method,” in Proceedings of the 9th International Symposium on Physical Measurements and Signature in Remote Sensing (ISPMSRS) (ISPRS Working Groups WG VII/1, 2005), Vol. XXXVI, pp. 328-330.

Y. F. Wang, Computational Methods for Inverse Problems and Their Applications (Higher Education Press, 2007).

Wen, Z. W.

Z. W. Wen and Y. F. Wang, “ A new trust region algorithm for image restoration,” Sci. China Ser. A Math., Phys., Astron. 48, 169-184 (2005).
[CrossRef]

Wright, S. J.

J. Nocedal and S. J. Wright, Numerical Optimization (Springer-Verlag, 1999).
[CrossRef]

Xiao, T. Y.

T. Y. Xiao, Sh. G. Yu, and Y. F. Wang, Numerical Methods for the Solution of Inverse Problems (Beijing: Science Press, 2003).

Yachida, M.

Yang, C. C.

Yang, H.

Y. F. Wang, S. Q. Ma, H. Yang, J. D. Wang, and X. W. Li, “On the effective inversion by imposing a priori information for retrieval of land surface parameters,” Sci. China (Ser. D) 52, 1-10 (2009).
[CrossRef]

Y. F. Wang, X. W. Li, S. Q. Ma, H. Yang, Z. Nashed, and Y. N. Guan, “BRDF model inversion of multiangular remote sensing: ill-posedness and interior point solution method,” in Proceedings of the 9th International Symposium on Physical Measurements and Signature in Remote Sensing (ISPMSRS) (ISPRS Working Groups WG VII/1, 2005), Vol. XXXVI, pp. 328-330.

Yu, Sh. G.

T. Y. Xiao, Sh. G. Yu, and Y. F. Wang, Numerical Methods for the Solution of Inverse Problems (Beijing: Science Press, 2003).

Yuan, Y.

Y. F. Wang and Y. Yuan, “Convergence and regularity of trust region methods for nonlinear ill-posed inverse problems,” Inverse Probl. 21, 821-838 (2005).
[CrossRef]

Yuan, Y. X.

Y. X. Yuan, Numerical Methods for Nonlinear Programming (Shanghai Science and Technology Publication, 1993).

Y. X. Yuan and W. Y. Sun, Theory and Methods for Optimization (Science Press, 1997).

Y. X. Yuan, “Subspace techniques for nonlinear optimization,” in Some Topics in Industrial and Applied Mathematics, R. Jeltsch, D. Q.Li, and I. H. Sloan, eds., Series in Contemporary Applied Mathematics CAM 8 (Higher Education Press, (2007), pp. 206-218.
[CrossRef]

Zhu, W. W.

W. W. Zhu, Nikolas P. Galatsanos, and A. K. Katsaggelos, “Regularized multichannel image restoration using cross-validation,” Graphical Models 57, 345-356 (1995).

Appl. Opt. (1)

Graphical Models (1)

W. W. Zhu, Nikolas P. Galatsanos, and A. K. Katsaggelos, “Regularized multichannel image restoration using cross-validation,” Graphical Models 57, 345-356 (1995).

IEEE Trans. Acoust., Speech Signal Process. (4)

M. A. Sezan and H. J. Trussell, “Use of a priori knowledge in multispectral image restoration,” in IEEE Trans. Acoust., Speech Signal Process. 3, 1429-1432 (1989).
[CrossRef]

N. P. Galatsanos and R. T. Chin, “Digital restoration of multichannel images,” IEEE Trans. Acoust., Speech Signal Process. 37, 415-421 (1989).
[CrossRef]

B. R. Hunt and O. Kubler, “Karhunen-Loeve multispectral image restoration, part I: theory,” IEEE Trans. Acoust., Speech Signal Process. ASSP-32, 592-600 (1984).
[CrossRef]

A. K. Katsaggelos and J. K. Paik, “Iterative color image restoration algorithms,” IEEE Trans. Acoust., Speech Signal Process. 2, 1028-1031 (1988).

IEEE Trans. Acoust., Speech, Signal Process. (1)

D. L. Angwin and H. Haufman, “Effects of modeling domains on recursive color image restoration,” in IEEE Trans. Acoust., Speech, Signal Process. 12, 1229-1231 (1987).

IEEE Trans. Circuits Syst. I Fundam. Theory Appl. (1)

T. W. S. Chow, X. D. Li, and K.-T. Ng, “Double-regularization approach for blind restoration of multichannel imagery,” IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 48, 1075-1085 (2001).
[CrossRef]

IEEE Trans. Circuits Syst. Video Technol. (1)

G. Angelopoulos and I. Pitas, “Multichannel Wiener filters in color image restoration,” IEEE Trans. Circuits Syst. Video Technol. 4, 83-87 (1994).
[CrossRef]

IEEE Trans. Image Process. (2)

R. R. Schultz and R. L. Stevenson, “Stochastic modeling and estimation of multispectral image data,” IEEE Trans. Image Process. 4, 1109-1119 (1995).
[CrossRef]

C. R. Vogel and M. E. Oman, “A fast, robust algorithm for total variation based reconstruction of noisy, blurred images,” IEEE Trans. Image Process. 7, 813-824 (1998).
[CrossRef]

IEEE Trans. Inf. Theory (1)

D. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289-1306 (2006).
[CrossRef]

IEEE Trans. Signal Process. (3)

Y. C. Eldar, A. Ben-Tal, and A. Nemirovski, “Robust mean-squared error estimation in the presence of model uncertainties,” IEEE Trans. Signal Process. 53, 168-181 (ISPRS Working Groups WG VII/1, 2005).
[CrossRef]

N. P. Galatsanos and R. T. Chin, “Restoration of color images by multichannel Kalman filtering,” IEEE Trans. Signal Process. 39, 2237-2252 (1991).
[CrossRef]

N. P. Galatsanos, A. K. Katsaggelos, R. T. Chin, and A. D. Hillery, “Least squares restoration of multichannel images,” IEEE Trans. Signal Process. 39, 2222-2236 (1991).
[CrossRef]

Inverse Probl. (2)

M. Hanke and J. Nagy, “Restoration of atmospherically blurred images by symmetric indefinite conjugate gradient techniques,” Inverse Probl. 12, 157-173 (1996).
[CrossRef]

Y. F. Wang and Y. Yuan, “Convergence and regularity of trust region methods for nonlinear ill-posed inverse problems,” Inverse Probl. 21, 821-838 (2005).
[CrossRef]

Inverse Probl. Sci. Eng. (1)

Y. F. Wang and S. Q. Ma, “Projected Barzilai-Borwein method for large scale nonnegative image restoration,” Inverse Probl. Sci. Eng. 15, 559-583 (2007).
[CrossRef]

J. Aerosol Sci. (1)

Y. F. Wang, S. F. Fan, and X. Feng, “Retrieval of the aerosol particle size distribution function by incorporating a priori information,” J. Aerosol Sci. 38885-901 (2007).
[CrossRef]

J. Electron. Imaging (1)

M.-C. Hong, T. Stathaki, and A. K. Katsasggelos, “Iterative regularized mixed norm multichannel image restoration,” J. Electron. Imaging 14, 013004 (2005).
[CrossRef]

J. Opt. Soc. Am. A (2)

Math. Program. (1)

A. Beck, A. Ben, and Y. C. Eldar, “Robust mean-squared error estimation of multiple signals in linear systems affected by model and noise uncertainties,” Math. Program. 107, 155-187(2006).
[CrossRef]

Numer. Linear Algebra Appl. (2)

D. P. O'Leary, “A generalized conjugate gradient algorithm for solving a class of quadratic programming problems,” Numer. Linear Algebra Appl. 34, 371-399 (1980).
[CrossRef]

M. Hanke, J. Nagy, and C. R. Vogel, “Quasi-Newton approach to nonnegative image restorations,” Numer. Linear Algebra Appl. 316, 223-236 (2000).
[CrossRef]

Opt. Eng. (1)

M.-C. Hong, T. Stathaki, and A. K. Katsasggelos, “Iterative regularized least-mean mixed-norm image restoration,” Opt. Eng. 41, 2525-2524 (2002).
[CrossRef]

Physica D (Amsterdam) (1)

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

Sci. China (Ser. D) (1)

Y. F. Wang, S. Q. Ma, H. Yang, J. D. Wang, and X. W. Li, “On the effective inversion by imposing a priori information for retrieval of land surface parameters,” Sci. China (Ser. D) 52, 1-10 (2009).
[CrossRef]

Sci. China Ser. A Math., Phys., Astron. (2)

Y. F. Wang, “On the regularity of trust region-CG algorithm: with application to image deconvolution problem,” Sci. China Ser. A Math., Phys., Astron. 46, 312-325 (2003).

Z. W. Wen and Y. F. Wang, “ A new trust region algorithm for image restoration,” Sci. China Ser. A Math., Phys., Astron. 48, 169-184 (2005).
[CrossRef]

SIAM J. Control Optim. (2)

D. P. Bertsekas, “Projected Newton methods for optimization problems with simple constraints,” SIAM J. Control Optim. 20, 221-246 (1982).
[CrossRef]

J. J. More and G. Toraldo, “On the solution of large quadratic programming problems with bound constraints,” SIAM J. Control Optim. 1, 93-113 (1991).
[CrossRef]

SIAM J. Sci. Comput. (1)

J. Bardsley and C. R. Vogel, “A nonnegatively constrained convex programming method for image reconstruction,” SIAM J. Sci. Comput. 25, 1326-1343 (2004).
[CrossRef]

Other (17)

P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion (Society for Industrial and Applied Mathematics, 1998).
[CrossRef]

C. R. Vogel, Computational Methods for Inverse Problems (Society for Applied and Industrial Mathematics, 2002).
[CrossRef]

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and Vorst H. Van der, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Society for Applied and Industrial Mathematics, 1994).
[CrossRef]

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

T. Y. Xiao, Sh. G. Yu, and Y. F. Wang, Numerical Methods for the Solution of Inverse Problems (Beijing: Science Press, 2003).

A. M. Tekalp and G. Pavlovic, “Space-variant and color image restoration using Kalman filtering,” in Multidimensional Signal Processing Workshop, Vol. 6 (IEEE, 1989), pp. 186-187
[CrossRef]

Y. F. Wang, X. W. Li, S. Q. Ma, H. Yang, Z. Nashed, and Y. N. Guan, “BRDF model inversion of multiangular remote sensing: ill-posedness and interior point solution method,” in Proceedings of the 9th International Symposium on Physical Measurements and Signature in Remote Sensing (ISPMSRS) (ISPRS Working Groups WG VII/1, 2005), Vol. XXXVI, pp. 328-330.

A. Bakushinsky and A. Goncharsky, Ill-Posed Problems: Theory and Applications (Kluwer Academic, 1994).
[CrossRef]

Y. F. Wang, Computational Methods for Inverse Problems and Their Applications (Higher Education Press, 2007).

J. Nagy and Z. Strakos, “Enforcing nonnegativity in image reconstruction algorithms,” Proc. SPIE 4121, 182-190(2000).

M. Rojas and T. Steihaug, “An interior-point trust-region-based method for large-scale nonnegative regularization,” CERFACS Tech. Rep. TR/PA/01/11 (Centre Européen de Recherche et de Formation Avancée en Calcul Scientifique, 2002).

R. Fletcher, Practical Methods of Optimization, 2nd ed.(Wiley,, 1987).

J. Nocedal and S. J. Wright, Numerical Optimization (Springer-Verlag, 1999).
[CrossRef]

Y. X. Yuan, Numerical Methods for Nonlinear Programming (Shanghai Science and Technology Publication, 1993).

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, 1998).
[CrossRef]

Y. X. Yuan and W. Y. Sun, Theory and Methods for Optimization (Science Press, 1997).

Y. X. Yuan, “Subspace techniques for nonlinear optimization,” in Some Topics in Industrial and Applied Mathematics, R. Jeltsch, D. Q.Li, and I. H. Sloan, eds., Series in Contemporary Applied Mathematics CAM 8 (Higher Education Press, (2007), pp. 206-218.
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Original input three channel image.

Fig. 2
Fig. 2

Blurred noisy image with noise level equaling 0.01.

Fig. 3
Fig. 3

Within-channel restoration: the restored image for ν = 0.005 and noise level = 0.01 .

Fig. 4
Fig. 4

Blurred noisy image with noise level equaling 0.01.

Fig. 5
Fig. 5

Between-channel restoration: the restored image for ν = 0.005 and noise level = 0.01 .

Fig. 6
Fig. 6

Within-channel restoration: the restored image for ν = 0.005 and noise level = 0.01 in the case p = 2 and q = 1 .

Fig. 7
Fig. 7

Between-channel restoration: the restored image for ν = 0.005 and noise level = 0.01 in the case p = 2 and q = 1 .

Fig. 8
Fig. 8

Within-channel restoration: the restored image for ν = 0.005 and noise level = 0.01 in the case p = 1.5 and q = 1 .

Fig. 9
Fig. 9

Between-channel restoration: the restored image for ν = 0.005 and noise level level = 0.01 in the case p = 1.5 and q = 1 .

Fig. 10
Fig. 10

Within-channel restoration: the restored image for ν = 0.005 and noise level = 0.01 in the case p = 1.6 and q = 1.1 .

Fig. 11
Fig. 11

Between-channel restoration: the restored image for ν = 0.005 and noise level = 0.01 in the case p = 1.6 and q = 1.1 .

Tables (2)

Tables Icon

Table 1 RMSEs and Variance Differences for R, G, and B Channels and CPU Time (seconds) of Our Regularizing Algorithm for the Quadratic Inversion Model

Tables Icon

Table 2 RMSEs for R, G, and B Channels and CPU Time (seconds) of Our Regularizing Algorithm for the General I p I q Inversion Model

Equations (39)

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

h n = h + n = K f + n ,
h n = [ h n , 1 h n , 2 h n , N ] , h = [ h 1 h 2 h N ] , f = [ f 1 f 2 f N ] , n = [ n 1 n 2 n N ] ,
K = [ K 11 K 12 K 1 N K 21 K 22 K 2 N K N 1 K N 2 K N N ] ,
h n h = n minimization .
min J [ f ] = 1 2 K f h n l p p + ν 2 L ( f f 0 ) l q q ,
min K f h n l 2 2 + ν Γ ( f ) ,
min ϕ ( x ) = c + b T x + 1 2 x T A x , s.t.   D x l ,
min Q [ s k ] = 1 2 ( A s k , s k ) + ( g k , s k ) , s.t.  s k j = 0 , j W k .
α k = min { 1 , min j W k , s k j < 0 x k j s k j } .
G s k * + g k j W k * λ j * = 0 ,
s k * j = 0 , j W k * ,
λ j * 0 , j W k * .
min J [ f ] , s.t.   f 0 ,
min ϕ [ f ] = 1 2 f T A f b T f , s.t.   f 0 ,
min ϕ s k [ s k + f k ] = 1 2 ( A s k , s k ) + ( g k , s k ) + c , s.t.   s k j = 0 , j W k ,
min Q [ s k ] = 1 2 ( A s k , s k ) + ( g k , s k ) , s.t.  s k j = 0 , j W k .
tol = r k l 2 2 r k + 1 l 2 2 r k + 1 l 2 2
min J [ f ] , s . t . f 0 ,
g ( f ) = 1 2 p K T [ | r 1 | p 1 sign ( r 1 ) | r 2 | p 1 sign ( r 2 ) | r m | p 1 sign ( r m ) ] + 1 2 ν q L T [ | f 1 f 1 0 | q 1 sign ( f 1 f 1 0 ) | f 2 - f 2 0 | q 1 sign ( f 2 f 2 0 ) | f n f n 0 | q 1 sign ( f n f n 0 ) ]
H ( f ) = 1 2 p ( p 1 ) K T diag ( | r 1 | p 2 , | r 2 | p 2 , | r m | p 2 ) K + 1 2 ν q ( q 1 ) L T diag ( | f 1 f 1 0 | q 2 , | f 2 f 2 0 | q 2 , , | f n f n 0 | q 2 ) L ,
min Ψ [ s ] = ( s , g k ) + 1 2 ( H k s , s ) ,
s.t.   s + f k 0 ,
tol = r k l p p r k + 1 l p p r k + 1 l p p
f k + 1 = f k + λ k d k ,
λ k = g k T d k / d k T A d k
k ( x ξ , y η ) = 1 2 π ρ ρ ¯ exp ( 1 2 ( x ξ ρ ) 2 1 2 ( y η ρ ¯ ) 2 ) ,
n = level N h × randn ( N 2 , 1 ) ,
K = [ K 11 0 0 0 K 22 0 0 0 K N N ] ,
K = [ a 11 K 11 a 12 K 12 a 13 K 1 N a 21 K 21 a 22 K 22 a 23 K 2 N a N 1 K N 1 a N 2 K N 2 a N 3 K N N ] ,
k ( x ξ , y η ) = k x ( x ξ ) k y ( y η ) .
K i j = K x , i j K y , i j .
K i j = K 0 K 1 K 1 N K 1 K 0 K 1 K 1 K N 1 K 1 K 0
K ˜ i j = F Λ F ,
rmse = 1 N M 2 i = 1 N M 2 ( ( K f * ) i h i ) 2 ( ( K f * ) i ) 2 ,
[ a 11 K 11 a 12 K 12 a 13 K 13 a 21 K 21 a 22 K 22 a 23 K 22 a 31 K 31 a 32 K 33 a 33 K 33 ] .
min J [ f ] , s.t.   f 0 ,
min Ψ [ s k ] , s.t.   s ˜ k j = 0 , j W k ,
grad k [ Ψ ] ϱ grad 0 [ Ψ ] 2 ,
( grad k [ Ψ ] ) i { ( grad k [ Ψ ] ) i if ( s k ) i > 0 min { ( grad k [ Ψ ] ) i , 0 } if ( s k ) i > 0 .

Metrics