Abstract

In this paper we present a new algorithm for restoring an object from multiple undersampled low-resolution (LR) images that are degraded by optical blur and additive white Gaussian noise. We formulate the multiframe superresolution problem as maximum a posteriori estimation. The prior knowledge that the object is sparse in some domain is incorporated in two ways: first we use the popular 1 norm as the regularization operator. Second, we model wavelet coefficients of natural objects using generalized Gaussian densities. The model parameters are learned from a set of training objects, and the regularization operator is derived from these parameters. We compare the results from our algorithms with an expectation-maximization (EM) algorithm for 1 norm minimization and also with the linear minimum-mean-squared error (LMMSE) estimator. Using only eight 4×4 pixel downsampled LR images the reconstruction errors of object estimates obtained from our algorithm are 5.5% smaller than by the EM method and 14.3% smaller than by the LMMSE method.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S.Chaudhuri, ed., Super-Resolution Imaging (Kluwer, 2001).
  2. K. Aizawa, T. Komatsu, and T. Saito, “A scheme for acquiring very high resolution images using multiple cameras,” in IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1992), Vol. 3, pp. 23-26.
  3. P. Shankar, W. Hassenplaugh, R. Morrison, R. Stack, and M. Neifeld, “Multiaperture imaging,” Appl. Opt. 45, 2871-2883 (2006).
    [CrossRef] [PubMed]
  4. A. Fruchter and R. Hook, “Drizzle: a method for the linear reconstruction of undersampled images,” Publ. Astron. Soc. Pac. 114, 144-152 (2002).
    [CrossRef]
  5. R. Tsai and T. Huang, “Multiframe image restoration and registration,” Advances in Computer Vision and Image Processing 1, 317-339 (1984).
  6. H. Andrews and B. Hunt, Digital Image Restoration (Prentice-Hall, 1977).
  7. M. Irani and S. Peleg, “Improving resolution by image registration,” Comput. Vis. Graph. Image Process. 53, 231-239 (1991).
  8. P. Vandewalle, L. Sbaiz, J. Vandewalle, and M. Vetterli, “How to take advantage of aliasing in bandlimited signals,” in Proceedings of the IEEE International Conference on Accoustics, Speech, and Signal Processing (IEEE, 2004), pp. 948-951.
  9. N. Nguyen, G. Golub, and P. Milanfar, “Blind restoration/superresolution with generalized cross-validation using Gauss-type quadrature rules,” in Proceedings of the 33rd Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, Calif., October 1999, pp. 1257-1261.
  10. R. Hardie, K. Bernard, and E. Armstrong, “Joint MAP registration and high-resolution image estimation using a sequence of undersampled images,” IEEE Trans. Image Process. 6, 1621-1633 (1997).
    [CrossRef] [PubMed]
  11. R. Hardie, K. Barnard, J. Bognar, E. Armstrong, and E. Watson, “High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system,” Opt. Eng. (Bellingham) 37, 247-260 (1998).
    [CrossRef]
  12. S. Baker and T. Kanade, “Limits on super-resolution and how to break them,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 1167-1183 (2002).
    [CrossRef]
  13. P. Shankar and M. Neifeld, “Multiframe superresolution of binary images,” Appl. Opt. 46, 1211-1222 (2007).
    [CrossRef] [PubMed]
  14. E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489-509 (2006).
    [CrossRef]
  15. S. Chen, D. Donoho, and M. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput. (USA) 20, 33-61 (1998).
    [CrossRef]
  16. E. Candes and J. Romberg, “Signal recovery from random projections,” Proc. SPIE 5678, 76-86 (2005).
    [CrossRef]
  17. I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. 57, 1413-1457 (2004).
    [CrossRef]
  18. E. Cands, M. Wakin, and S. Boyd, “Enhancing sparsity by reweighted L1 minimization,” Technical Report, California Institute of Technology, http://www.acm.caltech.edu/emmanuel/papers/rwll-oct2007.pdf.
  19. M. Duarte, M. Wakin, and R. Baraniuk, “Fast reconstruction of piecewise smooth signals from random projections,” Online Proceedings of the Workshop on Signal Processing with Adaptative Sparse Structured Representations, SPARS 2005, http://spars05.irisa.fr/ACTES/TS5-3.pdf.
  20. D. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inf. Theory 41, 613-627 (1995).
    [CrossRef]
  21. M. Figueiredo and R. Nowak, “An EM algorithm for wavelet-based image restoration,” IEEE Trans. Image Process. 12, 906916 (2003).
    [CrossRef]
  22. D. Wipf and B. Rao, “Sparse Bayesian learning for basis selection,” IEEE Trans. Signal Process. 52, 2153-2164 (2004).
    [CrossRef]
  23. C. Paige and M. Saunders, “LSQR: An algorithm for sparse linear equations and sparse least squares,” ACM Trans. Math. Softw. 8, 43-71 (1982).
    [CrossRef]
  24. C. Paige and M. Saunders, “LSQR: Sparse linear equations and least squares problems,” ACM Trans. Math. Softw. 8, 195-209 (1982).
    [CrossRef]
  25. S. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674-693 (1989).
    [CrossRef]
  26. M. A. Turk and A. P. Pentland, “Face recognition using eigenfaces,” in IEEE Proceedings on Computer Vision and Pattern Recognition (IEEE, 1991), pp. 586-591.
    [CrossRef]
  27. E. Candes, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207-1223 (2006).
    [CrossRef]
  28. M. Figueiredo, R. Nowak, and S. Wright, “Gradient projections for sparse reconstruction: Application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Signal Process. 4, 586-597 (1987).
  29. H. Barrett and K. Myers, Foundations of Image Science (Wiley Series in Pure and Applied Optics, 2004).
  30. M. Kilmer and D. O'Leary, “Choosing regularization parameters in iterative methods for ill-posed problems,” SIAM J. Matrix Anal. Appl. 22, 1204-1221 (2001).
    [CrossRef]
  31. P. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion (SIAM, 1998).
    [CrossRef]
  32. N. Valdivia and E. Williams, “Krylov subspace iterative methods for boundary element method based near-field acoustic holography,” J. Acoust. Soc. Am. 117, 711-724 (2005).
    [CrossRef] [PubMed]
  33. M. Jiang, L. Xia, G. Shou, and M. Tang, “Combination of the LSQR method and a genetic algorithm for solving the electrocardiography inverse problem,” Phys. Med. Biol. 52, 1277-1294 (2007).
    [CrossRef] [PubMed]
  34. B. Rao and K. Kreutz-Delgado, “An affine scaling methodology for best basis selection,” IEEE Trans. Signal Process. 47, 187-200 (1999).
    [CrossRef]
  35. P. Hansen, “Analysis of discrete ill-posed problems by means of the L-curve,” SIAM Rev. 34, 561-580 (1992).
    [CrossRef]
  36. S. Mallat, “A compact multiresolution representation: the wavelet model,” presented at the IEEE Workshop Computer Society on Computer Vision, Miami, Florida, December 2-7, 1987.
  37. M. Belge, M. Kilmer, and E. Miller. “Wavelet domain image restoration with adaptive edge-preserving regularization,” IEEE Trans. Image Process. 9, 597-608 (2000).
    [CrossRef]
  38. B. Jeffs and M. Gunsay, “Restoration of blurred star field images by maximally sparse optimization,” IEEE Trans. Image Process. 2, 202-211 (1993).
    [CrossRef] [PubMed]
  39. G. Harikumar and Y. Bresler, “A new algorithm for computing sparse solutions to linear inverse problems,” in Proceedings of the 1996 IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 1996), Vol. 3, pp. 1131-1334.
  40. J. Mannos and D. Sakrison, “The effects of visual fidelity criterion on the encoding of image,” IRE Trans. Inf. Theory 20, 525-536 (1974).
    [CrossRef]
  41. USC-SIPI image database, Signal and Image Processing Institute at the University of Southern California, http://sipi.usc.edu/database.
  42. A. Goshtasby, “Image registration by local approximation methods,” Image Vis. Comput. 6, 255-261 (1988).
    [CrossRef]

2007

P. Shankar and M. Neifeld, “Multiframe superresolution of binary images,” Appl. Opt. 46, 1211-1222 (2007).
[CrossRef] [PubMed]

M. Jiang, L. Xia, G. Shou, and M. Tang, “Combination of the LSQR method and a genetic algorithm for solving the electrocardiography inverse problem,” Phys. Med. Biol. 52, 1277-1294 (2007).
[CrossRef] [PubMed]

2006

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489-509 (2006).
[CrossRef]

P. Shankar, W. Hassenplaugh, R. Morrison, R. Stack, and M. Neifeld, “Multiaperture imaging,” Appl. Opt. 45, 2871-2883 (2006).
[CrossRef] [PubMed]

E. Candes, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207-1223 (2006).
[CrossRef]

2005

E. Candes and J. Romberg, “Signal recovery from random projections,” Proc. SPIE 5678, 76-86 (2005).
[CrossRef]

N. Valdivia and E. Williams, “Krylov subspace iterative methods for boundary element method based near-field acoustic holography,” J. Acoust. Soc. Am. 117, 711-724 (2005).
[CrossRef] [PubMed]

2004

I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. 57, 1413-1457 (2004).
[CrossRef]

D. Wipf and B. Rao, “Sparse Bayesian learning for basis selection,” IEEE Trans. Signal Process. 52, 2153-2164 (2004).
[CrossRef]

2003

M. Figueiredo and R. Nowak, “An EM algorithm for wavelet-based image restoration,” IEEE Trans. Image Process. 12, 906916 (2003).
[CrossRef]

2002

A. Fruchter and R. Hook, “Drizzle: a method for the linear reconstruction of undersampled images,” Publ. Astron. Soc. Pac. 114, 144-152 (2002).
[CrossRef]

S. Baker and T. Kanade, “Limits on super-resolution and how to break them,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 1167-1183 (2002).
[CrossRef]

2001

M. Kilmer and D. O'Leary, “Choosing regularization parameters in iterative methods for ill-posed problems,” SIAM J. Matrix Anal. Appl. 22, 1204-1221 (2001).
[CrossRef]

2000

M. Belge, M. Kilmer, and E. Miller. “Wavelet domain image restoration with adaptive edge-preserving regularization,” IEEE Trans. Image Process. 9, 597-608 (2000).
[CrossRef]

1999

B. Rao and K. Kreutz-Delgado, “An affine scaling methodology for best basis selection,” IEEE Trans. Signal Process. 47, 187-200 (1999).
[CrossRef]

1998

R. Hardie, K. Barnard, J. Bognar, E. Armstrong, and E. Watson, “High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system,” Opt. Eng. (Bellingham) 37, 247-260 (1998).
[CrossRef]

S. Chen, D. Donoho, and M. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput. (USA) 20, 33-61 (1998).
[CrossRef]

1997

R. Hardie, K. Bernard, and E. Armstrong, “Joint MAP registration and high-resolution image estimation using a sequence of undersampled images,” IEEE Trans. Image Process. 6, 1621-1633 (1997).
[CrossRef] [PubMed]

1995

D. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inf. Theory 41, 613-627 (1995).
[CrossRef]

1993

B. Jeffs and M. Gunsay, “Restoration of blurred star field images by maximally sparse optimization,” IEEE Trans. Image Process. 2, 202-211 (1993).
[CrossRef] [PubMed]

1992

P. Hansen, “Analysis of discrete ill-posed problems by means of the L-curve,” SIAM Rev. 34, 561-580 (1992).
[CrossRef]

1991

M. Irani and S. Peleg, “Improving resolution by image registration,” Comput. Vis. Graph. Image Process. 53, 231-239 (1991).

1989

S. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674-693 (1989).
[CrossRef]

1988

A. Goshtasby, “Image registration by local approximation methods,” Image Vis. Comput. 6, 255-261 (1988).
[CrossRef]

1987

M. Figueiredo, R. Nowak, and S. Wright, “Gradient projections for sparse reconstruction: Application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Signal Process. 4, 586-597 (1987).

1984

R. Tsai and T. Huang, “Multiframe image restoration and registration,” Advances in Computer Vision and Image Processing 1, 317-339 (1984).

1982

C. Paige and M. Saunders, “LSQR: An algorithm for sparse linear equations and sparse least squares,” ACM Trans. Math. Softw. 8, 43-71 (1982).
[CrossRef]

C. Paige and M. Saunders, “LSQR: Sparse linear equations and least squares problems,” ACM Trans. Math. Softw. 8, 195-209 (1982).
[CrossRef]

1974

J. Mannos and D. Sakrison, “The effects of visual fidelity criterion on the encoding of image,” IRE Trans. Inf. Theory 20, 525-536 (1974).
[CrossRef]

Aizawa, K.

K. Aizawa, T. Komatsu, and T. Saito, “A scheme for acquiring very high resolution images using multiple cameras,” in IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1992), Vol. 3, pp. 23-26.

Andrews, H.

H. Andrews and B. Hunt, Digital Image Restoration (Prentice-Hall, 1977).

Armstrong, E.

R. Hardie, K. Barnard, J. Bognar, E. Armstrong, and E. Watson, “High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system,” Opt. Eng. (Bellingham) 37, 247-260 (1998).
[CrossRef]

R. Hardie, K. Bernard, and E. Armstrong, “Joint MAP registration and high-resolution image estimation using a sequence of undersampled images,” IEEE Trans. Image Process. 6, 1621-1633 (1997).
[CrossRef] [PubMed]

Baker, S.

S. Baker and T. Kanade, “Limits on super-resolution and how to break them,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 1167-1183 (2002).
[CrossRef]

Baraniuk, R.

M. Duarte, M. Wakin, and R. Baraniuk, “Fast reconstruction of piecewise smooth signals from random projections,” Online Proceedings of the Workshop on Signal Processing with Adaptative Sparse Structured Representations, SPARS 2005, http://spars05.irisa.fr/ACTES/TS5-3.pdf.

Barnard, K.

R. Hardie, K. Barnard, J. Bognar, E. Armstrong, and E. Watson, “High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system,” Opt. Eng. (Bellingham) 37, 247-260 (1998).
[CrossRef]

Barrett, H.

H. Barrett and K. Myers, Foundations of Image Science (Wiley Series in Pure and Applied Optics, 2004).

Belge, M.

M. Belge, M. Kilmer, and E. Miller. “Wavelet domain image restoration with adaptive edge-preserving regularization,” IEEE Trans. Image Process. 9, 597-608 (2000).
[CrossRef]

Bernard, K.

R. Hardie, K. Bernard, and E. Armstrong, “Joint MAP registration and high-resolution image estimation using a sequence of undersampled images,” IEEE Trans. Image Process. 6, 1621-1633 (1997).
[CrossRef] [PubMed]

Bognar, J.

R. Hardie, K. Barnard, J. Bognar, E. Armstrong, and E. Watson, “High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system,” Opt. Eng. (Bellingham) 37, 247-260 (1998).
[CrossRef]

Boyd, S.

E. Cands, M. Wakin, and S. Boyd, “Enhancing sparsity by reweighted L1 minimization,” Technical Report, California Institute of Technology, http://www.acm.caltech.edu/emmanuel/papers/rwll-oct2007.pdf.

Bresler, Y.

G. Harikumar and Y. Bresler, “A new algorithm for computing sparse solutions to linear inverse problems,” in Proceedings of the 1996 IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 1996), Vol. 3, pp. 1131-1334.

Candes, E.

E. Candes, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207-1223 (2006).
[CrossRef]

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489-509 (2006).
[CrossRef]

E. Candes and J. Romberg, “Signal recovery from random projections,” Proc. SPIE 5678, 76-86 (2005).
[CrossRef]

Cands, E.

E. Cands, M. Wakin, and S. Boyd, “Enhancing sparsity by reweighted L1 minimization,” Technical Report, California Institute of Technology, http://www.acm.caltech.edu/emmanuel/papers/rwll-oct2007.pdf.

Chen, S.

S. Chen, D. Donoho, and M. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput. (USA) 20, 33-61 (1998).
[CrossRef]

Daubechies, I.

I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. 57, 1413-1457 (2004).
[CrossRef]

De Mol, C.

I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. 57, 1413-1457 (2004).
[CrossRef]

Defrise, M.

I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. 57, 1413-1457 (2004).
[CrossRef]

Donoho, D.

S. Chen, D. Donoho, and M. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput. (USA) 20, 33-61 (1998).
[CrossRef]

D. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inf. Theory 41, 613-627 (1995).
[CrossRef]

Duarte, M.

M. Duarte, M. Wakin, and R. Baraniuk, “Fast reconstruction of piecewise smooth signals from random projections,” Online Proceedings of the Workshop on Signal Processing with Adaptative Sparse Structured Representations, SPARS 2005, http://spars05.irisa.fr/ACTES/TS5-3.pdf.

Figueiredo, M.

M. Figueiredo and R. Nowak, “An EM algorithm for wavelet-based image restoration,” IEEE Trans. Image Process. 12, 906916 (2003).
[CrossRef]

M. Figueiredo, R. Nowak, and S. Wright, “Gradient projections for sparse reconstruction: Application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Signal Process. 4, 586-597 (1987).

Fruchter, A.

A. Fruchter and R. Hook, “Drizzle: a method for the linear reconstruction of undersampled images,” Publ. Astron. Soc. Pac. 114, 144-152 (2002).
[CrossRef]

Golub, G.

N. Nguyen, G. Golub, and P. Milanfar, “Blind restoration/superresolution with generalized cross-validation using Gauss-type quadrature rules,” in Proceedings of the 33rd Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, Calif., October 1999, pp. 1257-1261.

Goshtasby, A.

A. Goshtasby, “Image registration by local approximation methods,” Image Vis. Comput. 6, 255-261 (1988).
[CrossRef]

Gunsay, M.

B. Jeffs and M. Gunsay, “Restoration of blurred star field images by maximally sparse optimization,” IEEE Trans. Image Process. 2, 202-211 (1993).
[CrossRef] [PubMed]

Hansen, P.

P. Hansen, “Analysis of discrete ill-posed problems by means of the L-curve,” SIAM Rev. 34, 561-580 (1992).
[CrossRef]

P. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion (SIAM, 1998).
[CrossRef]

Hardie, R.

R. Hardie, K. Barnard, J. Bognar, E. Armstrong, and E. Watson, “High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system,” Opt. Eng. (Bellingham) 37, 247-260 (1998).
[CrossRef]

R. Hardie, K. Bernard, and E. Armstrong, “Joint MAP registration and high-resolution image estimation using a sequence of undersampled images,” IEEE Trans. Image Process. 6, 1621-1633 (1997).
[CrossRef] [PubMed]

Harikumar, G.

G. Harikumar and Y. Bresler, “A new algorithm for computing sparse solutions to linear inverse problems,” in Proceedings of the 1996 IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 1996), Vol. 3, pp. 1131-1334.

Hassenplaugh, W.

Hook, R.

A. Fruchter and R. Hook, “Drizzle: a method for the linear reconstruction of undersampled images,” Publ. Astron. Soc. Pac. 114, 144-152 (2002).
[CrossRef]

Huang, T.

R. Tsai and T. Huang, “Multiframe image restoration and registration,” Advances in Computer Vision and Image Processing 1, 317-339 (1984).

Hunt, B.

H. Andrews and B. Hunt, Digital Image Restoration (Prentice-Hall, 1977).

Irani, M.

M. Irani and S. Peleg, “Improving resolution by image registration,” Comput. Vis. Graph. Image Process. 53, 231-239 (1991).

Jeffs, B.

B. Jeffs and M. Gunsay, “Restoration of blurred star field images by maximally sparse optimization,” IEEE Trans. Image Process. 2, 202-211 (1993).
[CrossRef] [PubMed]

Jiang, M.

M. Jiang, L. Xia, G. Shou, and M. Tang, “Combination of the LSQR method and a genetic algorithm for solving the electrocardiography inverse problem,” Phys. Med. Biol. 52, 1277-1294 (2007).
[CrossRef] [PubMed]

Kanade, T.

S. Baker and T. Kanade, “Limits on super-resolution and how to break them,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 1167-1183 (2002).
[CrossRef]

Kilmer, M.

M. Kilmer and D. O'Leary, “Choosing regularization parameters in iterative methods for ill-posed problems,” SIAM J. Matrix Anal. Appl. 22, 1204-1221 (2001).
[CrossRef]

M. Belge, M. Kilmer, and E. Miller. “Wavelet domain image restoration with adaptive edge-preserving regularization,” IEEE Trans. Image Process. 9, 597-608 (2000).
[CrossRef]

Komatsu, T.

K. Aizawa, T. Komatsu, and T. Saito, “A scheme for acquiring very high resolution images using multiple cameras,” in IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1992), Vol. 3, pp. 23-26.

Kreutz-Delgado, K.

B. Rao and K. Kreutz-Delgado, “An affine scaling methodology for best basis selection,” IEEE Trans. Signal Process. 47, 187-200 (1999).
[CrossRef]

Mallat, S.

S. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674-693 (1989).
[CrossRef]

S. Mallat, “A compact multiresolution representation: the wavelet model,” presented at the IEEE Workshop Computer Society on Computer Vision, Miami, Florida, December 2-7, 1987.

Mannos, J.

J. Mannos and D. Sakrison, “The effects of visual fidelity criterion on the encoding of image,” IRE Trans. Inf. Theory 20, 525-536 (1974).
[CrossRef]

Milanfar, P.

N. Nguyen, G. Golub, and P. Milanfar, “Blind restoration/superresolution with generalized cross-validation using Gauss-type quadrature rules,” in Proceedings of the 33rd Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, Calif., October 1999, pp. 1257-1261.

Miller, E.

M. Belge, M. Kilmer, and E. Miller. “Wavelet domain image restoration with adaptive edge-preserving regularization,” IEEE Trans. Image Process. 9, 597-608 (2000).
[CrossRef]

Morrison, R.

Myers, K.

H. Barrett and K. Myers, Foundations of Image Science (Wiley Series in Pure and Applied Optics, 2004).

Neifeld, M.

Nguyen, N.

N. Nguyen, G. Golub, and P. Milanfar, “Blind restoration/superresolution with generalized cross-validation using Gauss-type quadrature rules,” in Proceedings of the 33rd Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, Calif., October 1999, pp. 1257-1261.

Nowak, R.

M. Figueiredo and R. Nowak, “An EM algorithm for wavelet-based image restoration,” IEEE Trans. Image Process. 12, 906916 (2003).
[CrossRef]

M. Figueiredo, R. Nowak, and S. Wright, “Gradient projections for sparse reconstruction: Application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Signal Process. 4, 586-597 (1987).

O'Leary, D.

M. Kilmer and D. O'Leary, “Choosing regularization parameters in iterative methods for ill-posed problems,” SIAM J. Matrix Anal. Appl. 22, 1204-1221 (2001).
[CrossRef]

Paige, C.

C. Paige and M. Saunders, “LSQR: An algorithm for sparse linear equations and sparse least squares,” ACM Trans. Math. Softw. 8, 43-71 (1982).
[CrossRef]

C. Paige and M. Saunders, “LSQR: Sparse linear equations and least squares problems,” ACM Trans. Math. Softw. 8, 195-209 (1982).
[CrossRef]

Peleg, S.

M. Irani and S. Peleg, “Improving resolution by image registration,” Comput. Vis. Graph. Image Process. 53, 231-239 (1991).

Pentland, A. P.

M. A. Turk and A. P. Pentland, “Face recognition using eigenfaces,” in IEEE Proceedings on Computer Vision and Pattern Recognition (IEEE, 1991), pp. 586-591.
[CrossRef]

Rao, B.

D. Wipf and B. Rao, “Sparse Bayesian learning for basis selection,” IEEE Trans. Signal Process. 52, 2153-2164 (2004).
[CrossRef]

B. Rao and K. Kreutz-Delgado, “An affine scaling methodology for best basis selection,” IEEE Trans. Signal Process. 47, 187-200 (1999).
[CrossRef]

Romberg, J.

E. Candes, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207-1223 (2006).
[CrossRef]

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489-509 (2006).
[CrossRef]

E. Candes and J. Romberg, “Signal recovery from random projections,” Proc. SPIE 5678, 76-86 (2005).
[CrossRef]

Saito, T.

K. Aizawa, T. Komatsu, and T. Saito, “A scheme for acquiring very high resolution images using multiple cameras,” in IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1992), Vol. 3, pp. 23-26.

Sakrison, D.

J. Mannos and D. Sakrison, “The effects of visual fidelity criterion on the encoding of image,” IRE Trans. Inf. Theory 20, 525-536 (1974).
[CrossRef]

Saunders, M.

S. Chen, D. Donoho, and M. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput. (USA) 20, 33-61 (1998).
[CrossRef]

C. Paige and M. Saunders, “LSQR: Sparse linear equations and least squares problems,” ACM Trans. Math. Softw. 8, 195-209 (1982).
[CrossRef]

C. Paige and M. Saunders, “LSQR: An algorithm for sparse linear equations and sparse least squares,” ACM Trans. Math. Softw. 8, 43-71 (1982).
[CrossRef]

Sbaiz, L.

P. Vandewalle, L. Sbaiz, J. Vandewalle, and M. Vetterli, “How to take advantage of aliasing in bandlimited signals,” in Proceedings of the IEEE International Conference on Accoustics, Speech, and Signal Processing (IEEE, 2004), pp. 948-951.

Shankar, P.

Shou, G.

M. Jiang, L. Xia, G. Shou, and M. Tang, “Combination of the LSQR method and a genetic algorithm for solving the electrocardiography inverse problem,” Phys. Med. Biol. 52, 1277-1294 (2007).
[CrossRef] [PubMed]

Stack, R.

Tang, M.

M. Jiang, L. Xia, G. Shou, and M. Tang, “Combination of the LSQR method and a genetic algorithm for solving the electrocardiography inverse problem,” Phys. Med. Biol. 52, 1277-1294 (2007).
[CrossRef] [PubMed]

Tao, T.

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489-509 (2006).
[CrossRef]

E. Candes, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207-1223 (2006).
[CrossRef]

Tsai, R.

R. Tsai and T. Huang, “Multiframe image restoration and registration,” Advances in Computer Vision and Image Processing 1, 317-339 (1984).

Turk, M. A.

M. A. Turk and A. P. Pentland, “Face recognition using eigenfaces,” in IEEE Proceedings on Computer Vision and Pattern Recognition (IEEE, 1991), pp. 586-591.
[CrossRef]

Valdivia, N.

N. Valdivia and E. Williams, “Krylov subspace iterative methods for boundary element method based near-field acoustic holography,” J. Acoust. Soc. Am. 117, 711-724 (2005).
[CrossRef] [PubMed]

Vandewalle, J.

P. Vandewalle, L. Sbaiz, J. Vandewalle, and M. Vetterli, “How to take advantage of aliasing in bandlimited signals,” in Proceedings of the IEEE International Conference on Accoustics, Speech, and Signal Processing (IEEE, 2004), pp. 948-951.

Vandewalle, P.

P. Vandewalle, L. Sbaiz, J. Vandewalle, and M. Vetterli, “How to take advantage of aliasing in bandlimited signals,” in Proceedings of the IEEE International Conference on Accoustics, Speech, and Signal Processing (IEEE, 2004), pp. 948-951.

Vetterli, M.

P. Vandewalle, L. Sbaiz, J. Vandewalle, and M. Vetterli, “How to take advantage of aliasing in bandlimited signals,” in Proceedings of the IEEE International Conference on Accoustics, Speech, and Signal Processing (IEEE, 2004), pp. 948-951.

Wakin, M.

E. Cands, M. Wakin, and S. Boyd, “Enhancing sparsity by reweighted L1 minimization,” Technical Report, California Institute of Technology, http://www.acm.caltech.edu/emmanuel/papers/rwll-oct2007.pdf.

M. Duarte, M. Wakin, and R. Baraniuk, “Fast reconstruction of piecewise smooth signals from random projections,” Online Proceedings of the Workshop on Signal Processing with Adaptative Sparse Structured Representations, SPARS 2005, http://spars05.irisa.fr/ACTES/TS5-3.pdf.

Watson, E.

R. Hardie, K. Barnard, J. Bognar, E. Armstrong, and E. Watson, “High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system,” Opt. Eng. (Bellingham) 37, 247-260 (1998).
[CrossRef]

Williams, E.

N. Valdivia and E. Williams, “Krylov subspace iterative methods for boundary element method based near-field acoustic holography,” J. Acoust. Soc. Am. 117, 711-724 (2005).
[CrossRef] [PubMed]

Wipf, D.

D. Wipf and B. Rao, “Sparse Bayesian learning for basis selection,” IEEE Trans. Signal Process. 52, 2153-2164 (2004).
[CrossRef]

Wright, S.

M. Figueiredo, R. Nowak, and S. Wright, “Gradient projections for sparse reconstruction: Application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Signal Process. 4, 586-597 (1987).

Xia, L.

M. Jiang, L. Xia, G. Shou, and M. Tang, “Combination of the LSQR method and a genetic algorithm for solving the electrocardiography inverse problem,” Phys. Med. Biol. 52, 1277-1294 (2007).
[CrossRef] [PubMed]

ACM Trans. Math. Softw.

C. Paige and M. Saunders, “LSQR: An algorithm for sparse linear equations and sparse least squares,” ACM Trans. Math. Softw. 8, 43-71 (1982).
[CrossRef]

C. Paige and M. Saunders, “LSQR: Sparse linear equations and least squares problems,” ACM Trans. Math. Softw. 8, 195-209 (1982).
[CrossRef]

Advances in Computer Vision and Image Processing

R. Tsai and T. Huang, “Multiframe image restoration and registration,” Advances in Computer Vision and Image Processing 1, 317-339 (1984).

Appl. Opt.

Commun. Pure Appl. Math.

I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. 57, 1413-1457 (2004).
[CrossRef]

E. Candes, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207-1223 (2006).
[CrossRef]

Comput. Vis. Graph. Image Process.

M. Irani and S. Peleg, “Improving resolution by image registration,” Comput. Vis. Graph. Image Process. 53, 231-239 (1991).

IEEE J. Sel. Top. Signal Process.

M. Figueiredo, R. Nowak, and S. Wright, “Gradient projections for sparse reconstruction: Application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Signal Process. 4, 586-597 (1987).

IEEE Trans. Image Process.

M. Figueiredo and R. Nowak, “An EM algorithm for wavelet-based image restoration,” IEEE Trans. Image Process. 12, 906916 (2003).
[CrossRef]

M. Belge, M. Kilmer, and E. Miller. “Wavelet domain image restoration with adaptive edge-preserving regularization,” IEEE Trans. Image Process. 9, 597-608 (2000).
[CrossRef]

B. Jeffs and M. Gunsay, “Restoration of blurred star field images by maximally sparse optimization,” IEEE Trans. Image Process. 2, 202-211 (1993).
[CrossRef] [PubMed]

R. Hardie, K. Bernard, and E. Armstrong, “Joint MAP registration and high-resolution image estimation using a sequence of undersampled images,” IEEE Trans. Image Process. 6, 1621-1633 (1997).
[CrossRef] [PubMed]

IEEE Trans. Inf. Theory

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489-509 (2006).
[CrossRef]

D. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inf. Theory 41, 613-627 (1995).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell.

S. Baker and T. Kanade, “Limits on super-resolution and how to break them,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 1167-1183 (2002).
[CrossRef]

S. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674-693 (1989).
[CrossRef]

IEEE Trans. Signal Process.

D. Wipf and B. Rao, “Sparse Bayesian learning for basis selection,” IEEE Trans. Signal Process. 52, 2153-2164 (2004).
[CrossRef]

B. Rao and K. Kreutz-Delgado, “An affine scaling methodology for best basis selection,” IEEE Trans. Signal Process. 47, 187-200 (1999).
[CrossRef]

Image Vis. Comput.

A. Goshtasby, “Image registration by local approximation methods,” Image Vis. Comput. 6, 255-261 (1988).
[CrossRef]

IRE Trans. Inf. Theory

J. Mannos and D. Sakrison, “The effects of visual fidelity criterion on the encoding of image,” IRE Trans. Inf. Theory 20, 525-536 (1974).
[CrossRef]

J. Acoust. Soc. Am.

N. Valdivia and E. Williams, “Krylov subspace iterative methods for boundary element method based near-field acoustic holography,” J. Acoust. Soc. Am. 117, 711-724 (2005).
[CrossRef] [PubMed]

Opt. Eng. (Bellingham)

R. Hardie, K. Barnard, J. Bognar, E. Armstrong, and E. Watson, “High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system,” Opt. Eng. (Bellingham) 37, 247-260 (1998).
[CrossRef]

Phys. Med. Biol.

M. Jiang, L. Xia, G. Shou, and M. Tang, “Combination of the LSQR method and a genetic algorithm for solving the electrocardiography inverse problem,” Phys. Med. Biol. 52, 1277-1294 (2007).
[CrossRef] [PubMed]

Proc. SPIE

E. Candes and J. Romberg, “Signal recovery from random projections,” Proc. SPIE 5678, 76-86 (2005).
[CrossRef]

Publ. Astron. Soc. Pac.

A. Fruchter and R. Hook, “Drizzle: a method for the linear reconstruction of undersampled images,” Publ. Astron. Soc. Pac. 114, 144-152 (2002).
[CrossRef]

SIAM J. Matrix Anal. Appl.

M. Kilmer and D. O'Leary, “Choosing regularization parameters in iterative methods for ill-posed problems,” SIAM J. Matrix Anal. Appl. 22, 1204-1221 (2001).
[CrossRef]

SIAM J. Sci. Comput. (USA)

S. Chen, D. Donoho, and M. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput. (USA) 20, 33-61 (1998).
[CrossRef]

SIAM Rev.

P. Hansen, “Analysis of discrete ill-posed problems by means of the L-curve,” SIAM Rev. 34, 561-580 (1992).
[CrossRef]

Other

S. Mallat, “A compact multiresolution representation: the wavelet model,” presented at the IEEE Workshop Computer Society on Computer Vision, Miami, Florida, December 2-7, 1987.

G. Harikumar and Y. Bresler, “A new algorithm for computing sparse solutions to linear inverse problems,” in Proceedings of the 1996 IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 1996), Vol. 3, pp. 1131-1334.

P. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion (SIAM, 1998).
[CrossRef]

H. Barrett and K. Myers, Foundations of Image Science (Wiley Series in Pure and Applied Optics, 2004).

M. A. Turk and A. P. Pentland, “Face recognition using eigenfaces,” in IEEE Proceedings on Computer Vision and Pattern Recognition (IEEE, 1991), pp. 586-591.
[CrossRef]

E. Cands, M. Wakin, and S. Boyd, “Enhancing sparsity by reweighted L1 minimization,” Technical Report, California Institute of Technology, http://www.acm.caltech.edu/emmanuel/papers/rwll-oct2007.pdf.

M. Duarte, M. Wakin, and R. Baraniuk, “Fast reconstruction of piecewise smooth signals from random projections,” Online Proceedings of the Workshop on Signal Processing with Adaptative Sparse Structured Representations, SPARS 2005, http://spars05.irisa.fr/ACTES/TS5-3.pdf.

S.Chaudhuri, ed., Super-Resolution Imaging (Kluwer, 2001).

K. Aizawa, T. Komatsu, and T. Saito, “A scheme for acquiring very high resolution images using multiple cameras,” in IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1992), Vol. 3, pp. 23-26.

P. Vandewalle, L. Sbaiz, J. Vandewalle, and M. Vetterli, “How to take advantage of aliasing in bandlimited signals,” in Proceedings of the IEEE International Conference on Accoustics, Speech, and Signal Processing (IEEE, 2004), pp. 948-951.

N. Nguyen, G. Golub, and P. Milanfar, “Blind restoration/superresolution with generalized cross-validation using Gauss-type quadrature rules,” in Proceedings of the 33rd Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, Calif., October 1999, pp. 1257-1261.

H. Andrews and B. Hunt, Digital Image Restoration (Prentice-Hall, 1977).

USC-SIPI image database, Signal and Image Processing Institute at the University of Southern California, http://sipi.usc.edu/database.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (20)

Fig. 1
Fig. 1

Example of a 128 × 128 pixel object in the (a) space domain with 16 384 nonzero pixels, and (b) wavelet domain with 4079 nonzero wavelet coefficients. The wavelet decomposition is performed at three resolution levels using the Daubechies-4 filter. The wavelet coefficients in each subband are scaled to enhance image contrast for display purpose.

Fig. 2
Fig. 2

Schematic of an imaging channel showing degradations in a typical multiframe imaging system. H c = S H c c d H o p t and H = H c T .

Fig. 3
Fig. 3

Examples of generalized Gaussian density functions. β = 2 refers to a Gaussian density and β = 1 refers to a Laplacian density.

Fig. 4
Fig. 4

Example object model in the wavelet domain using a generalized Gaussian density. (a) A 128 × 128 pixel natural object, and (b) histogram of wavelet coefficients from (a) in the LH subband of the 2nd decomposition level along with a GGD fit. The fit parameters are α = 0.383 and β = 0.685 . The Daubechies-4 filter is used for computing the wavelet transform.

Fig. 5
Fig. 5

Schematic of a 3-level wavelet decomposition and the associated GGD model parameters for each subband.

Fig. 6
Fig. 6

Example of the diffraction-limited optical point-spread function. The blur spot covers one object pixel. The measurement pixel width is twice the object pixel width.

Fig. 7
Fig. 7

Set of 26 sparse objects of size 128 × 128 pixels used for training. The number below each object is the percentage of nonzero wavelet coefficients.

Fig. 8
Fig. 8

Example demonstrating GGD model inadequacy for a very sparse wavelet subband. Only nonzero wavelet coefficients whose absolute value is greater than or equal to z m i n are used to obtain a GGD function fit. The knowledge of z m i n is also used in the sparse reconstruction algorithm.

Fig. 9
Fig. 9

Sample restored images at SNR = 20 dB by using algorithms described in this paper. (a) A 128 × 128 pixel object, and (b)–(d) three LR images degraded by optical blur with w = 0.5 and D = 2 pixel downsampling. Images were restored from K = 2 LR frames and using the (e) LMMSE, (f) EM, (g) 1 penalty, and (h) GGD penalty algorithms. They have RMSE values of 6.33%, 6,49%, 5.97%, and 6.08%, respectively. Images in (i), (j), (k), and (l) are reconstructed from K = 4 LR frames using LMMSE, EM, 1 and GGD penalty, respectively. They have RMSE values of 5.75%, 5.73%, 5.31%, and 5.43%, respectively.

Fig. 10
Fig. 10

Example of the evolution of the doubly iterative algorithm outlined in Subsection 3B for 1 penalty (solid curve) and GGD penalty (dashed curve), (a) Total objective cost ( C ( k ) ) , regularization cost ( λ ( k ) L ( k ) z ̂ ( k ) 2 ) , and estimation error ( z z ̂ ( k ) 2 ) versus iteration number k, (b) regularization parameters versus k, and (c) 0 (%) norm of estimates versus k.

Fig. 11
Fig. 11

Plot of reconstruction RMSE versus SNR for the case D = 2 pixel downsampling, w = 0.5 degree of optical blur, and (a) K = 1 , (b) K = 2 , (c) K = 3 , and (d) K = 4 LR images. The RMSE is calculated only over the pixels that are seen in all K images.

Fig. 12
Fig. 12

Plot of visually weighted RMSE versus SNR for the case D = 2 pixel downsampling, w = 0.5 degree of optical blur, and (a) K = 1 , (b) K = 2 , (c) K = 3 , and (d) K = 4 LR images.

Fig. 13
Fig. 13

The plot of 0 norm (%) of reconstructions versus SNR for the case D = 2 pixel downsampling, w = 0.5 degree of optical blur, and (a) K = 1 , (b) K = 2 , (c) K = 3 , and (d) K = 4 LR images.

Fig. 14
Fig. 14

Sample restored images at SNR = 50 dB by using algorithms described in this paper. (a) A 128 × 128 pixel object, and (b)–(d) three LR images that are degraded by optical blur with w = 1.0 and D = 4 pixel downsampling. Object estimates from K = 4 such LR images using the (e) LMMSE, (f) EM, (g) 1 penalty, and (h) GGD penalty algorithms. Images in (e), (f), (g), and (h) have RMSE values of 5.81%, 7.06%, 4.92%, and 5.09%, respectively. The estimates were obtained from K = 12 LR images by using the (i) LMMSE, (j) EM, (k) 1 , and (l) GGD penalty methods, respectively. They have RMSE values of 4.50%, 5.15%, 3.71%, and 3.71%, respectively.

Fig. 15
Fig. 15

Plot of reconstruction RMSE versus SNR for the case D = 4 pixel downsampling, w = 1.0 degree of optical blur, and (a) K = 4 , (b) K = 8 , and (c) K = 12 LR images.

Fig. 16
Fig. 16

Plot of visually weighted RMSE versus SNR for the case D = 4 pixel downsampling, w = 1.0 degree of optical blur, and (a) K = 4 , (b) K = 8 , and (c) K = 12 LR images.

Fig. 17
Fig. 17

Plot of 0 norm (%) of reconstruction versus SNR for the case D = 4 pixel downsampling, w = 1.0 degree of optical blur, and (a) K = 4 , (b) K = 8 , and (c) K = 12 LR images.

Fig. 18
Fig. 18

Snapshot of the experimental setup. The cameras are arranged in an affine geometry and the objects are displayed on the plasma monitor.

Fig. 19
Fig. 19

Sample reconstructions of a sparse object in space domain. (a) A captured image that is degraded by D = 2 pixel downsampling, (b) the least square (minimum L 2 norm) estimate, and (c) the space-domain 1 penalty reconstruction from K = 4 frames. (d) A captured image that is degraded by D = 4 pixel downsampling, (e) the least-squares estimate, and (f) the space-domain 1 penalty reconstruction from K = 4 frames.

Fig. 20
Fig. 20

Sample reconstructions of a sparse object in wavelet domain. (a) A captured image that is degraded by D = 2 pixel downsampling, and images reconstructed using K = 4 such frames and the (b) LMMSE, (c) EM, (d) 1 penalty, and (e) GGD penalty methods. (f) A captured image that is degraded by D = 4 pixel downsampling, and images reconstructed using K = 4 such frames and the (g) LMMSE, (h) EM, (i) 1 penalty, and (j) GGD penalty methods.

Tables (2)

Tables Icon

Table 1 GGD Function Parameters Obtained by Fitting the Histograms of Wavelet Coefficients of Training Objects Shown in Fig. 7 a

Tables Icon

Table 2 Point-Spread Functions

Equations (22)

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

y k = S H c c d H o p t T k f + n k .
y k = H k f + n k ,
y = H f + n ,
W f = C f H T [ H C f H T + σ 2 I ] 1 ,
y = H f + n = H W T W f = Q z + n ,
z ̂ = arg min z z 1
such that Q z y 2 2 ϵ ,
z ̂ = arg min z [ Q z y 2 2 + λ z 1 ] ,
z ̂ ( k + 1 ) = z ̂ ( k ) + Q T ( y Q z ̂ ( k ) ) = sgn ( z ̂ ( k + 1 ) ) × max ( z ̂ ( k + 1 ) τ , 0 ) .
z ̂ ( y , λ ) = arg min z [ Q z y 2 2 + λ L z 2 2 ] ,
z ̂ ( k + 1 ) = arg min z [ Q z y 2 2 + λ ( k + 1 ) L ( k ) z 2 2 ] ,
λ ( k + 1 ) = arg min λ L ( k ) z ̂ ( y , λ , k ) 2 2
such that Q z ̂ ( y , λ , k ) y 2 2 P σ 2 ,
z ̂ ( y , λ , k ) = arg min z [ Q z y 2 2 + λ L ( k ) z 2 2 ] .
C ( k ) = Q z ̂ ( k ) y 2 2 + λ ( k ) L ( k 1 ) z ̂ ( k ) 2 2 ,
a ̂ ( k + 1 ) = arg min a Q [ L ( k ) ] 1 a y 2 2 + λ ( k + 1 ) a 2 2 ,
z ̂ ( k + 1 ) = [ L ( k ) ] 1 a ̂ ( k + 1 ) .
L ( k ) z ̂ ( k + 1 ) 2 2 = L ( k ) z ̂ ( y , λ ( k + 1 ) ) 2 2 L ( k ) z ̂ ( y , 0 ) 2 2 = L ( k 1 ) z ̂ ( k ) 2 2 .
p ( z ; α , β ) = β 2 α Γ ( 1 β ) e ( z α ) β ,
z ̂ = max z P ( z y ) = max z P ( y z ) P ( z ) P ( y ) ( Bayes rule ) , = max z P ( y z ) P ( z ) ( y is fixed ) .
P ( z ) = i = 0 N 1 β i ̃ 2 α i ̃ Γ ( 1 β i ̃ ) e ( z i α i ̃ ) β i ̃ ,
z ̂ = arg min z [ y Q z 2 2 + λ i = 0 N 1 ( z i α i ̃ ) β i ̃ ] ,

Metrics