Abstract

Super-resolution image reconstruction, which has been a hot research topic in recent years, is a process to reconstruct high-resolution images from shifted, low-resolution, degraded observations. Among the available reconstruction frameworks, the maximum a posteriori (MAP) model is widely used. However, existing methods usually employ a fixed prior item and regularization parameter for the entire HR image, ignoring local spatially adaptive properties, and the large computation load caused by the solution of the large-scale ill-posed problem is another issue to be noted. In this paper, a block-based local spatially adaptive reconstruction algorithm is proposed. To reduce the large computation load and realize the local spatially adaptive process of the prior model and regularization parameter, first the target image is divided into several same-sized blocks and the structure tensor is used to analyze the local spatial properties of each block. Different property prior items and regularization parameters are then applied adaptively to different properties’ blocks. Experimental results show that the proposed method achieves better performance than methods with a fixed prior item and regularization parameter.

© 2011 Optical Society of America

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2009 (4)

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

Takeda, P. Milanfar, M. Protter, and M. Elad, “Super-resolution without explicit subpixel motion estimation,” IEEE Trans. Image Process. 18, pp. 1958–1975 (2009).
[CrossRef] [PubMed]

H. Ji and C. Fermuller, “Robust wavelet-based super-resolution reconstruction: theory and algorithm,” IEEE Trans. Pattern Anal. Mach. Intell. 31, 649–660 (2009).
[CrossRef] [PubMed]

H. Shen and L. Zhang, “A MAP-based algorithm for destriping and inpainting of remotely sensed images,” IEEE Trans. Geosci. Remote Sens. 47, 1492–1502 (2009)
[CrossRef]

2008 (2)

L. C. Pickup, D. P. Capel, S. J. Roberts, and A. Zisserman, “Bayesian methods for image super-resolution,” Comput. J. (UK) 52, 101–113 (2008).
[CrossRef]

X. Zhang and K.-M. Lam, “Image magnification based on a blockwise adaptive Markov random field model,” Image Vision Comput. 26, 1277–1284 (2008).
[CrossRef]

2007 (2)

H. Shen, L. Zhang, B. Huang, and P. Li, “A MAP approach for joint motion estimation, segmentation and super-resolution,” IEEE Trans. Image Process. 16, 479–490 (2007).
[CrossRef] [PubMed]

M. K. Ng, H. Shen, E. Y. Lam, and L. Zhang, “A total variation regularization based super-resolution reconstruction algorithm for digital video,” EURASIP J. Adv. Signal Process. Article ID 74585 (2007).
[CrossRef]

2006 (4)

P. Vandewalle, S. Susstrunk, and A. Vetterli, “A frequency domain approach to registration of aliased images with application to super-resolution,” EURASIP J. Appl. Signal Process. , Article ID 71459 (2006).
[CrossRef]

N. A. Woods, N. P. Galatsanos, and A. K. Katsaggelos, “Stochastic methods for joint registration, restoration, and interpolation of multiple undersampled images,” IEEE Trans. Image Process. 15, 201–213 (2006).
[CrossRef] [PubMed]

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

T. Brox, J. Weickert, B. Burgeth, and P. Mrázek, “Nonlinear structure tensors,” Image Vis. Comput. 24, 41–55 (2006).
[CrossRef]

2004 (5)

Z. Wang, A. C. Bovik, and H. R. Sheikh, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Proc. 13, 600–612 (2004).
[CrossRef]

Z. Lin and H. Shum, “Fundamental limits of reconstruction-based superresolution algorithms under local translation,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 83––97 (2004).
[CrossRef] [PubMed]

S. Farsiu, M. D. Robinson, M. Elad, and P. Milanfar, “Fast and robust multi-frame super resolution,” IEEE Trans. Image Process. 13, 1327–1344 (2004).
[CrossRef] [PubMed]

S. Farsiu, M. D. Robinson, M. Elad, and P. Milanfar, “Advances and challenges in super-resolution,” Int. J. Imaging Syst. Technol. 14, 47–57 (2004).
[CrossRef]

M. K. Ng, C. K. Sze, and S. P. Yung, “Wavelet algorithms for deblurring models,” Int. J. Imaging Syst. Technol. 14, 113–121(2004).
[CrossRef]

2003 (2)

R. H. Chan, T. F. Chan, L. Shen, and Z. Shen, “Wavelet algorithms for high-resolution image reconstruction,” SIAM J. Sci. Comput. 24, 1408–1432 (2003).
[CrossRef]

S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20, 21–36 (2003).
[CrossRef]

2002 (1)

R. Gonsalves and F. Khaghani, “Super resolution based on low-resolution, warped images,” Proc. SPIE 4790, 10–20(2002).

2000 (2)

M. S. Alam, J. G. Bognar, R. C. Hardie, and B. J. Yasuda, “Infrared image registration and high-resolution reconstruction using multiple translationally shifted aliased video frames,” IEEE Trans. Instrum. Meas. 49, 915–923 (2000).
[CrossRef]

N. Nguyen and P. Milanfar, “A wavelet-based interpolation restoration method for superresolution (wavelet superresolution),” Circuits, Systems, Signal Process. 19, 321–338 (2000).
[CrossRef]

1998 (1)

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

1997 (3)

M. Elad and A. Feuer, “Restoration of a single superresolution image from several blurred, noisy, and undersampled measured images,” IEEE Trans. Image Proc. 6, 1646–1658 (1997).
[CrossRef]

A. J. Patti, M. I. Sezan, and A. M. Tekalp, “Super resolution video reconstruction with arbitrary sampling lattices and nonzero aperture time,” IEEE Trans. Image Process. 6, 1064–1076(1997).
[CrossRef] [PubMed]

R. C. Hardie, K. J. Barnard, and E. 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]

1996 (1)

R. R. Schulz and R. L. Stevenson, “Extraction of high-resolution frames from video sequences,” IEEE Trans. Image Process. 5, 996–1011 (1996).
[CrossRef]

1993 (2)

M. Irani and S. Peleg, “Motion analysis for image enhancement resolution, occlusion, and transparency,” J. Visual Commun. Image Represent. 4, 324–335 (1993).
[CrossRef]

S. P. Kim and W. Y. Su, “Recursive high-resolution reconstruction of blurred multiframe images,” IEEE Trans. Image Process. 2, 534–539 (1993).
[CrossRef] [PubMed]

1992 (1)

H. Ur and D. Gross, “Improved resolution from sub-pixel shifted pictures,” CVGIP: Graph. Models Image Process 54, 181–186(1992).
[CrossRef]

1991 (1)

M. Irani and S. Peleg, “Improving resolution by image registration,” CVGIP: Graph. Models Image Process. 53, 231–239 (1991).
[CrossRef]

1990 (1)

S. P. Kim, N. K. Bose, and H. M. Valenzuela, “Recursive reconstruction of high resolution image from noisy undersampled multiframes,” IEEE Trans. Acoust. Speech Signal Process. 38, 1013–1027 (1990).
[CrossRef]

1989 (1)

1984 (1)

R. Y. Tsai and T. S. Huang, “Multi-frame image restoration and registration,” Adv. Comput. Vision Image Process. 1, 317–339(1984).

1982 (1)

T. Peli and J. S. Lim, “Adaptive filtering for image enhancement,” Opt. Eng. 21, 108–112 (1982)

1980 (1)

J. Lim, “Image restoration by short space spectral subtraction,” IEEE Trans. Acoust. Speech Signal Process. 28, 191–197(1980).
[CrossRef]

1972 (1)

T. G. Stockman, “Image processing in the context of a visual model,” Proc. IEEE 60, 828–842 (1972).
[CrossRef]

Alam, M. S.

M. S. Alam, J. G. Bognar, R. C. Hardie, and B. J. Yasuda, “Infrared image registration and high-resolution reconstruction using multiple translationally shifted aliased video frames,” IEEE Trans. Instrum. Meas. 49, 915–923 (2000).
[CrossRef]

Armstrong, E. E.

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

R. C. Hardie, K. J. Barnard, and E. 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]

Barnard, K. J.

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

R. C. Hardie, K. J. Barnard, and E. 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]

Bigun, J.

J. Bigun and G. H. Granlund, “Optimal orientation detection of linear symmetry,” Proceedings First International Conference on Computer Vision (IEEE, 1987), pp. 433–438.

Bognar, J. G.

M. S. Alam, J. G. Bognar, R. C. Hardie, and B. J. Yasuda, “Infrared image registration and high-resolution reconstruction using multiple translationally shifted aliased video frames,” IEEE Trans. Instrum. Meas. 49, 915–923 (2000).
[CrossRef]

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

Bose, N. K.

S. P. Kim, N. K. Bose, and H. M. Valenzuela, “Recursive reconstruction of high resolution image from noisy undersampled multiframes,” IEEE Trans. Acoust. Speech Signal Process. 38, 1013–1027 (1990).
[CrossRef]

N. K. Bose, H. C. Kim, and H. M. Valenzuela, “Recursive implementation of total least squares algorithm for image reconstruction from noisy, undersampled multiframes,” Proceedings of IEEE Conference on Acoustics, Speech and Signal Processing (IEEE, 1993), pp. 269–272.
[CrossRef]

Bovik, A. C.

Z. Wang, A. C. Bovik, and H. R. Sheikh, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Proc. 13, 600–612 (2004).
[CrossRef]

Brox, T.

T. Brox, J. Weickert, B. Burgeth, and P. Mrázek, “Nonlinear structure tensors,” Image Vis. Comput. 24, 41–55 (2006).
[CrossRef]

Burgeth, B.

T. Brox, J. Weickert, B. Burgeth, and P. Mrázek, “Nonlinear structure tensors,” Image Vis. Comput. 24, 41–55 (2006).
[CrossRef]

Capel, D. P.

L. C. Pickup, D. P. Capel, S. J. Roberts, and A. Zisserman, “Bayesian methods for image super-resolution,” Comput. J. (UK) 52, 101–113 (2008).
[CrossRef]

Chan, R. H.

R. H. Chan, T. F. Chan, L. Shen, and Z. Shen, “Wavelet algorithms for high-resolution image reconstruction,” SIAM J. Sci. Comput. 24, 1408–1432 (2003).
[CrossRef]

Chan, T. F.

R. H. Chan, T. F. Chan, L. Shen, and Z. Shen, “Wavelet algorithms for high-resolution image reconstruction,” SIAM J. Sci. Comput. 24, 1408–1432 (2003).
[CrossRef]

Chaudhuri, S.

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

Elad, M.

Takeda, P. Milanfar, M. Protter, and M. Elad, “Super-resolution without explicit subpixel motion estimation,” IEEE Trans. Image Process. 18, pp. 1958–1975 (2009).
[CrossRef] [PubMed]

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

S. Farsiu, M. D. Robinson, M. Elad, and P. Milanfar, “Fast and robust multi-frame super resolution,” IEEE Trans. Image Process. 13, 1327–1344 (2004).
[CrossRef] [PubMed]

S. Farsiu, M. D. Robinson, M. Elad, and P. Milanfar, “Advances and challenges in super-resolution,” Int. J. Imaging Syst. Technol. 14, 47–57 (2004).
[CrossRef]

M. Elad and A. Feuer, “Restoration of a single superresolution image from several blurred, noisy, and undersampled measured images,” IEEE Trans. Image Proc. 6, 1646–1658 (1997).
[CrossRef]

Farsiu, S.

S. Farsiu, M. D. Robinson, M. Elad, and P. Milanfar, “Advances and challenges in super-resolution,” Int. J. Imaging Syst. Technol. 14, 47–57 (2004).
[CrossRef]

S. Farsiu, M. D. Robinson, M. Elad, and P. Milanfar, “Fast and robust multi-frame super resolution,” IEEE Trans. Image Process. 13, 1327–1344 (2004).
[CrossRef] [PubMed]

Fermuller, C.

H. Ji and C. Fermuller, “Robust wavelet-based super-resolution reconstruction: theory and algorithm,” IEEE Trans. Pattern Anal. Mach. Intell. 31, 649–660 (2009).
[CrossRef] [PubMed]

Feuer, A.

M. Elad and A. Feuer, “Restoration of a single superresolution image from several blurred, noisy, and undersampled measured images,” IEEE Trans. Image Proc. 6, 1646–1658 (1997).
[CrossRef]

Forstner, W.

W. Forstner and E. Gulch, “A fast operator for detection and precise location of distinct points, corners and centres of circular features,” Proceedings ISPRS Intercommission Conference on Fast Processing of Photogrammetric Data (Academic, 1987), pp. 281–305.

Galatsanos, N. P.

N. A. Woods, N. P. Galatsanos, and A. K. Katsaggelos, “Stochastic methods for joint registration, restoration, and interpolation of multiple undersampled images,” IEEE Trans. Image Process. 15, 201–213 (2006).
[CrossRef] [PubMed]

Golub, G. H.

G. H. Golub and C. F. van Loan, Matrix Computation, 3rd ed. (Johns Hopkins University Press, 1996)

Gonsalves, R.

R. Gonsalves and F. Khaghani, “Super resolution based on low-resolution, warped images,” Proc. SPIE 4790, 10–20(2002).

Granlund, G. H.

J. Bigun and G. H. Granlund, “Optimal orientation detection of linear symmetry,” Proceedings First International Conference on Computer Vision (IEEE, 1987), pp. 433–438.

Gross, D.

H. Ur and D. Gross, “Improved resolution from sub-pixel shifted pictures,” CVGIP: Graph. Models Image Process 54, 181–186(1992).
[CrossRef]

Gulch, E.

W. Forstner and E. Gulch, “A fast operator for detection and precise location of distinct points, corners and centres of circular features,” Proceedings ISPRS Intercommission Conference on Fast Processing of Photogrammetric Data (Academic, 1987), pp. 281–305.

Hardie, R. C.

M. S. Alam, J. G. Bognar, R. C. Hardie, and B. J. Yasuda, “Infrared image registration and high-resolution reconstruction using multiple translationally shifted aliased video frames,” IEEE Trans. Instrum. Meas. 49, 915–923 (2000).
[CrossRef]

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

R. C. Hardie, K. J. Barnard, and E. 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]

Huang, B.

H. Shen, L. Zhang, B. Huang, and P. Li, “A MAP approach for joint motion estimation, segmentation and super-resolution,” IEEE Trans. Image Process. 16, 479–490 (2007).
[CrossRef] [PubMed]

Huang, T. S.

R. Y. Tsai and T. S. Huang, “Multi-frame image restoration and registration,” Adv. Comput. Vision Image Process. 1, 317–339(1984).

Irani, M.

M. Irani and S. Peleg, “Motion analysis for image enhancement resolution, occlusion, and transparency,” J. Visual Commun. Image Represent. 4, 324–335 (1993).
[CrossRef]

M. Irani and S. Peleg, “Improving resolution by image registration,” CVGIP: Graph. Models Image Process. 53, 231–239 (1991).
[CrossRef]

Ji, H.

H. Ji and C. Fermuller, “Robust wavelet-based super-resolution reconstruction: theory and algorithm,” IEEE Trans. Pattern Anal. Mach. Intell. 31, 649–660 (2009).
[CrossRef] [PubMed]

Kang, M. G.

S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20, 21–36 (2003).
[CrossRef]

Katsaggelos, A. K.

N. A. Woods, N. P. Galatsanos, and A. K. Katsaggelos, “Stochastic methods for joint registration, restoration, and interpolation of multiple undersampled images,” IEEE Trans. Image Process. 15, 201–213 (2006).
[CrossRef] [PubMed]

B. C. Tom and A. K. Katsaggelos, “Reconstruction of a high-resolution image by simultaneous registration, restoration, and interpolation of low-resolution images,” Proceedings of 1995 IEEE International Conference on Image Processing (IEEE, 1995), pp. 539–542.
[CrossRef]

A. K. Katsaggelos, R. Molina, and J. Mateos, Super Resolution of Images and Video (Morgan and Claypool, 2007).

Khaghani, F.

R. Gonsalves and F. Khaghani, “Super resolution based on low-resolution, warped images,” Proc. SPIE 4790, 10–20(2002).

Kim, H. C.

N. K. Bose, H. C. Kim, and H. M. Valenzuela, “Recursive implementation of total least squares algorithm for image reconstruction from noisy, undersampled multiframes,” Proceedings of IEEE Conference on Acoustics, Speech and Signal Processing (IEEE, 1993), pp. 269–272.
[CrossRef]

Kim, S. P.

S. P. Kim and W. Y. Su, “Recursive high-resolution reconstruction of blurred multiframe images,” IEEE Trans. Image Process. 2, 534–539 (1993).
[CrossRef] [PubMed]

S. P. Kim, N. K. Bose, and H. M. Valenzuela, “Recursive reconstruction of high resolution image from noisy undersampled multiframes,” IEEE Trans. Acoust. Speech Signal Process. 38, 1013–1027 (1990).
[CrossRef]

Lam, E. Y.

M. K. Ng, H. Shen, E. Y. Lam, and L. Zhang, “A total variation regularization based super-resolution reconstruction algorithm for digital video,” EURASIP J. Adv. Signal Process. Article ID 74585 (2007).
[CrossRef]

Lam, K.-M.

X. Zhang and K.-M. Lam, “Image magnification based on a blockwise adaptive Markov random field model,” Image Vision Comput. 26, 1277–1284 (2008).
[CrossRef]

Li, P.

H. Shen, L. Zhang, B. Huang, and P. Li, “A MAP approach for joint motion estimation, segmentation and super-resolution,” IEEE Trans. Image Process. 16, 479–490 (2007).
[CrossRef] [PubMed]

Lim, J.

J. Lim, “Image restoration by short space spectral subtraction,” IEEE Trans. Acoust. Speech Signal Process. 28, 191–197(1980).
[CrossRef]

Lim, J. S.

T. Peli and J. S. Lim, “Adaptive filtering for image enhancement,” Opt. Eng. 21, 108–112 (1982)

Lin, Z.

Z. Lin and H. Shum, “Fundamental limits of reconstruction-based superresolution algorithms under local translation,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 83––97 (2004).
[CrossRef] [PubMed]

Mateos, J.

A. K. Katsaggelos, R. Molina, and J. Mateos, Super Resolution of Images and Video (Morgan and Claypool, 2007).

Milanfar, P.

Takeda, P. Milanfar, M. Protter, and M. Elad, “Super-resolution without explicit subpixel motion estimation,” IEEE Trans. Image Process. 18, pp. 1958–1975 (2009).
[CrossRef] [PubMed]

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

S. Farsiu, M. D. Robinson, M. Elad, and P. Milanfar, “Fast and robust multi-frame super resolution,” IEEE Trans. Image Process. 13, 1327–1344 (2004).
[CrossRef] [PubMed]

S. Farsiu, M. D. Robinson, M. Elad, and P. Milanfar, “Advances and challenges in super-resolution,” Int. J. Imaging Syst. Technol. 14, 47–57 (2004).
[CrossRef]

N. Nguyen and P. Milanfar, “A wavelet-based interpolation restoration method for superresolution (wavelet superresolution),” Circuits, Systems, Signal Process. 19, 321–338 (2000).
[CrossRef]

P. Milanfar, Super Resolution imaging (CRC Press, 2010).

Molina, R.

A. K. Katsaggelos, R. Molina, and J. Mateos, Super Resolution of Images and Video (Morgan and Claypool, 2007).

Mrázek, P.

T. Brox, J. Weickert, B. Burgeth, and P. Mrázek, “Nonlinear structure tensors,” Image Vis. Comput. 24, 41–55 (2006).
[CrossRef]

Ng, M. K.

M. K. Ng, H. Shen, E. Y. Lam, and L. Zhang, “A total variation regularization based super-resolution reconstruction algorithm for digital video,” EURASIP J. Adv. Signal Process. Article ID 74585 (2007).
[CrossRef]

M. K. Ng, C. K. Sze, and S. P. Yung, “Wavelet algorithms for deblurring models,” Int. J. Imaging Syst. Technol. 14, 113–121(2004).
[CrossRef]

Nguyen, N.

N. Nguyen and P. Milanfar, “A wavelet-based interpolation restoration method for superresolution (wavelet superresolution),” Circuits, Systems, Signal Process. 19, 321–338 (2000).
[CrossRef]

Oskoui, P.

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]

Park, M. K.

S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20, 21–36 (2003).
[CrossRef]

Park, S. C.

S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20, 21–36 (2003).
[CrossRef]

Patti, A. J.

A. J. Patti, M. I. Sezan, and A. M. Tekalp, “Super resolution video reconstruction with arbitrary sampling lattices and nonzero aperture time,” IEEE Trans. Image Process. 6, 1064–1076(1997).
[CrossRef] [PubMed]

Peleg, S.

M. Irani and S. Peleg, “Motion analysis for image enhancement resolution, occlusion, and transparency,” J. Visual Commun. Image Represent. 4, 324–335 (1993).
[CrossRef]

M. Irani and S. Peleg, “Improving resolution by image registration,” CVGIP: Graph. Models Image Process. 53, 231–239 (1991).
[CrossRef]

Peli, T.

T. Peli and J. S. Lim, “Adaptive filtering for image enhancement,” Opt. Eng. 21, 108–112 (1982)

Pickup, L. C.

L. C. Pickup, D. P. Capel, S. J. Roberts, and A. Zisserman, “Bayesian methods for image super-resolution,” Comput. J. (UK) 52, 101–113 (2008).
[CrossRef]

L. C. Pickup, “Machine learning in multi-frame image super-resolution,” Ph.D. (University of Oxford, 2007).

Protter, M.

Takeda, P. Milanfar, M. Protter, and M. Elad, “Super-resolution without explicit subpixel motion estimation,” IEEE Trans. Image Process. 18, pp. 1958–1975 (2009).
[CrossRef] [PubMed]

M. Protter, M. Elad, H. Takeda, and P. Milanfar, “Generalizing the nonlocal-means to super-resolution reconstruction,” IEEE Trans. Image Process. 18, pp. 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]

Roberts, S. J.

L. C. Pickup, D. P. Capel, S. J. Roberts, and A. Zisserman, “Bayesian methods for image super-resolution,” Comput. J. (UK) 52, 101–113 (2008).
[CrossRef]

Robinson, M. D.

S. Farsiu, M. D. Robinson, M. Elad, and P. Milanfar, “Fast and robust multi-frame super resolution,” IEEE Trans. Image Process. 13, 1327–1344 (2004).
[CrossRef] [PubMed]

S. Farsiu, M. D. Robinson, M. Elad, and P. Milanfar, “Advances and challenges in super-resolution,” Int. J. Imaging Syst. Technol. 14, 47–57 (2004).
[CrossRef]

Schulz, R. R.

R. R. Schulz and R. L. Stevenson, “Extraction of high-resolution frames from video sequences,” IEEE Trans. Image Process. 5, 996–1011 (1996).
[CrossRef]

Sezan, M. I.

A. J. Patti, M. I. Sezan, and A. M. Tekalp, “Super resolution video reconstruction with arbitrary sampling lattices and nonzero aperture time,” IEEE Trans. Image Process. 6, 1064–1076(1997).
[CrossRef] [PubMed]

Sheikh, H. R.

Z. Wang, A. C. Bovik, and H. R. Sheikh, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Proc. 13, 600–612 (2004).
[CrossRef]

Shen, H.

H. Shen and L. Zhang, “A MAP-based algorithm for destriping and inpainting of remotely sensed images,” IEEE Trans. Geosci. Remote Sens. 47, 1492–1502 (2009)
[CrossRef]

H. Shen, L. Zhang, B. Huang, and P. Li, “A MAP approach for joint motion estimation, segmentation and super-resolution,” IEEE Trans. Image Process. 16, 479–490 (2007).
[CrossRef] [PubMed]

M. K. Ng, H. Shen, E. Y. Lam, and L. Zhang, “A total variation regularization based super-resolution reconstruction algorithm for digital video,” EURASIP J. Adv. Signal Process. Article ID 74585 (2007).
[CrossRef]

Shen, L.

R. H. Chan, T. F. Chan, L. Shen, and Z. Shen, “Wavelet algorithms for high-resolution image reconstruction,” SIAM J. Sci. Comput. 24, 1408–1432 (2003).
[CrossRef]

Shen, Z.

R. H. Chan, T. F. Chan, L. Shen, and Z. Shen, “Wavelet algorithms for high-resolution image reconstruction,” SIAM J. Sci. Comput. 24, 1408–1432 (2003).
[CrossRef]

Shum, H.

Z. Lin and H. Shum, “Fundamental limits of reconstruction-based superresolution algorithms under local translation,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 83––97 (2004).
[CrossRef] [PubMed]

Stark, H.

Stevenson, R. L.

R. R. Schulz and R. L. Stevenson, “Extraction of high-resolution frames from video sequences,” IEEE Trans. Image Process. 5, 996–1011 (1996).
[CrossRef]

Stockman, T. G.

T. G. Stockman, “Image processing in the context of a visual model,” Proc. IEEE 60, 828–842 (1972).
[CrossRef]

Su, H.

H. Su, L. Tang, D. Tretter, and J. Zhou, “A practical and adaptive framework for super-resolution,” Proceedings of IEEE International Conference on Image Processing 1 (IEEE, 2008), pp. 1236–1249.

Su, W. Y.

S. P. Kim and W. Y. Su, “Recursive high-resolution reconstruction of blurred multiframe images,” IEEE Trans. Image Process. 2, 534–539 (1993).
[CrossRef] [PubMed]

Susstrunk, S.

P. Vandewalle, S. Susstrunk, and A. Vetterli, “A frequency domain approach to registration of aliased images with application to super-resolution,” EURASIP J. Appl. Signal Process. , Article ID 71459 (2006).
[CrossRef]

Sze, C. K.

M. K. Ng, C. K. Sze, and S. P. Yung, “Wavelet algorithms for deblurring models,” Int. J. Imaging Syst. Technol. 14, 113–121(2004).
[CrossRef]

Takeda,

Takeda, P. Milanfar, M. Protter, and M. Elad, “Super-resolution without explicit subpixel motion estimation,” IEEE Trans. Image Process. 18, pp. 1958–1975 (2009).
[CrossRef] [PubMed]

Takeda, H.

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

Tang, L.

H. Su, L. Tang, D. Tretter, and J. Zhou, “A practical and adaptive framework for super-resolution,” Proceedings of IEEE International Conference on Image Processing 1 (IEEE, 2008), pp. 1236–1249.

Tekalp, A. M.

A. J. Patti, M. I. Sezan, and A. M. Tekalp, “Super resolution video reconstruction with arbitrary sampling lattices and nonzero aperture time,” IEEE Trans. Image Process. 6, 1064–1076(1997).
[CrossRef] [PubMed]

Tom, B. C.

B. C. Tom and A. K. Katsaggelos, “Reconstruction of a high-resolution image by simultaneous registration, restoration, and interpolation of low-resolution images,” Proceedings of 1995 IEEE International Conference on Image Processing (IEEE, 1995), pp. 539–542.
[CrossRef]

Tretter, D.

H. Su, L. Tang, D. Tretter, and J. Zhou, “A practical and adaptive framework for super-resolution,” Proceedings of IEEE International Conference on Image Processing 1 (IEEE, 2008), pp. 1236–1249.

Tsai, R. Y.

R. Y. Tsai and T. S. Huang, “Multi-frame image restoration and registration,” Adv. Comput. Vision Image Process. 1, 317–339(1984).

Ur, H.

H. Ur and D. Gross, “Improved resolution from sub-pixel shifted pictures,” CVGIP: Graph. Models Image Process 54, 181–186(1992).
[CrossRef]

Valenzuela, H. M.

S. P. Kim, N. K. Bose, and H. M. Valenzuela, “Recursive reconstruction of high resolution image from noisy undersampled multiframes,” IEEE Trans. Acoust. Speech Signal Process. 38, 1013–1027 (1990).
[CrossRef]

N. K. Bose, H. C. Kim, and H. M. Valenzuela, “Recursive implementation of total least squares algorithm for image reconstruction from noisy, undersampled multiframes,” Proceedings of IEEE Conference on Acoustics, Speech and Signal Processing (IEEE, 1993), pp. 269–272.
[CrossRef]

van Loan, C. F.

G. H. Golub and C. F. van Loan, Matrix Computation, 3rd ed. (Johns Hopkins University Press, 1996)

Vandewalle, P.

P. Vandewalle, S. Susstrunk, and A. Vetterli, “A frequency domain approach to registration of aliased images with application to super-resolution,” EURASIP J. Appl. Signal Process. , Article ID 71459 (2006).
[CrossRef]

Vetterli, A.

P. Vandewalle, S. Susstrunk, and A. Vetterli, “A frequency domain approach to registration of aliased images with application to super-resolution,” EURASIP J. Appl. Signal Process. , Article ID 71459 (2006).
[CrossRef]

Wang, Z.

Z. Wang, A. C. Bovik, and H. R. Sheikh, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Proc. 13, 600–612 (2004).
[CrossRef]

Watson, E. A.

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

Weickert, J.

T. Brox, J. Weickert, B. Burgeth, and P. Mrázek, “Nonlinear structure tensors,” Image Vis. Comput. 24, 41–55 (2006).
[CrossRef]

Woods, N. A.

N. A. Woods, N. P. Galatsanos, and A. K. Katsaggelos, “Stochastic methods for joint registration, restoration, and interpolation of multiple undersampled images,” IEEE Trans. Image Process. 15, 201–213 (2006).
[CrossRef] [PubMed]

Yasuda, B. J.

M. S. Alam, J. G. Bognar, R. C. Hardie, and B. J. Yasuda, “Infrared image registration and high-resolution reconstruction using multiple translationally shifted aliased video frames,” IEEE Trans. Instrum. Meas. 49, 915–923 (2000).
[CrossRef]

Yung, S. P.

M. K. Ng, C. K. Sze, and S. P. Yung, “Wavelet algorithms for deblurring models,” Int. J. Imaging Syst. Technol. 14, 113–121(2004).
[CrossRef]

Zhang, L.

H. Shen and L. Zhang, “A MAP-based algorithm for destriping and inpainting of remotely sensed images,” IEEE Trans. Geosci. Remote Sens. 47, 1492–1502 (2009)
[CrossRef]

M. K. Ng, H. Shen, E. Y. Lam, and L. Zhang, “A total variation regularization based super-resolution reconstruction algorithm for digital video,” EURASIP J. Adv. Signal Process. Article ID 74585 (2007).
[CrossRef]

H. Shen, L. Zhang, B. Huang, and P. Li, “A MAP approach for joint motion estimation, segmentation and super-resolution,” IEEE Trans. Image Process. 16, 479–490 (2007).
[CrossRef] [PubMed]

Zhang, X.

X. Zhang and K.-M. Lam, “Image magnification based on a blockwise adaptive Markov random field model,” Image Vision Comput. 26, 1277–1284 (2008).
[CrossRef]

Zhou, J.

H. Su, L. Tang, D. Tretter, and J. Zhou, “A practical and adaptive framework for super-resolution,” Proceedings of IEEE International Conference on Image Processing 1 (IEEE, 2008), pp. 1236–1249.

Zisserman, A.

L. C. Pickup, D. P. Capel, S. J. Roberts, and A. Zisserman, “Bayesian methods for image super-resolution,” Comput. J. (UK) 52, 101–113 (2008).
[CrossRef]

Adv. Comput. Vision Image Process. (1)

R. Y. Tsai and T. S. Huang, “Multi-frame image restoration and registration,” Adv. Comput. Vision Image Process. 1, 317–339(1984).

Circuits, Systems, Signal Process. (1)

N. Nguyen and P. Milanfar, “A wavelet-based interpolation restoration method for superresolution (wavelet superresolution),” Circuits, Systems, Signal Process. 19, 321–338 (2000).
[CrossRef]

Comput. J. (UK) (1)

L. C. Pickup, D. P. Capel, S. J. Roberts, and A. Zisserman, “Bayesian methods for image super-resolution,” Comput. J. (UK) 52, 101–113 (2008).
[CrossRef]

CVGIP: Graph. Models Image Process (1)

H. Ur and D. Gross, “Improved resolution from sub-pixel shifted pictures,” CVGIP: Graph. Models Image Process 54, 181–186(1992).
[CrossRef]

CVGIP: Graph. Models Image Process. (1)

M. Irani and S. Peleg, “Improving resolution by image registration,” CVGIP: Graph. Models Image Process. 53, 231–239 (1991).
[CrossRef]

EURASIP J. Adv. Signal Process. (1)

M. K. Ng, H. Shen, E. Y. Lam, and L. Zhang, “A total variation regularization based super-resolution reconstruction algorithm for digital video,” EURASIP J. Adv. Signal Process. Article ID 74585 (2007).
[CrossRef]

EURASIP J. Appl. Signal Process. (1)

P. Vandewalle, S. Susstrunk, and A. Vetterli, “A frequency domain approach to registration of aliased images with application to super-resolution,” EURASIP J. Appl. Signal Process. , Article ID 71459 (2006).
[CrossRef]

IEEE Signal Process. Mag. (1)

S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20, 21–36 (2003).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process. (2)

J. Lim, “Image restoration by short space spectral subtraction,” IEEE Trans. Acoust. Speech Signal Process. 28, 191–197(1980).
[CrossRef]

S. P. Kim, N. K. Bose, and H. M. Valenzuela, “Recursive reconstruction of high resolution image from noisy undersampled multiframes,” IEEE Trans. Acoust. Speech Signal Process. 38, 1013–1027 (1990).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

H. Shen and L. Zhang, “A MAP-based algorithm for destriping and inpainting of remotely sensed images,” IEEE Trans. Geosci. Remote Sens. 47, 1492–1502 (2009)
[CrossRef]

IEEE Trans. Image Proc. (2)

Z. Wang, A. C. Bovik, and H. R. Sheikh, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Proc. 13, 600–612 (2004).
[CrossRef]

M. Elad and A. Feuer, “Restoration of a single superresolution image from several blurred, noisy, and undersampled measured images,” IEEE Trans. Image Proc. 6, 1646–1658 (1997).
[CrossRef]

IEEE Trans. Image Process. (10)

R. R. Schulz and R. L. Stevenson, “Extraction of high-resolution frames from video sequences,” IEEE Trans. Image Process. 5, 996–1011 (1996).
[CrossRef]

R. C. Hardie, K. J. Barnard, and E. 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]

H. Shen, L. Zhang, B. Huang, and P. Li, “A MAP approach for joint motion estimation, segmentation and super-resolution,” IEEE Trans. Image Process. 16, 479–490 (2007).
[CrossRef] [PubMed]

N. A. Woods, N. P. Galatsanos, and A. K. Katsaggelos, “Stochastic methods for joint registration, restoration, and interpolation of multiple undersampled images,” IEEE Trans. Image Process. 15, 201–213 (2006).
[CrossRef] [PubMed]

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

Takeda, P. Milanfar, M. Protter, and M. Elad, “Super-resolution without explicit subpixel motion estimation,” IEEE Trans. Image Process. 18, pp. 1958–1975 (2009).
[CrossRef] [PubMed]

S. P. Kim and W. Y. Su, “Recursive high-resolution reconstruction of blurred multiframe images,” IEEE Trans. Image Process. 2, 534–539 (1993).
[CrossRef] [PubMed]

S. Farsiu, M. D. Robinson, M. Elad, and P. Milanfar, “Fast and robust multi-frame super resolution,” IEEE Trans. Image Process. 13, 1327–1344 (2004).
[CrossRef] [PubMed]

A. J. Patti, M. I. Sezan, and A. M. Tekalp, “Super resolution video reconstruction with arbitrary sampling lattices and nonzero aperture time,” IEEE Trans. Image Process. 6, 1064–1076(1997).
[CrossRef] [PubMed]

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

IEEE Trans. Instrum. Meas. (1)

M. S. Alam, J. G. Bognar, R. C. Hardie, and B. J. Yasuda, “Infrared image registration and high-resolution reconstruction using multiple translationally shifted aliased video frames,” IEEE Trans. Instrum. Meas. 49, 915–923 (2000).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (2)

H. Ji and C. Fermuller, “Robust wavelet-based super-resolution reconstruction: theory and algorithm,” IEEE Trans. Pattern Anal. Mach. Intell. 31, 649–660 (2009).
[CrossRef] [PubMed]

Z. Lin and H. Shum, “Fundamental limits of reconstruction-based superresolution algorithms under local translation,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 83––97 (2004).
[CrossRef] [PubMed]

Image Vis. Comput. (1)

T. Brox, J. Weickert, B. Burgeth, and P. Mrázek, “Nonlinear structure tensors,” Image Vis. Comput. 24, 41–55 (2006).
[CrossRef]

Image Vision Comput. (1)

X. Zhang and K.-M. Lam, “Image magnification based on a blockwise adaptive Markov random field model,” Image Vision Comput. 26, 1277–1284 (2008).
[CrossRef]

Int. J. Imaging Syst. Technol. (2)

M. K. Ng, C. K. Sze, and S. P. Yung, “Wavelet algorithms for deblurring models,” Int. J. Imaging Syst. Technol. 14, 113–121(2004).
[CrossRef]

S. Farsiu, M. D. Robinson, M. Elad, and P. Milanfar, “Advances and challenges in super-resolution,” Int. J. Imaging Syst. Technol. 14, 47–57 (2004).
[CrossRef]

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

J. Visual Commun. Image Represent. (1)

M. Irani and S. Peleg, “Motion analysis for image enhancement resolution, occlusion, and transparency,” J. Visual Commun. Image Represent. 4, 324–335 (1993).
[CrossRef]

Opt. Eng. (2)

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

T. Peli and J. S. Lim, “Adaptive filtering for image enhancement,” Opt. Eng. 21, 108–112 (1982)

Proc. IEEE (1)

T. G. Stockman, “Image processing in the context of a visual model,” Proc. IEEE 60, 828–842 (1972).
[CrossRef]

Proc. SPIE (1)

R. Gonsalves and F. Khaghani, “Super resolution based on low-resolution, warped images,” Proc. SPIE 4790, 10–20(2002).

SIAM J. Sci. Comput. (1)

R. H. Chan, T. F. Chan, L. Shen, and Z. Shen, “Wavelet algorithms for high-resolution image reconstruction,” SIAM J. Sci. Comput. 24, 1408–1432 (2003).
[CrossRef]

Other (11)

N. K. Bose, H. C. Kim, and H. M. Valenzuela, “Recursive implementation of total least squares algorithm for image reconstruction from noisy, undersampled multiframes,” Proceedings of IEEE Conference on Acoustics, Speech and Signal Processing (IEEE, 1993), pp. 269–272.
[CrossRef]

H. Su, L. Tang, D. Tretter, and J. Zhou, “A practical and adaptive framework for super-resolution,” Proceedings of IEEE International Conference on Image Processing 1 (IEEE, 2008), pp. 1236–1249.

B. C. Tom and A. K. Katsaggelos, “Reconstruction of a high-resolution image by simultaneous registration, restoration, and interpolation of low-resolution images,” Proceedings of 1995 IEEE International Conference on Image Processing (IEEE, 1995), pp. 539–542.
[CrossRef]

A. K. Katsaggelos, R. Molina, and J. Mateos, Super Resolution of Images and Video (Morgan and Claypool, 2007).

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

P. Milanfar, Super Resolution imaging (CRC Press, 2010).

L. C. Pickup, “Machine learning in multi-frame image super-resolution,” Ph.D. (University of Oxford, 2007).

G. H. Golub and C. F. van Loan, Matrix Computation, 3rd ed. (Johns Hopkins University Press, 1996)

W. Forstner and E. Gulch, “A fast operator for detection and precise location of distinct points, corners and centres of circular features,” Proceedings ISPRS Intercommission Conference on Fast Processing of Photogrammetric Data (Academic, 1987), pp. 281–305.

J. Bigun and G. H. Granlund, “Optimal orientation detection of linear symmetry,” Proceedings First International Conference on Computer Vision (IEEE, 1987), pp. 433–438.

http://users.soe.ucsc.edu/~milanfar/software/sr-datasets.html

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Figures (12)

Fig. 1
Fig. 1

Degradation process of the HR image.

Fig. 2
Fig. 2

Curve of the Huber function.

Fig. 3
Fig. 3

Reconstruction with different fixed regularization parameters. (a) Regularization parameter that is too large. (b) Regularization parameter that is too small.

Fig. 4
Fig. 4

Deblock mechanism.

Fig. 5
Fig. 5

Flow chart of our reconstruction framework.

Fig. 6
Fig. 6

Reconstruction results of the first experiment. (a) Original HR image. (b) LR image. (c) Bilinear interpolation. (d) Laplacian prior result. (e) Huber-MRF prior result. (f) Proposed LSA result.

Fig. 7
Fig. 7

Edge extraction results of the first experiment. (a) Laplacian prior result. (b) Huber-MRF prior result. (c) Proposed LSA result.

Fig. 8
Fig. 8

Reconstruction results of the second experiment. (a) Original HR image. (b) LR image. (c) Bilinear interpolation. (d) Laplacian prior result. (e) Huber-MRF prior result. (f) Proposed LSA result.

Fig. 9
Fig. 9

Reconstruction results of the third experiment. (a) LR image. (b) Bilinear interpolation. (c) Bi-cubic interpolation. (d) Laplacian prior result. (e) Huber-MRF prior result. (f) Proposed LSA result.

Fig. 10
Fig. 10

Change in the MSE and computation time versus block size in the first experiment (a) The change of the MSE value versus block size (b) The change of the computation time versus block size.

Fig. 11
Fig. 11

Change in the MSE value versus the threshold parameter T in the first experiment.

Fig. 12
Fig. 12

Change in the MSE value versus the number of LR images.

Tables (2)

Tables Icon

Table 1 MSE and SSIM Values of Different Reconstruction Methods in the First Experiment

Tables Icon

Table 2 MSE and SSIM Values of Different Reconstruction Methods in the Second Experiment

Equations (19)

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

y k = D B k M k x + n k k = 1 , 2 p ,
y 1 = D 1 B 1 M 1 x + n 1 y 2 = D 2 B 2 M 2 x + n 2 y p = D p B p M p x + n p } y = D B M x + n ,
x ^ = arg min { y D B M x 2 2 + λ U ( x ) } .
S ( x , y ) = I ( x , y ) I ( x , y ) T ,
I ( x , y ) = [ G x G y ] ,
S m = 1 n i = 1 n S i ( x , y ) = 1 n i = 1 n I i ( x , y ) I i ( x , y ) T m = 1 t m ,
δ m = | λ 1 m | + | λ 2 m | .
U ( x ) smooth = Q x 2 2 .
U ( x ) edge = i , j c C ρ ( d c ( x i , j ) ) .
ρ ( x ) = { x 2 | x | μ 2 μ | x | μ 2 | x | > μ ,
x i 1 , j 1 x i 1 , j x i 1 , j + 1 x i , j 1 x i , j x i , j + 1 x i + 1 , j 1 x i + 1 , j x i + 1 , j + 1 { d c 1 ( x i , j ) = x i 1 , j 2 x i , j + x i + 1 , j d c 2 ( x i , j ) = x i , j 1 2 x i , j + x i , j + 1 d c 3 ( x i , j ) = 1 2 ( x i 1 , j 1 2 x i , j + x i + 1 , j + 1 ) d c 4 ( x i , j ) = 1 2 ( x i 1 , j + 1 2 x i , j + x i + 1 , j 1 ) .
α ( δ m ) = c δ m + ε ,
x ^ = arg min { y D B M x 2 2 + { α ( δ m ) Q x 2 2 if δ m T α ( δ m ) i , j c C ρ ( d c ( x i , j ) ) if δ m > T } .
M T B T D T ( y D B M x ) + α ( δ m ) γ = 0 ,
x n + 1 = x n β n r n r n = M T B T D T ( y D B M x n ) + α ( δ m ) γ ,
β n = ( r n ) T r n ( r n ) T W ( r n ) ,
x n + 1 x n 2 x n 2 d .
MSE = 1 N x ^ x 2 ,
SSIM = ( 2 μ x μ x ^ + C 1 ) ( 2 σ x x ^ + C 2 ) ( μ x 2 + μ x ^ 2 + C 1 ) ( σ x 2 + σ x ^ 2 + C 2 ,

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