Abstract

A single-frame multichannel blind image deconvolution technique has been formulated recently as a blind source separation problem solved by independent component analysis (ICA). The attractive feature of this approach is that neither origin nor size of the spatially invariant blurring kernel has to be known. To enhance the statistical independence among the hidden variables, we employ multiscale analysis implemented by wavelet packets and use mutual information to locate a subband with the least dependent components, where the basis matrix is learned by means of standard ICA. We show that the proposed algorithm is capable of performing blind deconvolution of nonstationary signals that are not independent and identically distributed processes. The image poses these properties. The algorithm is tested on experimental data and compared with state-of-the-art single-frame blind image deconvolution algorithms. Our good experimental results demonstrate the viability of the proposed concept.

© 2007 Optical Society of America

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  17. A. Cichocki, "Blind source separation: new tools for extraction of source signals and denoising," in Independent Component Analyses, Wavelets, Unsupervised Smart Sensors, and Neural Networks III, H.Szu, ed., Proc. SPIE 5818, 11-25 (2005).
  18. A. Cichocki and P. Georgiev, "Blind source separationalgorithms with matrix constraints," IEICE Trans. Fundamentals E86-A, 522-531 (2003).
  19. T. Tanaka and A. Cichocki, "Subband decomposition independent component analysis and new performance criteria," in Proceedings of the IEEE Conference on Acoustics, Speech, and Signal Processing (IEEE, 2004), pp. 541-544.
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  24. C. L. Bryne, "Accelerating the EMML algorithm and related iterative algorithms by rescaled block-iterative methods," IEEE Trans. Image Process. 7, 100-109 (1998).
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  29. S. J. Orfanidis, Optimum Signal Processing: An Introduction, 2nd ed. (Macmillan, 1988).
  30. M. Zibulevsky, P. Kisilev, Y. Zeevi, and B. Pearlmutter, "Blind source separation via multinode sparse representation," in Advances in Neural Information Processing Systems (Morgan Kaufman, 2002), pp. 185-191.
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    [CrossRef] [PubMed]
  32. P. Georgiev, F. Theis, and A. Cichocki, "Sparse component analysis and blind source separation of underdetermined mixtures," IEEE Trans. Neural Netw. 16, 992-996 (2005).
    [CrossRef] [PubMed]
  33. Y. Li, S. Amari, A. Cichocki, D. W. C. Ho, and S. Xie, "Underdetermined blind source separation based on sparse representation," IEEE Trans. Signal Process. 54, 423-437 (2006).
    [CrossRef]
  34. P. Kisilev, M. Zibulevsky, and Y. Y. Zeevi, "Multiscale framework for blind separation of linearly mixed signals," J. Mach. Learn. Res. 4, 1339-1363 (2003).
  35. J. F. Cardoso, "Dependence, correlation and Gaussianity in independent component analysis," J. Mach. Learn. Res. 4, 1177-1203 (2003).
  36. D. R. Brillinger, Time Series Data Analysis and Theory (McGraw-Hill, 1981).
  37. P. McCullagh, Tensor Methods in Statistics (Chapman & Hall, 1995).
  38. P. Comon, "Independent component analysis, a new concept?" Signal Process. 36, 287-314 (1994).
    [CrossRef]
  39. J. F. Cardoso and A. Souloumiac, "Blind beamforming for non-Gaussian signals," IEE Proc. F, Radar Signal Process. 140, 362-370 (1993).
    [CrossRef]
  40. P. O. Hoyer, "Non-negative matrix factorization with sparseness constraints," J. Mach. Learn. Res. 5, 1457-1469 (2004).
  41. D. D. Lee and H. S. Seung, "Learning the parts of objects by non-negative matrix factorization," Nature 401, 788-791 (1999).
    [CrossRef] [PubMed]
  42. A. Cichocki, R. Zdunek, and S. Amari, "Csiszár's divergences for non-negative matrix factorization: family of new algorithms," Lect. Notes Comput. Sci. 3889, 32-39 (2006).
    [CrossRef]

2006 (6)

I. Kopriva and D. Nuzillard, "Non-negative matrix factorization approach to blind image deconvolution," Lect. Notes Comput. Sci. 3889, 966-973 (2006).
[CrossRef]

I. Kopriva, D. J. Garrood, and V. Borjanovic, "Single frame blind image deconvolution by non-negative sparse matrix factorization," Opt. Commun. 266, 456-464 (2006).
[CrossRef]

K. Zhang and L. W. Chan, "An adaptive method for subband decomposition ICA," Neural Comput. 18, 191-223 (2006).
[CrossRef]

K. Zhang and L. W. Chan, "Enhancement of source independence for blind source separation," Lect. Notes Comput. Sci. 3889, 731-738 (2006).
[CrossRef]

Y. Li, S. Amari, A. Cichocki, D. W. C. Ho, and S. Xie, "Underdetermined blind source separation based on sparse representation," IEEE Trans. Signal Process. 54, 423-437 (2006).
[CrossRef]

A. Cichocki, R. Zdunek, and S. Amari, "Csiszár's divergences for non-negative matrix factorization: family of new algorithms," Lect. Notes Comput. Sci. 3889, 32-39 (2006).
[CrossRef]

2005 (4)

P. Georgiev, F. Theis, and A. Cichocki, "Sparse component analysis and blind source separation of underdetermined mixtures," IEEE Trans. Neural Netw. 16, 992-996 (2005).
[CrossRef] [PubMed]

A. Cichocki, "Blind source separation: new tools for extraction of source signals and denoising," in Independent Component Analyses, Wavelets, Unsupervised Smart Sensors, and Neural Networks III, H.Szu, ed., Proc. SPIE 5818, 11-25 (2005).

I. Kopriva, "Single-frame multichannel blind deconvolution by nonnegative matrix factorization with sparseness constraints," Opt. Lett. 30, 3135-3137 (2005).
[CrossRef] [PubMed]

M. M. Bronstein, A. Bronstein, M. Zibulevsky, and Y. Y. Zeevi, "Blind deconvolution of images using optimal sparse representations," IEEE Trans. Image Process. 14, 726-736 (2005).
[CrossRef] [PubMed]

2004 (5)

M. Numata and N. Hamada, "Image restoration of multichannel blurred images by independent component analysis," in Proceedings of 2004 RISP International Workshop on Nonlinear Circuit and Signal Processing, Hawaii, March 5-7, 2004, pp. 197-200.

I. Kopriva, Q. Du, H. Szu, and W. Wasylkiwskyj, "Independent component analysis approach to image sharpening in the presence of atmospheric turbulence," Opt. Commun. 233, 7-14 (2004).
[CrossRef]

T. Tanaka and A. Cichocki, "Subband decomposition independent component analysis and new performance criteria," in Proceedings of the IEEE Conference on Acoustics, Speech, and Signal Processing (IEEE, 2004), pp. 541-544.

Y. Li, A. Cichocki, and S. Amari, "Analysis of sparse representation and blind source separation," Neural Comput. 16, 1193-1234 (2004).
[CrossRef] [PubMed]

P. O. Hoyer, "Non-negative matrix factorization with sparseness constraints," J. Mach. Learn. Res. 5, 1457-1469 (2004).

2003 (3)

P. Kisilev, M. Zibulevsky, and Y. Y. Zeevi, "Multiscale framework for blind separation of linearly mixed signals," J. Mach. Learn. Res. 4, 1339-1363 (2003).

J. F. Cardoso, "Dependence, correlation and Gaussianity in independent component analysis," J. Mach. Learn. Res. 4, 1177-1203 (2003).

A. Cichocki and P. Georgiev, "Blind source separationalgorithms with matrix constraints," IEICE Trans. Fundamentals E86-A, 522-531 (2003).

2002 (2)

A. Cichocki and S. Amari, Adaptive Blind Signal and Image Processing (Wiley, 2002).
[CrossRef]

M. Zibulevsky, P. Kisilev, Y. Zeevi, and B. Pearlmutter, "Blind source separation via multinode sparse representation," in Advances in Neural Information Processing Systems (Morgan Kaufman, 2002), pp. 185-191.

2001 (2)

A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis (Wiley Interscience, 2001).
[CrossRef]

S. Umeyama, "Blind deconvolution of blurred images by use of ICA," Electron. Commun. Jpn. Part III: Fund. Electron. Sci. 84, 1-9 (2001).

1999 (1)

D. D. Lee and H. S. Seung, "Learning the parts of objects by non-negative matrix factorization," Nature 401, 788-791 (1999).
[CrossRef] [PubMed]

1998 (2)

A. Hyvärinen, "Independent component analysis for time-dependent stochastic processes," in Proceedings of the International Conference on Artificial Neural Networks (Springer-Verlag Telos, 1998), pp. 541-546.

C. L. Bryne, "Accelerating the EMML algorithm and related iterative algorithms by rescaled block-iterative methods," IEEE Trans. Image Process. 7, 100-109 (1998).
[CrossRef]

1997 (2)

D. S. C. Biggs and M. Andrews, "Acceleration of iterative image restoration algorithms," Appl. Opt. 36, 1766-1775 (1997).
[CrossRef] [PubMed]

M. R. Banham and A. K. Katsaggelos, "Digital image restoration," IEEE Signal Process. Mag. 14, 24-41 (1997).
[CrossRef]

1996 (1)

D. Kundur and D. Hatzinakos, "Blind image deconvolution," IEEE Signal Process. Mag. 13, 43-64 (1996).
[CrossRef]

1995 (2)

1994 (3)

P. Comon, "Independent component analysis, a new concept?" Signal Process. 36, 287-314 (1994).
[CrossRef]

W. M. Lam and J. M. Shapiro, "A class of fast algorithms for the Peano-Hillbert space filling curve," in Proceedings of the IEEE International Conference Image Processing (ICIP-94) (IEEE, 1994), Vol. 1, pp. 638-641.
[CrossRef]

M. V. Wickerhauser, Adapted Wavelet Analysis from Theory to Software (Peters, 1994).

1993 (1)

J. F. Cardoso and A. Souloumiac, "Blind beamforming for non-Gaussian signals," IEE Proc. F, Radar Signal Process. 140, 362-370 (1993).
[CrossRef]

1991 (1)

R. L. Lagendijk and J. Biemond, Iterative Identification and Restoration of Images (Kluwer Academic, 1991).
[CrossRef]

1988 (2)

S. J. Orfanidis, Optimum Signal Processing: An Introduction, 2nd ed. (Macmillan, 1988).

J. G. Daugman, "Complete discrete 2-D Gabor transforms by neural networks for image analysis and compression," IEEE Trans. Acoust., Speech, Signal Process. 36, 1169-1179 (1988).
[CrossRef]

1981 (2)

D. R. Brillinger, Time Series Data Analysis and Theory (McGraw-Hill, 1981).

M. B. Priestley, Spectral Analysis and Time Series (Academic, 1981).

1974 (1)

L. B. Lucy, "An iterative technique for rectification of observed distribution," Astron. J. 79, 745-754 (1974).
[CrossRef]

1972 (1)

1962 (1)

A. M. Yaglom, Introduction to the Theory of Stationary Random Functions (Prentice Hall, 1962).

Amari, S.

Y. Li, S. Amari, A. Cichocki, D. W. C. Ho, and S. Xie, "Underdetermined blind source separation based on sparse representation," IEEE Trans. Signal Process. 54, 423-437 (2006).
[CrossRef]

A. Cichocki, R. Zdunek, and S. Amari, "Csiszár's divergences for non-negative matrix factorization: family of new algorithms," Lect. Notes Comput. Sci. 3889, 32-39 (2006).
[CrossRef]

Y. Li, A. Cichocki, and S. Amari, "Analysis of sparse representation and blind source separation," Neural Comput. 16, 1193-1234 (2004).
[CrossRef] [PubMed]

A. Cichocki and S. Amari, Adaptive Blind Signal and Image Processing (Wiley, 2002).
[CrossRef]

Andrews, M.

Banham, M. R.

M. R. Banham and A. K. Katsaggelos, "Digital image restoration," IEEE Signal Process. Mag. 14, 24-41 (1997).
[CrossRef]

Biemond, J.

R. L. Lagendijk and J. Biemond, Iterative Identification and Restoration of Images (Kluwer Academic, 1991).
[CrossRef]

Biggs, D. S. C.

Borjanovic, V.

I. Kopriva, D. J. Garrood, and V. Borjanovic, "Single frame blind image deconvolution by non-negative sparse matrix factorization," Opt. Commun. 266, 456-464 (2006).
[CrossRef]

Brillinger, D. R.

D. R. Brillinger, Time Series Data Analysis and Theory (McGraw-Hill, 1981).

Brinicombe, A. M.

Bronstein, A.

M. M. Bronstein, A. Bronstein, M. Zibulevsky, and Y. Y. Zeevi, "Blind deconvolution of images using optimal sparse representations," IEEE Trans. Image Process. 14, 726-736 (2005).
[CrossRef] [PubMed]

Bronstein, M. M.

M. M. Bronstein, A. Bronstein, M. Zibulevsky, and Y. Y. Zeevi, "Blind deconvolution of images using optimal sparse representations," IEEE Trans. Image Process. 14, 726-736 (2005).
[CrossRef] [PubMed]

Bryne, C. L.

C. L. Bryne, "Accelerating the EMML algorithm and related iterative algorithms by rescaled block-iterative methods," IEEE Trans. Image Process. 7, 100-109 (1998).
[CrossRef]

Cardoso, J. F.

J. F. Cardoso, "Dependence, correlation and Gaussianity in independent component analysis," J. Mach. Learn. Res. 4, 1177-1203 (2003).

J. F. Cardoso and A. Souloumiac, "Blind beamforming for non-Gaussian signals," IEE Proc. F, Radar Signal Process. 140, 362-370 (1993).
[CrossRef]

Chan, L. W.

K. Zhang and L. W. Chan, "Enhancement of source independence for blind source separation," Lect. Notes Comput. Sci. 3889, 731-738 (2006).
[CrossRef]

K. Zhang and L. W. Chan, "An adaptive method for subband decomposition ICA," Neural Comput. 18, 191-223 (2006).
[CrossRef]

Cichocki, A.

Y. Li, S. Amari, A. Cichocki, D. W. C. Ho, and S. Xie, "Underdetermined blind source separation based on sparse representation," IEEE Trans. Signal Process. 54, 423-437 (2006).
[CrossRef]

A. Cichocki, R. Zdunek, and S. Amari, "Csiszár's divergences for non-negative matrix factorization: family of new algorithms," Lect. Notes Comput. Sci. 3889, 32-39 (2006).
[CrossRef]

P. Georgiev, F. Theis, and A. Cichocki, "Sparse component analysis and blind source separation of underdetermined mixtures," IEEE Trans. Neural Netw. 16, 992-996 (2005).
[CrossRef] [PubMed]

A. Cichocki, "Blind source separation: new tools for extraction of source signals and denoising," in Independent Component Analyses, Wavelets, Unsupervised Smart Sensors, and Neural Networks III, H.Szu, ed., Proc. SPIE 5818, 11-25 (2005).

T. Tanaka and A. Cichocki, "Subband decomposition independent component analysis and new performance criteria," in Proceedings of the IEEE Conference on Acoustics, Speech, and Signal Processing (IEEE, 2004), pp. 541-544.

Y. Li, A. Cichocki, and S. Amari, "Analysis of sparse representation and blind source separation," Neural Comput. 16, 1193-1234 (2004).
[CrossRef] [PubMed]

A. Cichocki and P. Georgiev, "Blind source separationalgorithms with matrix constraints," IEICE Trans. Fundamentals E86-A, 522-531 (2003).

A. Cichocki and S. Amari, Adaptive Blind Signal and Image Processing (Wiley, 2002).
[CrossRef]

Comon, P.

P. Comon, "Independent component analysis, a new concept?" Signal Process. 36, 287-314 (1994).
[CrossRef]

Daugman, J. G.

J. G. Daugman, "Complete discrete 2-D Gabor transforms by neural networks for image analysis and compression," IEEE Trans. Acoust., Speech, Signal Process. 36, 1169-1179 (1988).
[CrossRef]

Du, Q.

I. Kopriva, Q. Du, H. Szu, and W. Wasylkiwskyj, "Independent component analysis approach to image sharpening in the presence of atmospheric turbulence," Opt. Commun. 233, 7-14 (2004).
[CrossRef]

Fish, D. A.

Garrood, D. J.

I. Kopriva, D. J. Garrood, and V. Borjanovic, "Single frame blind image deconvolution by non-negative sparse matrix factorization," Opt. Commun. 266, 456-464 (2006).
[CrossRef]

Georgiev, P.

P. Georgiev, F. Theis, and A. Cichocki, "Sparse component analysis and blind source separation of underdetermined mixtures," IEEE Trans. Neural Netw. 16, 992-996 (2005).
[CrossRef] [PubMed]

A. Cichocki and P. Georgiev, "Blind source separationalgorithms with matrix constraints," IEICE Trans. Fundamentals E86-A, 522-531 (2003).

Hamada, N.

M. Numata and N. Hamada, "Image restoration of multichannel blurred images by independent component analysis," in Proceedings of 2004 RISP International Workshop on Nonlinear Circuit and Signal Processing, Hawaii, March 5-7, 2004, pp. 197-200.

Hatzinakos, D.

D. Kundur and D. Hatzinakos, "Blind image deconvolution," IEEE Signal Process. Mag. 13, 43-64 (1996).
[CrossRef]

Ho, D. W. C.

Y. Li, S. Amari, A. Cichocki, D. W. C. Ho, and S. Xie, "Underdetermined blind source separation based on sparse representation," IEEE Trans. Signal Process. 54, 423-437 (2006).
[CrossRef]

Hoyer, P. O.

P. O. Hoyer, "Non-negative matrix factorization with sparseness constraints," J. Mach. Learn. Res. 5, 1457-1469 (2004).

Hyvärinen, A.

A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis (Wiley Interscience, 2001).
[CrossRef]

A. Hyvärinen, "Independent component analysis for time-dependent stochastic processes," in Proceedings of the International Conference on Artificial Neural Networks (Springer-Verlag Telos, 1998), pp. 541-546.

Karhunen, J.

A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis (Wiley Interscience, 2001).
[CrossRef]

Katsaggelos, A. K.

M. R. Banham and A. K. Katsaggelos, "Digital image restoration," IEEE Signal Process. Mag. 14, 24-41 (1997).
[CrossRef]

Kisilev, P.

P. Kisilev, M. Zibulevsky, and Y. Y. Zeevi, "Multiscale framework for blind separation of linearly mixed signals," J. Mach. Learn. Res. 4, 1339-1363 (2003).

M. Zibulevsky, P. Kisilev, Y. Zeevi, and B. Pearlmutter, "Blind source separation via multinode sparse representation," in Advances in Neural Information Processing Systems (Morgan Kaufman, 2002), pp. 185-191.

Kopriva, I.

I. Kopriva, D. J. Garrood, and V. Borjanovic, "Single frame blind image deconvolution by non-negative sparse matrix factorization," Opt. Commun. 266, 456-464 (2006).
[CrossRef]

I. Kopriva and D. Nuzillard, "Non-negative matrix factorization approach to blind image deconvolution," Lect. Notes Comput. Sci. 3889, 966-973 (2006).
[CrossRef]

I. Kopriva, "Single-frame multichannel blind deconvolution by nonnegative matrix factorization with sparseness constraints," Opt. Lett. 30, 3135-3137 (2005).
[CrossRef] [PubMed]

I. Kopriva, Q. Du, H. Szu, and W. Wasylkiwskyj, "Independent component analysis approach to image sharpening in the presence of atmospheric turbulence," Opt. Commun. 233, 7-14 (2004).
[CrossRef]

Kundur, D.

D. Kundur and D. Hatzinakos, "Blind image deconvolution," IEEE Signal Process. Mag. 13, 43-64 (1996).
[CrossRef]

Lagendijk, R. L.

R. L. Lagendijk and J. Biemond, Iterative Identification and Restoration of Images (Kluwer Academic, 1991).
[CrossRef]

Lam, W. M.

W. M. Lam and J. M. Shapiro, "A class of fast algorithms for the Peano-Hillbert space filling curve," in Proceedings of the IEEE International Conference Image Processing (ICIP-94) (IEEE, 1994), Vol. 1, pp. 638-641.
[CrossRef]

Lee, D. D.

D. D. Lee and H. S. Seung, "Learning the parts of objects by non-negative matrix factorization," Nature 401, 788-791 (1999).
[CrossRef] [PubMed]

Li, Y.

Y. Li, S. Amari, A. Cichocki, D. W. C. Ho, and S. Xie, "Underdetermined blind source separation based on sparse representation," IEEE Trans. Signal Process. 54, 423-437 (2006).
[CrossRef]

Y. Li, A. Cichocki, and S. Amari, "Analysis of sparse representation and blind source separation," Neural Comput. 16, 1193-1234 (2004).
[CrossRef] [PubMed]

Lucy, L. B.

L. B. Lucy, "An iterative technique for rectification of observed distribution," Astron. J. 79, 745-754 (1974).
[CrossRef]

McCullagh, P.

P. McCullagh, Tensor Methods in Statistics (Chapman & Hall, 1995).

Numata, M.

M. Numata and N. Hamada, "Image restoration of multichannel blurred images by independent component analysis," in Proceedings of 2004 RISP International Workshop on Nonlinear Circuit and Signal Processing, Hawaii, March 5-7, 2004, pp. 197-200.

Nuzillard, D.

I. Kopriva and D. Nuzillard, "Non-negative matrix factorization approach to blind image deconvolution," Lect. Notes Comput. Sci. 3889, 966-973 (2006).
[CrossRef]

Oja, E.

A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis (Wiley Interscience, 2001).
[CrossRef]

Orfanidis, S. J.

S. J. Orfanidis, Optimum Signal Processing: An Introduction, 2nd ed. (Macmillan, 1988).

Pearlmutter, B.

M. Zibulevsky, P. Kisilev, Y. Zeevi, and B. Pearlmutter, "Blind source separation via multinode sparse representation," in Advances in Neural Information Processing Systems (Morgan Kaufman, 2002), pp. 185-191.

Pike, E. R.

Priestley, M. B.

M. B. Priestley, Spectral Analysis and Time Series (Academic, 1981).

Richardson, W. H.

Seung, H. S.

D. D. Lee and H. S. Seung, "Learning the parts of objects by non-negative matrix factorization," Nature 401, 788-791 (1999).
[CrossRef] [PubMed]

Shapiro, J. M.

W. M. Lam and J. M. Shapiro, "A class of fast algorithms for the Peano-Hillbert space filling curve," in Proceedings of the IEEE International Conference Image Processing (ICIP-94) (IEEE, 1994), Vol. 1, pp. 638-641.
[CrossRef]

Souloumiac, A.

J. F. Cardoso and A. Souloumiac, "Blind beamforming for non-Gaussian signals," IEE Proc. F, Radar Signal Process. 140, 362-370 (1993).
[CrossRef]

Szu, H.

I. Kopriva, Q. Du, H. Szu, and W. Wasylkiwskyj, "Independent component analysis approach to image sharpening in the presence of atmospheric turbulence," Opt. Commun. 233, 7-14 (2004).
[CrossRef]

Tanaka, T.

T. Tanaka and A. Cichocki, "Subband decomposition independent component analysis and new performance criteria," in Proceedings of the IEEE Conference on Acoustics, Speech, and Signal Processing (IEEE, 2004), pp. 541-544.

Theis, F.

P. Georgiev, F. Theis, and A. Cichocki, "Sparse component analysis and blind source separation of underdetermined mixtures," IEEE Trans. Neural Netw. 16, 992-996 (2005).
[CrossRef] [PubMed]

Umeyama, S.

S. Umeyama, "Blind deconvolution of blurred images by use of ICA," Electron. Commun. Jpn. Part III: Fund. Electron. Sci. 84, 1-9 (2001).

Walker, J. G.

Wasylkiwskyj, W.

I. Kopriva, Q. Du, H. Szu, and W. Wasylkiwskyj, "Independent component analysis approach to image sharpening in the presence of atmospheric turbulence," Opt. Commun. 233, 7-14 (2004).
[CrossRef]

Wickerhauser, M. V.

M. V. Wickerhauser, Adapted Wavelet Analysis from Theory to Software (Peters, 1994).

Xie, S.

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Figures (13)

Fig. 1
Fig. 1

Gabor filters for two spatial frequencies, Ω = 2 , and four orientations, Q = 4 . The first two rows show real and imaginary parts of 2D Gabor filters for ω = 1 , and the last two rows show them for ω = 2 . Each column shows one of the four orientations.

Fig. 2
Fig. 2

Normalized singular values of the sample data covariance matrices of the multichannel images: crosses, defocused sub-Gaussian image shown in Fig. 3; circles, defocused super-Gaussian image shown in Fig. 9.

Fig. 3
Fig. 3

Nonsparse (sub-Gaussian) image degraded by out-of-focus blur obtained by a digital camera in manually defocused mode.

Fig. 4
Fig. 4

Multichannel version of the degraded image shown in Fig. 3, produced by the 2D Gabor filter bank shown in Fig. 1.

Fig. 5
Fig. 5

Nonsparse image reconstructed with the multiscale SDICA algorithm.

Fig. 6
Fig. 6

Nonsparse (sub-Gaussian) image reconstructed by direct application of the JADE algorithm to the linear multichannel model [expression (10)].

Fig. 7
Fig. 7

Nonsparse image reconstructed by the blind Richardson–Lucy algorithm after five iterations with a circular blurring kernel with radius of R = 3   pixels .

Fig. 8
Fig. 8

Nonsparse image reconstructed with the single-frame multichannel NMF algorithm.

Fig. 9
Fig. 9

Sparse (super-Gaussian) image degraded by out-of-focus blur obtained by a digital camera in manually defocused mode. Image was acquired under low-light-level conditions.

Fig. 10
Fig. 10

Sparse image reconstructed with the multiscale SDICA algorithm.

Fig. 11
Fig. 11

Sparse (super-Gaussian) image reconstructed by direct application of the JADE algorithm to the linear multichannel model [expression (10)].

Fig. 12
Fig. 12

Sparse image reconstructed by the blind Richardson–Lucy algorithm after five iterations with a circular blurring kernel with radius of R = 3   pixels .

Fig. 13
Fig. 13

Sparse image reconstructed with the single-frame multichannel NMF algorithm.

Equations (23)

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g ( x , y ) = s = K K t = K K h ( s , t ) f ( x + s , y + t ) ,
g = H f ,
H ̂ i + 1 ( k ) = [ ( f ̂ ( k 1 ) ) T ( g ( H ̂ i k f ̂ ( k 1 ) ) ) ] H ̂ i ( k ) ,
f ̂ i + 1 ( k ) = [ f ̂ i ( k ) ( H ( k ) T ( g ( H ( k ) f ̂ i ( k ) ) ) ) ] .
R ( x , y ) = G ( x , y ) cos ( π σ φ ( x , y ) ) ,
I ( x , y ) = G ( x , y ) sin ( π σ φ ( x , y ) ) ,
G ( x , y ) = exp ( x 2 + y 2 2 σ 2 ) ,
φ ( x , y ) = x cos ( π Q q ) + y sin ( π Q q ) , q = 0 , 1 , , Q 1 .
f ( x + s , y + t ) = f ( x , y ) + s f x ( x , y ) + t f y ( x , y ) + s 2 f x x ( x , y ) + t 2 f y y ( x , y ) + .
g 1 ( x , y ) = a 11 f ( x , y ) + a 12 f x ( x , y ) + a 13 f y ( x , y ) + a 14 f x x ( x , y ) + a 15 f y y ( x , y ) + ,
g l + 1 ( x , y ) = a ( l + 1 ) 1 f ( x , y ) + a ( l + 1 ) 2 f x ( x , y ) + a ( l + 1 ) 3 f y ( x , y ) + a ( l + 1 ) 4 f x x ( x , y ) + a ( l + 1 ) 5 f y y ( x , y ) + ,
f ( p ) = r = 0 R b ( r ) ϵ ( p r ) ,
f ( p ) = r = 0 R b ( p , r ) ϵ ( p r ) .
f ̇ ( p ) r = 0 R d b ( p , r ) d p ϵ ( p r ) .
[ R ̂ G G ] i j = 1 M N m = 1 M N g i ( m ) g j ( m ) , i , j { 1 , 2 , , L + 1 } .
P ̂ = max p ( t = 1 p σ t 2 t = 1 L + 1 σ t 2 < ϵ ) ,
WP ( G ) = A WP ( F ) .
f k n j ( ξ ) = l c k n l j φ j l ( ξ ) ,
g k n j ( ξ ) = l y k n l j φ j l ( ξ ) .
y l = A c l .
G k j ( ξ ) = A F k j ( ξ ) ,
I ̂ c j ( g 1 j , g 2 j , , g L + 1 j ) 1 4 1 k < l L + 1 k l cum 2 ( g k j , g l j ) + 1 24 1 k < l L + 1 k l ( cum 2 ( g k j , g k j , g l j ) + cum 2 ( g k j , g l j , g l j ) ) + 1 48 1 k < l L + 1 k l ( cum 2 ( g k j , g k j , g k j , g l j ) + cum 2 ( g k j , g k j , g l j , g l j ) + cum 2 ( g k j , g l j , g l j , g l j ) ) .
F ̂ = W ̂ G .

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