Abstract

Sparse representation of astronomical images is discussed. It is shown that a significant gain in sparsity is achieved when particular mixed dictionaries are used for approximating these types of images with greedy selection strategies. Experiments are conducted to confirm (i) the effectiveness at producing sparse representations and (ii) competitiveness, with respect to the time required to process large images. The latter is a consequence of the suitability of the proposed dictionaries for approximating images in partitions of small blocks. This feature makes it possible to apply the effective greedy selection technique called orthogonal matching pursuit, up to some block size. For blocks exceeding that size, a refinement of the original matching pursuit approach is considered. The resulting method is termed “self-projected matching pursuit,” because it is shown to be effective for implementing, via matching pursuit itself, the optional backprojection intermediate steps in that approach.

© 2013 Optical Society of America

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References

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  1. S. Fischer, G. Cristóbal, and R. Redondo, “Sparse overcomplete Gabor wavelet representation based on local competitions,” IEEE Trans. Image Process. 15, 265–272 (2006).
    [CrossRef]
  2. J. Mairal, M. Eldar, and G. Sapiro, “Sparse representation for color image restoration,” IEEE Trans. Image Process. 17, 53–69 (2008).
    [CrossRef]
  3. L. P. Yaroslavsky, G. Shabat, B. G. Salomon, I. A. Ideses, and B. Fishbain, “Nonuniform sampling, image recovery from sparse data and the discrete sampling theorem,” J. Opt. Soc. Am. A 26, 566–575 (2009).
    [CrossRef]
  4. J. Wright, Yi Ma, J. Mairal, G. Sapiro, T. S. Huang, and S. Yan, “Sparse representation for computer vision and pattern recognition,” Proc. IEEE 98, 1031–1044 (2010).
    [CrossRef]
  5. J.-L. Starck, F. Murtagh, and J. M. Fadili, Sparse Image and Signal Processing (Cambridge University, 2010).
  6. E. Candès and M. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
    [CrossRef]
  7. J. Romberg, “Imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 14–20 (2008).
    [CrossRef]
  8. R. Baraniuk, “More is less: signal processing and the data deluge,” Science 331, 717–719 (2011).
    [CrossRef]
  9. Z. Xu and E. Y. Lam, “Image reconstruction using spectroscopic and hyperspectral information for compressive terahertz imaging,” J. Opt. Soc. Am. A 27, 1638–1646 (2010).
    [CrossRef]
  10. A. Fannjiang and H.-C. Tseng, “Compressive imaging of subwavelength structures: periodic rough surfaces,” J. Opt. Soc. Am. A 29, 617–626 (2012).
    [CrossRef]
  11. J. Bobin, J.-L. Stack, and R. Ottensamer, “Compressed sensing in astronomy,,” IEEE J. Sel. Top. Signal Process. 2, 718–726 (2008).
    [CrossRef]
  12. E. Candès, Y. Eldar, D. Needell, and P. Randall, “Compressed sensing with coherent and redundant dictionaries,” Appl. Comput. Harmon. Anal. 31, 59–73 (2011).
    [CrossRef]
  13. J. Bowley and L. Rebollo-Neira, “Sparsity and something else: an approach to encrypted image folding,” IEEE Signal Process. Lett. 18, 189–192 (2011).
    [CrossRef]
  14. L. Rebollo-Neira, J. Bowley, A. Constantinides, and A. Plastino, “Self contained encrypted image folding,” Phys. A 391, 5858–5870 (2012).
    [CrossRef]
  15. http://http://www.eso.org/public/ .
  16. http://hubblesite.org/ .
  17. R. Young, An Introduction to Nonharmonic Fourier Series(Academic, 1980).
  18. S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput. 20, 33–61 (1998).
    [CrossRef]
  19. S. G. Mallat and Z. Zhang, “Matching pursuits with time-frequency dictionaries,” IEEE Trans. Signal Process. 41, 3397–3415 (1993).
    [CrossRef]
  20. L. K. Jones, “On a conjecture of Huber concerning the convergence of projection pursuit regression,” Ann. Statist. 15, 880–882 (1987).
    [CrossRef]
  21. Y. C. Pati, R. Rezaiifar, and P. S. Krishnaprasad, “Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition,” in Proceedings of 27th Asilomar Conference on Signals, Systems and Computers (IEEE, 1993), pp. 40–44.
  22. M. Andrle and L. Rebollo-Neira, “Cardinal B-spline dictionaries on a compact interval,” Appl. Comput. Harmon. Anal. 18, 336–346 (2005).
    [CrossRef]
  23. L. Rebollo-Neira and Z. Xu, “Sparse signal representation by adaptive non-uniform B-spline dictionaries on a compact interval,” Signal Process. 90, 2308–2313 (2010).
    [CrossRef]
  24. C. de Boor, A Practical Guide to Splines Applied Mathematical Sciences, Vol. 27 (Springer-Verlag, 1978).
  25. L. Rebollo-Neira and D. Lowe, “Optimized orthogonal matching pursuit approach,” IEEE Signal Process. Lett. 9, 137–140 (2002).
    [CrossRef]
  26. M. Andrle and L. Rebollo-Neira, “A swapping-based refinement of orthogonal matching pursuit strategies,” Signal Process. 86, 480–495 (2006).
    [CrossRef]
  27. http://www.nonlinear-approx/info .
  28. Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
    [CrossRef]
  29. I. Daubechies, Ten Lectures on Wavelets (SIAM, 1992).

2012 (2)

L. Rebollo-Neira, J. Bowley, A. Constantinides, and A. Plastino, “Self contained encrypted image folding,” Phys. A 391, 5858–5870 (2012).
[CrossRef]

A. Fannjiang and H.-C. Tseng, “Compressive imaging of subwavelength structures: periodic rough surfaces,” J. Opt. Soc. Am. A 29, 617–626 (2012).
[CrossRef]

2011 (3)

E. Candès, Y. Eldar, D. Needell, and P. Randall, “Compressed sensing with coherent and redundant dictionaries,” Appl. Comput. Harmon. Anal. 31, 59–73 (2011).
[CrossRef]

J. Bowley and L. Rebollo-Neira, “Sparsity and something else: an approach to encrypted image folding,” IEEE Signal Process. Lett. 18, 189–192 (2011).
[CrossRef]

R. Baraniuk, “More is less: signal processing and the data deluge,” Science 331, 717–719 (2011).
[CrossRef]

2010 (3)

J. Wright, Yi Ma, J. Mairal, G. Sapiro, T. S. Huang, and S. Yan, “Sparse representation for computer vision and pattern recognition,” Proc. IEEE 98, 1031–1044 (2010).
[CrossRef]

L. Rebollo-Neira and Z. Xu, “Sparse signal representation by adaptive non-uniform B-spline dictionaries on a compact interval,” Signal Process. 90, 2308–2313 (2010).
[CrossRef]

Z. Xu and E. Y. Lam, “Image reconstruction using spectroscopic and hyperspectral information for compressive terahertz imaging,” J. Opt. Soc. Am. A 27, 1638–1646 (2010).
[CrossRef]

2009 (1)

2008 (4)

J. Mairal, M. Eldar, and G. Sapiro, “Sparse representation for color image restoration,” IEEE Trans. Image Process. 17, 53–69 (2008).
[CrossRef]

E. Candès and M. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[CrossRef]

J. Romberg, “Imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 14–20 (2008).
[CrossRef]

J. Bobin, J.-L. Stack, and R. Ottensamer, “Compressed sensing in astronomy,,” IEEE J. Sel. Top. Signal Process. 2, 718–726 (2008).
[CrossRef]

2006 (2)

S. Fischer, G. Cristóbal, and R. Redondo, “Sparse overcomplete Gabor wavelet representation based on local competitions,” IEEE Trans. Image Process. 15, 265–272 (2006).
[CrossRef]

M. Andrle and L. Rebollo-Neira, “A swapping-based refinement of orthogonal matching pursuit strategies,” Signal Process. 86, 480–495 (2006).
[CrossRef]

2005 (1)

M. Andrle and L. Rebollo-Neira, “Cardinal B-spline dictionaries on a compact interval,” Appl. Comput. Harmon. Anal. 18, 336–346 (2005).
[CrossRef]

2004 (1)

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

2002 (1)

L. Rebollo-Neira and D. Lowe, “Optimized orthogonal matching pursuit approach,” IEEE Signal Process. Lett. 9, 137–140 (2002).
[CrossRef]

1998 (1)

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

1993 (1)

S. G. Mallat and Z. Zhang, “Matching pursuits with time-frequency dictionaries,” IEEE Trans. Signal Process. 41, 3397–3415 (1993).
[CrossRef]

1987 (1)

L. K. Jones, “On a conjecture of Huber concerning the convergence of projection pursuit regression,” Ann. Statist. 15, 880–882 (1987).
[CrossRef]

Andrle, M.

M. Andrle and L. Rebollo-Neira, “A swapping-based refinement of orthogonal matching pursuit strategies,” Signal Process. 86, 480–495 (2006).
[CrossRef]

M. Andrle and L. Rebollo-Neira, “Cardinal B-spline dictionaries on a compact interval,” Appl. Comput. Harmon. Anal. 18, 336–346 (2005).
[CrossRef]

Baraniuk, R.

R. Baraniuk, “More is less: signal processing and the data deluge,” Science 331, 717–719 (2011).
[CrossRef]

Bobin, J.

J. Bobin, J.-L. Stack, and R. Ottensamer, “Compressed sensing in astronomy,,” IEEE J. Sel. Top. Signal Process. 2, 718–726 (2008).
[CrossRef]

Bovik, A. C.

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

Bowley, J.

L. Rebollo-Neira, J. Bowley, A. Constantinides, and A. Plastino, “Self contained encrypted image folding,” Phys. A 391, 5858–5870 (2012).
[CrossRef]

J. Bowley and L. Rebollo-Neira, “Sparsity and something else: an approach to encrypted image folding,” IEEE Signal Process. Lett. 18, 189–192 (2011).
[CrossRef]

Candès, E.

E. Candès, Y. Eldar, D. Needell, and P. Randall, “Compressed sensing with coherent and redundant dictionaries,” Appl. Comput. Harmon. Anal. 31, 59–73 (2011).
[CrossRef]

E. Candès and M. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[CrossRef]

Chen, S. S.

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

Constantinides, A.

L. Rebollo-Neira, J. Bowley, A. Constantinides, and A. Plastino, “Self contained encrypted image folding,” Phys. A 391, 5858–5870 (2012).
[CrossRef]

Cristóbal, G.

S. Fischer, G. Cristóbal, and R. Redondo, “Sparse overcomplete Gabor wavelet representation based on local competitions,” IEEE Trans. Image Process. 15, 265–272 (2006).
[CrossRef]

Daubechies, I.

I. Daubechies, Ten Lectures on Wavelets (SIAM, 1992).

de Boor, C.

C. de Boor, A Practical Guide to Splines Applied Mathematical Sciences, Vol. 27 (Springer-Verlag, 1978).

Donoho, D. L.

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

Eldar, M.

J. Mairal, M. Eldar, and G. Sapiro, “Sparse representation for color image restoration,” IEEE Trans. Image Process. 17, 53–69 (2008).
[CrossRef]

Eldar, Y.

E. Candès, Y. Eldar, D. Needell, and P. Randall, “Compressed sensing with coherent and redundant dictionaries,” Appl. Comput. Harmon. Anal. 31, 59–73 (2011).
[CrossRef]

Fadili, J. M.

J.-L. Starck, F. Murtagh, and J. M. Fadili, Sparse Image and Signal Processing (Cambridge University, 2010).

Fannjiang, A.

Fischer, S.

S. Fischer, G. Cristóbal, and R. Redondo, “Sparse overcomplete Gabor wavelet representation based on local competitions,” IEEE Trans. Image Process. 15, 265–272 (2006).
[CrossRef]

Fishbain, B.

Huang, T. S.

J. Wright, Yi Ma, J. Mairal, G. Sapiro, T. S. Huang, and S. Yan, “Sparse representation for computer vision and pattern recognition,” Proc. IEEE 98, 1031–1044 (2010).
[CrossRef]

Ideses, I. A.

Jones, L. K.

L. K. Jones, “On a conjecture of Huber concerning the convergence of projection pursuit regression,” Ann. Statist. 15, 880–882 (1987).
[CrossRef]

Krishnaprasad, P. S.

Y. C. Pati, R. Rezaiifar, and P. S. Krishnaprasad, “Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition,” in Proceedings of 27th Asilomar Conference on Signals, Systems and Computers (IEEE, 1993), pp. 40–44.

Lam, E. Y.

Lowe, D.

L. Rebollo-Neira and D. Lowe, “Optimized orthogonal matching pursuit approach,” IEEE Signal Process. Lett. 9, 137–140 (2002).
[CrossRef]

Ma, Yi

J. Wright, Yi Ma, J. Mairal, G. Sapiro, T. S. Huang, and S. Yan, “Sparse representation for computer vision and pattern recognition,” Proc. IEEE 98, 1031–1044 (2010).
[CrossRef]

Mairal, J.

J. Wright, Yi Ma, J. Mairal, G. Sapiro, T. S. Huang, and S. Yan, “Sparse representation for computer vision and pattern recognition,” Proc. IEEE 98, 1031–1044 (2010).
[CrossRef]

J. Mairal, M. Eldar, and G. Sapiro, “Sparse representation for color image restoration,” IEEE Trans. Image Process. 17, 53–69 (2008).
[CrossRef]

Mallat, S. G.

S. G. Mallat and Z. Zhang, “Matching pursuits with time-frequency dictionaries,” IEEE Trans. Signal Process. 41, 3397–3415 (1993).
[CrossRef]

Murtagh, F.

J.-L. Starck, F. Murtagh, and J. M. Fadili, Sparse Image and Signal Processing (Cambridge University, 2010).

Needell, D.

E. Candès, Y. Eldar, D. Needell, and P. Randall, “Compressed sensing with coherent and redundant dictionaries,” Appl. Comput. Harmon. Anal. 31, 59–73 (2011).
[CrossRef]

Ottensamer, R.

J. Bobin, J.-L. Stack, and R. Ottensamer, “Compressed sensing in astronomy,,” IEEE J. Sel. Top. Signal Process. 2, 718–726 (2008).
[CrossRef]

Pati, Y. C.

Y. C. Pati, R. Rezaiifar, and P. S. Krishnaprasad, “Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition,” in Proceedings of 27th Asilomar Conference on Signals, Systems and Computers (IEEE, 1993), pp. 40–44.

Plastino, A.

L. Rebollo-Neira, J. Bowley, A. Constantinides, and A. Plastino, “Self contained encrypted image folding,” Phys. A 391, 5858–5870 (2012).
[CrossRef]

Randall, P.

E. Candès, Y. Eldar, D. Needell, and P. Randall, “Compressed sensing with coherent and redundant dictionaries,” Appl. Comput. Harmon. Anal. 31, 59–73 (2011).
[CrossRef]

Rebollo-Neira, L.

L. Rebollo-Neira, J. Bowley, A. Constantinides, and A. Plastino, “Self contained encrypted image folding,” Phys. A 391, 5858–5870 (2012).
[CrossRef]

J. Bowley and L. Rebollo-Neira, “Sparsity and something else: an approach to encrypted image folding,” IEEE Signal Process. Lett. 18, 189–192 (2011).
[CrossRef]

L. Rebollo-Neira and Z. Xu, “Sparse signal representation by adaptive non-uniform B-spline dictionaries on a compact interval,” Signal Process. 90, 2308–2313 (2010).
[CrossRef]

M. Andrle and L. Rebollo-Neira, “A swapping-based refinement of orthogonal matching pursuit strategies,” Signal Process. 86, 480–495 (2006).
[CrossRef]

M. Andrle and L. Rebollo-Neira, “Cardinal B-spline dictionaries on a compact interval,” Appl. Comput. Harmon. Anal. 18, 336–346 (2005).
[CrossRef]

L. Rebollo-Neira and D. Lowe, “Optimized orthogonal matching pursuit approach,” IEEE Signal Process. Lett. 9, 137–140 (2002).
[CrossRef]

Redondo, R.

S. Fischer, G. Cristóbal, and R. Redondo, “Sparse overcomplete Gabor wavelet representation based on local competitions,” IEEE Trans. Image Process. 15, 265–272 (2006).
[CrossRef]

Rezaiifar, R.

Y. C. Pati, R. Rezaiifar, and P. S. Krishnaprasad, “Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition,” in Proceedings of 27th Asilomar Conference on Signals, Systems and Computers (IEEE, 1993), pp. 40–44.

Romberg, J.

J. Romberg, “Imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 14–20 (2008).
[CrossRef]

Salomon, B. G.

Sapiro, G.

J. Wright, Yi Ma, J. Mairal, G. Sapiro, T. S. Huang, and S. Yan, “Sparse representation for computer vision and pattern recognition,” Proc. IEEE 98, 1031–1044 (2010).
[CrossRef]

J. Mairal, M. Eldar, and G. Sapiro, “Sparse representation for color image restoration,” IEEE Trans. Image Process. 17, 53–69 (2008).
[CrossRef]

Saunders, M. A.

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

Shabat, G.

Sheikh, H. R.

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

Simoncelli, E. P.

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

Stack, J.-L.

J. Bobin, J.-L. Stack, and R. Ottensamer, “Compressed sensing in astronomy,,” IEEE J. Sel. Top. Signal Process. 2, 718–726 (2008).
[CrossRef]

Starck, J.-L.

J.-L. Starck, F. Murtagh, and J. M. Fadili, Sparse Image and Signal Processing (Cambridge University, 2010).

Tseng, H.-C.

Wakin, M.

E. Candès and M. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[CrossRef]

Wang, Z.

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

Wright, J.

J. Wright, Yi Ma, J. Mairal, G. Sapiro, T. S. Huang, and S. Yan, “Sparse representation for computer vision and pattern recognition,” Proc. IEEE 98, 1031–1044 (2010).
[CrossRef]

Xu, Z.

L. Rebollo-Neira and Z. Xu, “Sparse signal representation by adaptive non-uniform B-spline dictionaries on a compact interval,” Signal Process. 90, 2308–2313 (2010).
[CrossRef]

Z. Xu and E. Y. Lam, “Image reconstruction using spectroscopic and hyperspectral information for compressive terahertz imaging,” J. Opt. Soc. Am. A 27, 1638–1646 (2010).
[CrossRef]

Yan, S.

J. Wright, Yi Ma, J. Mairal, G. Sapiro, T. S. Huang, and S. Yan, “Sparse representation for computer vision and pattern recognition,” Proc. IEEE 98, 1031–1044 (2010).
[CrossRef]

Yaroslavsky, L. P.

Young, R.

R. Young, An Introduction to Nonharmonic Fourier Series(Academic, 1980).

Zhang, Z.

S. G. Mallat and Z. Zhang, “Matching pursuits with time-frequency dictionaries,” IEEE Trans. Signal Process. 41, 3397–3415 (1993).
[CrossRef]

Ann. Statist. (1)

L. K. Jones, “On a conjecture of Huber concerning the convergence of projection pursuit regression,” Ann. Statist. 15, 880–882 (1987).
[CrossRef]

Appl. Comput. Harmon. Anal. (2)

E. Candès, Y. Eldar, D. Needell, and P. Randall, “Compressed sensing with coherent and redundant dictionaries,” Appl. Comput. Harmon. Anal. 31, 59–73 (2011).
[CrossRef]

M. Andrle and L. Rebollo-Neira, “Cardinal B-spline dictionaries on a compact interval,” Appl. Comput. Harmon. Anal. 18, 336–346 (2005).
[CrossRef]

IEEE J. Sel. Top. Signal Process. (1)

J. Bobin, J.-L. Stack, and R. Ottensamer, “Compressed sensing in astronomy,,” IEEE J. Sel. Top. Signal Process. 2, 718–726 (2008).
[CrossRef]

IEEE Signal Process. Lett. (2)

J. Bowley and L. Rebollo-Neira, “Sparsity and something else: an approach to encrypted image folding,” IEEE Signal Process. Lett. 18, 189–192 (2011).
[CrossRef]

L. Rebollo-Neira and D. Lowe, “Optimized orthogonal matching pursuit approach,” IEEE Signal Process. Lett. 9, 137–140 (2002).
[CrossRef]

IEEE Signal Process. Mag. (2)

E. Candès and M. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[CrossRef]

J. Romberg, “Imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 14–20 (2008).
[CrossRef]

IEEE Trans. Image Process. (3)

S. Fischer, G. Cristóbal, and R. Redondo, “Sparse overcomplete Gabor wavelet representation based on local competitions,” IEEE Trans. Image Process. 15, 265–272 (2006).
[CrossRef]

J. Mairal, M. Eldar, and G. Sapiro, “Sparse representation for color image restoration,” IEEE Trans. Image Process. 17, 53–69 (2008).
[CrossRef]

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

IEEE Trans. Signal Process. (1)

S. G. Mallat and Z. Zhang, “Matching pursuits with time-frequency dictionaries,” IEEE Trans. Signal Process. 41, 3397–3415 (1993).
[CrossRef]

J. Opt. Soc. Am. A (3)

Phys. A (1)

L. Rebollo-Neira, J. Bowley, A. Constantinides, and A. Plastino, “Self contained encrypted image folding,” Phys. A 391, 5858–5870 (2012).
[CrossRef]

Proc. IEEE (1)

J. Wright, Yi Ma, J. Mairal, G. Sapiro, T. S. Huang, and S. Yan, “Sparse representation for computer vision and pattern recognition,” Proc. IEEE 98, 1031–1044 (2010).
[CrossRef]

Science (1)

R. Baraniuk, “More is less: signal processing and the data deluge,” Science 331, 717–719 (2011).
[CrossRef]

SIAM J. Sci. Comput. (1)

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

Signal Process. (2)

M. Andrle and L. Rebollo-Neira, “A swapping-based refinement of orthogonal matching pursuit strategies,” Signal Process. 86, 480–495 (2006).
[CrossRef]

L. Rebollo-Neira and Z. Xu, “Sparse signal representation by adaptive non-uniform B-spline dictionaries on a compact interval,” Signal Process. 90, 2308–2313 (2010).
[CrossRef]

Other (8)

C. de Boor, A Practical Guide to Splines Applied Mathematical Sciences, Vol. 27 (Springer-Verlag, 1978).

http://www.nonlinear-approx/info .

J.-L. Starck, F. Murtagh, and J. M. Fadili, Sparse Image and Signal Processing (Cambridge University, 2010).

I. Daubechies, Ten Lectures on Wavelets (SIAM, 1992).

Y. C. Pati, R. Rezaiifar, and P. S. Krishnaprasad, “Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition,” in Proceedings of 27th Asilomar Conference on Signals, Systems and Computers (IEEE, 1993), pp. 40–44.

http://http://www.eso.org/public/ .

http://hubblesite.org/ .

R. Young, An Introduction to Nonharmonic Fourier Series(Academic, 1980).

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Figures (5)

Fig. 1.
Fig. 1.

Chirp signal approximated up to error ρ=0.001f by (i) K=683 orthogonal DC components taken from Eq. (7) with N=M=2000, (ii) K=286 atoms taken from Eq. (7) with M=2N=4000 using OMP or K=1638 using MP, and (iii) K=300 atoms taken from Eq. (7) with M=2N=4000 using SPMP with p=10 or K=286 with p=3.

Fig. 2.
Fig. 2.

Prototype atoms as defined in Eqs. (12) and (13). The redundant discrete B-spline component of the dictionary is constructed by translation of these atoms, applying the cutoff approach at the boundaries.

Fig. 3.
Fig. 3.

First image is the nebula Orion (Messier 42 Ref. eso1006), and the second the spiral galaxy NGC 4945 Ref. eso0931). Both images are taken from ESO [15] at publication and screen resolutions. The corresponding sizes (in pixels) are 4000×3252 and 1280×1574 (nebula) 4000×4000 and 1280×1280 (galaxy).

Fig. 4.
Fig. 4.

SR versus partition of side length 8, 16, 24, 32, 40, and 48 yielded by OMP2D, SPMP2D with projection steps 1 (SPMP2D1) and 10 (SPMP2D10), MP2D, and the DCT. The constant dotted line corresponds to the DWT result and is plotted only for visual comparison, since the DWT is applied to the whole image. The left and right graphs correspond to the nebula and galaxy of Fig. 3, respectively. The top graphs correspond to the higher resolution 4000×3252 pixels and 4000×4000 pixels, respectively. The bottom graphs correspond to the lower resolution 1280×1574 and 1280×1280 pixel images, respectively.

Fig. 5.
Fig. 5.

Two top graphs show the prototype atoms defining the RDW dictionary. The two bottom graphs show the prototype atoms of random shape defining a realization of the RR dictionary.

Tables (3)

Tables Icon

Table 1. SR and Execution Time, in Seconds, for Approximating a Complete Image with Different Approaches and Different Sized Blocks Partitioning the Imagea

Tables Icon

Table 2. Average SR (SR¯) and Average Processing Time (t¯) per Image (in Seconds) for Approximating, up to a PSNR of 45 dB, a Set of 55 Images from the HST Web Sitea

Tables Icon

Table 3. Mean (SR¯) and Variance (σSR) of the SR Obtained with Different Mixed Dictionaries, by Partitioning the Images into Blocks of Size 8, 16, 24, and 32 and Applying the OMP2D Approach with RDCT-RDBS, RDCT-RDW, and RDCT-RR Dictionariesa

Equations (22)

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

fK=n=1Kc(n)dn,whereK<N.
Rk=dn,Rkdn+Rk+1,n=1,,M,
Rk2=|dn,Rk|2+Rk+12,n=1,,M.
f=fk+Rk+1
fk=n=1kdn,Rndn.
fk=n=1kck(n)dn+R˜k,
n=1kck(n)dn=P^Vkf,withVk=span{dn}n=1k.
D1={vn;vn(i)=wncos(π(2i1)(n1)2M),i=1,,N}n=1M,
IK=n=1KcK(n)Dn,
Bm(x)=1m!i=0m(1)i(mi)(xi)+m1,
B2l(x)={xlif0x<l2xliflx<2l0otherwise,
B4l(x)={x36l3if0x<lx32l3+2x2l22xl+23iflx<2lx32l34x2l2+10xl223if2lx<3lx36l3+2x2l28xl+323if3lx<4l0otherwise.
Ds={biYms(ji)|N;j=1,,L}i=1Ms,m=2,4,s=2,,9,
Y2s={B2l,l=1,2,3fors=2,3,4,(respectively)d1B2l,l=2,3fors=5,6,(respectively),
Y4s={B42fors=7d1B42fors=8d2B42fors=9.
k+1x,k+1y=argmaxn=1,,Mxm=1,,My|i=1j=1Nx,Nydnx(i)Rk(i,j)dmy(j)|withRk(i,j)=I(i,j)n=1kck(n)dnxx(i)dnyy(j).
P^VkI=n=1kAnBnk,IF=n=1kck(n)An,
I=h=1HIh,
IKh(i,j)=n=1KhcKh(n)dnxx(i)dnyy(j),i,j=1,Nh.
SR=number of pixelsnumber of coeffients=HNh2h=1HKh.
Izk=n=1kczk(n)dnxxdnyy,z=1,2,3,
k+1x,k+1y=argmaxn=1,,Mxm=1,,Myz=13|i=1j=1Nx,Nydnx(i)Rzk(i,j)dmy(j)|withRzk(i,j)=I(i,j)n=1kczk(n)dnxx(i)dnyy(j),z=1,2,3.

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