Abstract

We show that, in contrast to popular belief, sub-wavelength information can be recovered from the far-field of an optical image, thereby overcoming the loss of information embedded in decaying evanescent waves. The only requirement is that the image is known to be sparse, a specific but very general and wide-spread property of signals which occur almost everywhere in nature. The reconstruction method relies on newly-developed compressed sensing techniques, which we adapt to optical super-resolution and sub-wavelength imaging. Our approach exhibits robustness to noise and imperfections. We provide an experimental proof-of-principle by demonstrating image recovery at a spatial resolution 5-times higher than the finest resolution defined by a spatial filter. The technique is general, and can be extended beyond optical microscopy, for example, to atomic force microscopes, scanning-tunneling microscopes, and other imaging systems.

© 2009 Optical Society of America

Full Article  |  PDF Article

Errata

Snir Gazit, Alexander Szameit, Yonina C. Eldar, and Mordechai Segev, "Super-resolution and reconstruction of sparse sub-wavelength images: erratum," Opt. Express 18, 26631-26631 (2010)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-18-25-26631

References

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  35. T. Blu, P. L. Dragotti, M. Vetterli, P. Marziliano, and L. Coulot, "Sparse sampling of signal innovations," IEEE Signal Process. Mag. 25, 31-40 (2008).
    [CrossRef]
  36. D. L. Donoho and J. Tanner, "Sparse nonnegative solution of underdetermined linear equations by linear programming," Proc. Natl. Acad. Sci. 102, 9446-9451 (2005).
    [CrossRef] [PubMed]
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    [CrossRef]

2009

F. M. Huang and N. I. Zheludev, "Super-resolution without evanescent waves," Nano Lett. 9, 1249-1254 (2009).
[CrossRef] [PubMed]

S.W. Hell, R. Schmidt, and A. Egner, "Diffraction-unlimited three-dimensional optical nanoscopy with opposing lenses," Nat. Photon. 3, 381-387 (2009).
[CrossRef]

Y. C. Eldar, "Compressed sensing of analog signals in shift-invariant spaces," IEEE Trans. Signal Process. 57, 2986-2997 (2009).
[CrossRef]

M. Mishali and Y. C. Eldar, "Blind multi-band signal reconstruction: Compressed sensing for analog signals," IEEE Trans. Signal Process. 57, 993-1009 (2009).
[CrossRef]

Y. C. Eldar and T. Michaeli, "Beyond bandlimited sampling," IEEE Signal Proc. Mag. 26, 48-68 (2009).
[CrossRef]

2008

T. Blu, P. L. Dragotti, M. Vetterli, P. Marziliano, and L. Coulot, "Sparse sampling of signal innovations," IEEE Signal Process. Mag. 25, 31-40 (2008).
[CrossRef]

E. J. Candes and M. B. Wakin, "An introduction to compressive sampling," IEEE Signal Process. Mag. 25, 21-30 (2008).
[CrossRef]

A. M. Bruckstein, M. Elad, and M. Zibulevsky, "On the uniqueness of nonnegative sparse solutions to underdetermined systems of equations," IEEE Trans. Inf. Theory 54, 4813-4820 (2008).
[CrossRef]

A. Ashok, P. K. Baheti, and M. A. Neifeld, "Compressive imaging system design using task-specific information," Appl. Opt. 47, 4457-4471 (2008).
[CrossRef] [PubMed]

N. I. Zheludev, "What diffraction limit?" Nat. Mater. 7, 420-422 (2008).
[CrossRef] [PubMed]

2007

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, "Far-field optical hyperlens magnifying sub-diffraction-limited objects," Science 315, 1686 (2007).
[CrossRef] [PubMed]

I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, "Magnifying superlens in the visible frequency range," Science 315, 1699-1701 (2007).
[CrossRef] [PubMed]

2006

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

E. J. Candes and T. Tao, "Near-optimal signal recovery from random projections: Universal encoding strategies?" IEEE Trans. Inf. Theory 52, 5406-5425 (2006).
[CrossRef]

A. Salandrino and N. Engheta, "Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations," Phys. Rev. B 74, 075103 (2006).
[CrossRef]

Z. Jacob, L. V. Alexeyev, and E. Narimanov, "Optical hyperlens: far-field imaging beyond the diffraction limit," Opt. Express 14, 8247-8256 (2006).
[CrossRef] [PubMed]

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

2005

D. L. Donoho and J. Tanner, "Sparse nonnegative solution of underdetermined linear equations by linear programming," Proc. Natl. Acad. Sci. 102, 9446-9451 (2005).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-diffraction-limited optical imaging with a silver superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

2003

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, "Myosin v walks hand-overhand: Single fluorophore imaging with 1.5nm localization," Science 300, 2061-2065 (2003).
[CrossRef] [PubMed]

D. L. Donoho and M. Elad, "Optimally sparse representation in general (nonorthogonal) dictionaries via l1 minimization," Proc. Natl. Acad. Sci. 100, 2197-2201 (2003).
[CrossRef]

2002

M. Vetterli, P. Marziliano, and T. Blu, "Sampling signals with finite rate of innovation," IEEE Trans. Sig. Proc. 50, 1417-1428 (2002).
[CrossRef]

2001

V. A. Mandelshtam, "FDM: the Filter Diagonalization Method for data processing in NMR experiments," Prog. Nucl. Mag. Res. Sp. 38, 159-196 (2001).
[CrossRef]

2000

J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

1998

T. W. Ebbesen, H. G. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolf, "Extraordinary optical transmission through subwavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

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

1991

E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, and R. L. Kostelak, "Breaking the diffraction barrier: optical microscopy on a nanometric scale," Science 251, 1468-1470 (1991).
[CrossRef] [PubMed]

1984

A. Lewis, M. Isaacson, A. Harotunian, and A. Muray, "Development of a 500°a spatial-resolution light microscope: I. light is efficiently transmitted through l/16 diameter apertures," Ultramicroscopy 13, 227-232 (1984).
[CrossRef]

1975

A. Papoulis, "A new algorithm in spectral analysis and band-limited extrapolation," IEEE Trans. Circuits Syst. 22, 735-742 (1975).
[CrossRef]

1974

R. W. Gerchberg, "Super-resolution through error energy reduction," J. Mod. Opt. 21, 709-720 (1974).

1972

E. A. Ash and G. Nicholls, "Super-resolution aperture scanning microscope," Nature 237, 510-512 (1972).
[CrossRef] [PubMed]

Alexeyev, L. V.

Ash, E. A.

E. A. Ash and G. Nicholls, "Super-resolution aperture scanning microscope," Nature 237, 510-512 (1972).
[CrossRef] [PubMed]

Ashok, A.

Baheti, P. K.

Ben-Haim, Z.

Z. Ben-Haim, Y. C. Eldar, and M. Elad, "Near-oracle performance of basis pursuit under random noise," IEEE Trans. Signal Process. (submitted).

Betzig, E.

E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, and R. L. Kostelak, "Breaking the diffraction barrier: optical microscopy on a nanometric scale," Science 251, 1468-1470 (1991).
[CrossRef] [PubMed]

Blu, T.

T. Blu, P. L. Dragotti, M. Vetterli, P. Marziliano, and L. Coulot, "Sparse sampling of signal innovations," IEEE Signal Process. Mag. 25, 31-40 (2008).
[CrossRef]

M. Vetterli, P. Marziliano, and T. Blu, "Sampling signals with finite rate of innovation," IEEE Trans. Sig. Proc. 50, 1417-1428 (2002).
[CrossRef]

Bruckstein, A. M.

A. M. Bruckstein, M. Elad, and M. Zibulevsky, "On the uniqueness of nonnegative sparse solutions to underdetermined systems of equations," IEEE Trans. Inf. Theory 54, 4813-4820 (2008).
[CrossRef]

Candes, E. J.

E. J. Candes and M. B. Wakin, "An introduction to compressive sampling," IEEE Signal Process. Mag. 25, 21-30 (2008).
[CrossRef]

E. J. Candes and T. Tao, "Near-optimal signal recovery from random projections: Universal encoding strategies?" IEEE Trans. Inf. Theory 52, 5406-5425 (2006).
[CrossRef]

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

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]

Coulot, L.

T. Blu, P. L. Dragotti, M. Vetterli, P. Marziliano, and L. Coulot, "Sparse sampling of signal innovations," IEEE Signal Process. Mag. 25, 31-40 (2008).
[CrossRef]

Davis, C. C.

I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, "Magnifying superlens in the visible frequency range," Science 315, 1699-1701 (2007).
[CrossRef] [PubMed]

Donoho, D. L.

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

D. L. Donoho and J. Tanner, "Sparse nonnegative solution of underdetermined linear equations by linear programming," Proc. Natl. Acad. Sci. 102, 9446-9451 (2005).
[CrossRef] [PubMed]

D. L. Donoho and M. Elad, "Optimally sparse representation in general (nonorthogonal) dictionaries via l1 minimization," Proc. Natl. Acad. Sci. 100, 2197-2201 (2003).
[CrossRef]

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

Dragotti, P. L.

T. Blu, P. L. Dragotti, M. Vetterli, P. Marziliano, and L. Coulot, "Sparse sampling of signal innovations," IEEE Signal Process. Mag. 25, 31-40 (2008).
[CrossRef]

Ebbesen, T. W.

T. W. Ebbesen, H. G. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolf, "Extraordinary optical transmission through subwavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Egner, A.

S.W. Hell, R. Schmidt, and A. Egner, "Diffraction-unlimited three-dimensional optical nanoscopy with opposing lenses," Nat. Photon. 3, 381-387 (2009).
[CrossRef]

Elad, M.

A. M. Bruckstein, M. Elad, and M. Zibulevsky, "On the uniqueness of nonnegative sparse solutions to underdetermined systems of equations," IEEE Trans. Inf. Theory 54, 4813-4820 (2008).
[CrossRef]

D. L. Donoho and M. Elad, "Optimally sparse representation in general (nonorthogonal) dictionaries via l1 minimization," Proc. Natl. Acad. Sci. 100, 2197-2201 (2003).
[CrossRef]

Z. Ben-Haim, Y. C. Eldar, and M. Elad, "Near-oracle performance of basis pursuit under random noise," IEEE Trans. Signal Process. (submitted).

Eldar, Y. C.

Y. C. Eldar and T. Michaeli, "Beyond bandlimited sampling," IEEE Signal Proc. Mag. 26, 48-68 (2009).
[CrossRef]

Y. C. Eldar, "Compressed sensing of analog signals in shift-invariant spaces," IEEE Trans. Signal Process. 57, 2986-2997 (2009).
[CrossRef]

M. Mishali and Y. C. Eldar, "Blind multi-band signal reconstruction: Compressed sensing for analog signals," IEEE Trans. Signal Process. 57, 993-1009 (2009).
[CrossRef]

Z. Ben-Haim, Y. C. Eldar, and M. Elad, "Near-oracle performance of basis pursuit under random noise," IEEE Trans. Signal Process. (submitted).

Engheta, N.

A. Salandrino and N. Engheta, "Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations," Phys. Rev. B 74, 075103 (2006).
[CrossRef]

Fang, N.

N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-diffraction-limited optical imaging with a silver superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Forkey, J. N.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, "Myosin v walks hand-overhand: Single fluorophore imaging with 1.5nm localization," Science 300, 2061-2065 (2003).
[CrossRef] [PubMed]

Gerchberg, R. W.

R. W. Gerchberg, "Super-resolution through error energy reduction," J. Mod. Opt. 21, 709-720 (1974).

Ghaemi, H. F.

T. W. Ebbesen, H. G. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolf, "Extraordinary optical transmission through subwavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Goldman, Y. E.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, "Myosin v walks hand-overhand: Single fluorophore imaging with 1.5nm localization," Science 300, 2061-2065 (2003).
[CrossRef] [PubMed]

Ha, T.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, "Myosin v walks hand-overhand: Single fluorophore imaging with 1.5nm localization," Science 300, 2061-2065 (2003).
[CrossRef] [PubMed]

Harotunian, A.

A. Lewis, M. Isaacson, A. Harotunian, and A. Muray, "Development of a 500°a spatial-resolution light microscope: I. light is efficiently transmitted through l/16 diameter apertures," Ultramicroscopy 13, 227-232 (1984).
[CrossRef]

Harris, T. D.

E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, and R. L. Kostelak, "Breaking the diffraction barrier: optical microscopy on a nanometric scale," Science 251, 1468-1470 (1991).
[CrossRef] [PubMed]

Hell, S.W.

S.W. Hell, R. Schmidt, and A. Egner, "Diffraction-unlimited three-dimensional optical nanoscopy with opposing lenses," Nat. Photon. 3, 381-387 (2009).
[CrossRef]

Huang, F. M.

F. M. Huang and N. I. Zheludev, "Super-resolution without evanescent waves," Nano Lett. 9, 1249-1254 (2009).
[CrossRef] [PubMed]

Hung, Y. J.

I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, "Magnifying superlens in the visible frequency range," Science 315, 1699-1701 (2007).
[CrossRef] [PubMed]

Isaacson, M.

A. Lewis, M. Isaacson, A. Harotunian, and A. Muray, "Development of a 500°a spatial-resolution light microscope: I. light is efficiently transmitted through l/16 diameter apertures," Ultramicroscopy 13, 227-232 (1984).
[CrossRef]

Jacob, Z.

Kostelak, R. L.

E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, and R. L. Kostelak, "Breaking the diffraction barrier: optical microscopy on a nanometric scale," Science 251, 1468-1470 (1991).
[CrossRef] [PubMed]

Lee, H.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, "Far-field optical hyperlens magnifying sub-diffraction-limited objects," Science 315, 1686 (2007).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-diffraction-limited optical imaging with a silver superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Lewis, A.

A. Lewis, M. Isaacson, A. Harotunian, and A. Muray, "Development of a 500°a spatial-resolution light microscope: I. light is efficiently transmitted through l/16 diameter apertures," Ultramicroscopy 13, 227-232 (1984).
[CrossRef]

Lezec, H. G.

T. W. Ebbesen, H. G. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolf, "Extraordinary optical transmission through subwavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Liu, Z.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, "Far-field optical hyperlens magnifying sub-diffraction-limited objects," Science 315, 1686 (2007).
[CrossRef] [PubMed]

Mandelshtam, V. A.

V. A. Mandelshtam, "FDM: the Filter Diagonalization Method for data processing in NMR experiments," Prog. Nucl. Mag. Res. Sp. 38, 159-196 (2001).
[CrossRef]

Marziliano, P.

T. Blu, P. L. Dragotti, M. Vetterli, P. Marziliano, and L. Coulot, "Sparse sampling of signal innovations," IEEE Signal Process. Mag. 25, 31-40 (2008).
[CrossRef]

M. Vetterli, P. Marziliano, and T. Blu, "Sampling signals with finite rate of innovation," IEEE Trans. Sig. Proc. 50, 1417-1428 (2002).
[CrossRef]

McKinney, S. A.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, "Myosin v walks hand-overhand: Single fluorophore imaging with 1.5nm localization," Science 300, 2061-2065 (2003).
[CrossRef] [PubMed]

Michaeli, T.

Y. C. Eldar and T. Michaeli, "Beyond bandlimited sampling," IEEE Signal Proc. Mag. 26, 48-68 (2009).
[CrossRef]

Mishali, M.

M. Mishali and Y. C. Eldar, "Blind multi-band signal reconstruction: Compressed sensing for analog signals," IEEE Trans. Signal Process. 57, 993-1009 (2009).
[CrossRef]

Muray, A.

A. Lewis, M. Isaacson, A. Harotunian, and A. Muray, "Development of a 500°a spatial-resolution light microscope: I. light is efficiently transmitted through l/16 diameter apertures," Ultramicroscopy 13, 227-232 (1984).
[CrossRef]

Narimanov, E.

Neifeld, M. A.

Nicholls, G.

E. A. Ash and G. Nicholls, "Super-resolution aperture scanning microscope," Nature 237, 510-512 (1972).
[CrossRef] [PubMed]

Papoulis, A.

A. Papoulis, "A new algorithm in spectral analysis and band-limited extrapolation," IEEE Trans. Circuits Syst. 22, 735-742 (1975).
[CrossRef]

Pendry, J. B.

J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

Romberg, J.

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

Salandrino, A.

A. Salandrino and N. Engheta, "Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations," Phys. Rev. B 74, 075103 (2006).
[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]

Schmidt, R.

S.W. Hell, R. Schmidt, and A. Egner, "Diffraction-unlimited three-dimensional optical nanoscopy with opposing lenses," Nat. Photon. 3, 381-387 (2009).
[CrossRef]

Selvin, P. R.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, "Myosin v walks hand-overhand: Single fluorophore imaging with 1.5nm localization," Science 300, 2061-2065 (2003).
[CrossRef] [PubMed]

Smolyaninov, I. I.

I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, "Magnifying superlens in the visible frequency range," Science 315, 1699-1701 (2007).
[CrossRef] [PubMed]

Sun, C.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, "Far-field optical hyperlens magnifying sub-diffraction-limited objects," Science 315, 1686 (2007).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-diffraction-limited optical imaging with a silver superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Tanner, J.

D. L. Donoho and J. Tanner, "Sparse nonnegative solution of underdetermined linear equations by linear programming," Proc. Natl. Acad. Sci. 102, 9446-9451 (2005).
[CrossRef] [PubMed]

Tao, T.

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

E. J. Candes and T. Tao, "Near-optimal signal recovery from random projections: Universal encoding strategies?" IEEE Trans. Inf. Theory 52, 5406-5425 (2006).
[CrossRef]

Thio, T.

T. W. Ebbesen, H. G. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolf, "Extraordinary optical transmission through subwavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Trautman, J. K.

E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, and R. L. Kostelak, "Breaking the diffraction barrier: optical microscopy on a nanometric scale," Science 251, 1468-1470 (1991).
[CrossRef] [PubMed]

Vetterli, M.

T. Blu, P. L. Dragotti, M. Vetterli, P. Marziliano, and L. Coulot, "Sparse sampling of signal innovations," IEEE Signal Process. Mag. 25, 31-40 (2008).
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M. Vetterli, P. Marziliano, and T. Blu, "Sampling signals with finite rate of innovation," IEEE Trans. Sig. Proc. 50, 1417-1428 (2002).
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E. J. Candes and M. B. Wakin, "An introduction to compressive sampling," IEEE Signal Process. Mag. 25, 21-30 (2008).
[CrossRef]

Weiner, J. S.

E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, and R. L. Kostelak, "Breaking the diffraction barrier: optical microscopy on a nanometric scale," Science 251, 1468-1470 (1991).
[CrossRef] [PubMed]

Wolf, P. A.

T. W. Ebbesen, H. G. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolf, "Extraordinary optical transmission through subwavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Xiong, Y.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, "Far-field optical hyperlens magnifying sub-diffraction-limited objects," Science 315, 1686 (2007).
[CrossRef] [PubMed]

Yildiz, A.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, "Myosin v walks hand-overhand: Single fluorophore imaging with 1.5nm localization," Science 300, 2061-2065 (2003).
[CrossRef] [PubMed]

Zhang, X.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, "Far-field optical hyperlens magnifying sub-diffraction-limited objects," Science 315, 1686 (2007).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-diffraction-limited optical imaging with a silver superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

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F. M. Huang and N. I. Zheludev, "Super-resolution without evanescent waves," Nano Lett. 9, 1249-1254 (2009).
[CrossRef] [PubMed]

N. I. Zheludev, "What diffraction limit?" Nat. Mater. 7, 420-422 (2008).
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Zibulevsky, M.

A. M. Bruckstein, M. Elad, and M. Zibulevsky, "On the uniqueness of nonnegative sparse solutions to underdetermined systems of equations," IEEE Trans. Inf. Theory 54, 4813-4820 (2008).
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Appl. Opt.

IEEE Signal Proc. Mag.

Y. C. Eldar and T. Michaeli, "Beyond bandlimited sampling," IEEE Signal Proc. Mag. 26, 48-68 (2009).
[CrossRef]

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T. Blu, P. L. Dragotti, M. Vetterli, P. Marziliano, and L. Coulot, "Sparse sampling of signal innovations," IEEE Signal Process. Mag. 25, 31-40 (2008).
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E. J. Candes and M. B. Wakin, "An introduction to compressive sampling," IEEE Signal Process. Mag. 25, 21-30 (2008).
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IEEE Trans. Circuits Syst.

A. Papoulis, "A new algorithm in spectral analysis and band-limited extrapolation," IEEE Trans. Circuits Syst. 22, 735-742 (1975).
[CrossRef]

IEEE Trans. Inf. Theory

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

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

E. J. Candes and T. Tao, "Near-optimal signal recovery from random projections: Universal encoding strategies?" IEEE Trans. Inf. Theory 52, 5406-5425 (2006).
[CrossRef]

A. M. Bruckstein, M. Elad, and M. Zibulevsky, "On the uniqueness of nonnegative sparse solutions to underdetermined systems of equations," IEEE Trans. Inf. Theory 54, 4813-4820 (2008).
[CrossRef]

IEEE Trans. Sig. Proc.

M. Vetterli, P. Marziliano, and T. Blu, "Sampling signals with finite rate of innovation," IEEE Trans. Sig. Proc. 50, 1417-1428 (2002).
[CrossRef]

IEEE Trans. Signal Process.

Z. Ben-Haim, Y. C. Eldar, and M. Elad, "Near-oracle performance of basis pursuit under random noise," IEEE Trans. Signal Process. (submitted).

Y. C. Eldar, "Compressed sensing of analog signals in shift-invariant spaces," IEEE Trans. Signal Process. 57, 2986-2997 (2009).
[CrossRef]

M. Mishali and Y. C. Eldar, "Blind multi-band signal reconstruction: Compressed sensing for analog signals," IEEE Trans. Signal Process. 57, 993-1009 (2009).
[CrossRef]

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R. W. Gerchberg, "Super-resolution through error energy reduction," J. Mod. Opt. 21, 709-720 (1974).

Nano Lett.

F. M. Huang and N. I. Zheludev, "Super-resolution without evanescent waves," Nano Lett. 9, 1249-1254 (2009).
[CrossRef] [PubMed]

Nat. Mater.

N. I. Zheludev, "What diffraction limit?" Nat. Mater. 7, 420-422 (2008).
[CrossRef] [PubMed]

Nat. Photon.

S.W. Hell, R. Schmidt, and A. Egner, "Diffraction-unlimited three-dimensional optical nanoscopy with opposing lenses," Nat. Photon. 3, 381-387 (2009).
[CrossRef]

Nature

T. W. Ebbesen, H. G. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolf, "Extraordinary optical transmission through subwavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

E. A. Ash and G. Nicholls, "Super-resolution aperture scanning microscope," Nature 237, 510-512 (1972).
[CrossRef] [PubMed]

Opt. Express

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A. Salandrino and N. Engheta, "Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations," Phys. Rev. B 74, 075103 (2006).
[CrossRef]

Phys. Rev. Lett.

J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci.

D. L. Donoho and J. Tanner, "Sparse nonnegative solution of underdetermined linear equations by linear programming," Proc. Natl. Acad. Sci. 102, 9446-9451 (2005).
[CrossRef] [PubMed]

D. L. Donoho and M. Elad, "Optimally sparse representation in general (nonorthogonal) dictionaries via l1 minimization," Proc. Natl. Acad. Sci. 100, 2197-2201 (2003).
[CrossRef]

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V. A. Mandelshtam, "FDM: the Filter Diagonalization Method for data processing in NMR experiments," Prog. Nucl. Mag. Res. Sp. 38, 159-196 (2001).
[CrossRef]

Science

N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-diffraction-limited optical imaging with a silver superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, and R. L. Kostelak, "Breaking the diffraction barrier: optical microscopy on a nanometric scale," Science 251, 1468-1470 (1991).
[CrossRef] [PubMed]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, "Far-field optical hyperlens magnifying sub-diffraction-limited objects," Science 315, 1686 (2007).
[CrossRef] [PubMed]

I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, "Magnifying superlens in the visible frequency range," Science 315, 1699-1701 (2007).
[CrossRef] [PubMed]

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, "Myosin v walks hand-overhand: Single fluorophore imaging with 1.5nm localization," Science 300, 2061-2065 (2003).
[CrossRef] [PubMed]

SIAM J. Sci. Comput.

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

Ultramicroscopy

A. Lewis, M. Isaacson, A. Harotunian, and A. Muray, "Development of a 500°a spatial-resolution light microscope: I. light is efficiently transmitted through l/16 diameter apertures," Ultramicroscopy 13, 227-232 (1984).
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O. Katz, Y. Bromberg, and Y. Silberberg, "Ghost imaging via compressed sensing," in "Frontiers in Optics (FiO)," (2009).

M. Mishali and Y. C. Eldar, "From theory to practice: Sub-nyquist sampling of sparse wideband analog signals," arXiv [0902.4291v1] (2009).

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Figures (10)

Fig. 1.
Fig. 1.

Theoretical reconstruction of one-dimensional sub-wavelength information (amplitude and phase). (a) The original function, which we want to reconstruct. (b) The Fourier (plane-wave) spectrum of the original information shown in (a). The vertical red lines indicate the width of the low-pass filter, which for sub-wavelength information is 2/λ. (c) The distorted image obtained by an inverse Fourier transform on the filtered spectrum; the features are highly blurred. (d) The low-pass-filtered spectrum; a large fraction of the frequency contents is lost. (e,f) Reconstructed image (e) and its spectrum (f) using CS-methods based on the sparsity of the original information. The function is reconstructed perfectly in both real space and Fourier space, including the phase information. Our algorithm is robust against noise. (g,h) Adding 1% noise to the filtered spectrum (not shown here), we are still able to reconstruct the original information at high quality in both real space (g) and Fourier space (h). Amplitude and intensity are given in arbitrary units (a.u.), because the system does not depend on the light intensity.

Fig. 2.
Fig. 2.

Theoretical reconstruction of two-dimensional sub-wavelength information. (a,b) The original information consists of an arrangement of circles, forming the Star of David (a), and its respective Fourier transform (b). (c,d) After some propagation distance, all spatial frequencies above 1/λ are lost (d), so that the actual observed image is strongly blurred and the fine features cannot be resolved anymore (c). (e,f) Applying our CS algorithm reveals the underlying sub-wavelength structure in the real space (e), since the Fourier spectrum is fully restored (f).

Fig. 3.
Fig. 3.

Experimental setup for the proof-of-concept experiments. The laser beam is collimated using lenses L1 and L2, before the sample is illuminated. The signal is then Fourier transformed using lens L3, low-pass filtered by the slit and again Fourier transformed into the real plane by lens L4. Another lens L5 performs an additional Fourier transform, which is recorded by a camera. In order to measure the phase distribution, a probe beam is super-imposed (using the beam splitter BS) on the signal in order to create interference fringes. In an alternative setup, the information can be directly taken in the real plane, so that the camera is positioned directly behind lens L4. .

Fig. 4.
Fig. 4.

Experimental proof-of-concept: reconstruction of amplitude information. (a,b,c) The original information consisting of three vertical stripes (a), its Fourier spectrum (b), and a horizontal cross-section of the amplitude, taken through the real-space information (c). (d,e,f) Using the optical slit, the signal is low-pass filtered at the vertical red lines, yielding a highly blurred image (d). The Fourier spectrum now contains now only the lowest frequencies (e), which cause the mergence of the three stripes (in real-space) into one, as seen in the horizontal cross section (f). (g,h,k) Reconstruction using CS methods yields a high quality recovered information (g) and its respective Fourier spectrum (h). The strong correspondence between original and recovery is clearly visible in the horizontal cross section (k).

Fig. 5.
Fig. 5.

Experimental proof-of-concept: reconstruction of amplitude + phase information . An important feature of our proposed algorithm is the ability to recover both amplitude and phase, which is essential for pictorial information carried upon electromagnetic waves. (a,b,c) The original information consisting of three vertical stripes (a), its Fourier spectrum (b), and a horizontal cross-section of the amplitude, taken through the real-space information, revealing that the two stripes on the right are π-phase shifted with respect to the stripe on the left (c). (d,e,f) Using the optical slit, the signal is low-pass filtered at the vertical red lines, yielding a highly blurred image consisting of two distinct lobes (d). The Fourier spectrum now contains now only the lowest frequencies (e), which cause the mergence of the two stripes on the right, as seen in the horizontal cross section (f). (g,h,k) Reconstruction using CS methods yields a high quality recovered information (g) and its respective Fourier spectrum (h). The strong correspondence between original and recovery is clearly visible in the horizontal cross section (k).

Fig. 6.
Fig. 6.

(a) Mutual coherence of the lowpass Fourier matrix. (b) Reconstruction guarantees for BP. The maximal sparsity level that ensures exact reconstruction remains very low even for relatively high values of α. The case S=2 corresponding to two spikes is illustrated by the horizontal line.

Fig. 7.
Fig. 7.

(a) Reconstruction of an in-phase signal. The first row corresponds to the original information in real-space and in Fourier space. The following rows are reconstruction using Basis Pursuit with different cutoff frequencies, as indicated by the red LPF. Green corresponds to the original signal while blue is the reconstructed signal. In this example, these sequences overlap completely matrix. (b) Reconstruction of a multi-phase signal. The first row corresponds to the original sampled information in real space and Fourier space. The following rows are reconstruction using Basis Pursuit with different cutoff frequencies, as indicated by the red LPF. Green corresponds to the original signal while blue is the reconstructed signal. In this example, a high cutoff frequency is needed in order to obtain good recovery.

Fig. 8.
Fig. 8.

Probability of support recovery as a function of the SNR using the annihilating filter method.

Fig. 9.
Fig. 9.

(a) Probability of support recovery as a function of the SNR using the NLHT algorithm. (b) Reconstruction results of the NLHT algorithm for multiple-phase spikes.

Fig. 10.
Fig. 10.

Comparison between the performance of the Gerchberg-Papoulis extrapolation algorithm and our CS approach. The comparison is made on our experimental data (of Fig.4 of the paper) (a,b,c) The filtered information, blurred to a single stripe (a), its cut Fourier spectrum (b), and a horizontal cross-section of the amplitude, taken through the real-space information (c). (d,e,f) Reconstruction using GP-extrapolation methods yields a distorted recovery with little resemblance to the original data (d) and an incorrect Fourier spectrum (e). The recovery error is most apparent in the horizontal cross section (f). (g,h,k) Reconstruction using CS methods yields a high quality recovered information (g) and its respective Fourier spectrum (h). The strong correspondence between original and recovered image is clearly visible in the horizontal cross section (k).

Tables (1)

Tables Icon

Algorithm 1 Non Local hard Thresholding.

Equations (42)

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

E x y z = 0 = e { f ( x , y ) e iwt }
g x y = F k x k y H k x k y z e i ( k x x + k y y ) dk x dk y
p , q = p H q = Σ i p i * q i ,
( P 0 ) min d d 0 subject to y = Ψ ˜ x = Ψ ˜ Φ d = W d .
y = Wd 1 = Wd 2 .
W ( d 1 d 2 ) = W z = 0 ,
d 1 0 + d 2 0 d 1 d 2 0 .
μ ( A ) = max j , i j = a i H a j a i 2 a j 2 ,
Spark ( A ) 1 + 1 μ .
d 0 1 2 ( 1 + 1 μ ( W ) ) ,
( P 1 ) min d d 1 subiect to ψ ˜ x = ψ ˜ Φ d = W d .
g ( x ) = Σ = 0 N 1 d a ( x Δ ) .
A ( v ) = a ( x ) e j 2 π vx dx ,
D ( e j 2 π Δ v ) = Σ = 0 N 1 d e j 2 π Δ v .
D [ k ] = Σ = 0 N 1 d e j 2 π k N .
d = 1 N Σ k = ( N 1 ) 2 ( N 1 ) 2 D [ k ] e j 2 π k N .
G ( v ) = Σ = 0 N 1 d a ( x Δ ) e j 2 π vx dx
= Σ = 0 N 1 d e j 2 π Δ v a ( x ) e j 2 π v x dx
= D ( e j 2 π Δ v ) A ( v ) .
H ( v ) = { 1 v v c 0 else .
G FF ( v ) = G ( v ) H ( v ) = { D ( e j 2 π Δ v ) A ( v ) v v c 0 else .
k max = α N ,
c k = D ( e j 2 π k N ) A ( k N ) = D [ k ] A ( k N ) , k k max .
b k = c k A ( k N ) = D [ k ] , k k max .
g FF ( x ) = Σ = 0 N 1 d ( a s ) ( x )
a ( x ) = a ( x ) ,
s ( x ) = 2 α sinc ( 2 α x ) ,
( f 1 f 2 ) ( x ) = f 1 ( x χ ) f 2 ( χ ) d χ .
g FF ( x = k ) = d ˜ [ k ] = Σ = 0 l = N 1 d a ( k χ ) s ( χ ) , 0 k N 1 .
g FF = ( x = k τ ) = d ˜ [ k ] = Σ = 0 = N 1 d α ( k τ χ ) s ( χ ) d χ .
d ˜ = Md
M k = a ( k τ χ ) s ( χ ) d χ .
N F ˜ d ˜ = F ˜ M F ˜ H D .
P kk = { A ( k N ) e j 2 π τ k N , k k max 0 , k > k max .
d k = D ( e j 2 π k N ) , k k max ,
M = 2 k max + 1 = 2 α N + 1
( P 0 ) min d d 0 subject to b = Fd .
M N 2 β .
( P 1 ) min d d 1 subject to b = F d .
[ b 1 b 2 b k max b 0 b 1 b k max + 1 b k max 2 b k max 3 b 1 ] [ h 1 h 2 h S ] = [ b 0 b 1 b k max 1 ] .
( P 1 ) min d d 1 s . t b Fd 2 ε .
min d ̂ d ̂ 1 subject to b F d ̂ 2 , d ̂ [ ] = 0 , S .

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