## 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**OSA Recommended Articles**

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### References

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- E. Hecht, Optics (Addison-Wesley, 1998).

- M. Saleh and B. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

- E. A. Ash and G. Nicholls, “Super-resolution aperture scanning microscope,” Nature 237, 510–512 (1972).

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

[Crossref] - 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] - 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] - F. M. Huang and N. I. Zheludev, “Super-resolution without evanescent waves,” Nano Lett. 9, 1249–1254 (2009).

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

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

[Crossref] - 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-over-hand: Single fluorophore imaging with 1.5nm localization,” Science 300, 2061–2065 (2003).

[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] - N. I. Zheludev, “What diffraction limit?” Nat. Mater. 7, 420–422 (2008).

[Crossref] [PubMed] - J. W. Goodman, Introduction to Fourier optics(Englewood, CO: Roberts & Co. Publishers, 2005), 3rd ed.

- A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. 22, 735–742 (1975).

[Crossref] - R. W. Gerchberg, “Super-resolution through error energy reduction,” J. Mod. Opt. 21, 709–720 (1974).

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

[Crossref] - D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).

[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] -
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] - O. Katz, Y. Bromberg, and Y. Silberberg, “Ghost imaging via compressed sensing,” in “Frontiers in Optics (FiO),” (2009).

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

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

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

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

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

- 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] - Y. C. Eldar and T. Michaeli, “Beyond bandlimited sampling,” IEEE Signal Proc. Mag. 26, 48–68 (2009).

[Crossref] - 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] - 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] - 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]

#### 2009 (5)

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 (5)

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 (2)

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 (5)

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 (2)

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 (2)

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin v walks hand-over-hand: 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 (1)

M. Vetterli, P. Marziliano, and T. Blu, “Sampling signals with finite rate of innovation,” IEEE Trans. Sig. Proc. 50, 1417–1428 (2002).

[Crossref]

#### 2001 (1)

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 (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).

[Crossref]
[PubMed]

#### 1998 (2)

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 (1)

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 (1)

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

[Crossref]

#### 1975 (1)

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. 22, 735–742 (1975).

[Crossref]

#### 1974 (1)

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

#### 1972 (1)

E. A. Ash and G. Nicholls, “Super-resolution aperture scanning microscope,” Nature 237, 510–512 (1972).

[Crossref]
[PubMed]

#### Alexeyev, L. V.

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]

#### Ash, E. A.

E. A. Ash and G. Nicholls, “Super-resolution aperture scanning microscope,” Nature 237, 510–512 (1972).

[Crossref]
[PubMed]

#### Ashok, A.

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]

#### Baheti, P. K.

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]

#### 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]

#### Bromberg, Y.

O. Katz, Y. Bromberg, and Y. Silberberg, “Ghost imaging via compressed sensing,” in “Frontiers in Optics (FiO),” (2009).

#### 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, 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]

#### 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.

[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).

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

#### 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-over-hand: 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-over-hand: Single fluorophore imaging with 1.5nm localization,” Science 300, 2061–2065 (2003).

[Crossref]
[PubMed]

#### Goodman, J. W.

J. W. Goodman, Introduction to Fourier optics(Englewood, CO: Roberts & Co. Publishers, 2005), 3rd ed.

#### Ha, T.

[Crossref]
[PubMed]

#### Harotunian, A.

A. Lewis, M. Isaacson, A. Harotunian, and A. Muray, “Development of a 500å 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]

#### Hecht, E.

E. Hecht, Optics (Addison-Wesley, 1998).

#### 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 500å spatial-resolution light-microscope: I. light is efficiently transmitted through l/16 diameter apertures,” Ultramicroscopy 13, 227–232 (1984).

[Crossref]

#### Jacob, Z.

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]

#### Katz, O.

O. Katz, Y. Bromberg, and Y. Silberberg, “Ghost imaging via compressed sensing,” in “Frontiers in Optics (FiO),” (2009).

#### Kostelak, R. L.

[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.

[Crossref]

#### Lezec, H. G.

[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.

[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.

[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]

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

#### Muray, A.

[Crossref]

#### Narimanov, E.

[Crossref]
[PubMed]

#### Neifeld, M. A.

[Crossref]
[PubMed]

#### 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]

#### Saleh, M.

M. Saleh and B. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

#### Saunders, M. A.

[Crossref]

#### Schmidt, R.

[Crossref]

#### Selvin, P. R.

[Crossref]
[PubMed]

#### Silberberg, Y.

O. Katz, Y. Bromberg, and Y. Silberberg, “Ghost imaging via compressed sensing,” in “Frontiers in Optics (FiO),” (2009).

#### Smolyaninov, I. I.

[Crossref]
[PubMed]

#### Sun, C.

[Crossref]
[PubMed]

[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.

[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]

#### Teich, B.

M. Saleh and B. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

#### Thio, T.

[Crossref]

#### Trautman, J. K.

[Crossref]
[PubMed]

#### Vetterli, M.

[Crossref]

[Crossref]

#### Wakin, M. B.

E. J. Candes and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25, 21–30 (2008).

[Crossref]

#### Weiner, J. S.

[Crossref]
[PubMed]

#### Wolf, P. A.

[Crossref]

#### Xiong, Y.

[Crossref]
[PubMed]

#### Yildiz, A.

[Crossref]
[PubMed]

#### Zhang, X.

[Crossref]
[PubMed]

[Crossref]
[PubMed]

#### Zheludev, N. I.

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).

[Crossref]
[PubMed]

#### Zibulevsky, M.

[Crossref]

#### Appl. Opt. (1)

[Crossref]
[PubMed]

#### IEEE Signal Proc. Mag. (1)

[Crossref]

#### IEEE Signal Process. Mag. (2)

[Crossref]

[Crossref]

#### IEEE Trans. Circuits Syst. (1)

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 (4)

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).

[Crossref]

[Crossref]

[Crossref]

[Crossref]

#### IEEE Trans. Sig. Proc. (1)

[Crossref]

#### IEEE Trans. Signal Process. (2)

Y. C. Eldar, “Compressed sensing of analog signals in shift-invariant spaces,” IEEE Trans. Signal Process. 57, 2986–2997 (2009).

[Crossref]

[Crossref]

#### J. Mod. Opt. (1)

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

#### Nano Lett. (1)

[Crossref]
[PubMed]

#### Nat. Mater. (1)

N. I. Zheludev, “What diffraction limit?” Nat. Mater. 7, 420–422 (2008).

[Crossref]
[PubMed]

#### Nat. Photon. (1)

[Crossref]

#### Nature (2)

[Crossref]

[Crossref]
[PubMed]

#### Opt. Express (1)

[Crossref]
[PubMed]

#### Phys. Rev. B (1)

[Crossref]

#### Phys. Rev. Lett. (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).

[Crossref]
[PubMed]

#### Proc. Natl. Acad. Sci. (2)

[Crossref]

[Crossref]
[PubMed]

#### Prog. Nucl. Mag. Res. Sp. (1)

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 (5)

[Crossref]
[PubMed]

[Crossref]
[PubMed]

[Crossref]
[PubMed]

[Crossref]
[PubMed]

[Crossref]
[PubMed]

#### SIAM J. Sci. Comput. (1)

[Crossref]

#### Ultramicroscopy (1)

[Crossref]

#### Other (6)

E. Hecht, Optics (Addison-Wesley, 1998).

M. Saleh and B. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

J. W. Goodman, Introduction to Fourier optics(Englewood, CO: Roberts & Co. Publishers, 2005), 3rd ed.

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. 2.**

**Fig. 3.**

**Fig. 4.**

**Fig. 5.**

**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

**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.**

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

**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.**

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)

**Algorithm 1** Non Local hard Thresholding.

### Equations (42)

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