Abstract

We demonstrate theoretically and experimentally the reconstruction of images borne on incoherent light at a resolution greatly exceeding the finest resolution defined by the NA of the system. Our method relies on compressed sensing techniques, which assume that the object is sparse in a known basis, and only that. The approach is robust against noise and can be used for reconstructing subwavelength images through measurements taken in the optical far field.

© 2010 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Englewood, 2005).
  2. A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, Ultramicroscopy 13, 227 (1984).
    [CrossRef]
  3. E. Betzig, J. J. Trautman, T. D. Harris, J. S. Weiner, and R. L. Kostelak, Science 251, 1468 (1991).
    [CrossRef] [PubMed]
  4. F. M. Huang, T. S. Kao, V. A. Fedotov, Y. Chen, and N. I. Zheludev, Nano Lett. 8, 2469 (2008).
    [CrossRef] [PubMed]
  5. A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, Science 300, 2061 (2003).
    [CrossRef] [PubMed]
  6. S. W. Hell, R. Schmidt, and A. Egner, Nature Photon. 3, 381 (2009).
    [CrossRef]
  7. J. Pendry, Phys. Rev. Lett. 85, 3966 (2000).
    [CrossRef] [PubMed]
  8. N. Fang, H. Lee, C. Sun, and X. Zhang, Science 308, 534 (2005).
    [CrossRef] [PubMed]
  9. Z. Jacob, L. V. Alekseyev, and E. Narimanov, Opt. Express 14, 8247 (2006).
    [CrossRef] [PubMed]
  10. A. Salandrino and N. Engheta, Phys. Rev. B 74, 075103 (2006).
    [CrossRef]
  11. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, Science 315, 1686 (2007).
    [CrossRef] [PubMed]
  12. I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, Science 315, 1699 (2007).
    [CrossRef] [PubMed]
  13. A. Papoulis, IEEE Trans. Circuits Syst. 22, 735 (1975).
    [CrossRef]
  14. R. W. Gerchberg, J. Mod. Opt. 21, 709 (1974).
  15. B. R. Hunt, Int. J. Imaging Syst. Technol. 6, 297 (1995).
    [CrossRef]
  16. S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, Opt. Express 17, 23920 (2009).
    [CrossRef]
  17. E. J. Candes and T. Tao, IEEE Trans. Inf. Theory 52, 5406 (2006).
    [CrossRef]
  18. D. L. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).
    [CrossRef]
  19. P. Sementilli, B. R. Hunt, and M. S. Nadar, J. Opt. Soc. Am. A 10, 2265 (1993).
    [CrossRef]
  20. A. M. Bruckstein, D. L. Donoho, and M. Elad, SIAM Rev. 51, 34 (2009).
    [CrossRef]
  21. Y. C. Eldar, IEEE Trans. Signal Process. 57, 2986 (2009).
    [CrossRef]
  22. M. Mishali and Y. C. Eldar, IEEE J. Sel. Top. Signal Process. 4, 375 (2010).
    [CrossRef]
  23. J. A. Tropp, IEEE Trans Info. Theory 50, 2231 (2004).
    [CrossRef]
  24. D. L. Donoho, M. Elad, and V. M. Temlyakov, IEEE Trans. Inf. Theory 52, 6 (2006).
    [CrossRef]
  25. S. S. Chen, D. L. Donoho, and M. A. Saunders, SIAM J. Sci. Comput. (USA) 20, 33 (1998).
    [CrossRef]

2010 (1)

M. Mishali and Y. C. Eldar, IEEE J. Sel. Top. Signal Process. 4, 375 (2010).
[CrossRef]

2009 (4)

A. M. Bruckstein, D. L. Donoho, and M. Elad, SIAM Rev. 51, 34 (2009).
[CrossRef]

Y. C. Eldar, IEEE Trans. Signal Process. 57, 2986 (2009).
[CrossRef]

S. W. Hell, R. Schmidt, and A. Egner, Nature Photon. 3, 381 (2009).
[CrossRef]

S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, Opt. Express 17, 23920 (2009).
[CrossRef]

2008 (1)

F. M. Huang, T. S. Kao, V. A. Fedotov, Y. Chen, and N. I. Zheludev, Nano Lett. 8, 2469 (2008).
[CrossRef] [PubMed]

2007 (2)

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, Science 315, 1686 (2007).
[CrossRef] [PubMed]

I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, Science 315, 1699 (2007).
[CrossRef] [PubMed]

2006 (5)

E. J. Candes and T. Tao, IEEE Trans. Inf. Theory 52, 5406 (2006).
[CrossRef]

D. L. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).
[CrossRef]

Z. Jacob, L. V. Alekseyev, and E. Narimanov, Opt. Express 14, 8247 (2006).
[CrossRef] [PubMed]

A. Salandrino and N. Engheta, Phys. Rev. B 74, 075103 (2006).
[CrossRef]

D. L. Donoho, M. Elad, and V. M. Temlyakov, IEEE Trans. Inf. Theory 52, 6 (2006).
[CrossRef]

2005 (2)

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Englewood, 2005).

N. Fang, H. Lee, C. Sun, and X. Zhang, Science 308, 534 (2005).
[CrossRef] [PubMed]

2004 (1)

J. A. Tropp, IEEE Trans Info. Theory 50, 2231 (2004).
[CrossRef]

2003 (1)

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, Science 300, 2061 (2003).
[CrossRef] [PubMed]

2000 (1)

J. Pendry, Phys. Rev. Lett. 85, 3966 (2000).
[CrossRef] [PubMed]

1998 (1)

S. S. Chen, D. L. Donoho, and M. A. Saunders, SIAM J. Sci. Comput. (USA) 20, 33 (1998).
[CrossRef]

1995 (1)

B. R. Hunt, Int. J. Imaging Syst. Technol. 6, 297 (1995).
[CrossRef]

1993 (1)

1991 (1)

E. Betzig, J. J. Trautman, T. D. Harris, J. S. Weiner, and R. L. Kostelak, Science 251, 1468 (1991).
[CrossRef] [PubMed]

1984 (1)

A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, Ultramicroscopy 13, 227 (1984).
[CrossRef]

1975 (1)

A. Papoulis, IEEE Trans. Circuits Syst. 22, 735 (1975).
[CrossRef]

1974 (1)

R. W. Gerchberg, J. Mod. Opt. 21, 709 (1974).

Alekseyev, L. V.

Betzig, E.

E. Betzig, J. J. Trautman, T. D. Harris, J. S. Weiner, and R. L. Kostelak, Science 251, 1468 (1991).
[CrossRef] [PubMed]

Bruckstein, A. M.

A. M. Bruckstein, D. L. Donoho, and M. Elad, SIAM Rev. 51, 34 (2009).
[CrossRef]

Candes, E. J.

E. J. Candes and T. Tao, IEEE Trans. Inf. Theory 52, 5406 (2006).
[CrossRef]

Chen, S. S.

S. S. Chen, D. L. Donoho, and M. A. Saunders, SIAM J. Sci. Comput. (USA) 20, 33 (1998).
[CrossRef]

Chen, Y.

F. M. Huang, T. S. Kao, V. A. Fedotov, Y. Chen, and N. I. Zheludev, Nano Lett. 8, 2469 (2008).
[CrossRef] [PubMed]

Davis, C. C.

I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, Science 315, 1699 (2007).
[CrossRef] [PubMed]

Donoho, D. L.

A. M. Bruckstein, D. L. Donoho, and M. Elad, SIAM Rev. 51, 34 (2009).
[CrossRef]

D. L. Donoho, M. Elad, and V. M. Temlyakov, IEEE Trans. Inf. Theory 52, 6 (2006).
[CrossRef]

D. L. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).
[CrossRef]

S. S. Chen, D. L. Donoho, and M. A. Saunders, SIAM J. Sci. Comput. (USA) 20, 33 (1998).
[CrossRef]

Egner, A.

S. W. Hell, R. Schmidt, and A. Egner, Nature Photon. 3, 381 (2009).
[CrossRef]

Elad, M.

A. M. Bruckstein, D. L. Donoho, and M. Elad, SIAM Rev. 51, 34 (2009).
[CrossRef]

D. L. Donoho, M. Elad, and V. M. Temlyakov, IEEE Trans. Inf. Theory 52, 6 (2006).
[CrossRef]

Eldar, Y. C.

M. Mishali and Y. C. Eldar, IEEE J. Sel. Top. Signal Process. 4, 375 (2010).
[CrossRef]

Y. C. Eldar, IEEE Trans. Signal Process. 57, 2986 (2009).
[CrossRef]

S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, Opt. Express 17, 23920 (2009).
[CrossRef]

Engheta, N.

A. Salandrino and N. Engheta, Phys. Rev. B 74, 075103 (2006).
[CrossRef]

Fang, N.

N. Fang, H. Lee, C. Sun, and X. Zhang, Science 308, 534 (2005).
[CrossRef] [PubMed]

Fedotov, V. A.

F. M. Huang, T. S. Kao, V. A. Fedotov, Y. Chen, and N. I. Zheludev, Nano Lett. 8, 2469 (2008).
[CrossRef] [PubMed]

Forkey, J. N.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, Science 300, 2061 (2003).
[CrossRef] [PubMed]

Gazit, S.

Gerchberg, R. W.

R. W. Gerchberg, J. Mod. Opt. 21, 709 (1974).

Goldman, Y. E.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, Science 300, 2061 (2003).
[CrossRef] [PubMed]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Englewood, 2005).

Ha, T.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, Science 300, 2061 (2003).
[CrossRef] [PubMed]

Harootunian, A.

A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, Ultramicroscopy 13, 227 (1984).
[CrossRef]

Harris, T. D.

E. Betzig, J. J. Trautman, T. D. Harris, J. S. Weiner, and R. L. Kostelak, Science 251, 1468 (1991).
[CrossRef] [PubMed]

Hell, S. W.

S. W. Hell, R. Schmidt, and A. Egner, Nature Photon. 3, 381 (2009).
[CrossRef]

Huang, F. M.

F. M. Huang, T. S. Kao, V. A. Fedotov, Y. Chen, and N. I. Zheludev, Nano Lett. 8, 2469 (2008).
[CrossRef] [PubMed]

Hung, Y. J.

I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, Science 315, 1699 (2007).
[CrossRef] [PubMed]

Hunt, B. R.

Isaacson, M.

A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, Ultramicroscopy 13, 227 (1984).
[CrossRef]

Jacob, Z.

Kao, T. S.

F. M. Huang, T. S. Kao, V. A. Fedotov, Y. Chen, and N. I. Zheludev, Nano Lett. 8, 2469 (2008).
[CrossRef] [PubMed]

Kostelak, R. L.

E. Betzig, J. J. Trautman, T. D. Harris, J. S. Weiner, and R. L. Kostelak, Science 251, 1468 (1991).
[CrossRef] [PubMed]

Lee, H.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, Science 315, 1686 (2007).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, Science 308, 534 (2005).
[CrossRef] [PubMed]

Lewis, A.

A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, Ultramicroscopy 13, 227 (1984).
[CrossRef]

Liu, Z.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, Science 315, 1686 (2007).
[CrossRef] [PubMed]

McKinney, S. A.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, Science 300, 2061 (2003).
[CrossRef] [PubMed]

Mishali, M.

M. Mishali and Y. C. Eldar, IEEE J. Sel. Top. Signal Process. 4, 375 (2010).
[CrossRef]

Muray, A.

A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, Ultramicroscopy 13, 227 (1984).
[CrossRef]

Nadar, M. S.

Narimanov, E.

Papoulis, A.

A. Papoulis, IEEE Trans. Circuits Syst. 22, 735 (1975).
[CrossRef]

Pendry, J.

J. Pendry, Phys. Rev. Lett. 85, 3966 (2000).
[CrossRef] [PubMed]

Salandrino, A.

A. Salandrino and N. Engheta, Phys. Rev. B 74, 075103 (2006).
[CrossRef]

Saunders, M. A.

S. S. Chen, D. L. Donoho, and M. A. Saunders, SIAM J. Sci. Comput. (USA) 20, 33 (1998).
[CrossRef]

Schmidt, R.

S. W. Hell, R. Schmidt, and A. Egner, Nature Photon. 3, 381 (2009).
[CrossRef]

Segev, M.

Selvin, P. R.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, Science 300, 2061 (2003).
[CrossRef] [PubMed]

Sementilli, P.

Smolyaninov, I. I.

I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, Science 315, 1699 (2007).
[CrossRef] [PubMed]

Sun, C.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, Science 315, 1686 (2007).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, Science 308, 534 (2005).
[CrossRef] [PubMed]

Szameit, A.

Tao, T.

E. J. Candes and T. Tao, IEEE Trans. Inf. Theory 52, 5406 (2006).
[CrossRef]

Temlyakov, V. M.

D. L. Donoho, M. Elad, and V. M. Temlyakov, IEEE Trans. Inf. Theory 52, 6 (2006).
[CrossRef]

Trautman, J. J.

E. Betzig, J. J. Trautman, T. D. Harris, J. S. Weiner, and R. L. Kostelak, Science 251, 1468 (1991).
[CrossRef] [PubMed]

Tropp, J. A.

J. A. Tropp, IEEE Trans Info. Theory 50, 2231 (2004).
[CrossRef]

Weiner, J. S.

E. Betzig, J. J. Trautman, T. D. Harris, J. S. Weiner, and R. L. Kostelak, Science 251, 1468 (1991).
[CrossRef] [PubMed]

Xiong, Y.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, 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, Science 300, 2061 (2003).
[CrossRef] [PubMed]

Zhang, X.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, Science 315, 1686 (2007).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, Science 308, 534 (2005).
[CrossRef] [PubMed]

Zheludev, N. I.

F. M. Huang, T. S. Kao, V. A. Fedotov, Y. Chen, and N. I. Zheludev, Nano Lett. 8, 2469 (2008).
[CrossRef] [PubMed]

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

M. Mishali and Y. C. Eldar, IEEE J. Sel. Top. Signal Process. 4, 375 (2010).
[CrossRef]

IEEE Trans Info. Theory (1)

J. A. Tropp, IEEE Trans Info. Theory 50, 2231 (2004).
[CrossRef]

IEEE Trans. Circuits Syst. (1)

A. Papoulis, IEEE Trans. Circuits Syst. 22, 735 (1975).
[CrossRef]

IEEE Trans. Inf. Theory (3)

E. J. Candes and T. Tao, IEEE Trans. Inf. Theory 52, 5406 (2006).
[CrossRef]

D. L. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).
[CrossRef]

D. L. Donoho, M. Elad, and V. M. Temlyakov, IEEE Trans. Inf. Theory 52, 6 (2006).
[CrossRef]

IEEE Trans. Signal Process. (1)

Y. C. Eldar, IEEE Trans. Signal Process. 57, 2986 (2009).
[CrossRef]

Int. J. Imaging Syst. Technol. (1)

B. R. Hunt, Int. J. Imaging Syst. Technol. 6, 297 (1995).
[CrossRef]

J. Mod. Opt. (1)

R. W. Gerchberg, J. Mod. Opt. 21, 709 (1974).

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

Nano Lett. (1)

F. M. Huang, T. S. Kao, V. A. Fedotov, Y. Chen, and N. I. Zheludev, Nano Lett. 8, 2469 (2008).
[CrossRef] [PubMed]

Nature Photon. (1)

S. W. Hell, R. Schmidt, and A. Egner, Nature Photon. 3, 381 (2009).
[CrossRef]

Opt. Express (2)

Phys. Rev. B (1)

A. Salandrino and N. Engheta, Phys. Rev. B 74, 075103 (2006).
[CrossRef]

Phys. Rev. Lett. (1)

J. Pendry, Phys. Rev. Lett. 85, 3966 (2000).
[CrossRef] [PubMed]

Science (5)

N. Fang, H. Lee, C. Sun, and X. Zhang, Science 308, 534 (2005).
[CrossRef] [PubMed]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, Science 315, 1686 (2007).
[CrossRef] [PubMed]

I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, Science 315, 1699 (2007).
[CrossRef] [PubMed]

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, Science 300, 2061 (2003).
[CrossRef] [PubMed]

E. Betzig, J. J. Trautman, T. D. Harris, J. S. Weiner, and R. L. Kostelak, Science 251, 1468 (1991).
[CrossRef] [PubMed]

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

S. S. Chen, D. L. Donoho, and M. A. Saunders, SIAM J. Sci. Comput. (USA) 20, 33 (1998).
[CrossRef]

SIAM Rev. (1)

A. M. Bruckstein, D. L. Donoho, and M. Elad, SIAM Rev. 51, 34 (2009).
[CrossRef]

Ultramicroscopy (1)

A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, Ultramicroscopy 13, 227 (1984).
[CrossRef]

Other (1)

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Englewood, 2005).

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Figures (4)

Fig. 1
Fig. 1

(a) Simulated 2D OTF. (b) Our measured 1D OTF (only positive frequencies shown).

Fig. 2
Fig. 2

Theoretical example demonstrating reconstruction of a 1D image carried by incoherent light. (a) 1D image comprising of rectangles 2 / ( 5 ν c ) wide ( λ / 5 , if ν c = 2 / λ ). (b) Spatial power spectrum of (a). (c) Smeared (filtered) image. (d) Spatial power spectrum of the filtered image of (c), with the triangle marking the OTF. (e) CS-reconstructed image. (f) Spatial power spectrum of the reconstructed image. Note the reconstructed frequencies in (f), residing far beyond ν c . Although showing only the magnitude of the spectrum, the phase of the Fourier transform was also reconstructed well.

Fig. 3
Fig. 3

Experimental setup.

Fig. 4
Fig. 4

Experiments. (a) Original 2D image of three stripes. (b) Smeared (filtered) image. (c) CS-reconstructed image. The weak “ghost” rectangles in (c) arise from slight mismatch between the locations of the basis functions in the model and the true locations of the stripes; this happens because the blurred image is sampled by the pixels of the camera, leading to quantization error.

Equations (2)

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u out ( x , y ) = u in h ic ( x , y ) ,
x ̂ = arg x   min x 1   subject to { ( Σ T W Σ ) 1 / 2 < ε , x 0 , }

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