Abstract

We demonstrate that sub-wavelength optical images borne on partially-spatially-incoherent light can be recovered, from their far-field or from the blurred image, given the prior knowledge that the image is sparse, and only that. The reconstruction method relies on the recently demonstrated sparsity-based sub-wavelength imaging. However, for partially-spatially-incoherent light, the relation between the measurements and the image is quadratic, yielding non-convex measurement equations that do not conform to previously used techniques. Consequently, we demonstrate new algorithmic methodology, referred to as quadratic compressed sensing, which can be applied to a range of other problems involving information recovery from partial correlation measurements, including when the correlation function has local dependencies. Specifically for microscopy, this method can be readily extended to white light microscopes with the additional knowledge of the light source spectrum.

© 2011 OSA

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References

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  1. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  2. E. A. Ash and G. Nicholls, “Super-resolution aperture scanning microscope,” Nature 237(5357), 510–512 (1972).
    [CrossRef] [PubMed]
  3. A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, “Development of a 500 A spatial resolution light microscope: I. light is efficiently transmitted through λ/16 diameter apertures,” Ultramicroscopy 13(3), 227–231 (1984).
    [CrossRef]
  4. 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(5000), 1468–1470 (1991).
    [CrossRef] [PubMed]
  5. S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Opt. Lett. 19(11), 780–782 (1994).
    [CrossRef] [PubMed]
  6. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006).
    [CrossRef] [PubMed]
  7. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
    [CrossRef] [PubMed]
  8. I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315(5819), 1699–1701 (2007).
    [CrossRef] [PubMed]
  9. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature 391(6668), 667–669 (1998).
    [CrossRef]
  10. F. M. Huang and N. I. Zheludev, “Super-resolution without evanescent waves,” Nano Lett. 9(3), 1249–1254 (2009).
    [CrossRef] [PubMed]
  11. S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express 17(26), 23920–23946 (2009).
    [CrossRef] [PubMed]
  12. E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
    [CrossRef]
  13. A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Rev. 51(1), 34–81 (2009).
    [CrossRef]
  14. A. Szameit, Y. Shechtman, H. Dana, S. Steiner, S. Gazit, T. Cohen-Hyams, E. Bullkich, O. Cohen, Y. C. Eldar, S. Shoham, E. B. Kley, and M. Segev, “Far-field microscopy of sparse subwavelength objects,” arXiv:1010.0631 [physics.optics] (2010).
  15. Y. Shechtman, S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse images carried by incoherent light,” Opt. Lett. 35(8), 1148–1150 (2010).
    [CrossRef] [PubMed]
  16. J. W. Goodman, Statistical Optics (Wiley, 1985).
  17. M. Fazel, H. Hindi, and S. Boyd, “Log-det heuristic for matrix rank minimization with applications to Hankel and Euclidean distance matrices,” in Proceedings of Am. Control Conf. (2003).
  18. E. J. Candes (Departments of Mathematics and of Statistics, Stanford University, Stanford CA 94305), Y. C. Eldar, T. Strohmer, and V. Voroninski are preparing a manuscript to be called “Phase retrieval via matrix completion.”
  19. D. Gross, Y. K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. 105(15), 150401 (2010).
    [CrossRef] [PubMed]
  20. A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106(10), 100401 (2011).
    [CrossRef] [PubMed]

2011

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106(10), 100401 (2011).
[CrossRef] [PubMed]

2010

D. Gross, Y. K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. 105(15), 150401 (2010).
[CrossRef] [PubMed]

Y. Shechtman, S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse images carried by incoherent light,” Opt. Lett. 35(8), 1148–1150 (2010).
[CrossRef] [PubMed]

2009

S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express 17(26), 23920–23946 (2009).
[CrossRef] [PubMed]

F. M. Huang and N. I. Zheludev, “Super-resolution without evanescent waves,” Nano Lett. 9(3), 1249–1254 (2009).
[CrossRef] [PubMed]

A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Rev. 51(1), 34–81 (2009).
[CrossRef]

2007

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[CrossRef] [PubMed]

I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315(5819), 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(2), 489–509 (2006).
[CrossRef]

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006).
[CrossRef] [PubMed]

1998

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature 391(6668), 667–669 (1998).
[CrossRef]

1994

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(5000), 1468–1470 (1991).
[CrossRef] [PubMed]

1984

A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, “Development of a 500 A spatial resolution light microscope: I. light is efficiently transmitted through λ/16 diameter apertures,” Ultramicroscopy 13(3), 227–231 (1984).
[CrossRef]

1972

E. A. Ash and G. Nicholls, “Super-resolution aperture scanning microscope,” Nature 237(5357), 510–512 (1972).
[CrossRef] [PubMed]

Alekseyev, L. V.

Almeida, M. P.

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106(10), 100401 (2011).
[CrossRef] [PubMed]

Ash, E. A.

E. A. Ash and G. Nicholls, “Super-resolution aperture scanning microscope,” Nature 237(5357), 510–512 (1972).
[CrossRef] [PubMed]

Becker, S.

D. Gross, Y. K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. 105(15), 150401 (2010).
[CrossRef] [PubMed]

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(5000), 1468–1470 (1991).
[CrossRef] [PubMed]

Broome, M. A.

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106(10), 100401 (2011).
[CrossRef] [PubMed]

Bruckstein, A. M.

A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Rev. 51(1), 34–81 (2009).
[CrossRef]

Candes, E. 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(2), 489–509 (2006).
[CrossRef]

Davis, C. C.

I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315(5819), 1699–1701 (2007).
[CrossRef] [PubMed]

Donoho, D. L.

A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Rev. 51(1), 34–81 (2009).
[CrossRef]

Ebbesen, T. W.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature 391(6668), 667–669 (1998).
[CrossRef]

Eisert, J.

D. Gross, Y. K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. 105(15), 150401 (2010).
[CrossRef] [PubMed]

Elad, M.

A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Rev. 51(1), 34–81 (2009).
[CrossRef]

Eldar, Y. C.

Fedrizzi, A.

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106(10), 100401 (2011).
[CrossRef] [PubMed]

Flammia, S. T.

D. Gross, Y. K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. 105(15), 150401 (2010).
[CrossRef] [PubMed]

Gazit, S.

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature 391(6668), 667–669 (1998).
[CrossRef]

Gross, D.

D. Gross, Y. K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. 105(15), 150401 (2010).
[CrossRef] [PubMed]

Harootunian, A.

A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, “Development of a 500 A spatial resolution light microscope: I. light is efficiently transmitted through λ/16 diameter apertures,” Ultramicroscopy 13(3), 227–231 (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(5000), 1468–1470 (1991).
[CrossRef] [PubMed]

Hell, S. W.

Huang, F. M.

F. M. Huang and N. I. Zheludev, “Super-resolution without evanescent waves,” Nano Lett. 9(3), 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(5819), 1699–1701 (2007).
[CrossRef] [PubMed]

Isaacson, M.

A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, “Development of a 500 A spatial resolution light microscope: I. light is efficiently transmitted through λ/16 diameter apertures,” Ultramicroscopy 13(3), 227–231 (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(5000), 1468–1470 (1991).
[CrossRef] [PubMed]

Kosut, R. L.

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106(10), 100401 (2011).
[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(5819), 1686 (2007).
[CrossRef] [PubMed]

Lewis, A.

A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, “Development of a 500 A spatial resolution light microscope: I. light is efficiently transmitted through λ/16 diameter apertures,” Ultramicroscopy 13(3), 227–231 (1984).
[CrossRef]

Lezec, H. J.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature 391(6668), 667–669 (1998).
[CrossRef]

Liu, Y. K.

D. Gross, Y. K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. 105(15), 150401 (2010).
[CrossRef] [PubMed]

Liu, Z.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[CrossRef] [PubMed]

Mohseni, M.

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106(10), 100401 (2011).
[CrossRef] [PubMed]

Muray, A.

A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, “Development of a 500 A spatial resolution light microscope: I. light is efficiently transmitted through λ/16 diameter apertures,” Ultramicroscopy 13(3), 227–231 (1984).
[CrossRef]

Narimanov, E.

Nicholls, G.

E. A. Ash and G. Nicholls, “Super-resolution aperture scanning microscope,” Nature 237(5357), 510–512 (1972).
[CrossRef] [PubMed]

Rabitz, H.

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106(10), 100401 (2011).
[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(2), 489–509 (2006).
[CrossRef]

Segev, M.

Shabani, A.

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106(10), 100401 (2011).
[CrossRef] [PubMed]

Shechtman, Y.

Smolyaninov, I. I.

I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315(5819), 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(5819), 1686 (2007).
[CrossRef] [PubMed]

Szameit, A.

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(2), 489–509 (2006).
[CrossRef]

Thio, T.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature 391(6668), 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(5000), 1468–1470 (1991).
[CrossRef] [PubMed]

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(5000), 1468–1470 (1991).
[CrossRef] [PubMed]

White, A. G.

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106(10), 100401 (2011).
[CrossRef] [PubMed]

Wichmann, J.

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature 391(6668), 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(5819), 1686 (2007).
[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(5819), 1686 (2007).
[CrossRef] [PubMed]

Zheludev, N. I.

F. M. Huang and N. I. Zheludev, “Super-resolution without evanescent waves,” Nano Lett. 9(3), 1249–1254 (2009).
[CrossRef] [PubMed]

IEEE Trans. Inf. Theory

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[CrossRef]

Nano Lett.

F. M. Huang and N. I. Zheludev, “Super-resolution without evanescent waves,” Nano Lett. 9(3), 1249–1254 (2009).
[CrossRef] [PubMed]

Nature

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature 391(6668), 667–669 (1998).
[CrossRef]

E. A. Ash and G. Nicholls, “Super-resolution aperture scanning microscope,” Nature 237(5357), 510–512 (1972).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

D. Gross, Y. K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. 105(15), 150401 (2010).
[CrossRef] [PubMed]

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106(10), 100401 (2011).
[CrossRef] [PubMed]

Science

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(5000), 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(5819), 1686 (2007).
[CrossRef] [PubMed]

I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315(5819), 1699–1701 (2007).
[CrossRef] [PubMed]

SIAM Rev.

A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Rev. 51(1), 34–81 (2009).
[CrossRef]

Ultramicroscopy

A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, “Development of a 500 A spatial resolution light microscope: I. light is efficiently transmitted through λ/16 diameter apertures,” Ultramicroscopy 13(3), 227–231 (1984).
[CrossRef]

Other

A. Szameit, Y. Shechtman, H. Dana, S. Steiner, S. Gazit, T. Cohen-Hyams, E. Bullkich, O. Cohen, Y. C. Eldar, S. Shoham, E. B. Kley, and M. Segev, “Far-field microscopy of sparse subwavelength objects,” arXiv:1010.0631 [physics.optics] (2010).

J. W. Goodman, Statistical Optics (Wiley, 1985).

M. Fazel, H. Hindi, and S. Boyd, “Log-det heuristic for matrix rank minimization with applications to Hankel and Euclidean distance matrices,” in Proceedings of Am. Control Conf. (2003).

E. J. Candes (Departments of Mathematics and of Statistics, Stanford University, Stanford CA 94305), Y. C. Eldar, T. Strohmer, and V. Voroninski are preparing a manuscript to be called “Phase retrieval via matrix completion.”

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

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Figures (8)

Fig. 1
Fig. 1

Setting for imaging using partially spatially coherent light.

Fig. 2
Fig. 2

Illustration of the effect of mutual-intensity width on output intensity, for 2 equal phase (a) or inverse phase (b) spikes, separated by 0.55λ (same/inverse phase). In (c) and (d) the spikes are 1λ apart. Coherent light corresponds to a mutual-intensity width of infinity; incoherent light corresponds to mutual-intensity width of 0. The partially coherent mutual-intensity width here is 0.55λ FWHM. The resolution is better for equal phase spikes when using incoherent light (a), and for inverse phase spikes – using coherent light (b). Note that the effect of the coherence of the light is much stronger for closely (sub-λ) separated peaks, and that the partially-incoherent image in that case (dash-dotted in (a) and (b)) is considerably different than both the coherent and incoherent images.

Fig. 3
Fig. 3

Algorithm 1.

Fig. 4
Fig. 4

Example subset of Mu matrices (dark indicates high values), corresponding to a Gaussian mutual-intensity function, and an impulse response of the form of sinc(x). Shown are the matrices M50 (a), M60 (b),and M70 (c).

Fig. 5
Fig. 5

Reconstruction of a sparse object consisting of 4 spikes at different phases ( ± π), distanced 0.5λ apart. The mutual intensity function of the light impinging on the object is a Gaussian with FWHM of 0.55λ. The coherent impulse response of the optical system is taken as h ( x ) = 1 x h 0 sin ( 2 π x λ ) . (a) Sparsity-based reconstruction, which yields reconstruction that is practically identical to the true input field (dashed-blue). (b) Reconstruction under the same assumptions, without using the sparsity prior Both reconstructions (dotted green, marked with *) are consistent with the measurements (red). The sparsity-based method of (a) yields a reconstruction virtually identical to the initial object, whereas the reconstruction that does not use sparsity, shown in (b), cannot retrieve any of the fine features of the object. The corresponding bandwidth extrapolation is shown in (c) – the Fourier spectrum of the recovered intensity (dashed green) contains information much higher than the output intensity, which is limited by the physical cutoff frequency at 2/λ (in blue).

Fig. 6
Fig. 6

Reconstruction error as a function of noise in the measurements, for sparsity-based reconstruction (marked with *) and for non-sparsity based reconstruction (dashed). The advantage of sparsity based reconstruction is clear, especially under low noise levels.

Fig. 7
Fig. 7

Reconstruction error as a function of the number of peaks. The peaks are equally spaced inside a range of 1.3λ. Each point is averaged over 100 reconstructions.

Fig. 8
Fig. 8

Reconstruction error as a function of the number of samples in the image plane. The samples are equally spaced. Each point in the plot is averaged over 100 reconstructions.

Equations (9)

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

I ( u ) = h ( u η 1 ) h * ( u η 2 ) J ( η 1 , η 2 ) d η 1 d η 2
J ( η 1 , η 2 ) = A ( η 1 , t ) u ( η 1 , t ) A * ( η 2 , t ) u * ( η 2 , t ) = = A ( η 1 ) A * ( η 2 ) u ( η 1 , t ) u * ( η 2 , t ) = A ( η 1 ) A * ( η 2 ) B ( η 1 , η 2 ) .
I ( u ) = h ( u η 1 ) h * ( u η 2 ) A ( η 1 ) A * ( η 2 ) B ( | η 1 η 2 | ) d η 1 d η 2 .
I ( u ) = i j h u ( η i ) h u * ( η j ) B ( η i η j ) A ( η i ) A * ( η j )
y u = a * M u a
( P )     min a a 0                               s . t . | a * M u a y u | 2 ε         u U a * a y * y
argmin X     log det ( X + δ I )     s . t . a ( b | X a b | 2 ) 1 / 2 ζ | T r a c e ( M u X ) y u | 2 ε T r a c e ( X ) t r ( y y T ) X 0
M u = h u h * u . * B
ε rec = a rec a true 2 a true 2

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