Abstract

Speckle suppression is one of the most important tasks in the image transmission through turbid media. Insufficient speckle suppression requires an additional procedure such as temporal ensemble averaging over multiple exposures. In this paper, we consider the image recovery process based on the so-called transmission matrix (TM) of turbid media for the image transmission through the media. We show that the speckle left unremoved in the TM-based image recovery can be suppressed effectively via sparse representation (SR). SR is a relatively new signal reconstruction framework which works well even for ill-conditioned problems. This is the first study to show the benefit of using the SR as compared to the phase conjugation (PC) a de facto standard method to date for TM-based imaging through turbid media including a live cell through tissue slice.

© 2014 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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2013 (1)

T. R. Hillman, T. Yamauchi, W. Choi, R. R. Dasari, M. S. Feld, Y. Park, and Z. Yaqoob, “Digital optical phase conjugation for delivering two-dimensional images through turbid media,” Sci Rep3, 1909 (2013).
[CrossRef] [PubMed]

2012 (3)

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics6(5), 283–292 (2012).
[CrossRef]

V. Studer, J. Bobin, M. Chahid, H. S. Mousavi, E. Candes, and M. Dahan, “Compressive fluorescence microscopy for biological and hyperspectral imaging,” Proc. Natl. Acad. Sci. U.S.A.109(26), E1679–E1687 (2012).
[CrossRef] [PubMed]

L. Zhu, W. Zhang, D. Elnatan, and B. Huang, “Faster STORM using compressed sensing,” Nat. Methods9(7), 721–723 (2012).
[CrossRef] [PubMed]

2011 (4)

Y. Choi, T. D. Yang, C. Fang-Yen, P. Kang, K. J. Lee, R. R. Dasari, M. S. Feld, and W. Choi, “Overcoming the diffraction limit using multiple light scattering in a highly disordered medium,” Phys. Rev. Lett.107(2), 023902 (2011).
[CrossRef] [PubMed]

E. J. Candès, Y. C. Eldar, D. Needell, and P. Randall, “Compressed sensing with coherent and redundant dictionaries,” Appl. Comput. Harmon. Anal.31(1), 59–73 (2011).
[CrossRef]

J. Yang and Y. Zhang, “Alternating direction algorithms for L1-problems in compressive sensing,” SIAM J. Sci. Comput.33(1), 250–278 (2011).
[CrossRef]

Y. Choi, M. Kim, C. Yoon, T. D. Yang, K. J. Lee, and W. Choi, “Synthetic aperture microscopy for high resolution imaging through a turbid medium,” Opt. Lett.36(21), 4263–4265 (2011).
[CrossRef] [PubMed]

2010 (4)

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics4(5), 320–322 (2010).
[CrossRef]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett.104(10), 100601 (2010).
[CrossRef] [PubMed]

M. Cui and C. Yang, “Implementation of a digital optical phase conjugation system and its application to study the robustness of turbidity suppression by phase conjugation,” Opt. Express18(4), 3444–3455 (2010).
[CrossRef] [PubMed]

S. M. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat Commun1(6), 81 (2010).
[CrossRef] [PubMed]

2009 (2)

D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express17(15), 13040–13049 (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]

2008 (1)

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics2(2), 110–115 (2008).
[CrossRef] [PubMed]

2007 (1)

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med.58(6), 1182–1195 (2007).
[CrossRef] [PubMed]

2005 (1)

1988 (1)

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett.61(20), 2328–2331 (1988).
[CrossRef] [PubMed]

1976 (1)

Bobin, J.

V. Studer, J. Bobin, M. Chahid, H. S. Mousavi, E. Candes, and M. Dahan, “Compressive fluorescence microscopy for biological and hyperspectral imaging,” Proc. Natl. Acad. Sci. U.S.A.109(26), E1679–E1687 (2012).
[CrossRef] [PubMed]

Boccara, A. C.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett.104(10), 100601 (2010).
[CrossRef] [PubMed]

S. M. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat Commun1(6), 81 (2010).
[CrossRef] [PubMed]

Brady, D. J.

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.

V. Studer, J. Bobin, M. Chahid, H. S. Mousavi, E. Candes, and M. Dahan, “Compressive fluorescence microscopy for biological and hyperspectral imaging,” Proc. Natl. Acad. Sci. U.S.A.109(26), E1679–E1687 (2012).
[CrossRef] [PubMed]

Candès, E. J.

E. J. Candès, Y. C. Eldar, D. Needell, and P. Randall, “Compressed sensing with coherent and redundant dictionaries,” Appl. Comput. Harmon. Anal.31(1), 59–73 (2011).
[CrossRef]

Carminati, R.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett.104(10), 100601 (2010).
[CrossRef] [PubMed]

Chahid, M.

V. Studer, J. Bobin, M. Chahid, H. S. Mousavi, E. Candes, and M. Dahan, “Compressive fluorescence microscopy for biological and hyperspectral imaging,” Proc. Natl. Acad. Sci. U.S.A.109(26), E1679–E1687 (2012).
[CrossRef] [PubMed]

Choi, K.

Choi, W.

T. R. Hillman, T. Yamauchi, W. Choi, R. R. Dasari, M. S. Feld, Y. Park, and Z. Yaqoob, “Digital optical phase conjugation for delivering two-dimensional images through turbid media,” Sci Rep3, 1909 (2013).
[CrossRef] [PubMed]

Y. Choi, T. D. Yang, C. Fang-Yen, P. Kang, K. J. Lee, R. R. Dasari, M. S. Feld, and W. Choi, “Overcoming the diffraction limit using multiple light scattering in a highly disordered medium,” Phys. Rev. Lett.107(2), 023902 (2011).
[CrossRef] [PubMed]

Y. Choi, M. Kim, C. Yoon, T. D. Yang, K. J. Lee, and W. Choi, “Synthetic aperture microscopy for high resolution imaging through a turbid medium,” Opt. Lett.36(21), 4263–4265 (2011).
[CrossRef] [PubMed]

Choi, Y.

Y. Choi, T. D. Yang, C. Fang-Yen, P. Kang, K. J. Lee, R. R. Dasari, M. S. Feld, and W. Choi, “Overcoming the diffraction limit using multiple light scattering in a highly disordered medium,” Phys. Rev. Lett.107(2), 023902 (2011).
[CrossRef] [PubMed]

Y. Choi, M. Kim, C. Yoon, T. D. Yang, K. J. Lee, and W. Choi, “Synthetic aperture microscopy for high resolution imaging through a turbid medium,” Opt. Lett.36(21), 4263–4265 (2011).
[CrossRef] [PubMed]

Cui, M.

Dahan, M.

V. Studer, J. Bobin, M. Chahid, H. S. Mousavi, E. Candes, and M. Dahan, “Compressive fluorescence microscopy for biological and hyperspectral imaging,” Proc. Natl. Acad. Sci. U.S.A.109(26), E1679–E1687 (2012).
[CrossRef] [PubMed]

Dasari, R. R.

T. R. Hillman, T. Yamauchi, W. Choi, R. R. Dasari, M. S. Feld, Y. Park, and Z. Yaqoob, “Digital optical phase conjugation for delivering two-dimensional images through turbid media,” Sci Rep3, 1909 (2013).
[CrossRef] [PubMed]

Y. Choi, T. D. Yang, C. Fang-Yen, P. Kang, K. J. Lee, R. R. Dasari, M. S. Feld, and W. Choi, “Overcoming the diffraction limit using multiple light scattering in a highly disordered medium,” Phys. Rev. Lett.107(2), 023902 (2011).
[CrossRef] [PubMed]

T. Ikeda, G. Popescu, R. R. Dasari, and M. S. Feld, “Hilbert phase microscopy for investigating fast dynamics in transparent systems,” Opt. Lett.30(10), 1165–1167 (2005).
[CrossRef] [PubMed]

Donoho, D.

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med.58(6), 1182–1195 (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]

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.

E. J. Candès, Y. C. Eldar, D. Needell, and P. Randall, “Compressed sensing with coherent and redundant dictionaries,” Appl. Comput. Harmon. Anal.31(1), 59–73 (2011).
[CrossRef]

Elnatan, D.

L. Zhu, W. Zhang, D. Elnatan, and B. Huang, “Faster STORM using compressed sensing,” Nat. Methods9(7), 721–723 (2012).
[CrossRef] [PubMed]

Fang-Yen, C.

Y. Choi, T. D. Yang, C. Fang-Yen, P. Kang, K. J. Lee, R. R. Dasari, M. S. Feld, and W. Choi, “Overcoming the diffraction limit using multiple light scattering in a highly disordered medium,” Phys. Rev. Lett.107(2), 023902 (2011).
[CrossRef] [PubMed]

Feld, M. S.

T. R. Hillman, T. Yamauchi, W. Choi, R. R. Dasari, M. S. Feld, Y. Park, and Z. Yaqoob, “Digital optical phase conjugation for delivering two-dimensional images through turbid media,” Sci Rep3, 1909 (2013).
[CrossRef] [PubMed]

Y. Choi, T. D. Yang, C. Fang-Yen, P. Kang, K. J. Lee, R. R. Dasari, M. S. Feld, and W. Choi, “Overcoming the diffraction limit using multiple light scattering in a highly disordered medium,” Phys. Rev. Lett.107(2), 023902 (2011).
[CrossRef] [PubMed]

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics2(2), 110–115 (2008).
[CrossRef] [PubMed]

T. Ikeda, G. Popescu, R. R. Dasari, and M. S. Feld, “Hilbert phase microscopy for investigating fast dynamics in transparent systems,” Opt. Lett.30(10), 1165–1167 (2005).
[CrossRef] [PubMed]

Feng, S.

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett.61(20), 2328–2331 (1988).
[CrossRef] [PubMed]

Fink, M.

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics6(5), 283–292 (2012).
[CrossRef]

S. M. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat Commun1(6), 81 (2010).
[CrossRef] [PubMed]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett.104(10), 100601 (2010).
[CrossRef] [PubMed]

Freund, I.

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett.61(20), 2328–2331 (1988).
[CrossRef] [PubMed]

Gigan, S.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett.104(10), 100601 (2010).
[CrossRef] [PubMed]

S. M. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat Commun1(6), 81 (2010).
[CrossRef] [PubMed]

Goodman, J. W.

Hillman, T. R.

T. R. Hillman, T. Yamauchi, W. Choi, R. R. Dasari, M. S. Feld, Y. Park, and Z. Yaqoob, “Digital optical phase conjugation for delivering two-dimensional images through turbid media,” Sci Rep3, 1909 (2013).
[CrossRef] [PubMed]

Horisaki, R.

Huang, B.

L. Zhu, W. Zhang, D. Elnatan, and B. Huang, “Faster STORM using compressed sensing,” Nat. Methods9(7), 721–723 (2012).
[CrossRef] [PubMed]

Ikeda, T.

Kang, P.

Y. Choi, T. D. Yang, C. Fang-Yen, P. Kang, K. J. Lee, R. R. Dasari, M. S. Feld, and W. Choi, “Overcoming the diffraction limit using multiple light scattering in a highly disordered medium,” Phys. Rev. Lett.107(2), 023902 (2011).
[CrossRef] [PubMed]

Kim, M.

Lagendijk, A.

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics6(5), 283–292 (2012).
[CrossRef]

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics4(5), 320–322 (2010).
[CrossRef]

Lee, K. J.

Y. Choi, M. Kim, C. Yoon, T. D. Yang, K. J. Lee, and W. Choi, “Synthetic aperture microscopy for high resolution imaging through a turbid medium,” Opt. Lett.36(21), 4263–4265 (2011).
[CrossRef] [PubMed]

Y. Choi, T. D. Yang, C. Fang-Yen, P. Kang, K. J. Lee, R. R. Dasari, M. S. Feld, and W. Choi, “Overcoming the diffraction limit using multiple light scattering in a highly disordered medium,” Phys. Rev. Lett.107(2), 023902 (2011).
[CrossRef] [PubMed]

Lerosey, G.

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics6(5), 283–292 (2012).
[CrossRef]

S. M. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat Commun1(6), 81 (2010).
[CrossRef] [PubMed]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett.104(10), 100601 (2010).
[CrossRef] [PubMed]

Lim, S.

Lustig, M.

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med.58(6), 1182–1195 (2007).
[CrossRef] [PubMed]

Marks, D. L.

Mosk, A. P.

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics6(5), 283–292 (2012).
[CrossRef]

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics4(5), 320–322 (2010).
[CrossRef]

Mousavi, H. S.

V. Studer, J. Bobin, M. Chahid, H. S. Mousavi, E. Candes, and M. Dahan, “Compressive fluorescence microscopy for biological and hyperspectral imaging,” Proc. Natl. Acad. Sci. U.S.A.109(26), E1679–E1687 (2012).
[CrossRef] [PubMed]

Needell, D.

E. J. Candès, Y. C. Eldar, D. Needell, and P. Randall, “Compressed sensing with coherent and redundant dictionaries,” Appl. Comput. Harmon. Anal.31(1), 59–73 (2011).
[CrossRef]

Park, Y.

T. R. Hillman, T. Yamauchi, W. Choi, R. R. Dasari, M. S. Feld, Y. Park, and Z. Yaqoob, “Digital optical phase conjugation for delivering two-dimensional images through turbid media,” Sci Rep3, 1909 (2013).
[CrossRef] [PubMed]

Pauly, J. M.

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med.58(6), 1182–1195 (2007).
[CrossRef] [PubMed]

Popescu, G.

Popoff, S. M.

S. M. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat Commun1(6), 81 (2010).
[CrossRef] [PubMed]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett.104(10), 100601 (2010).
[CrossRef] [PubMed]

Psaltis, D.

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics2(2), 110–115 (2008).
[CrossRef] [PubMed]

Randall, P.

E. J. Candès, Y. C. Eldar, D. Needell, and P. Randall, “Compressed sensing with coherent and redundant dictionaries,” Appl. Comput. Harmon. Anal.31(1), 59–73 (2011).
[CrossRef]

Rosenbluh, M.

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett.61(20), 2328–2331 (1988).
[CrossRef] [PubMed]

Studer, V.

V. Studer, J. Bobin, M. Chahid, H. S. Mousavi, E. Candes, and M. Dahan, “Compressive fluorescence microscopy for biological and hyperspectral imaging,” Proc. Natl. Acad. Sci. U.S.A.109(26), E1679–E1687 (2012).
[CrossRef] [PubMed]

Vellekoop, I. M.

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics4(5), 320–322 (2010).
[CrossRef]

Yamauchi, T.

T. R. Hillman, T. Yamauchi, W. Choi, R. R. Dasari, M. S. Feld, Y. Park, and Z. Yaqoob, “Digital optical phase conjugation for delivering two-dimensional images through turbid media,” Sci Rep3, 1909 (2013).
[CrossRef] [PubMed]

Yang, C.

Yang, J.

J. Yang and Y. Zhang, “Alternating direction algorithms for L1-problems in compressive sensing,” SIAM J. Sci. Comput.33(1), 250–278 (2011).
[CrossRef]

Yang, T. D.

Y. Choi, T. D. Yang, C. Fang-Yen, P. Kang, K. J. Lee, R. R. Dasari, M. S. Feld, and W. Choi, “Overcoming the diffraction limit using multiple light scattering in a highly disordered medium,” Phys. Rev. Lett.107(2), 023902 (2011).
[CrossRef] [PubMed]

Y. Choi, M. Kim, C. Yoon, T. D. Yang, K. J. Lee, and W. Choi, “Synthetic aperture microscopy for high resolution imaging through a turbid medium,” Opt. Lett.36(21), 4263–4265 (2011).
[CrossRef] [PubMed]

Yaqoob, Z.

T. R. Hillman, T. Yamauchi, W. Choi, R. R. Dasari, M. S. Feld, Y. Park, and Z. Yaqoob, “Digital optical phase conjugation for delivering two-dimensional images through turbid media,” Sci Rep3, 1909 (2013).
[CrossRef] [PubMed]

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics2(2), 110–115 (2008).
[CrossRef] [PubMed]

Yoon, C.

Zhang, W.

L. Zhu, W. Zhang, D. Elnatan, and B. Huang, “Faster STORM using compressed sensing,” Nat. Methods9(7), 721–723 (2012).
[CrossRef] [PubMed]

Zhang, Y.

J. Yang and Y. Zhang, “Alternating direction algorithms for L1-problems in compressive sensing,” SIAM J. Sci. Comput.33(1), 250–278 (2011).
[CrossRef]

Zhu, L.

L. Zhu, W. Zhang, D. Elnatan, and B. Huang, “Faster STORM using compressed sensing,” Nat. Methods9(7), 721–723 (2012).
[CrossRef] [PubMed]

Appl. Comput. Harmon. Anal. (1)

E. J. Candès, Y. C. Eldar, D. Needell, and P. Randall, “Compressed sensing with coherent and redundant dictionaries,” Appl. Comput. Harmon. Anal.31(1), 59–73 (2011).
[CrossRef]

J. Opt. Soc. Am. (1)

Magn. Reson. Med. (1)

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med.58(6), 1182–1195 (2007).
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Nat Commun (1)

S. M. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat Commun1(6), 81 (2010).
[CrossRef] [PubMed]

Nat. Methods (1)

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[CrossRef] [PubMed]

Nat. Photonics (3)

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[CrossRef]

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[CrossRef]

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[CrossRef] [PubMed]

Opt. Express (2)

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Proc. Natl. Acad. Sci. U.S.A. (1)

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[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Experimental schematics. (a) Recording of the TM. SP: sample plane, BS: beam splitter, SB: sample beam, RB: reference beam. (b) The recorded TM. Only the intensities of the SFs are shown here. (c) Recording of the OSF. SLM: spatial light modulator. (d) The recorded OSF. Only the intensities of the SFs are shown here. Scale bar: 10 μm.

Fig. 2
Fig. 2

Estimated angular spectra using (a) PC, (b) PINV, and (c) SSE, respectively. Here, M = 4389, N = 20000 and K is less than 147. All angular spectra are represented in log scale for better visibility.

Fig. 3
Fig. 3

Recovered amplitude images averaged over one, three, five, and seven samples and cross sections of them averaged over seven samples using (a) PC, (b) PINV, and (c) SSE, respectively. Here, M = 4389, N = 20000 and K is less than 147. Scale bar: 10 μm. Constrast-to-noise ratios (CNRs) are calculated in the subsets (red arrow lines) of the cross sections.

Fig. 4
Fig. 4

PSNR performances of PC, PINV, and SSE.

Fig. 5
Fig. 5

Recovered quantitative phase images of a live cell averaged over one, three, five, and seven samples using (a) PC, (b) PINV, and (c) SSE, respectively. Here, M = 696, N = 5000. Color bar: phase in radians. Scale bar: 10 μm.

Equations (8)

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o(x,y)= k x k y a( k x , k y )p(x,y; k x , k y )
y(x,y)= k x k y a( k x , k y )t(x,y; k x , k y ) +n(x,y)
o ^ (x,y)= k x k y a ^ ( k x , k y )p(x,y; k x , k y )
=o(x,y)+ k x k y e( k x , k y )p(x,y; k x , k y )
y=Ta+n
a ^ PC = T * y
= T * Ta+ T * n
a ^ SSE = argmin a a 1 subject to y=Ta.

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