Abstract

We demonstrate the use of phase-space imaging for 3D localization of multiple point sources inside scattering material. The effect of scattering is to spread angular (spatial frequency) information, which can be measured by phase space imaging. We derive a multi-slice forward model for homogenous volumetric scattering, then develop a reconstruction algorithm that exploits sparsity in order to further constrain the problem. By using 4D measurements for 3D reconstruction, the dimensionality mismatch provides significant robustness to multiple scattering, with either static or dynamic diffusers. Experimentally, our high-resolution 4D phase-space data is collected by a spectrogram setup, with results successfully recovering the 3D positions of multiple LEDs embedded in turbid scattering media.

© 2015 Optical Society of America

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2015 (1)

2014 (6)

2013 (1)

B. Bhaskar, G. Tang, and B. Recht, “Atomic norm denoising with applications to line spectral estimation,” IEEE Trans. Signal Process. 61, 5987–5999 (2013).
[Crossref]

2012 (5)

L. Waller, G. Situ, and J. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
[Crossref]

O. Katz, E. Small, and Y. Silberberg, “Looking around corners and through thin turbid layers in real time with scattered incoherent light,” Nature Photonics 6, 549–553 (2012).
[Crossref]

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics 6, 583–587 (2012).
[Crossref]

E. Small, O. Katz, and Y. Silberberg, “Spatiotemporal focusing through a thin scattering layer,” Opt. Express 20, 5189–5195 (2012).
[Crossref] [PubMed]

A. M. Maiden, M. J. Humphry, and J. M. Rodenburg, “Ptychographic transmission microscopy in three dimensions using a multi-slice approach,” J. Opt. Soc. Am. A 29, 1606–1614 (2012).
[Crossref]

2011 (3)

2010 (6)

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. Express 18, 3444–3455 (2010).
[Crossref] [PubMed]

F. E. Robles and A. Wax, “Separating the scattering and absorption coefficients using the real and imaginary parts of the refractive index with low-coherence interferometry,” Opt. Lett. 35, 2843–2845 (2010).
[Crossref] [PubMed]

C.-L. Hsieh, Y. Pu, R. Grange, G. Laporte, and D. Psaltis, “Imaging through turbid layers by scanning the phase conjugated second harmonic radiation from a nanoparticle,” Opt. Express 18, 20723–20731 (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, 100601 (2010).
[Crossref] [PubMed]

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7, 141–147 (2010).
[Crossref]

E. J. McDowell, M. Cui, I. M. Vellekoop, V. Senekerimyan, Z. Yaqoob, and C. Yang, “Turbidity suppression from the ballistic to the diffusive regime in biological tissues using optical phase conjugation,” J. Biomed. Opt. 15, 025004 (2010).
[Crossref] [PubMed]

2009 (2)

2008 (1)

2007 (2)

2006 (1)

M. Levoy, “Light fields and computational imaging,” Computer 39, 46–55 (2006).
[Crossref]

2004 (1)

M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas, “Synthetic aperture confocal imaging,” ACM Trans. Graph. 23, 825–834 (2004).
[Crossref]

2003 (1)

1998 (1)

1994 (1)

R. Tibshirani, “Regression shrinkage and selection via the lasso,” J. Royal Stat. Soc. Ser. B 58, 267–288 (1994).

1991 (2)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and et al., “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

L. Wang, P. Ho, C. Liu, G. Zhang, and R. Alfano, “Ballistic 2-d imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[Crossref] [PubMed]

1990 (1)

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[Crossref] [PubMed]

1980 (1)

H. Bartelt, K. Brenner, and A. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[Crossref]

1978 (1)

M. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[Crossref]

1968 (2)

1966 (1)

J. Goodman, W. Huntley, D. Jackson, and M. Lehmann, “Wavefront-reconstruction imaging through random media,” Appl. Phys. Lett. 8, 311–313 (1966).
[Crossref]

1957 (1)

J. M. Cowley and A. F. Moodie, “The scattering of electrons by atoms and crystals. I. A new theoretical approach,” Acta Crystallog. 10, 609–619 (1957).
[Crossref]

Abookasis, D.

D. Abookasis and T. Moshe, “Feasibility study of hidden flow imaging based on laser speckle technique using multiperspectives contrast images,” Opt. Lasers Eng. 62, 38–45 (2014).
[Crossref]

J. Rosen and D. Abookasis, “Seeing through biological tissues using the fly eye principle,” Opt. Express 11, 3605–3611 (2003).
[Crossref] [PubMed]

Agrawal, G.

G. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, 2001).

Alfano, R.

L. Wang, P. Ho, C. Liu, G. Zhang, and R. Alfano, “Ballistic 2-d imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[Crossref] [PubMed]

Alonso, M. A.

Bartelt, H.

H. Bartelt, K. Brenner, and A. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[Crossref]

Bastiaans, M.

M. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[Crossref]

Betzig, E.

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7, 141–147 (2010).
[Crossref]

Bhaskar, B.

B. Bhaskar, G. Tang, and B. Recht, “Atomic norm denoising with applications to line spectral estimation,” IEEE Trans. Signal Process. 61, 5987–5999 (2013).
[Crossref]

G. Tang, B. Bhaskar, and B. Recht, “Sparse recovery over continuous dictionaries-just discretize,” in Asilomar Conference on Signals, Systems and Computers, 2013), pp. 1043–1047.

Boccara, A.

S. Gigan, S. Popoff, A. Liutkus, D. Martina, O. Katz, G. Chardon, R. Carminati, G. Lerosey, M. Fink, A. Boccara, I. Carron, and L. Daudet, “Image transmission through a scattering medium: Inverse problem and sparsity-based imaging,” in “13th Workshop on Information Optics (WIO, 2014)”, pp. 1–3.

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, 100601 (2010).
[Crossref] [PubMed]

Bolas, M.

M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas, “Synthetic aperture confocal imaging,” ACM Trans. Graph. 23, 825–834 (2004).
[Crossref]

Bredif, M.

R. Ng, M. Levoy, M. Bredif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Tech. Rep. CTSR 2005-02, (Stanford, 2005).

Brenner, K.

H. Bartelt, K. Brenner, and A. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[Crossref]

Brown, T. G.

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, 100601 (2010).
[Crossref] [PubMed]

S. Gigan, S. Popoff, A. Liutkus, D. Martina, O. Katz, G. Chardon, R. Carminati, G. Lerosey, M. Fink, A. Boccara, I. Carron, and L. Daudet, “Image transmission through a scattering medium: Inverse problem and sparsity-based imaging,” in “13th Workshop on Information Optics (WIO, 2014)”, pp. 1–3.

Carron, I.

S. Gigan, S. Popoff, A. Liutkus, D. Martina, O. Katz, G. Chardon, R. Carminati, G. Lerosey, M. Fink, A. Boccara, I. Carron, and L. Daudet, “Image transmission through a scattering medium: Inverse problem and sparsity-based imaging,” in “13th Workshop on Information Optics (WIO, 2014)”, pp. 1–3.

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and et al., “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Chardon, G.

S. Gigan, S. Popoff, A. Liutkus, D. Martina, O. Katz, G. Chardon, R. Carminati, G. Lerosey, M. Fink, A. Boccara, I. Carron, and L. Daudet, “Image transmission through a scattering medium: Inverse problem and sparsity-based imaging,” in “13th Workshop on Information Optics (WIO, 2014)”, pp. 1–3.

Chen, B.

M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas, “Synthetic aperture confocal imaging,” ACM Trans. Graph. 23, 825–834 (2004).
[Crossref]

Chen, H. H.

C.-K. Liang, T.-H. Lin, B.-Y. Wong, C. Liu, and H. H. Chen, “Programmable aperture photography: multiplexed light field acquisition,” in “ACM Transactions on Graphics (TOG),” (ACM, 2008), vol. 27, p. 55.
[Crossref]

Cho, S.

Choi, W.

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics 6, 583–587 (2012).
[Crossref]

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics 6, 583–587 (2012).
[Crossref]

Choi, Y.

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics 6, 583–587 (2012).
[Crossref]

Christofferson, J.

Cowley, J. M.

J. M. Cowley and A. F. Moodie, “The scattering of electrons by atoms and crystals. I. A new theoretical approach,” Acta Crystallog. 10, 609–619 (1957).
[Crossref]

Cui, M.

E. J. McDowell, M. Cui, I. M. Vellekoop, V. Senekerimyan, Z. Yaqoob, and C. Yang, “Turbidity suppression from the ballistic to the diffusive regime in biological tissues using optical phase conjugation,” J. Biomed. Opt. 15, 025004 (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. Express 18, 3444–3455 (2010).
[Crossref] [PubMed]

Daudet, L.

S. Gigan, S. Popoff, A. Liutkus, D. Martina, O. Katz, G. Chardon, R. Carminati, G. Lerosey, M. Fink, A. Boccara, I. Carron, and L. Daudet, “Image transmission through a scattering medium: Inverse problem and sparsity-based imaging,” in “13th Workshop on Information Optics (WIO, 2014)”, pp. 1–3.

Denk, W.

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[Crossref] [PubMed]

Duval, G.

R. Ng, M. Levoy, M. Bredif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Tech. Rep. CTSR 2005-02, (Stanford, 2005).

Dylov, D. V.

Eriksson, B.

Farsiu, S.

Figueiredo, M.

S. Wright, R. Nowak, and M. Figueiredo, “Sparse reconstruction by separable approximation,” IEEE Trans. Sign. Process. 57, 2479–2493 (2009).
[Crossref]

Fink, M.

O. Katz, P. Heidmann, M. Fink, and S. Gigan, “Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations,” Nat. Photonics 8, 784–790 (2014).
[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, 100601 (2010).
[Crossref] [PubMed]

S. Gigan, S. Popoff, A. Liutkus, D. Martina, O. Katz, G. Chardon, R. Carminati, G. Lerosey, M. Fink, A. Boccara, I. Carron, and L. Daudet, “Image transmission through a scattering medium: Inverse problem and sparsity-based imaging,” in “13th Workshop on Information Optics (WIO, 2014)”, pp. 1–3.

Fleischer, J.

L. Waller, G. Situ, and J. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
[Crossref]

Fleischer, J. W.

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and et al., “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Friedlander, B.

Gaskill, J. D.

Gigan, S.

O. Katz, P. Heidmann, M. Fink, and S. Gigan, “Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations,” Nat. Photonics 8, 784–790 (2014).
[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, 100601 (2010).
[Crossref] [PubMed]

S. Gigan, S. Popoff, A. Liutkus, D. Martina, O. Katz, G. Chardon, R. Carminati, G. Lerosey, M. Fink, A. Boccara, I. Carron, and L. Daudet, “Image transmission through a scattering medium: Inverse problem and sparsity-based imaging,” in “13th Workshop on Information Optics (WIO, 2014)”, pp. 1–3.

Goodman, J.

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F. Robles, C. Wilson, G. Grant, and A. Wax, “Molecular imaging true-colour spectroscopic optical coherence tomography,” Nat. Photonics 5, 744–747 (2011).
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D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and et al., “Optical coherence tomography,” Science 254, 1178–1181 (1991).
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R. Ng, M. Levoy, M. Bredif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Tech. Rep. CTSR 2005-02, (Stanford, 2005).

Hee, M. R.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and et al., “Optical coherence tomography,” Science 254, 1178–1181 (1991).
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O. Katz, P. Heidmann, M. Fink, and S. Gigan, “Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations,” Nat. Photonics 8, 784–790 (2014).
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M. Testorf, B. Hennelly, and J. Ojeda-Castañeda, Phase-Space Optics: Fundamentals and Applications (McGraw-Hill Professional, 2009).

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L. Wang, P. Ho, C. Liu, G. Zhang, and R. Alfano, “Ballistic 2-d imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
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Horowitz, M.

M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas, “Synthetic aperture confocal imaging,” ACM Trans. Graph. 23, 825–834 (2004).
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R. Ng, M. Levoy, M. Bredif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Tech. Rep. CTSR 2005-02, (Stanford, 2005).

Hsieh, C.-L.

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and et al., “Optical coherence tomography,” Science 254, 1178–1181 (1991).
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Huntley, W.

J. Goodman, W. Huntley, D. Jackson, and M. Lehmann, “Wavefront-reconstruction imaging through random media,” Appl. Phys. Lett. 8, 311–313 (1966).
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H. Nagahara, C. Zhou, T. Watanabe, H. Ishiguro, and S. K. Nayar, “Programmable aperture camera using LCoS,” in “Computer Vision–ECCV 2010,” (Springer, 2010), pp. 337–350.
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J. Goodman, W. Huntley, D. Jackson, and M. Lehmann, “Wavefront-reconstruction imaging through random media,” Appl. Phys. Lett. 8, 311–313 (1966).
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N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7, 141–147 (2010).
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O. Katz, P. Heidmann, M. Fink, and S. Gigan, “Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations,” Nat. Photonics 8, 784–790 (2014).
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O. Katz, E. Small, and Y. Silberberg, “Looking around corners and through thin turbid layers in real time with scattered incoherent light,” Nature Photonics 6, 549–553 (2012).
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E. Small, O. Katz, and Y. Silberberg, “Spatiotemporal focusing through a thin scattering layer,” Opt. Express 20, 5189–5195 (2012).
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S. Gigan, S. Popoff, A. Liutkus, D. Martina, O. Katz, G. Chardon, R. Carminati, G. Lerosey, M. Fink, A. Boccara, I. Carron, and L. Daudet, “Image transmission through a scattering medium: Inverse problem and sparsity-based imaging,” in “13th Workshop on Information Optics (WIO, 2014)”, pp. 1–3.

Kim, J.

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics 6, 583–587 (2012).
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J. Kim, D. Lanman, Y. Mukaigawa, and R. Raskar, “Descattering transmission via angular filtering,” in Computer Vision ECCV 2010, vol. 6311 of Lecture Notes in Computer Science, K. Daniilidis, P. Maragos, and N. Paragios, eds. (Springer, 2010), pp. 86–99.

Kim, M.

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics 6, 583–587 (2012).
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Lanman, D.

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Laporte, G.

Lehmann, M.

J. Goodman, W. Huntley, D. Jackson, and M. Lehmann, “Wavefront-reconstruction imaging through random media,” Appl. Phys. Lett. 8, 311–313 (1966).
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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, 100601 (2010).
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M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas, “Synthetic aperture confocal imaging,” ACM Trans. Graph. 23, 825–834 (2004).
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Z. Zhang and M. Levoy, “Wigner distributions and how they relate to the light field,” in “IEEE International Conference on Computational Photography (ICCP, 2009”, pp. 1–10.

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Liang, C.-K.

C.-K. Liang, T.-H. Lin, B.-Y. Wong, C. Liu, and H. H. Chen, “Programmable aperture photography: multiplexed light field acquisition,” in “ACM Transactions on Graphics (TOG),” (ACM, 2008), vol. 27, p. 55.
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D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and et al., “Optical coherence tomography,” Science 254, 1178–1181 (1991).
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C.-K. Liang, T.-H. Lin, B.-Y. Wong, C. Liu, and H. H. Chen, “Programmable aperture photography: multiplexed light field acquisition,” in “ACM Transactions on Graphics (TOG),” (ACM, 2008), vol. 27, p. 55.
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L. Wang, P. Ho, C. Liu, G. Zhang, and R. Alfano, “Ballistic 2-d imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
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C.-K. Liang, T.-H. Lin, B.-Y. Wong, C. Liu, and H. H. Chen, “Programmable aperture photography: multiplexed light field acquisition,” in “ACM Transactions on Graphics (TOG),” (ACM, 2008), vol. 27, p. 55.
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S. Gigan, S. Popoff, A. Liutkus, D. Martina, O. Katz, G. Chardon, R. Carminati, G. Lerosey, M. Fink, A. Boccara, I. Carron, and L. Daudet, “Image transmission through a scattering medium: Inverse problem and sparsity-based imaging,” in “13th Workshop on Information Optics (WIO, 2014)”, pp. 1–3.

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H. Bartelt, K. Brenner, and A. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
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McDowall, I.

M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas, “Synthetic aperture confocal imaging,” ACM Trans. Graph. 23, 825–834 (2004).
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E. J. McDowell, M. Cui, I. M. Vellekoop, V. Senekerimyan, Z. Yaqoob, and C. Yang, “Turbidity suppression from the ballistic to the diffusive regime in biological tissues using optical phase conjugation,” J. Biomed. Opt. 15, 025004 (2010).
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Milanfar, P.

Milkie, D. E.

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7, 141–147 (2010).
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J. M. Cowley and A. F. Moodie, “The scattering of electrons by atoms and crystals. I. A new theoretical approach,” Acta Crystallog. 10, 609–619 (1957).
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Nagahara, H.

H. Nagahara, C. Zhou, T. Watanabe, H. Ishiguro, and S. K. Nayar, “Programmable aperture camera using LCoS,” in “Computer Vision–ECCV 2010,” (Springer, 2010), pp. 337–350.
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Nayar, S. K.

H. Nagahara, C. Zhou, T. Watanabe, H. Ishiguro, and S. K. Nayar, “Programmable aperture camera using LCoS,” in “Computer Vision–ECCV 2010,” (Springer, 2010), pp. 337–350.
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R. Ng, M. Levoy, M. Bredif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Tech. Rep. CTSR 2005-02, (Stanford, 2005).

Nowak, R.

Ojeda-Castañeda, J.

M. Testorf, B. Hennelly, and J. Ojeda-Castañeda, Phase-Space Optics: Fundamentals and Applications (McGraw-Hill Professional, 2009).

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Park, Q.-H.

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics 6, 583–587 (2012).
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Pennington, K.

Popoff, S.

S. Gigan, S. Popoff, A. Liutkus, D. Martina, O. Katz, G. Chardon, R. Carminati, G. Lerosey, M. Fink, A. Boccara, I. Carron, and L. Daudet, “Image transmission through a scattering medium: Inverse problem and sparsity-based imaging,” in “13th Workshop on Information Optics (WIO, 2014)”, pp. 1–3.

Popoff, S. M.

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, 100601 (2010).
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Pu, Y.

Puliafito, C. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and et al., “Optical coherence tomography,” Science 254, 1178–1181 (1991).
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Raskar, R.

J. Kim, D. Lanman, Y. Mukaigawa, and R. Raskar, “Descattering transmission via angular filtering,” in Computer Vision ECCV 2010, vol. 6311 of Lecture Notes in Computer Science, K. Daniilidis, P. Maragos, and N. Paragios, eds. (Springer, 2010), pp. 86–99.

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B. Bhaskar, G. Tang, and B. Recht, “Atomic norm denoising with applications to line spectral estimation,” IEEE Trans. Signal Process. 61, 5987–5999 (2013).
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G. Tang, B. Bhaskar, and B. Recht, “Sparse recovery over continuous dictionaries-just discretize,” in Asilomar Conference on Signals, Systems and Computers, 2013), pp. 1043–1047.

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Robles, F. E.

Rodenburg, J. M.

Rosen, J.

Schuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and et al., “Optical coherence tomography,” Science 254, 1178–1181 (1991).
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E. J. McDowell, M. Cui, I. M. Vellekoop, V. Senekerimyan, Z. Yaqoob, and C. Yang, “Turbidity suppression from the ballistic to the diffusive regime in biological tissues using optical phase conjugation,” J. Biomed. Opt. 15, 025004 (2010).
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Shakouri, A.

Sharma, K. A.

Silberberg, Y.

E. Small, O. Katz, and Y. Silberberg, “Spatiotemporal focusing through a thin scattering layer,” Opt. Express 20, 5189–5195 (2012).
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O. Katz, E. Small, and Y. Silberberg, “Looking around corners and through thin turbid layers in real time with scattered incoherent light,” Nature Photonics 6, 549–553 (2012).
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Situ, G.

L. Waller, G. Situ, and J. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
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O. Katz, E. Small, and Y. Silberberg, “Looking around corners and through thin turbid layers in real time with scattered incoherent light,” Nature Photonics 6, 549–553 (2012).
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E. Small, O. Katz, and Y. Silberberg, “Spatiotemporal focusing through a thin scattering layer,” Opt. Express 20, 5189–5195 (2012).
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D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and et al., “Optical coherence tomography,” Science 254, 1178–1181 (1991).
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W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
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D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and et al., “Optical coherence tomography,” Science 254, 1178–1181 (1991).
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Takeda, M.

Tang, G.

B. Bhaskar, G. Tang, and B. Recht, “Atomic norm denoising with applications to line spectral estimation,” IEEE Trans. Signal Process. 61, 5987–5999 (2013).
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G. Tang, B. Bhaskar, and B. Recht, “Sparse recovery over continuous dictionaries-just discretize,” in Asilomar Conference on Signals, Systems and Computers, 2013), pp. 1043–1047.

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M. Testorf, B. Hennelly, and J. Ojeda-Castañeda, Phase-Space Optics: Fundamentals and Applications (McGraw-Hill Professional, 2009).

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R. Tibshirani, “Regression shrinkage and selection via the lasso,” J. Royal Stat. Soc. Ser. B 58, 267–288 (1994).

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M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas, “Synthetic aperture confocal imaging,” ACM Trans. Graph. 23, 825–834 (2004).
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Vellekoop, I.

Vellekoop, I. M.

E. J. McDowell, M. Cui, I. M. Vellekoop, V. Senekerimyan, Z. Yaqoob, and C. Yang, “Turbidity suppression from the ballistic to the diffusive regime in biological tissues using optical phase conjugation,” J. Biomed. Opt. 15, 025004 (2010).
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Wang, L.

L. Wang, P. Ho, C. Liu, G. Zhang, and R. Alfano, “Ballistic 2-d imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
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H. Nagahara, C. Zhou, T. Watanabe, H. Ishiguro, and S. K. Nayar, “Programmable aperture camera using LCoS,” in “Computer Vision–ECCV 2010,” (Springer, 2010), pp. 337–350.
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W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
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F. Robles, C. Wilson, G. Grant, and A. Wax, “Molecular imaging true-colour spectroscopic optical coherence tomography,” Nat. Photonics 5, 744–747 (2011).
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C.-K. Liang, T.-H. Lin, B.-Y. Wong, C. Liu, and H. H. Chen, “Programmable aperture photography: multiplexed light field acquisition,” in “ACM Transactions on Graphics (TOG),” (ACM, 2008), vol. 27, p. 55.
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E. J. McDowell, M. Cui, I. M. Vellekoop, V. Senekerimyan, Z. Yaqoob, and C. Yang, “Turbidity suppression from the ballistic to the diffusive regime in biological tissues using optical phase conjugation,” J. Biomed. Opt. 15, 025004 (2010).
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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. Express 18, 3444–3455 (2010).
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E. J. McDowell, M. Cui, I. M. Vellekoop, V. Senekerimyan, Z. Yaqoob, and C. Yang, “Turbidity suppression from the ballistic to the diffusive regime in biological tissues using optical phase conjugation,” J. Biomed. Opt. 15, 025004 (2010).
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M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics 6, 583–587 (2012).
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H. Nagahara, C. Zhou, T. Watanabe, H. Ishiguro, and S. K. Nayar, “Programmable aperture camera using LCoS,” in “Computer Vision–ECCV 2010,” (Springer, 2010), pp. 337–350.
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ACM Trans. Graph. (1)

M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas, “Synthetic aperture confocal imaging,” ACM Trans. Graph. 23, 825–834 (2004).
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J. M. Cowley and A. F. Moodie, “The scattering of electrons by atoms and crystals. I. A new theoretical approach,” Acta Crystallog. 10, 609–619 (1957).
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Adv. Opt. Photon. (1)

Appl. Opt. (1)

Appl. Phys. Lett. (1)

J. Goodman, W. Huntley, D. Jackson, and M. Lehmann, “Wavefront-reconstruction imaging through random media,” Appl. Phys. Lett. 8, 311–313 (1966).
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Computer (1)

M. Levoy, “Light fields and computational imaging,” Computer 39, 46–55 (2006).
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IEEE Trans. Sign. Process. (1)

S. Wright, R. Nowak, and M. Figueiredo, “Sparse reconstruction by separable approximation,” IEEE Trans. Sign. Process. 57, 2479–2493 (2009).
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IEEE Trans. Signal Process. (1)

B. Bhaskar, G. Tang, and B. Recht, “Atomic norm denoising with applications to line spectral estimation,” IEEE Trans. Signal Process. 61, 5987–5999 (2013).
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J. Biomed. Opt. (1)

E. J. McDowell, M. Cui, I. M. Vellekoop, V. Senekerimyan, Z. Yaqoob, and C. Yang, “Turbidity suppression from the ballistic to the diffusive regime in biological tissues using optical phase conjugation,” J. Biomed. Opt. 15, 025004 (2010).
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J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

J. Royal Stat. Soc. Ser. B (1)

R. Tibshirani, “Regression shrinkage and selection via the lasso,” J. Royal Stat. Soc. Ser. B 58, 267–288 (1994).

Nat. Methods (1)

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7, 141–147 (2010).
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Nat. Photonics (4)

L. Waller, G. Situ, and J. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
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M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics 6, 583–587 (2012).
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O. Katz, P. Heidmann, M. Fink, and S. Gigan, “Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations,” Nat. Photonics 8, 784–790 (2014).
[Crossref]

F. Robles, C. Wilson, G. Grant, and A. Wax, “Molecular imaging true-colour spectroscopic optical coherence tomography,” Nat. Photonics 5, 744–747 (2011).
[Crossref]

Nature Photonics (1)

O. Katz, E. Small, and Y. Silberberg, “Looking around corners and through thin turbid layers in real time with scattered incoherent light,” Nature Photonics 6, 549–553 (2012).
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Opt. Commun. (2)

H. Bartelt, K. Brenner, and A. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
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M. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
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I. Moon and B. Javidi, “Three-dimensional visualization of objects in scattering medium by use of computational integral imaging,” Opt. Express 16, 13080–13089 (2008).
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K. T. Takasaki and J. W. Fleischer, “Phase-space measurement for depth-resolved memory-effect imaging,” Opt. Express 22, 31426–31433 (2014).
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F. Robles, R. Graf, and A. Wax, “Dual window method for processing spectroscopic optical coherence tomography signals with simultaneously high spectral and temporal resolution,” Opt. Express 17, 6799–6812 (2009).
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Figures (3)

Fig. 1
Fig. 1

Phase space measurements for three point sources at different depths, with and without scattering media. (a) The 3D positions of the point sources. (b) A phase-space slice and intensity cut-line for the case of no scattering. Each point source creates a line whose intercept with ux = 0 defines its lateral position and slope defines its depth. (c) A phase-space slice and intensity cut-line for the case of point sources in volumetric scattering material. The lines blur in proportion to point source’s depth inside the scattering material.

Fig. 2
Fig. 2

Experimental localization of LEDs at different depths, with and without rotating diffusers between them. (a) The setup uses an object consisting of 3 LEDs and a spectrogram measurement system that employs a DMD in Fourier space to scan the aperture. (b–d) Intensity, phase space (along green line) and recovered LED positions for the case (b) without scattering, (c) with diffuser 3 only, and (d) with all three diffusers. The algorithm successfully recovers the position and depth of each LED in all cases.

Fig. 3
Fig. 3

4D phase-space experiments with and without volumetric scattering. The first column contains the 2D intensity images that are captured by a traditional camera and the second column shows some example phase space 2D slices (along the red lines in the intensity images). The third column shows the recovered xz positions of point sources (y not shown) and the last column shows reconstructed 3D positions of the LEDs. Each color dot corresponds to a successful LED position recovery, while red dots are failures due to occlusions by the finite size of LEDs.

Equations (11)

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W ( r , u ) = f ˜ * ( u + u / 2 ) f ˜ ( u u / 2 ) e i 2 π u r d 2 u ,
I ( r , u ) = | f ˜ ( u ) a ( u u ) e i 2 π u r d 2 u | 2 ,
𝒫 Δ z W ( r , u ) = W ( r , u ) δ ( r r λ Δ z u / n r ) d 2 r ,
d u x d x = n r λ z ,
𝒟 σ , N W ( r , u ) = W ( r , u ) N 12 π σ 2 exp ( N 12 σ 2 ( u u ) 2 ) d 2 u .
W N ( r , u ) = ( 𝒟 σ , N 𝒫 Δ z = / N ) N W s ( r , u ) .
W ( r , u ) = lim N W N ( r , u ) = n r 2 2 π λ 2 2 σ 2 exp ( n r 2 2 λ 2 2 σ 2 ( r r s λ u / n r ) 2 ) .
W ( r , u ) = n r 2 2 π λ 2 ( z d z s ) 2 σ 2 exp ( n r 2 2 λ 2 ( z d z s ) 2 σ 2 ( r r s + λ ( z d z d z s n r ) u ) 2 ) .
𝒜 = { a ( r , u ; r s , z s ) = n r 2 2 π λ 2 ( z d z s ) 2 σ 2 e n r 2 2 λ 2 ( z d z s ) 2 σ 2 ( r r s + λ ( z d ( z d z s ) / n r ) u ) 2 , r s , z s } ,
I ^ ( r , u ) = r s , z s c ( r s , z s ) a ( r , u ; r s , z s ) ,
min c 0 r , u | I ( r , u ) I ^ ( r , u ) | 2 + μ r s , z s | c ( r s , z s ) | ,

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