Abstract

Smart control of light propagation through highly scattering media is a much desired goal with major technological implications. Since interaction of light with highly scattering media results in partial or complete depletion of ballistic photons, it is in principle impossible to transmit images through distances longer than the extinction length. Nevertheless, different methods for image transmission, focusing, and imaging through scattering media by means of wavefront control have been published over the past few years. In this paper we show that single-pixel optical systems, based on compressive detection, can also overcome the fundamental limitation imposed by multiple scattering to successfully transmit information. But, in contrast with the recently introduced schemes that use the transmission matrix technique, our approach does not require any a-priori calibration process that ultimately makes the present method suitable to use with dynamic scattering media. This represents an advantage over previous methods that rely on optical feedback wavefront control, especially for short speckle decorrelation times.

© 2014 Optical Society of America

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2014 (2)

Y. Choi, C. Yoon, M. Kim, W. Choi, and W. Choi, “Optical Imaging With the Use of a Scattering Lens,” IEEE J. Sel. Top. Quantum Electron. 20, 6800213 (2014).

M. Plöschner, B. Straka, K. Dholakia, and T. Cizmár, ”GPU accelerated toolbox for real-time beam-shaping in multimode fibres,” Opt. Express 22, 2933–2947 (2014).
[CrossRef] [PubMed]

2013 (5)

J. Hunt, T. Discroll, A. Mrozack, G. Lipworth, M. Reynolds, D. Brady, and D. R. Smith, “Metamaterial Apertures for Computational Imaging,” Science 339, 310–313 (2013).
[CrossRef] [PubMed]

M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7, 919–924 (2013).
[CrossRef]

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D Computational Imaging with Single-Pixel Detectors,” Science 340, 844–847 (2013).
[CrossRef] [PubMed]

G. A. Howland and J. C. Howell, “Efficient High-Dimensional Entanglement Imaging with a Compressive-Sensing Double-Pixel Camera,” Phys. Rev. X 3, 011013 (2013).

K. Chung, J. Wallace, S. Y. Kim, S. Kalyanasundaram, A. S. Andalman, T. J. Davidson, J. J. Mirzabekov, K. A. Zalocusky, J. Mattis, A. K. Denisin, S. Pak, H. Bernstein, C. Ramakrishnan, L. Grosenick, V. Gradinaru, and K. Deisseroth, “Structural and molecular interrogation of intact biological systems,” Nature (London) 497, 332–337 (2013).
[CrossRef]

2012 (7)

M. J. Booth, D. Dbarre, and A. Jesacher, “Adaptive optics for biomedical microscopy,” Opt. Photon. News 23, 22 (2012).
[CrossRef]

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6, 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. USA 109, E1679–E1687 (2012).
[CrossRef] [PubMed]

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature (London) 491, 232–234 (2012).
[CrossRef]

T. Cizmar and K. Dholakia, “Exploiting multimode waveguides for pure fibre-based imaging,” Nature Commun. 3, 1027 (2012).
[CrossRef]

D. B. Conkey, A. M. Caravaca-Aguirre, and R. Piestun, “High-speed scattering medium characterization with application to focusing light through turbid media,” Opt. Express 20, 1733–1740 (2012).
[CrossRef] [PubMed]

P. Clemente, V. Durán, E. Tajahuerce, V. Torres-Company, and J. Lancis, “Single-pixel digital ghost holography,” Phys. Rev. A 86, 041803 (2012).
[CrossRef]

2011 (4)

F. Magalhães, F. M. Araújo, M. V. Correia, M. Abolbashari, and F. Farahi, “ Active illumination single-pixel camera based on compressive sensing,” Appl. Opt. 50, 405–414 (2011).
[CrossRef] [PubMed]

S. M. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Controlling light through optical disordered media: transmission matrix approach,” New J. Phys. 13, 123021 (2011).
[CrossRef]

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5, 372–377 (2011).
[CrossRef]

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, “ Spatio-temporal focusing of an ultrafast pulse through a multiply scattering medium,” Nat. Commun. 2, 447–455 (2011).
[CrossRef] [PubMed]

2010 (4)

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque materia,” Nat. Commun. 1, 1038(2010).
[CrossRef]

J. Katz and J. Sheng, “Applications of Holography in Fluid Mechanics and Particle Dynamics,” Annu. Rev. Fluid Mech. 42, 531–555 (2010).
[CrossRef]

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
[CrossRef]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and G. 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]

2009 (2)

Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009).
[CrossRef]

M. Wenner, “The most transparent research,” Nature Medicine 15, 1106–1109 (2009).
[CrossRef] [PubMed]

2008 (2)

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-Pixel Imaging via Compressive Sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

E. J. Candes and M. B. Wakin, “An Introduction To Compressive Sampling,” IEEE Signal Process. Mag. 25, 21–30 (2008).
[CrossRef]

1976 (1)

Abolbashari, M.

Andalman, A. S.

K. Chung, J. Wallace, S. Y. Kim, S. Kalyanasundaram, A. S. Andalman, T. J. Davidson, J. J. Mirzabekov, K. A. Zalocusky, J. Mattis, A. K. Denisin, S. Pak, H. Bernstein, C. Ramakrishnan, L. Grosenick, V. Gradinaru, and K. Deisseroth, “Structural and molecular interrogation of intact biological systems,” Nature (London) 497, 332–337 (2013).
[CrossRef]

Araújo, F. M.

Austin, D. R.

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, “ Spatio-temporal focusing of an ultrafast pulse through a multiply scattering medium,” Nat. Commun. 2, 447–455 (2011).
[CrossRef] [PubMed]

Baraniuk, R. G.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-Pixel Imaging via Compressive Sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

Bernstein, H.

K. Chung, J. Wallace, S. Y. Kim, S. Kalyanasundaram, A. S. Andalman, T. J. Davidson, J. J. Mirzabekov, K. A. Zalocusky, J. Mattis, A. K. Denisin, S. Pak, H. Bernstein, C. Ramakrishnan, L. Grosenick, V. Gradinaru, and K. Deisseroth, “Structural and molecular interrogation of intact biological systems,” Nature (London) 497, 332–337 (2013).
[CrossRef]

Bertolotti, J.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature (London) 491, 232–234 (2012).
[CrossRef]

Blum, C.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature (London) 491, 232–234 (2012).
[CrossRef]

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. USA 109, E1679–E1687 (2012).
[CrossRef] [PubMed]

Boccara, A. C.

S. M. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Controlling light through optical disordered media: transmission matrix approach,” New J. Phys. 13, 123021 (2011).
[CrossRef]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque materia,” Nat. Commun. 1, 1038(2010).
[CrossRef]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and G. 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]

Bondareff, P.

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, “ Spatio-temporal focusing of an ultrafast pulse through a multiply scattering medium,” Nat. Commun. 2, 447–455 (2011).
[CrossRef] [PubMed]

Booth, M. J.

M. J. Booth, D. Dbarre, and A. Jesacher, “Adaptive optics for biomedical microscopy,” Opt. Photon. News 23, 22 (2012).
[CrossRef]

Bowman, A.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D Computational Imaging with Single-Pixel Detectors,” Science 340, 844–847 (2013).
[CrossRef] [PubMed]

Bowman, R.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D Computational Imaging with Single-Pixel Detectors,” Science 340, 844–847 (2013).
[CrossRef] [PubMed]

Brady, D.

J. Hunt, T. Discroll, A. Mrozack, G. Lipworth, M. Reynolds, D. Brady, and D. R. Smith, “Metamaterial Apertures for Computational Imaging,” Science 339, 310–313 (2013).
[CrossRef] [PubMed]

Bromberg, Y.

M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7, 919–924 (2013).
[CrossRef]

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5, 372–377 (2011).
[CrossRef]

Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (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. USA 109, E1679–E1687 (2012).
[CrossRef] [PubMed]

Candes, E. J.

E. J. Candes and M. B. Wakin, “An Introduction To Compressive Sampling,” IEEE Signal Process. Mag. 25, 21–30 (2008).
[CrossRef]

Caravaca-Aguirre, A. M.

Carminati, R.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and G. 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]

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. USA 109, E1679–E1687 (2012).
[CrossRef] [PubMed]

Chatel, B.

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, “ Spatio-temporal focusing of an ultrafast pulse through a multiply scattering medium,” Nat. Commun. 2, 447–455 (2011).
[CrossRef] [PubMed]

Choi, W.

Y. Choi, C. Yoon, M. Kim, W. Choi, and W. Choi, “Optical Imaging With the Use of a Scattering Lens,” IEEE J. Sel. Top. Quantum Electron. 20, 6800213 (2014).

Y. Choi, C. Yoon, M. Kim, W. Choi, and W. Choi, “Optical Imaging With the Use of a Scattering Lens,” IEEE J. Sel. Top. Quantum Electron. 20, 6800213 (2014).

R. T. 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. Rep.3, (2013).
[CrossRef] [PubMed]

Choi, Y.

Y. Choi, C. Yoon, M. Kim, W. Choi, and W. Choi, “Optical Imaging With the Use of a Scattering Lens,” IEEE J. Sel. Top. Quantum Electron. 20, 6800213 (2014).

Chung, K.

K. Chung, J. Wallace, S. Y. Kim, S. Kalyanasundaram, A. S. Andalman, T. J. Davidson, J. J. Mirzabekov, K. A. Zalocusky, J. Mattis, A. K. Denisin, S. Pak, H. Bernstein, C. Ramakrishnan, L. Grosenick, V. Gradinaru, and K. Deisseroth, “Structural and molecular interrogation of intact biological systems,” Nature (London) 497, 332–337 (2013).
[CrossRef]

Cizmar, T.

T. Cizmar and K. Dholakia, “Exploiting multimode waveguides for pure fibre-based imaging,” Nature Commun. 3, 1027 (2012).
[CrossRef]

Cizmár, T.

Clemente, P.

P. Clemente, V. Durán, E. Tajahuerce, V. Torres-Company, and J. Lancis, “Single-pixel digital ghost holography,” Phys. Rev. A 86, 041803 (2012).
[CrossRef]

Conkey, D. B.

Correia, M. V.

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. USA 109, E1679–E1687 (2012).
[CrossRef] [PubMed]

Dasari, R. R.

R. T. 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. Rep.3, (2013).
[CrossRef] [PubMed]

Davenport, M. A.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-Pixel Imaging via Compressive Sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

Davidson, N.

M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7, 919–924 (2013).
[CrossRef]

Davidson, T. J.

K. Chung, J. Wallace, S. Y. Kim, S. Kalyanasundaram, A. S. Andalman, T. J. Davidson, J. J. Mirzabekov, K. A. Zalocusky, J. Mattis, A. K. Denisin, S. Pak, H. Bernstein, C. Ramakrishnan, L. Grosenick, V. Gradinaru, and K. Deisseroth, “Structural and molecular interrogation of intact biological systems,” Nature (London) 497, 332–337 (2013).
[CrossRef]

Dbarre, D.

M. J. Booth, D. Dbarre, and A. Jesacher, “Adaptive optics for biomedical microscopy,” Opt. Photon. News 23, 22 (2012).
[CrossRef]

Deisseroth, K.

K. Chung, J. Wallace, S. Y. Kim, S. Kalyanasundaram, A. S. Andalman, T. J. Davidson, J. J. Mirzabekov, K. A. Zalocusky, J. Mattis, A. K. Denisin, S. Pak, H. Bernstein, C. Ramakrishnan, L. Grosenick, V. Gradinaru, and K. Deisseroth, “Structural and molecular interrogation of intact biological systems,” Nature (London) 497, 332–337 (2013).
[CrossRef]

Denisin, A. K.

K. Chung, J. Wallace, S. Y. Kim, S. Kalyanasundaram, A. S. Andalman, T. J. Davidson, J. J. Mirzabekov, K. A. Zalocusky, J. Mattis, A. K. Denisin, S. Pak, H. Bernstein, C. Ramakrishnan, L. Grosenick, V. Gradinaru, and K. Deisseroth, “Structural and molecular interrogation of intact biological systems,” Nature (London) 497, 332–337 (2013).
[CrossRef]

Dholakia, K.

Discroll, T.

J. Hunt, T. Discroll, A. Mrozack, G. Lipworth, M. Reynolds, D. Brady, and D. R. Smith, “Metamaterial Apertures for Computational Imaging,” Science 339, 310–313 (2013).
[CrossRef] [PubMed]

Duarte, M. F.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-Pixel Imaging via Compressive Sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

Durán, V.

P. Clemente, V. Durán, E. Tajahuerce, V. Torres-Company, and J. Lancis, “Single-pixel digital ghost holography,” Phys. Rev. A 86, 041803 (2012).
[CrossRef]

Edgar, M. P.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D Computational Imaging with Single-Pixel Detectors,” Science 340, 844–847 (2013).
[CrossRef] [PubMed]

Farahi, F.

Feld, M. S.

R. T. 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. Rep.3, (2013).
[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. Photonics 6, 283–292 (2012).
[CrossRef]

S. M. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Controlling light through optical disordered media: transmission matrix approach,” New J. Phys. 13, 123021 (2011).
[CrossRef]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque materia,” Nat. Commun. 1, 1038(2010).
[CrossRef]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and G. 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]

Friesem, A. A.

M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7, 919–924 (2013).
[CrossRef]

Gigan, G.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and G. 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]

Gigan, S.

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, “ Spatio-temporal focusing of an ultrafast pulse through a multiply scattering medium,” Nat. Commun. 2, 447–455 (2011).
[CrossRef] [PubMed]

S. M. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Controlling light through optical disordered media: transmission matrix approach,” New J. Phys. 13, 123021 (2011).
[CrossRef]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque materia,” Nat. Commun. 1, 1038(2010).
[CrossRef]

Goodman, J. W.

Gradinaru, V.

K. Chung, J. Wallace, S. Y. Kim, S. Kalyanasundaram, A. S. Andalman, T. J. Davidson, J. J. Mirzabekov, K. A. Zalocusky, J. Mattis, A. K. Denisin, S. Pak, H. Bernstein, C. Ramakrishnan, L. Grosenick, V. Gradinaru, and K. Deisseroth, “Structural and molecular interrogation of intact biological systems,” Nature (London) 497, 332–337 (2013).
[CrossRef]

Grosenick, L.

K. Chung, J. Wallace, S. Y. Kim, S. Kalyanasundaram, A. S. Andalman, T. J. Davidson, J. J. Mirzabekov, K. A. Zalocusky, J. Mattis, A. K. Denisin, S. Pak, H. Bernstein, C. Ramakrishnan, L. Grosenick, V. Gradinaru, and K. Deisseroth, “Structural and molecular interrogation of intact biological systems,” Nature (London) 497, 332–337 (2013).
[CrossRef]

Hillman, R. T.

R. T. 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. Rep.3, (2013).
[CrossRef] [PubMed]

Howell, J. C.

G. A. Howland and J. C. Howell, “Efficient High-Dimensional Entanglement Imaging with a Compressive-Sensing Double-Pixel Camera,” Phys. Rev. X 3, 011013 (2013).

Howland, G. A.

G. A. Howland and J. C. Howell, “Efficient High-Dimensional Entanglement Imaging with a Compressive-Sensing Double-Pixel Camera,” Phys. Rev. X 3, 011013 (2013).

Hunt, J.

J. Hunt, T. Discroll, A. Mrozack, G. Lipworth, M. Reynolds, D. Brady, and D. R. Smith, “Metamaterial Apertures for Computational Imaging,” Science 339, 310–313 (2013).
[CrossRef] [PubMed]

Jesacher, A.

M. J. Booth, D. Dbarre, and A. Jesacher, “Adaptive optics for biomedical microscopy,” Opt. Photon. News 23, 22 (2012).
[CrossRef]

Kalyanasundaram, S.

K. Chung, J. Wallace, S. Y. Kim, S. Kalyanasundaram, A. S. Andalman, T. J. Davidson, J. J. Mirzabekov, K. A. Zalocusky, J. Mattis, A. K. Denisin, S. Pak, H. Bernstein, C. Ramakrishnan, L. Grosenick, V. Gradinaru, and K. Deisseroth, “Structural and molecular interrogation of intact biological systems,” Nature (London) 497, 332–337 (2013).
[CrossRef]

Katz, J.

J. Katz and J. Sheng, “Applications of Holography in Fluid Mechanics and Particle Dynamics,” Annu. Rev. Fluid Mech. 42, 531–555 (2010).
[CrossRef]

Katz, O.

M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7, 919–924 (2013).
[CrossRef]

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5, 372–377 (2011).
[CrossRef]

Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009).
[CrossRef]

Kelly, K. F.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-Pixel Imaging via Compressive Sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

Kim, M.

Y. Choi, C. Yoon, M. Kim, W. Choi, and W. Choi, “Optical Imaging With the Use of a Scattering Lens,” IEEE J. Sel. Top. Quantum Electron. 20, 6800213 (2014).

Kim, S. Y.

K. Chung, J. Wallace, S. Y. Kim, S. Kalyanasundaram, A. S. Andalman, T. J. Davidson, J. J. Mirzabekov, K. A. Zalocusky, J. Mattis, A. K. Denisin, S. Pak, H. Bernstein, C. Ramakrishnan, L. Grosenick, V. Gradinaru, and K. Deisseroth, “Structural and molecular interrogation of intact biological systems,” Nature (London) 497, 332–337 (2013).
[CrossRef]

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. Photonics 6, 283–292 (2012).
[CrossRef]

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature (London) 491, 232–234 (2012).
[CrossRef]

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
[CrossRef]

Lancis, J.

P. Clemente, V. Durán, E. Tajahuerce, V. Torres-Company, and J. Lancis, “Single-pixel digital ghost holography,” Phys. Rev. A 86, 041803 (2012).
[CrossRef]

Laska, J. N.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-Pixel Imaging via Compressive Sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

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. Photonics 6, 283–292 (2012).
[CrossRef]

S. M. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Controlling light through optical disordered media: transmission matrix approach,” New J. Phys. 13, 123021 (2011).
[CrossRef]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque materia,” Nat. Commun. 1, 1038(2010).
[CrossRef]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and G. 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]

Lipworth, G.

J. Hunt, T. Discroll, A. Mrozack, G. Lipworth, M. Reynolds, D. Brady, and D. R. Smith, “Metamaterial Apertures for Computational Imaging,” Science 339, 310–313 (2013).
[CrossRef] [PubMed]

Magalhães, F.

Mattis, J.

K. Chung, J. Wallace, S. Y. Kim, S. Kalyanasundaram, A. S. Andalman, T. J. Davidson, J. J. Mirzabekov, K. A. Zalocusky, J. Mattis, A. K. Denisin, S. Pak, H. Bernstein, C. Ramakrishnan, L. Grosenick, V. Gradinaru, and K. Deisseroth, “Structural and molecular interrogation of intact biological systems,” Nature (London) 497, 332–337 (2013).
[CrossRef]

McCabe, D. J.

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, “ Spatio-temporal focusing of an ultrafast pulse through a multiply scattering medium,” Nat. Commun. 2, 447–455 (2011).
[CrossRef] [PubMed]

Mirzabekov, J. J.

K. Chung, J. Wallace, S. Y. Kim, S. Kalyanasundaram, A. S. Andalman, T. J. Davidson, J. J. Mirzabekov, K. A. Zalocusky, J. Mattis, A. K. Denisin, S. Pak, H. Bernstein, C. Ramakrishnan, L. Grosenick, V. Gradinaru, and K. Deisseroth, “Structural and molecular interrogation of intact biological systems,” Nature (London) 497, 332–337 (2013).
[CrossRef]

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. Photonics 6, 283–292 (2012).
[CrossRef]

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature (London) 491, 232–234 (2012).
[CrossRef]

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 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. USA 109, E1679–E1687 (2012).
[CrossRef] [PubMed]

Mrozack, A.

J. Hunt, T. Discroll, A. Mrozack, G. Lipworth, M. Reynolds, D. Brady, and D. R. Smith, “Metamaterial Apertures for Computational Imaging,” Science 339, 310–313 (2013).
[CrossRef] [PubMed]

Nixon, M.

M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7, 919–924 (2013).
[CrossRef]

Padgett, M. J.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D Computational Imaging with Single-Pixel Detectors,” Science 340, 844–847 (2013).
[CrossRef] [PubMed]

Pak, S.

K. Chung, J. Wallace, S. Y. Kim, S. Kalyanasundaram, A. S. Andalman, T. J. Davidson, J. J. Mirzabekov, K. A. Zalocusky, J. Mattis, A. K. Denisin, S. Pak, H. Bernstein, C. Ramakrishnan, L. Grosenick, V. Gradinaru, and K. Deisseroth, “Structural and molecular interrogation of intact biological systems,” Nature (London) 497, 332–337 (2013).
[CrossRef]

Park, Y.

R. T. 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. Rep.3, (2013).
[CrossRef] [PubMed]

Piestun, R.

Plöschner, M.

Popoff, S.

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque materia,” Nat. Commun. 1, 1038(2010).
[CrossRef]

Popoff, S. M.

S. M. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Controlling light through optical disordered media: transmission matrix approach,” New J. Phys. 13, 123021 (2011).
[CrossRef]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and G. 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]

Ramakrishnan, C.

K. Chung, J. Wallace, S. Y. Kim, S. Kalyanasundaram, A. S. Andalman, T. J. Davidson, J. J. Mirzabekov, K. A. Zalocusky, J. Mattis, A. K. Denisin, S. Pak, H. Bernstein, C. Ramakrishnan, L. Grosenick, V. Gradinaru, and K. Deisseroth, “Structural and molecular interrogation of intact biological systems,” Nature (London) 497, 332–337 (2013).
[CrossRef]

Reynolds, M.

J. Hunt, T. Discroll, A. Mrozack, G. Lipworth, M. Reynolds, D. Brady, and D. R. Smith, “Metamaterial Apertures for Computational Imaging,” Science 339, 310–313 (2013).
[CrossRef] [PubMed]

Sheng, J.

J. Katz and J. Sheng, “Applications of Holography in Fluid Mechanics and Particle Dynamics,” Annu. Rev. Fluid Mech. 42, 531–555 (2010).
[CrossRef]

Silberberg, Y.

M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7, 919–924 (2013).
[CrossRef]

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5, 372–377 (2011).
[CrossRef]

Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009).
[CrossRef]

Small, E.

M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7, 919–924 (2013).
[CrossRef]

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5, 372–377 (2011).
[CrossRef]

Smith, D. R.

J. Hunt, T. Discroll, A. Mrozack, G. Lipworth, M. Reynolds, D. Brady, and D. R. Smith, “Metamaterial Apertures for Computational Imaging,” Science 339, 310–313 (2013).
[CrossRef] [PubMed]

Straka, B.

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. USA 109, E1679–E1687 (2012).
[CrossRef] [PubMed]

Sun, B.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D Computational Imaging with Single-Pixel Detectors,” Science 340, 844–847 (2013).
[CrossRef] [PubMed]

Sun, T.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-Pixel Imaging via Compressive Sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

Tajahuerce, E.

P. Clemente, V. Durán, E. Tajahuerce, V. Torres-Company, and J. Lancis, “Single-pixel digital ghost holography,” Phys. Rev. A 86, 041803 (2012).
[CrossRef]

Tajalli, A.

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, “ Spatio-temporal focusing of an ultrafast pulse through a multiply scattering medium,” Nat. Commun. 2, 447–455 (2011).
[CrossRef] [PubMed]

Takhar, D.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-Pixel Imaging via Compressive Sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

Torres-Company, V.

P. Clemente, V. Durán, E. Tajahuerce, V. Torres-Company, and J. Lancis, “Single-pixel digital ghost holography,” Phys. Rev. A 86, 041803 (2012).
[CrossRef]

van Putten, E. G.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature (London) 491, 232–234 (2012).
[CrossRef]

Vellekoop, I. M.

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
[CrossRef]

Vittert, L. E.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D Computational Imaging with Single-Pixel Detectors,” Science 340, 844–847 (2013).
[CrossRef] [PubMed]

Vos, W. L.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature (London) 491, 232–234 (2012).
[CrossRef]

Wakin, M. B.

E. J. Candes and M. B. Wakin, “An Introduction To Compressive Sampling,” IEEE Signal Process. Mag. 25, 21–30 (2008).
[CrossRef]

Wallace, J.

K. Chung, J. Wallace, S. Y. Kim, S. Kalyanasundaram, A. S. Andalman, T. J. Davidson, J. J. Mirzabekov, K. A. Zalocusky, J. Mattis, A. K. Denisin, S. Pak, H. Bernstein, C. Ramakrishnan, L. Grosenick, V. Gradinaru, and K. Deisseroth, “Structural and molecular interrogation of intact biological systems,” Nature (London) 497, 332–337 (2013).
[CrossRef]

Walmsley, I. A.

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, “ Spatio-temporal focusing of an ultrafast pulse through a multiply scattering medium,” Nat. Commun. 2, 447–455 (2011).
[CrossRef] [PubMed]

Welsh, S.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D Computational Imaging with Single-Pixel Detectors,” Science 340, 844–847 (2013).
[CrossRef] [PubMed]

Wenner, M.

M. Wenner, “The most transparent research,” Nature Medicine 15, 1106–1109 (2009).
[CrossRef] [PubMed]

Yamauchi, T.

R. T. 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. Rep.3, (2013).
[CrossRef] [PubMed]

Yaqoob, Z.

R. T. 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. Rep.3, (2013).
[CrossRef] [PubMed]

Yoon, C.

Y. Choi, C. Yoon, M. Kim, W. Choi, and W. Choi, “Optical Imaging With the Use of a Scattering Lens,” IEEE J. Sel. Top. Quantum Electron. 20, 6800213 (2014).

Zalocusky, K. A.

K. Chung, J. Wallace, S. Y. Kim, S. Kalyanasundaram, A. S. Andalman, T. J. Davidson, J. J. Mirzabekov, K. A. Zalocusky, J. Mattis, A. K. Denisin, S. Pak, H. Bernstein, C. Ramakrishnan, L. Grosenick, V. Gradinaru, and K. Deisseroth, “Structural and molecular interrogation of intact biological systems,” Nature (London) 497, 332–337 (2013).
[CrossRef]

Annu. Rev. Fluid Mech. (1)

J. Katz and J. Sheng, “Applications of Holography in Fluid Mechanics and Particle Dynamics,” Annu. Rev. Fluid Mech. 42, 531–555 (2010).
[CrossRef]

Appl. Opt. (1)

IEEE J. Sel. Top. Quantum Electron. (1)

Y. Choi, C. Yoon, M. Kim, W. Choi, and W. Choi, “Optical Imaging With the Use of a Scattering Lens,” IEEE J. Sel. Top. Quantum Electron. 20, 6800213 (2014).

IEEE Signal Process. Mag. (2)

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-Pixel Imaging via Compressive Sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

E. J. Candes and M. B. Wakin, “An Introduction To Compressive Sampling,” IEEE Signal Process. Mag. 25, 21–30 (2008).
[CrossRef]

J. Opt. Soc. Am. (1)

Nat. Commun. (2)

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, “ Spatio-temporal focusing of an ultrafast pulse through a multiply scattering medium,” Nat. Commun. 2, 447–455 (2011).
[CrossRef] [PubMed]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque materia,” Nat. Commun. 1, 1038(2010).
[CrossRef]

Nat. Photonics (4)

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6, 283–292 (2012).
[CrossRef]

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
[CrossRef]

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5, 372–377 (2011).
[CrossRef]

M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7, 919–924 (2013).
[CrossRef]

Nature (London) (2)

K. Chung, J. Wallace, S. Y. Kim, S. Kalyanasundaram, A. S. Andalman, T. J. Davidson, J. J. Mirzabekov, K. A. Zalocusky, J. Mattis, A. K. Denisin, S. Pak, H. Bernstein, C. Ramakrishnan, L. Grosenick, V. Gradinaru, and K. Deisseroth, “Structural and molecular interrogation of intact biological systems,” Nature (London) 497, 332–337 (2013).
[CrossRef]

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature (London) 491, 232–234 (2012).
[CrossRef]

Nature Commun. (1)

T. Cizmar and K. Dholakia, “Exploiting multimode waveguides for pure fibre-based imaging,” Nature Commun. 3, 1027 (2012).
[CrossRef]

Nature Medicine (1)

M. Wenner, “The most transparent research,” Nature Medicine 15, 1106–1109 (2009).
[CrossRef] [PubMed]

New J. Phys. (1)

S. M. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Controlling light through optical disordered media: transmission matrix approach,” New J. Phys. 13, 123021 (2011).
[CrossRef]

Opt. Express (2)

Opt. Photon. News (1)

M. J. Booth, D. Dbarre, and A. Jesacher, “Adaptive optics for biomedical microscopy,” Opt. Photon. News 23, 22 (2012).
[CrossRef]

Phys. Rev. A (2)

Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009).
[CrossRef]

P. Clemente, V. Durán, E. Tajahuerce, V. Torres-Company, and J. Lancis, “Single-pixel digital ghost holography,” Phys. Rev. A 86, 041803 (2012).
[CrossRef]

Phys. Rev. Lett. (1)

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and G. 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]

Phys. Rev. X (1)

G. A. Howland and J. C. Howell, “Efficient High-Dimensional Entanglement Imaging with a Compressive-Sensing Double-Pixel Camera,” Phys. Rev. X 3, 011013 (2013).

Proc. Natl. Acad. Sci. USA (1)

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. USA 109, E1679–E1687 (2012).
[CrossRef] [PubMed]

Science (2)

J. Hunt, T. Discroll, A. Mrozack, G. Lipworth, M. Reynolds, D. Brady, and D. R. Smith, “Metamaterial Apertures for Computational Imaging,” Science 339, 310–313 (2013).
[CrossRef] [PubMed]

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D Computational Imaging with Single-Pixel Detectors,” Science 340, 844–847 (2013).
[CrossRef] [PubMed]

Other (2)

R. T. 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. Rep.3, (2013).
[CrossRef] [PubMed]

E. J. Candes, http://www.stat.stanford.edu/candes/l1magic .

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of the experimental set-up. (a) Set of microstructured patterns is projected onto the sample, which is placed in front of the scattering layer. The transmitted light undergoes strong scattering and a fraction is recorded by a photodiode. Within single-pixel imaging schemes, we need to project several patterns to form the image with measurements captured sequentially. (b) Photograph of the scattering layer covering partially a text located at a distance 10mm from the diffuser. In this way we show that the portion of the text under the diffuser is completely hidden. (c) Plot of two thousand projections derived from the photodetected signal in terms of the input modes.

Fig. 2
Fig. 2

Seeing through scattering media with a single-pixel camera. (a) The Cheshire Cat is placed in front of a scattering layer that completely hides it. We show the image of the object as seen by a charge-coupled-device camera (Stingray-F-145 with square pixel size of 6.45 × 6.45μm2). The image contains not information at all on the shape of the object. (b) Image of the retrieved object from the photodetected values recorded by the photodiode. The good quality of the reconstructed image confirms that disorder-assisted imaging can indeed recover fine details of the object even though a sensor without spatial resolution is used.

Fig. 3
Fig. 3

Disorder-assisted compressive microscopy. (a) Schematic diagram of the optical setup. LS laser source; L lens; BS beam splitter; SLM programmable spatial light modulator; D iris diaphram; MO microscope objective; OBJ sample (stained onion cells); SM scattering medium; CCD charge-coupled device camera; PD Photodiode. (b) Plot of the quality of the recovered images in dB as a function of the compression ratio. We show as insets the image of the onion cells retrieved from the photodetected signals generated by a randomized selection (24% and 50%) of the 4096 input modes, along with the unsampled image (CR = 1) for comparison.

Fig. 4
Fig. 4

Image reconstruction through a dynamic scattering medium. (a) Image of the sample (a typical resolution chart) reconstructed from the photodetected signal showing the successful image retrieving ability of our single-pixel imaging setup through a moving diffuser. (b) Snapshot of the speckle field generated by the object placed in front of a moving diffuser. At the right side we plot the detected signal with an integration time smaller than the typical temporal drift of the speckle grains corresponding to both a pixel of the camera (top) and a region of 64 × 64 pixels (bottom).

Fig. 5
Fig. 5

Charting the statistical properties of the scattering layer. (a) Histogram of the absolute value of the transmission matrix elements assessed by experimental means. The plot fairly matches the Rayleigh distribution, as we expected. (b) Histogram of the summations An evaluated from the data in (a). The small ratio between the standard deviation σA and the mean value 〈A〉 ensures that the integrated contributions coming from the different input modes are nearly equally weighted. (c) Relative dispersion of the An contributions in terms of the number of output modes integrated by the area of the sensor. (d) Histogram of the summations Bnn′ evaluated from the data in (a) assigning randomly a uniform phase distribution between 0 and 2π. Note that in order to make easier the comparison of results, the scale of the abscissa axis in the graphs (b) and (d) is the same.

Equations (4)

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y = MI in = ( H 1 11 H 1 n n H N 11 H N n n ) I in ,
I out = m = 1 S I m out = m = 1 s | n = 1 N k m n E n in | 2 = m = 1 S n = 1 N | k m n | 2 | E n in | 2 + m = 1 S n = 1 N n = 1 n n N k m n k m n * E n in E n i n * .
I out = n = 1 N A n | E n in | 2 + n = 1 N n = 1 n n N B n n E n in E n i n * .
I out n = 1 N | E n in | 2 .

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