Abstract

Fluorescence tomography is a well-established methodology able to provide structural and functional information on the measured object. At optical wavelengths, the unpredictable scattering of light is often considered a problem to overcome, rather than a feature to exploit. Advances in disordered photonics have shed new light on possibilities offered by opaque materials, treating them as autocorrelation lenses able to create images and focus light. In this Letter, we propose tomography through disorder, introducing a modified Fourier-slice theorem, the cornerstone of the computed tomography, aiming to reconstruct a three-dimensional fluorescent sample hidden behind an opaque curtain.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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A. C. Kak, M. Slaney, and G. Wang, Med. Phys. 29, 107 (2002).
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M. Jang, Y. Horie, A. Shibukawa, J. Brake, Y. Liu, S. M. Kamali, A. Arbabi, H. Ruan, A. Faraon, and C. Yang, Nat. Photonics 12, 84 (2018).
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S. Mukherjee, A. Vijayakumar, M. Kumar, and J. Rosen, Sci. Rep. 8, 1 (2018).
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Lagendijk, A.

J. Bertolotti, E. G. Van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, Nature 491, 232 (2012).
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I. M. Vellekoop, A. Lagendijk, and A. Mosk, Nat. Photonics 4, 320 (2010).
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Lee, K.

K. Lee and Y. Park, Nat. Commun. 7, 13359 (2016).
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Leonetti, M.

D. Di Battista, D. Ancora, G. Zacharakis, G. Ruocco, and M. Leonetti, Opt. Express 26, 15594 (2018).
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D. Di Battista, D. Ancora, M. Leonetti, and G. Zacharakis, Appl. Phys. Lett. 109, 121110 (2016).
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D. Ancora, D. Di Battista, G. Giasafaki, S. E. Psycharakis, E. Liapis, J. Ripoll, and G. Zacharakis, Sci. Rep. 7, 11854 (2017).
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Liu, Y.

M. Jang, Y. Horie, A. Shibukawa, J. Brake, Y. Liu, S. M. Kamali, A. Arbabi, H. Ruan, A. Faraon, and C. Yang, Nat. Photonics 12, 84 (2018).
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Lu, D.

Marakis, E.

Marchesini, S.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. Spence, Phys. Rev. B 68, 140101 (2003).
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Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, IEEE Signal Process. Mag. 32(3), 87 (2015).
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J. A. Rodriguez, R. Xu, C.-C. Chen, Y. Zou, and J. Miao, J. Appl. Crystallogr. 46, 312 (2013).
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Mosk, A.

I. M. Vellekoop, A. Lagendijk, and A. Mosk, Nat. Photonics 4, 320 (2010).
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S. A. Goorden, M. Horstmann, A. P. Mosk, B. Škorić, and P. W. Pinkse, Optica 1, 421 (2014).
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J. Bertolotti, E. G. Van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, Nature 491, 232 (2012).
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S. Mukherjee, A. Vijayakumar, M. Kumar, and J. Rosen, Sci. Rep. 8, 1 (2018).
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Noy, A.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. Spence, Phys. Rev. B 68, 140101 (2003).
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Osten, W.

Park, Y.

K. Lee and Y. Park, Nat. Commun. 7, 13359 (2016).
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Peng, X.

Pinkse, P. W.

Psycharakis, S. E.

D. Ancora, D. Di Battista, G. Giasafaki, S. E. Psycharakis, E. Liapis, J. Ripoll, and G. Zacharakis, Sci. Rep. 7, 11854 (2017).
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Ripoll, J.

D. Ancora, D. Di Battista, G. Giasafaki, S. E. Psycharakis, E. Liapis, J. Ripoll, and G. Zacharakis, Sci. Rep. 7, 11854 (2017).
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J. A. Rodriguez, R. Xu, C.-C. Chen, Y. Zou, and J. Miao, J. Appl. Crystallogr. 46, 312 (2013).
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Rosen, J.

S. Mukherjee, A. Vijayakumar, M. Kumar, and J. Rosen, Sci. Rep. 8, 1 (2018).
[Crossref]

Rosenbluh, M.

I. Freund, M. Rosenbluh, and S. Feng, Phys. Rev. Lett. 61, 2328 (1988).
[Crossref]

Ruan, H.

M. Jang, Y. Horie, A. Shibukawa, J. Brake, Y. Liu, S. M. Kamali, A. Arbabi, H. Ruan, A. Faraon, and C. Yang, Nat. Photonics 12, 84 (2018).
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Ruocco, G.

Scarcelli, G.

Segev, M.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, IEEE Signal Process. Mag. 32(3), 87 (2015).
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Shao, X.

Sharpe, J.

J. Sharpe, Annu. Rev. Biomed. Eng. 6, 209 (2004).
[Crossref]

Shechtman, Y.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, IEEE Signal Process. Mag. 32(3), 87 (2015).
[Crossref]

Shibukawa, A.

M. Jang, Y. Horie, A. Shibukawa, J. Brake, Y. Liu, S. M. Kamali, A. Arbabi, H. Ruan, A. Faraon, and C. Yang, Nat. Photonics 12, 84 (2018).
[Crossref]

Škoric, B.

Slaney, M.

A. C. Kak, M. Slaney, and G. Wang, Med. Phys. 29, 107 (2002).
[Crossref]

Spence, J. C.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. Spence, Phys. Rev. B 68, 140101 (2003).
[Crossref]

Tanida, J.

Trull, A. K.

Tzortzakis, S.

van der Horst, J.

Van Putten, E. G.

J. Bertolotti, E. G. Van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, Nature 491, 232 (2012).
[Crossref]

van Vliet, L. J.

Vellekoop, I. M.

I. M. Vellekoop, A. Lagendijk, and A. Mosk, Nat. Photonics 4, 320 (2010).
[Crossref]

Vijayakumar, A.

S. Mukherjee, A. Vijayakumar, M. Kumar, and J. Rosen, Sci. Rep. 8, 1 (2018).
[Crossref]

Vos, W. L.

J. Bertolotti, E. G. Van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, Nature 491, 232 (2012).
[Crossref]

Waller, L.

Wang, G.

A. C. Kak, M. Slaney, and G. Wang, Med. Phys. 29, 107 (2002).
[Crossref]

Weierstall, U.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. Spence, Phys. Rev. B 68, 140101 (2003).
[Crossref]

Wiersma, D. S.

D. S. Wiersma, Nat. Photonics 7, 188 (2013).
[Crossref]

Wu, T.

Xu, R.

J. A. Rodriguez, R. Xu, C.-C. Chen, Y. Zou, and J. Miao, J. Appl. Crystallogr. 46, 312 (2013).
[Crossref]

Yang, C.

M. Jang, Y. Horie, A. Shibukawa, J. Brake, Y. Liu, S. M. Kamali, A. Arbabi, H. Ruan, A. Faraon, and C. Yang, Nat. Photonics 12, 84 (2018).
[Crossref]

Zacharakis, G.

D. Di Battista, D. Ancora, G. Zacharakis, G. Ruocco, and M. Leonetti, Opt. Express 26, 15594 (2018).
[Crossref]

D. Ancora, D. Di Battista, G. Giasafaki, S. E. Psycharakis, E. Liapis, J. Ripoll, and G. Zacharakis, Sci. Rep. 7, 11854 (2017).
[Crossref]

D. Di Battista, D. Ancora, M. Leonetti, and G. Zacharakis, Appl. Phys. Lett. 109, 121110 (2016).
[Crossref]

D. Di Battista, D. Ancora, H. Zhang, K. Lemonaki, E. Marakis, E. Liapis, S. Tzortzakis, and G. Zacharakis, Optica 3, 1237 (2016).
[Crossref]

Zhang, H.

Zou, Y.

J. A. Rodriguez, R. Xu, C.-C. Chen, Y. Zou, and J. Miao, J. Appl. Crystallogr. 46, 312 (2013).
[Crossref]

Annu. Rev. Biomed. Eng. (1)

J. Sharpe, Annu. Rev. Biomed. Eng. 6, 209 (2004).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

D. Di Battista, D. Ancora, M. Leonetti, and G. Zacharakis, Appl. Phys. Lett. 109, 121110 (2016).
[Crossref]

EURASIP J. Audio Speech Music Proc. (1)

G. Farahani, EURASIP J. Audio Speech Music Proc. 2017, 13 (2017).
[Crossref]

IEEE Signal Process. Mag. (1)

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, IEEE Signal Process. Mag. 32(3), 87 (2015).
[Crossref]

J. Appl. Crystallogr. (1)

J. A. Rodriguez, R. Xu, C.-C. Chen, Y. Zou, and J. Miao, J. Appl. Crystallogr. 46, 312 (2013).
[Crossref]

Med. Phys. (1)

A. C. Kak, M. Slaney, and G. Wang, Med. Phys. 29, 107 (2002).
[Crossref]

Nat. Commun. (2)

K. Lee and Y. Park, Nat. Commun. 7, 13359 (2016).
[Crossref]

T. Čižmár and K. Dholakia, Nat. Commun. 3, 1027 (2012).
[Crossref]

Nat. Photonics (4)

M. Jang, Y. Horie, A. Shibukawa, J. Brake, Y. Liu, S. M. Kamali, A. Arbabi, H. Ruan, A. Faraon, and C. Yang, Nat. Photonics 12, 84 (2018).
[Crossref]

D. S. Wiersma, Nat. Photonics 7, 188 (2013).
[Crossref]

O. Katz, P. Heidmann, M. Fink, and S. Gigan, Nat. Photonics 8, 784 (2014).
[Crossref]

I. M. Vellekoop, A. Lagendijk, and A. Mosk, Nat. Photonics 4, 320 (2010).
[Crossref]

Nature (1)

J. Bertolotti, E. G. Van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, Nature 491, 232 (2012).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Optica (4)

Phys. Rev. B (1)

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. Spence, Phys. Rev. B 68, 140101 (2003).
[Crossref]

Phys. Rev. Lett. (1)

I. Freund, M. Rosenbluh, and S. Feng, Phys. Rev. Lett. 61, 2328 (1988).
[Crossref]

Sci. Rep. (2)

S. Mukherjee, A. Vijayakumar, M. Kumar, and J. Rosen, Sci. Rep. 8, 1 (2018).
[Crossref]

D. Ancora, D. Di Battista, G. Giasafaki, S. E. Psycharakis, E. Liapis, J. Ripoll, and G. Zacharakis, Sci. Rep. 7, 11854 (2017).
[Crossref]

Other (2)

D. Ancora, D. Di Battista, A. M. Vidal, S. Avtzi, G. Zacharakis, and A. Bassi (2020). https://doi.org/10.6084/m9.figshare.11901927.v1
[Crossref]

J. Hsieh, Computed Tomography: Principles, Design, Artifacts, and Recent Advances (SPIE, 2003), Vol. 114.

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Figures (4)

Fig. 1.
Fig. 1. Sketch of the experimental setup for the imaging of a three-dimensional fluorescent object $ {\cal O} $ obscured behind an opaque diffuser (D) and a narrowband filter (F). The fluorescence is excited with a LED (L), and a camera (C) detects the scrambled wavefront. The setup is simple and does not have lenses nor objectives, but rather relies on speckle autocorrelation properties.
Fig. 2.
Fig. 2. (a) Speckle patterns acquired rotating the object at angle $ \varphi $ . $ {S_\varphi } $ is corrected by its low-pass envelope to resolve speckle fluctuations. (b) Autocorrelation sinogram $ {X_\varphi } $ calculated for each $ {S_\varphi } $ . To visualize the sinogram, we slice the stack $ {X_\varphi } $ vertically (red dot) and horizontally (blue dot), showing the results in the corresponding dot-labeled images on the right.
Fig. 3.
Fig. 3. Volume (a), 3D rendering of the autocorrelation sinogram $ {X_\varphi } $ previously shown in Fig. 2(b). Gray arrow (1), inversion of the $ {X_\varphi } $ via SIRT. Volume (b), result of the SIRT inversion of $ {X_\varphi } $ , giving rise to the reconstruction of the autocorrelation $ {{\cal X}^ \bullet } = {{\cal R}^{ - 1}}\{ {X_\varphi }\} $ . Gray arrow (2), 3D phase retrieval algorithm. Volume (3), output of the phase-retrieved fluorescence reconstruction after the recovery of the phase connected to the autocorrelation (operation 2).
Fig. 4.
Fig. 4. (a) Frontal view (at $ \varphi = 0^ \circ $ ) of the ground truth reconstruction of the object $ {\cal O} $ measured with a normal OPT approach. The location of the fluorescent signal is color encoded to assess the depth at which the signal is emitted. (b) Tomography of the same sample hidden behind the diffuser. The two “legs” are reconstructed at the right depths. (c) Tomographic section of the ground truth along the dashed line of panel (a). (d) Tomographic section of the retrieved volume (b) at the same height as in the dashed line of panel (a). We can notice the correct location of the two fluorescent objects. The object extends approximately 31.1 mm along the horizontal axis.

Equations (6)

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S φ S φ = ( O φ P S F φ ) ( O φ P S F φ ) = ( O φ O φ ) ( P S F φ P S F φ ) = O φ O φ .
X φ ( ξ , ε ) S φ S φ = F 1 { F { S φ ( x , y ) } 2 } ,
F { X } | k z = 0 = O ( x , y , z ) e i 2 π ( x k x + y k y ) d x d y d z 2 = O ( x , y ) e i 2 π ( x k x + y k y ) d x d y 2 ,
F { X } | k z = 0 = F { O } 2 = F { X } ,
H I O : O i + 1 = { O i i f ( x , y , z ) γ O i β O i i f ( x , y , z ) γ ,
E R : O i + 1 = { O i i f ( x , y , z ) γ 0 i f ( x , y , z ) γ .

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