Abstract

Light passing through scattering media will be strongly scattered and diffused into complex speckle pattern, which contains almost all the spatial information and spectral information of the objects. Although various methods have been proposed to recover the spatial information of the hidden objects, it is still a challenge to simultaneously obtain their spectral information. Here, we present an effective approach to realize spectral imaging through scattering media by combining the spectra retrieval and the speckle-correlation. Compared to the traditional imaging spectrometer, our approach is more flexible in the choice of core element. In this paper, we have demonstrated employing the frosted glass as the core element to achieve spectral imaging. Obtaining the spectral information and spatial information are demonstrated via numerical simulations. Experiment results further demonstrate the performance of our scheme in spectral imaging through scattering media. The spectral imaging based on scattering media is well suited for new type spectral imaging applications.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref] [PubMed]
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2017 (4)

2016 (1)

E. Edrei and G. Scarcelli, “Memory-effect based deconvolution microscopy for super-resolution imaging through scattering media,” Sci. Rep. 6(1), 33558 (2016).
[Crossref] [PubMed]

2015 (2)

P. Lai, L. Wang, J. W. Tay, and L. V. Wang, “Photoacoustically guided wavefront shaping for enhanced optical focusing in scattering media,” Nat. Photonics 9(2), 126–132 (2015).
[Crossref] [PubMed]

T. Chaigne, O. Katz, A. C. Boccara, M. Fink, E. Bossy, and S. Gigan, “Controlling light in scattering media noninvasively using the photo-acoustic transmission-matrix,” Nat. Photonics 9, 126–132 (2015).

2014 (2)

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(10), 784–790 (2014).
[Crossref]

A. K. Singh, D. N. Naik, G. Pedrini, M. Takeda, and W. Osten, “Looking through a diffuser and around an opaque surface: a holographic approach,” Opt. Express 22(7), 7694–7701 (2014).
[Crossref] [PubMed]

2013 (3)

B. Redding, S. M. Popoff, and H. Cao, “All-fiber spectrometer based on speckle pattern reconstruction,” Opt. Express 21(5), 6584–6600 (2013).
[Crossref] [PubMed]

H. Park and K. B. Crozier, “Multispectral imaging with vertical silicon nanowires,” Sci. Rep. 3(1), 2460 (2013).
[Crossref] [PubMed]

B. Redding, S. F. Liew, R. Sarma, and H. Cao, “Compact spectrometer based on a disordered photonic chip,” Nat. Photonics 7(9), 746–751 (2013).
[Crossref]

2012 (2)

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 491(7423), 232–234 (2012).
[Crossref] [PubMed]

O. Katz, E. Small, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5, 372–377 (2012).
[Crossref]

2010 (3)

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

V. Ntziachristos, “Going deeper than microscopy: the optical imaging frontier in biology,” Nat. Methods 7(8), 603–614 (2010).
[Crossref] [PubMed]

I. M. Vellekoop and C. M. Aegerter, “Scattered light fluorescence microscopy: imaging through turbid layers,” Opt. Lett. 35(8), 1245–1247 (2010).
[Crossref] [PubMed]

2007 (1)

1990 (1)

I. Freund, “Looking through walls and around corners,” Physica A 168(1), 49–65 (1990).
[Crossref]

1988 (2)

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

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

1982 (1)

Aegerter, C. M.

Akselrod, G. M.

J. W. Stewart, G. M. Akselrod, D. R. Smith, and M. H. Mikkelsen, “Toward multispectral imaging with colloidal metasurface pixels,” Adv. Mater. 29(6), 1602971 (2017).
[Crossref] [PubMed]

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 491(7423), 232–234 (2012).
[Crossref] [PubMed]

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 491(7423), 232–234 (2012).
[Crossref] [PubMed]

Boccara, A. C.

T. Chaigne, O. Katz, A. C. Boccara, M. Fink, E. Bossy, and S. Gigan, “Controlling light in scattering media noninvasively using the photo-acoustic transmission-matrix,” Nat. Photonics 9, 126–132 (2015).

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

Bossy, E.

T. Chaigne, O. Katz, A. C. Boccara, M. Fink, E. Bossy, and S. Gigan, “Controlling light in scattering media noninvasively using the photo-acoustic transmission-matrix,” Nat. Photonics 9, 126–132 (2015).

Cao, H.

B. Redding, S. F. Liew, R. Sarma, and H. Cao, “Compact spectrometer based on a disordered photonic chip,” Nat. Photonics 7(9), 746–751 (2013).
[Crossref]

B. Redding, S. M. Popoff, and H. Cao, “All-fiber spectrometer based on speckle pattern reconstruction,” Opt. Express 21(5), 6584–6600 (2013).
[Crossref] [PubMed]

Carminati, R.

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

Chaigne, T.

T. Chaigne, O. Katz, A. C. Boccara, M. Fink, E. Bossy, and S. Gigan, “Controlling light in scattering media noninvasively using the photo-acoustic transmission-matrix,” Nat. Photonics 9, 126–132 (2015).

Crozier, K. B.

H. Park and K. B. Crozier, “Multispectral imaging with vertical silicon nanowires,” Sci. Rep. 3(1), 2460 (2013).
[Crossref] [PubMed]

Dang, C.

Edrei, E.

E. Edrei and G. Scarcelli, “Memory-effect based deconvolution microscopy for super-resolution imaging through scattering media,” Sci. Rep. 6(1), 33558 (2016).
[Crossref] [PubMed]

Feng, S.

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

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

Fienup, J. R.

Fink, M.

T. Chaigne, O. Katz, A. C. Boccara, M. Fink, E. Bossy, and S. Gigan, “Controlling light in scattering media noninvasively using the photo-acoustic transmission-matrix,” Nat. Photonics 9, 126–132 (2015).

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(10), 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(10), 100601 (2010).
[Crossref] [PubMed]

French, R.

Freund, I.

I. Freund, “Looking through walls and around corners,” Physica A 168(1), 49–65 (1990).
[Crossref]

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

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

Gigan, S.

R. French, S. Gigan, and O. L. Muskens, “Speckle-based hyperspectral imaging combining multiple scattering and compressive sensing in nanowire mats,” Opt. Lett. 42(9), 1820–1823 (2017).
[Crossref] [PubMed]

T. Chaigne, O. Katz, A. C. Boccara, M. Fink, E. Bossy, and S. Gigan, “Controlling light in scattering media noninvasively using the photo-acoustic transmission-matrix,” Nat. Photonics 9, 126–132 (2015).

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(10), 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(10), 100601 (2010).
[Crossref] [PubMed]

Heidmann, P.

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(10), 784–790 (2014).
[Crossref]

Ho, H. P.

Huang, W.

Huang, X. L.

Katz, O.

T. Chaigne, O. Katz, A. C. Boccara, M. Fink, E. Bossy, and S. Gigan, “Controlling light in scattering media noninvasively using the photo-acoustic transmission-matrix,” Nat. Photonics 9, 126–132 (2015).

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(10), 784–790 (2014).
[Crossref]

O. Katz, E. Small, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5, 372–377 (2012).
[Crossref]

Lagendijk, A.

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 491(7423), 232–234 (2012).
[Crossref] [PubMed]

Lai, P.

P. Lai, L. Wang, J. W. Tay, and L. V. Wang, “Photoacoustically guided wavefront shaping for enhanced optical focusing in scattering media,” Nat. Photonics 9(2), 126–132 (2015).
[Crossref] [PubMed]

Lerosey, G.

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

Li, X. A.

Liew, S. F.

B. Redding, S. F. Liew, R. Sarma, and H. Cao, “Compact spectrometer based on a disordered photonic chip,” Nat. Photonics 7(9), 746–751 (2013).
[Crossref]

Mikkelsen, M. H.

J. W. Stewart, G. M. Akselrod, D. R. Smith, and M. H. Mikkelsen, “Toward multispectral imaging with colloidal metasurface pixels,” Adv. Mater. 29(6), 1602971 (2017).
[Crossref] [PubMed]

Mosk, A. P.

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 491(7423), 232–234 (2012).
[Crossref] [PubMed]

I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32(16), 2309–2311 (2007).
[Crossref] [PubMed]

Muskens, O. L.

Naik, D. N.

Ntziachristos, V.

V. Ntziachristos, “Going deeper than microscopy: the optical imaging frontier in biology,” Nat. Methods 7(8), 603–614 (2010).
[Crossref] [PubMed]

Osten, W.

Park, H.

H. Park and K. B. Crozier, “Multispectral imaging with vertical silicon nanowires,” Sci. Rep. 3(1), 2460 (2013).
[Crossref] [PubMed]

Pedrini, G.

Peng, J. X.

Popoff, S. M.

B. Redding, S. M. Popoff, and H. Cao, “All-fiber spectrometer based on speckle pattern reconstruction,” Opt. Express 21(5), 6584–6600 (2013).
[Crossref] [PubMed]

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

Redding, B.

B. Redding, S. M. Popoff, and H. Cao, “All-fiber spectrometer based on speckle pattern reconstruction,” Opt. Express 21(5), 6584–6600 (2013).
[Crossref] [PubMed]

B. Redding, S. F. Liew, R. Sarma, and H. Cao, “Compact spectrometer based on a disordered photonic chip,” Nat. Photonics 7(9), 746–751 (2013).
[Crossref]

Rosenbluh, M.

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

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

Sahoo, S. K.

Sarma, R.

B. Redding, S. F. Liew, R. Sarma, and H. Cao, “Compact spectrometer based on a disordered photonic chip,” Nat. Photonics 7(9), 746–751 (2013).
[Crossref]

Scarcelli, G.

E. Edrei and G. Scarcelli, “Memory-effect based deconvolution microscopy for super-resolution imaging through scattering media,” Sci. Rep. 6(1), 33558 (2016).
[Crossref] [PubMed]

Shen, X.

Silberberg, Y.

O. Katz, E. Small, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5, 372–377 (2012).
[Crossref]

Singh, A. K.

Small, E.

O. Katz, E. Small, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5, 372–377 (2012).
[Crossref]

Smith, D. R.

J. W. Stewart, G. M. Akselrod, D. R. Smith, and M. H. Mikkelsen, “Toward multispectral imaging with colloidal metasurface pixels,” Adv. Mater. 29(6), 1602971 (2017).
[Crossref] [PubMed]

Stewart, J. W.

J. W. Stewart, G. M. Akselrod, D. R. Smith, and M. H. Mikkelsen, “Toward multispectral imaging with colloidal metasurface pixels,” Adv. Mater. 29(6), 1602971 (2017).
[Crossref] [PubMed]

Takeda, M.

Tang, D.

Tay, J. W.

P. Lai, L. Wang, J. W. Tay, and L. V. Wang, “Photoacoustically guided wavefront shaping for enhanced optical focusing in scattering media,” Nat. Photonics 9(2), 126–132 (2015).
[Crossref] [PubMed]

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 491(7423), 232–234 (2012).
[Crossref] [PubMed]

Vellekoop, I. M.

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 491(7423), 232–234 (2012).
[Crossref] [PubMed]

Wang, L.

P. Lai, L. Wang, J. W. Tay, and L. V. Wang, “Photoacoustically guided wavefront shaping for enhanced optical focusing in scattering media,” Nat. Photonics 9(2), 126–132 (2015).
[Crossref] [PubMed]

Wang, L. V.

P. Lai, L. Wang, J. W. Tay, and L. V. Wang, “Photoacoustically guided wavefront shaping for enhanced optical focusing in scattering media,” Nat. Photonics 9(2), 126–132 (2015).
[Crossref] [PubMed]

Yang, T.

Zhou, X. H.

Adv. Mater. (1)

J. W. Stewart, G. M. Akselrod, D. R. Smith, and M. H. Mikkelsen, “Toward multispectral imaging with colloidal metasurface pixels,” Adv. Mater. 29(6), 1602971 (2017).
[Crossref] [PubMed]

Appl. Opt. (1)

Nat. Methods (1)

V. Ntziachristos, “Going deeper than microscopy: the optical imaging frontier in biology,” Nat. Methods 7(8), 603–614 (2010).
[Crossref] [PubMed]

Nat. Photonics (5)

O. Katz, E. Small, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5, 372–377 (2012).
[Crossref]

T. Chaigne, O. Katz, A. C. Boccara, M. Fink, E. Bossy, and S. Gigan, “Controlling light in scattering media noninvasively using the photo-acoustic transmission-matrix,” Nat. Photonics 9, 126–132 (2015).

B. Redding, S. F. Liew, R. Sarma, and H. Cao, “Compact spectrometer based on a disordered photonic chip,” Nat. Photonics 7(9), 746–751 (2013).
[Crossref]

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(10), 784–790 (2014).
[Crossref]

P. Lai, L. Wang, J. W. Tay, and L. V. Wang, “Photoacoustically guided wavefront shaping for enhanced optical focusing in scattering media,” Nat. Photonics 9(2), 126–132 (2015).
[Crossref] [PubMed]

Nature (1)

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 491(7423), 232–234 (2012).
[Crossref] [PubMed]

Opt. Express (2)

Opt. Lett. (4)

Optica (1)

Phys. Rev. Lett. (3)

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

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

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

Physica A (1)

I. Freund, “Looking through walls and around corners,” Physica A 168(1), 49–65 (1990).
[Crossref]

Sci. Rep. (2)

H. Park and K. B. Crozier, “Multispectral imaging with vertical silicon nanowires,” Sci. Rep. 3(1), 2460 (2013).
[Crossref] [PubMed]

E. Edrei and G. Scarcelli, “Memory-effect based deconvolution microscopy for super-resolution imaging through scattering media,” Sci. Rep. 6(1), 33558 (2016).
[Crossref] [PubMed]

Other (4)

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

S. P. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University Press, 2004).

P. M. Morse and H. Feshbach, Methods Of Theoretical Physics, Part 1 (McGraw-Hill, 1953).

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2007).

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Figures (8)

Fig. 1
Fig. 1 (a) Schematic of spectral imaging through scattering media: BS, beam splitter, OB, objective, SMF, single-mode fiber, C1 and C2, collimator. (b) The process of calibration matrix.
Fig. 2
Fig. 2 Numerical simulation results of spectral imaging through the scattering media. (a) First row: speckle patterns captured by camera1 in the condition of the 610nm spectrum illumination. Second row: speckle patterns captured by camera2 in the condition of the 610nm spectrum illumination. Third row: object reconstruction from the first row of (a). Forth row: spectral retrieval from the second row of (a). (b)–(d) Similar to (a) but for different spectral source illuminations. Scale bars: 120 pixels in the first row and second row of (a), (b), (c) and (d), and 12 pixels in the third row of (a), (b), (c) and (d).
Fig. 3
Fig. 3 Performance of spectral retrieval with increasing noise.
Fig. 4
Fig. 4 Experimental setup and calibration matrices of spectral imaging through scattering media. (a) Experimental optical setup: 1, Xenon lamp light source; 2, monochromator; 3, collimator; 4, BS; 5, object; 6, scattering media1; 7, camera1; 8, OB; 9, SMF; 10, C2; 11, scattering media2; 12, camera1. (b,c) Calibration matrix showing the intensity distributions of different wavelengths through the experiment system from 445 to 495 nm and from 610 to 660 nm.
Fig. 5
Fig. 5 Reconstructing spatial and spectral information. (a) First row: Reconstructed spatial intensities from the speckle pattern captured by camera1. Second row: Spectra recovered from the speckle pattern captured by camera2. (b)–(d) Similar to (a) but for different source illuminations or different objects, respectively. Scale bars: 52 um in the first row of (a), (b), (c) and (d).
Fig. 6
Fig. 6 Experimental results of spectral retrieval. (a,b) Spectral retrieval of a different range of wavelengths for narrow spectrum; (c,d) Comparing the results of spectral retrieval with the probes signals. (Narrow-band light sources are used to individually illuminate the sample.)
Fig. 7
Fig. 7 Experimental results of speckle-correlation imaging. (a) First row: original object. Second row: reconstruction image from speckle patterns in the condition of 1 nm FWHM. Third row: reconstruction image from speckle patterns in the condition of 16 nm FWHM. (b)–(e) Similar to (a) but for different objects. Scale bars: 130 um in the first row of (a), (b), (c) and (d), and 195 um in the second row and third row of (a), (b), (c) and (d).
Fig. 8
Fig. 8 Schematic diagram of scattering system.

Tables (2)

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Table 1 Primary Simulation Parameters

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Table 2 Quantitative Comparison for Experiment Results (Correlation)

Equations (10)

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s 0 = arg min s I Ψ s 2 .
s 0 = V D 1 U T I .
I ( u ) = O ( u ) S ( u ) .
I ( u ) I ( u ) = [ O ( u ) S ( u ) ] [ O ( u ) S ( u ) ] [ O ( u ) O ( u ) ] + C .
A ( x , y ) = I ( x , y ) I ( x , y ) = F 1 { | F { I ( x , y ) } | 2 } .
I ( u ) I ( u ) = [ O ( u ) S ( u ) ] [ O ( u ) S ( u ) ] = [ O ( u ) O ( u ) ] * [ i = 1 M S λ i ( u ) S λ i ( u ) + i = 1 M j i M S λ i ( u ) S λ j ( u ) ] .
E ( ρ I , λ ) = A E ( ρ O , λ ) e i k 2 d ( ρ T ρ O ) 2 P u p . ( ρ T , λ ) T ( ρ T , λ ) e i k 2 ( S O d ) ( ρ I ρ T ) 2 d ρ O d ρ T
I ( ρ I , λ ) = A 2 | E ( ρ O , λ ) e i k 2 d ( ρ T ρ O ) 2 P u p . ( ρ T , λ ) t ( ρ T , λ ) e i k h ( ρ T ) e i k 2 ( S O d ) ( ρ I ρ T ) 2 d ρ O d ρ T | 2 .
I ( ρ I , λ ) = A 2 | E ( ρ O , λ ) β ( ρ I , ρ T , ρ O , λ ) d ρ O d ρ T | 2 .
I ( ρ I ) = I ( ρ I , λ ) d λ .

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