Abstract

Recently introduced angular-memory-effect based techniques enable non-invasive imaging of objects hidden behind thin scattering layers. However, both the speckle-correlation and the bispectrum analysis are based on the statistical average of large amounts of speckle grains, which determines that they can hardly access the important information of the point-spread-function (PSF) of a highly scattering imaging system. Here, inspired by notions used in astronomy, we present a phase-diversity speckle imaging scheme, based on recording a sequence of intensity speckle patterns at various imaging planes, and experimentally demonstrate that in addition to being able to retrieve the image of hidden objects, we can also simultaneously estimate the pupil function and the PSF of a highly scattering imaging system without any guide-star nor reference.

© 2017 Optical Society of America

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References

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2017 (2)

S. Rotter and S. Gigan, “Light fields in complex media: Mesoscopic scattering meets wave control,” Rev. Mod. Phys. 89, 015005 (2017).

A. K. Singh, D. N. Naik, G. Pedrini, M. Takeda, and W. Osten, “Exploiting scattering media for exploring 3D objects,” Light Sci. Appl. 6, e16219 (2017).

2016 (2)

T. Wu, O. Katz, X. Shao, and S. Gigan, “Single-shot diffraction-limited imaging through scattering layers via bispectrum analysis,” Opt. Lett. 41(21), 5003–5006 (2016).
[PubMed]

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

2015 (4)

J. Yoon, K. Lee, J. Park, and Y. Park, “Measuring optical transmission matrices by wavefront shaping,” Opt. Express 23(8), 10158–10167 (2015).
[PubMed]

I. M. Vellekoop, “Feedback-based wavefront shaping,” Opt. Express 23(9), 12189–12206 (2015).
[PubMed]

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).
[PubMed]

R. Horstmeyer, H. Ruan, and C. Yang, “Guidestar-assisted wavefront-shaping methods for focusing light into biological tissue,” Nat. Photonics 9, 563–571 (2015).
[PubMed]

2014 (2)

T. Chaigne, J. Gateau, O. Katz, E. Bossy, and S. Gigan, “Light focusing and two-dimensional imaging through scattering media using the photoacoustic transmission matrix with an ultrasound array,” Opt. Lett. 39(9), 2664–2667 (2014).
[PubMed]

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).

2012 (3)

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

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

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).
[PubMed]

2010 (2)

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

2007 (1)

1998 (1)

C. R. Vogel, T. Chan, and R. Plemmons, “Fast algorithms for phase diversity-based blind deconvolution,” Adaptive Optical System Technologies, Parts 1 and 2 3353, 994– 1005 (1998).

1993 (1)

O. Von der Lühe, “Speckle imaging of solar small scale structure I. Methods,” Astron. Astrophys. 268, 374–390 (1993).

1992 (1)

1988 (2)

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61(7), 834–837 (1988).
[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).
[PubMed]

1983 (1)

1982 (1)

Aegerter, C. M.

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).
[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).
[PubMed]

Boccara, A. C.

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

Bossy, E.

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).
[PubMed]

Chaigne, T.

Chan, T.

C. R. Vogel, T. Chan, and R. Plemmons, “Fast algorithms for phase diversity-based blind deconvolution,” Adaptive Optical System Technologies, Parts 1 and 2 3353, 994– 1005 (1998).

Edrei, E.

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

Feng, S.

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61(7), 834–837 (1988).
[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).
[PubMed]

Fienup, J. R.

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).

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).

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).
[PubMed]

Freund, I.

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

Gateau, J.

Gigan, S.

S. Rotter and S. Gigan, “Light fields in complex media: Mesoscopic scattering meets wave control,” Rev. Mod. Phys. 89, 015005 (2017).

T. Wu, O. Katz, X. Shao, and S. Gigan, “Single-shot diffraction-limited imaging through scattering layers via bispectrum analysis,” Opt. Lett. 41(21), 5003–5006 (2016).
[PubMed]

T. Chaigne, J. Gateau, O. Katz, E. Bossy, and S. Gigan, “Light focusing and two-dimensional imaging through scattering media using the photoacoustic transmission matrix with an ultrasound array,” Opt. Lett. 39(9), 2664–2667 (2014).
[PubMed]

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).

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).
[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, 784–790 (2014).

Horstmeyer, R.

R. Horstmeyer, H. Ruan, and C. Yang, “Guidestar-assisted wavefront-shaping methods for focusing light into biological tissue,” Nat. Photonics 9, 563–571 (2015).
[PubMed]

Kane, C.

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61(7), 834–837 (1988).
[PubMed]

Katz, O.

T. Wu, O. Katz, X. Shao, and S. Gigan, “Single-shot diffraction-limited imaging through scattering layers via bispectrum analysis,” Opt. Lett. 41(21), 5003–5006 (2016).
[PubMed]

T. Chaigne, J. Gateau, O. Katz, E. Bossy, and S. Gigan, “Light focusing and two-dimensional imaging through scattering media using the photoacoustic transmission matrix with an ultrasound array,” Opt. Lett. 39(9), 2664–2667 (2014).
[PubMed]

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).

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

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).

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).
[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).
[PubMed]

Lee, K.

Lee, P. A.

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61(7), 834–837 (1988).
[PubMed]

Lerosey, G.

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

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).
[PubMed]

Lohmann, A. W.

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).

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).
[PubMed]

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

Naik, D. N.

A. K. Singh, D. N. Naik, G. Pedrini, M. Takeda, and W. Osten, “Exploiting scattering media for exploring 3D objects,” Light Sci. Appl. 6, e16219 (2017).

Osten, W.

A. K. Singh, D. N. Naik, G. Pedrini, M. Takeda, and W. Osten, “Exploiting scattering media for exploring 3D objects,” Light Sci. Appl. 6, e16219 (2017).

Park, J.

Park, Y.

Paxman, R. G.

Pedrini, G.

A. K. Singh, D. N. Naik, G. Pedrini, M. Takeda, and W. Osten, “Exploiting scattering media for exploring 3D objects,” Light Sci. Appl. 6, e16219 (2017).

Plemmons, R.

C. R. Vogel, T. Chan, and R. Plemmons, “Fast algorithms for phase diversity-based blind deconvolution,” Adaptive Optical System Technologies, Parts 1 and 2 3353, 994– 1005 (1998).

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

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).
[PubMed]

Rotter, S.

S. Rotter and S. Gigan, “Light fields in complex media: Mesoscopic scattering meets wave control,” Rev. Mod. Phys. 89, 015005 (2017).

Ruan, H.

R. Horstmeyer, H. Ruan, and C. Yang, “Guidestar-assisted wavefront-shaping methods for focusing light into biological tissue,” Nat. Photonics 9, 563–571 (2015).
[PubMed]

Scarcelli, G.

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

Schulz, T. J.

Shao, X.

Silberberg, Y.

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

Singh, A. K.

A. K. Singh, D. N. Naik, G. Pedrini, M. Takeda, and W. Osten, “Exploiting scattering media for exploring 3D objects,” Light Sci. Appl. 6, e16219 (2017).

Small, E.

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

Stone, A. D.

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61(7), 834–837 (1988).
[PubMed]

Takeda, M.

A. K. Singh, D. N. Naik, G. Pedrini, M. Takeda, and W. Osten, “Exploiting scattering media for exploring 3D objects,” Light Sci. Appl. 6, e16219 (2017).

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).
[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).
[PubMed]

Vellekoop, I. M.

Vogel, C. R.

C. R. Vogel, T. Chan, and R. Plemmons, “Fast algorithms for phase diversity-based blind deconvolution,” Adaptive Optical System Technologies, Parts 1 and 2 3353, 994– 1005 (1998).

Von der Lühe, O.

O. Von der Lühe, “Speckle imaging of solar small scale structure I. Methods,” Astron. Astrophys. 268, 374–390 (1993).

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).
[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).
[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).
[PubMed]

Weigelt, G.

Wirnitzer, B.

Wu, T.

Yang, C.

R. Horstmeyer, H. Ruan, and C. Yang, “Guidestar-assisted wavefront-shaping methods for focusing light into biological tissue,” Nat. Photonics 9, 563–571 (2015).
[PubMed]

Yoon, J.

Adaptive Optical System Technologies, Parts 1 and 2 (1)

C. R. Vogel, T. Chan, and R. Plemmons, “Fast algorithms for phase diversity-based blind deconvolution,” Adaptive Optical System Technologies, Parts 1 and 2 3353, 994– 1005 (1998).

Appl. Opt. (2)

Astron. Astrophys. (1)

O. Von der Lühe, “Speckle imaging of solar small scale structure I. Methods,” Astron. Astrophys. 268, 374–390 (1993).

J. Opt. Soc. Am. A (1)

Light Sci. Appl. (1)

A. K. Singh, D. N. Naik, G. Pedrini, M. Takeda, and W. Osten, “Exploiting scattering media for exploring 3D objects,” Light Sci. Appl. 6, e16219 (2017).

Nat. Photonics (5)

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).

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).

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

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).
[PubMed]

R. Horstmeyer, H. Ruan, and C. Yang, “Guidestar-assisted wavefront-shaping methods for focusing light into biological tissue,” Nat. Photonics 9, 563–571 (2015).
[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).
[PubMed]

Opt. Express (2)

Opt. Lett. (4)

Phys. Rev. Lett. (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).
[PubMed]

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61(7), 834–837 (1988).
[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).
[PubMed]

Rev. Mod. Phys. (1)

S. Rotter and S. Gigan, “Light fields in complex media: Mesoscopic scattering meets wave control,” Rev. Mod. Phys. 89, 015005 (2017).

Sci. Rep. (1)

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

Other (5)

T. Wu, “codes.zip,” figshare (2017), https://figshare.com/s/ac47b433be00ef4ef893 .

T. Wu, “data.zip,” figshare (2017), https://figshare.com/s/4969d19518a7680d07ec .

J. W. Goodman, Introduction to Fourier Optics (Roberts and Company Publishers, 2005).

J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. (Springer, 2006).

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

Supplementary Material (2)

NameDescription
» Code 1       Matlabs codes of phase-diversity
» Dataset 1       Experimental data of phase-diversity

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Figures (8)

Fig. 1
Fig. 1 Concept and numerical simulations: Jointly imaging hidden objects and estimating the PSF of a highly scattering imaging system. (a) An object is illuminated by a spatially incoherent light. The scattered light is recorded by a camera at various positions with a fixed interval δ . P0 is the initial position; (b) Simulated diversity speckle patterns, in which the selected sub-region is marked by a dashed box; (c) Retrieved image and diffraction-limited image; (d) Estimated sub-PSF and true sub-PSF (only the intensity is shown). Scale bar: 400 camera pixels in (b) and 10 camera pixels in (c) and (d).
Fig. 2
Fig. 2 Detailed information of the reconstruction of the speckle patterns in phase-diversity speckle imaging. Only a small region of speckle patterns is sufficient to reconstruct. Additional apodization function is required to smooth the edges of each sub-image. Scale bar: 10 camera pixels.
Fig. 3
Fig. 3 Estimating the sub-PSF of scattering imaging system with different objects. (a), (b) and (c) are respectively the corresponding simulated results with the objects “digit 2”, “letter H” and “double stars”. The first row shows the retrieved images and the second one is the estimated sub-PSFs. Scale bar: 10 camera pixels.
Fig. 4
Fig. 4 Experimental set-up. A light-emitting diode is used as the spatially incoherent source, and a narrow band-pass filter is mounted on the camera to ensure the high contrast of the speckle patterns. The camera is moved step-by-step with a linear translation stage. L: Lens.
Fig. 5
Fig. 5 Experimental results of phase-diversity speckle imaging. (a) Original image of hidden object, digit “2”; (b) Raw diversity camera images, in which three independent groups of sub-images are selected and reconstructed, respectively; (c) First column: estimated random local phase values, corresponding to different groups; second column: estimated sub-PSFs; third column: retrieved images of the hidden object from the three groups of speckle patterns. Scale bar: 500 camera pixels in (a) and (b); 10 camera pixels in (c).
Fig. 6
Fig. 6 Experimental results of imaging hidden object by simple deconvolution with the estimated sub-PSFs from phase-diversity speckle imaging method. (a) Original image of hidden object, digit “3”, placed at the same position as digit “2”; (b) Raw camera image. Selecting the same three areas, where the sub-PSFs are estimated in advance; (c) First column: selected three sub-images, smoothed by the Hanning window; second column: retrieved results from three sub-images by deconvolution. Middle images are the corresponding sub-PSFs estimated by phase-diversity speckle imaging method as in Fig. 5. Scale bar: 500 camera pixels in (a) and (b); 10 camera pixels in (c).
Fig. 7
Fig. 7 Deconvolution results with the sub-PSFs estimated from different groups of sub-images. (a) Sub-image of group 1; (b) Deconvolution result by using its corresponding sub-PSF; (c) and (d) are the results reconstructed with the sub-PSFs of group 2 and group 3, respectively. Scale bar: 10 camera pixels.
Fig. 8
Fig. 8 Impulse response of a highly scattering imaging system.

Equations (14)

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I n ( x ) = O ( x ) S n ( x ) + w n ( x )
H n ( f ) = | H n ( f ) | exp { i [ ϕ ( f ) + θ n ( f ) ] }
θ n ( f ) = n δ π λ v ( v + n δ ) | f | 2
L [ ϕ ( f ) ] = f [ | k = 0 N - 1 I ˜ k ( f ) S ˜ k * ( f ) | 2 j = 0 N - 1 | S ˜ j ( f ) | 2 + σ n = 0 N - 1 | I ˜ n ( f ) | 2 ]
S n ( x ) = | F 1 { H n ( f ) } | 2
O ( x ) = F 1 { n = 0 N - 1 I ˜ n ( f ) S ˜ n * ( f ) k = 0 N - 1 | S ˜ k ( f ) | 2 + σ }
U s _ f r o n t = 1 j λ z 1 exp { j k 2 z 1 [ ( x ε ) 2 + ( y η ) 2 ] }
U s _ b a c k = U s _ f r o n t P ( x , y )
P ( x , y ) = | P ( x , y ) | exp [ j φ ( x , y ) ]
h ( u , v ; ε , η ) = 1 j λ z 2 U s _ b a c k exp { j k 2 z 2 [ ( u x ) 2 + ( v y ) 2 ] } d x d y
h ( u , v ; ε , η ) = 1 λ 2 z 1 z 2 exp [ j k 2 z 1 ( ε 2 + η 2 ) ] exp [ j k 2 z 2 ( u 2 + v 2 ) ] × P ( x , y ) exp { j k 2 [ ( 1 z 1 + 1 z 2 ) ( x 2 + y 2 ) ] } × exp { j k [ x ( ε z 1 + u z 2 ) + y ( η z 1 + v z 2 ) ] } d x d y
h ( u , v ; ε , η ) H ( x , y ) exp { j k [ x ( ε z 1 + u z 2 ) + y ( η z 1 + v z 2 ) ] } d x d y
H ( x , y ) = | P ( x , y ) | exp { j k 2 [ ( 1 z 1 + 1 z 2 ) ( x 2 + y 2 ) ] + j φ ( x , y ) } = | H ( x , y ) | exp [ j ϕ ( x , y ) ]
H n ( x , y ) = | H n ( x , y ) | exp { j k 2 [ ( 1 z 1 + 1 z 2 + n Δ z ) ( x 2 + y 2 ) ] + j φ ( x , y ) } = | H n ( x , y ) | exp { j k 2 [ ( 1 z 1 + 1 z 2 n Δ z z 2 ( z 2 + n Δ z ) ) ( x 2 + y 2 ) ] + j φ ( x , y ) } = | H n ( x , y ) | exp { j k 2 [ ( 1 z 1 + 1 z 2 ) ( x 2 + y 2 ) ] + j φ ( x , y ) + j k 2 [ n Δ z z 2 ( z 2 + n Δ z ) ] ( x 2 + y 2 ) } = | H n ( x , y ) | exp { j [ ϕ ( x , y ) + θ n ( x , y ) ] }

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