Abstract

Objects imaged through thin scattering media can be reconstructed with the knowledge of the complex transmission function of the diffuser. We demonstrate image reconstruction of static and dynamic objects with numerical phase conjugation in a lensless setup. Data is acquired by single shot intensity capture of an object coherently illuminated and obscured by an inhomogeneous medium, i.e. light diffracted at a specimen is scattered by a polycarbonate diffuser and the resulting speckle field is recorded. As a preparational step, which has to be performed only one time before imaging, the complex speckle field diffracted by the diffuser to the camera chip is measured interferometrically, which allows to reconstruct the transmission function of the diffuser. After insertion of the specimen, the speckle field in the camera plane changes, and the complex field of the sample can be reconstructed from the new intensity distribution. After initial interferometric measurement of the diffuser field, the method is robust with respect to a subsequent misalignment of the diffuser. The method can be extended to image objects placed between a pair of thin scattering plates. Since the object information is contained in a single speckle intensity pattern, it is possible to image dynamic processes at video rate.

© 2014 Optical Society of America

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References

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  1. E. N. Leith and J. Upatnieks, “Holographic imagery through diffusing media,” J. Opt. Soc. Am. 56, 523 (1966).
    [Crossref]
  2. J. W. Goodman, W. H. Huntley, D. W. Jackson, and M. Lehmann, “Wavefront reconstruction imaging through random media,” Appl. Phys. Lett. 8, 311–313 (1966).
    [Crossref]
  3. H. Kogelnik and K. S. Pennington, “Holographic imaging through a random medium,” J. Opt. Soc. Am. 58(2), 273–274 (1968).
    [Crossref]
  4. I. Freund, “Looking through walls and around corners,” Physica A 168, 49–65 (1990).
    [Crossref]
  5. R. T. Tyson, Principles of Adaptive Optics (Academic, 2010).
    [Crossref]
  6. I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32(16), 2309–2311 (2007).
    [Crossref] [PubMed]
  7. I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photon. 4, 320–322 (2010).
    [Crossref]
  8. 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. Photon. 7, 919–924 (2013).
    [Crossref]
  9. E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett. 106, 193905 (2011).
    [Crossref] [PubMed]
  10. I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
    [Crossref] [PubMed]
  11. O. Katz, E. Small, and Y. Silberberg, “Looking around corners and through thin turbid layers in real time with scattered incoherent light,” Nat. Photon. 6, 549–553 (2012).
    [Crossref]
  12. 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]
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    [Crossref] [PubMed]
  14. S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
    [Crossref] [PubMed]
  15. 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]
  16. A. Kumar 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, 7694–7701 (2014).
    [Crossref]
  17. S. Li and J. Zhong, “Dynamic imaging through turbid media based on digital holography,” J. Opt. Soc. Am. A 31, 480–486 (2014).
    [Crossref]
  18. B. C. Kress and P. Meyrueis, Applied Digital Optics (Wiley, 2009).
    [Crossref]
  19. D. Malacara, Optical Shop Testing (Wiley, 2007).
    [Crossref]
  20. S. Bernet, W. Harm, A. Jesacher, and M. Ritsch-Marte, “Lensless digital holography with diffuse illumination through a pseudo-random phase mask,” Opt. Express 19, 25113–25124 (2011).
    [Crossref]

2014 (2)

2013 (1)

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. Photon. 7, 919–924 (2013).
[Crossref]

2012 (3)

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, “Looking around corners and through thin turbid layers in real time with scattered incoherent light,” Nat. Photon. 6, 549–553 (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]

2011 (2)

S. Bernet, W. Harm, A. Jesacher, and M. Ritsch-Marte, “Lensless digital holography with diffuse illumination through a pseudo-random phase mask,” Opt. Express 19, 25113–25124 (2011).
[Crossref]

E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett. 106, 193905 (2011).
[Crossref] [PubMed]

2010 (3)

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

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

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

2007 (1)

1990 (1)

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

1988 (1)

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

1968 (1)

1966 (2)

E. N. Leith and J. Upatnieks, “Holographic imagery through diffusing media,” J. Opt. Soc. Am. 56, 523 (1966).
[Crossref]

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

Akbulut, D.

E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett. 106, 193905 (2011).
[Crossref] [PubMed]

Bernet, S.

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]

E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett. 106, 193905 (2011).
[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.

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

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. Photon. 7, 919–924 (2013).
[Crossref]

Caravaca-Aguirre, A. M.

Carminati, R.

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

Conkey, D. B.

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. Photon. 7, 919–924 (2013).
[Crossref]

Feng, S.

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

Fink, M.

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

Freund, I.

I. Freund, “Looking through walls and around corners,” Physica A 168, 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, 2328–2331 (1988).
[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. Photon. 7, 919–924 (2013).
[Crossref]

Gigan, S.

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

Goodman, J. W.

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

Grange, R.

Harm, W.

Hsieh, C.

Huntley, W. H.

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

Jackson, D. W.

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

Jesacher, A.

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. Photon. 7, 919–924 (2013).
[Crossref]

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

Kogelnik, H.

Kress, B. C.

B. C. Kress and P. Meyrueis, Applied Digital Optics (Wiley, 2009).
[Crossref]

Kumar Singh, A.

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]

E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett. 106, 193905 (2011).
[Crossref] [PubMed]

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

Laporte, G.

Lehmann, M.

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

Leith, E. N.

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

Li, S.

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley, 2007).
[Crossref]

Meyrueis, P.

B. C. Kress and P. Meyrueis, Applied Digital Optics (Wiley, 2009).
[Crossref]

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]

E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett. 106, 193905 (2011).
[Crossref] [PubMed]

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

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

Naik, D. N.

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. Photon. 7, 919–924 (2013).
[Crossref]

Osten, W.

Pedrini, G.

Pennington, K. S.

Piestun, R.

Popoff, S. M.

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

Psaltis, D.

Pu, Y.

Ritsch-Marte, M.

Rosenbluh, M.

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

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. Photon. 7, 919–924 (2013).
[Crossref]

O. Katz, E. Small, and Y. Silberberg, “Looking around corners and through thin turbid layers in real time with scattered incoherent light,” Nat. Photon. 6, 549–553 (2012).
[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. Photon. 7, 919–924 (2013).
[Crossref]

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

Takeda, M.

Tyson, R. T.

R. T. Tyson, Principles of Adaptive Optics (Academic, 2010).
[Crossref]

Upatnieks, J.

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]

E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett. 106, 193905 (2011).
[Crossref] [PubMed]

Vellekoop, I. M.

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

I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32(16), 2309–2311 (2007).
[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 491(7423), 232–234 (2012).
[Crossref] [PubMed]

E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett. 106, 193905 (2011).
[Crossref] [PubMed]

Zhong, J.

Appl. Phys. Lett. (1)

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

J. Opt. Soc. Am. (2)

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

Nat. Photon. (3)

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

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photon. 4, 320–322 (2010).
[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. Photon. 7, 919–924 (2013).
[Crossref]

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 (4)

Opt. Lett. (1)

Phys. Rev. Lett. (3)

E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett. 106, 193905 (2011).
[Crossref] [PubMed]

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 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, 100601 (2010).
[Crossref] [PubMed]

Physica A (1)

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

Other (3)

R. T. Tyson, Principles of Adaptive Optics (Academic, 2010).
[Crossref]

B. C. Kress and P. Meyrueis, Applied Digital Optics (Wiley, 2009).
[Crossref]

D. Malacara, Optical Shop Testing (Wiley, 2007).
[Crossref]

Supplementary Material (2)

» Media 1: MP4 (9189 KB)     
» Media 2: MP4 (7826 KB)     

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Figures (6)

Fig. 1
Fig. 1 A monochromatic plane wave is partially scattered by an object and is incident on a thin diffusive medium. The resulting speckle pattern is captured on a digital sensor. The transverse planes relevant for object reconstruction are distinguished with primes and double primes. Note that U′(x, y) corresponds to the field at the right side of the diffuser.
Fig. 2
Fig. 2 (a) Illustration of the experimental configuration for lensless imaging through a diffuse medium. The vertical arrow indicates the reference wave of a Mach-Zehnder interferometer recombined with a beam splitter (BS) for measuring the diffuser’s refractive index distribution. (b) Conventional photography (Canon EOS 550d) of an insect placed 2 cm behind a polycarbonate diffuser with white LED illumination.
Fig. 3
Fig. 3 Comparison of the proposed method to digital inline holography. (a) Inline holography (inverted contrast) without diffuser inserted. (b) Inline holography with diffuser inserted. (c) Object reconstruction by numerical reversal of the scattering process with diffuser inserted. (d) Image reconstruction in a 12 mm out of focus plane (see Media 1 for numerical refocusing of the imaging volume from the diffuser plane to the object plane with a dynamic sample). (e) Amplitude profiles along the colored blue, red, and green lines in (a), (c), and (d), respectively.
Fig. 4
Fig. 4 Reconstructed amplitude distributions (a) before misalignment of the diffuser, (b) after misalignment (and without numerical compensation) of 10 μm, (c) with numerical compensation of the misalignment, (d) with numerical compensation of a 0.2 mm shift. In this case the background subtraction is not performed, resulting in a bright-field image.
Fig. 5
Fig. 5 (a) Illustration of the experimental configuration for lensless imaging of an insect placed between two diffusive media. The vertical arrow indicates the reference wave of a Mach-Zehner interferometer for measuring the diffuser’s refractive index distributions (b) Conventional photography of an insect placed between a pair of polycarbonate diffusers with white LED illumination.
Fig. 6
Fig. 6 Amplitude distributions with lensless imaging of an insect placed between a pair of polycarbonate diffusers. (a) Inline holography. (b) Object reconstruction by numerical reversal of the scattering processes. In (c) artefacts from the diffusers are reduced at the expense of a less suppressed background. Media 2 shows a visualization of the reconstruction routine and background normalized amplitude images of a dynamic sample.

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

A obj exp ( i ϕ obj ) = 𝒫 obj d { A obj exp ( i ϕ obj ) } exp ( i ϕ d ) ,
A obj exp ( i ϕ obj ) = 𝒫 d s { A obj exp ( i ϕ obj ) } ,
A obj exp ( i ϕ d ) = 𝒫 d s { exp ( i ϕ d ) } ,
I CCD | A obj exp [ i ϕ obj ] + A d exp [ i ϕ d ] | 2 = A d 2 + A obj 2 + A d A obj { exp [ i ( ϕ d ϕ obj ) ] + exp [ i ( ϕ d ϕ obj ) ] } .
A obj exp ( i ϕ obj ) I CCD A d 2 A d exp ( i ϕ d ) A obj exp [ i ( 2 ϕ d ϕ obj ) ] Noise .
A obj exp ( i ϕ obj ) = 𝒫 s d { A obj exp ( i ϕ obj ) } .
ϕ d = arg { 𝒫 s d [ A d exp ( i ϕ d ) ] } .
A obj exp ( i ϕ obj ) = 𝒫 d obj { A obj exp ( i ϕ obj ) exp ( i ϕ d ) } ,
A obj exp ( i ϕ obj ) = I CCD A d 2 2 A d 2 exp ( i ϕ d 2 ) .
A obj exp ( i ϕ obj ) = 𝒫 d obj { A obj exp ( i ϕ obj ) } ,
A obj exp ( i ϕ obj ) = 𝒫 d obj { A obj exp ( i ϕ obj ) exp ( i ϕ d 1 ) } .
C ( x c , y c ) = I ref ( x , y ) I shift ( x + x c , y + y c ) d x d y .
A obj exp ( i ϕ obj ) = I CCD exp ( i ϕ d ) .

Metrics