Abstract

We propose and demonstrate a new concept of incoherent digital holography termed coded aperture correlation holography (COACH). In COACH, the hologram of an object is formed by the interference of light diffracted from the object, with light diffracted from the same object, but that passes through a coded phase mask (CPM). Another hologram is recorded for a point object, under identical conditions and with the same CPM. This hologram is called the point spread function (PSF) hologram. The reconstructed image is obtained by correlating the object hologram with the PSF hologram. The image reconstruction of multiplane object using COACH was compared with that of other equivalent imaging systems, and has been found to possess a higher axial resolution compared to Fresnel incoherent correlation holography.

© 2016 Optical Society of America

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Corrections

24 June 2016: A correction was made to the author listing.


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References

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  1. B. M. Oliver, “Sparkling spots and random diffraction,” Proc. IEEE Lett. 51(1), 220–221 (1963).
  2. P. S. Considine, “Effects of Coherence on Imaging Systems,” J. Opt. Soc. Am. 56(8), 1001–1009 (1966).
    [Crossref]
  3. J. P. Mills and B. J. Thompson, “Effect of aberrations and apodization on the performance of coherent optical systems. II. Imaging,” J. Opt. Soc. Am. A 3(5), 704–716 (1986).
    [Crossref]
  4. J. B. Pawley, Handbook of biological and confocal microscopy, (Plenum, 1990) Chap. 1.
  5. M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198(2), 82–87 (2000).
    [Crossref] [PubMed]
  6. M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Sevenfold improvement of axial resolution in 3D wide-field microscopy using two objective lenses,” Proc. SPIE 2412, 147–156 (1995).
    [Crossref]
  7. G. Indebetouw and P. Klysubun, “Imaging through scattering media with depth resolution by use of low-coherence gating in spatiotemporal digital holography,” Opt. Lett. 25(4), 212–214 (2000).
    [Crossref] [PubMed]
  8. O. Mudanyali, D. Tseng, C. Oh, S. O. Isikman, I. Sencan, W. Bishara, C. Oztoprak, S. Seo, B. Khademhosseini, and A. Ozcan, “Compact, light-weight and cost-effective microscope based on lensless incoherent holography for telemedicine applications,” Lab Chip 10(11), 1417–1428 (2010).
    [Crossref] [PubMed]
  9. J. Rosen and G. Brooker, “Digital spatially incoherent Fresnel holography,” Opt. Lett. 32(8), 912–914 (2007).
    [Crossref] [PubMed]
  10. J. Rosen and G. Brooker, “Fluorescence incoherent color holography,” Opt. Express 15(5), 2244–2250 (2007).
    [Crossref] [PubMed]
  11. J. Rosen, N. Siegel, and G. Brooker, “Theoretical and experimental demonstration of resolution beyond the Rayleigh limit by FINCH fluorescence microscopic imaging,” Opt. Express 19(27), 26249–26268 (2011).
    [Crossref] [PubMed]
  12. J. Rosen and R. Kelner, “Modified Lagrange invariants and their role in determining transverse and axial imaging resolutions of self-interference incoherent holographic systems,” Opt. Express 22(23), 29048–29066 (2014).
    [Crossref] [PubMed]
  13. R. Kelner, B. Katz, and J. Rosen, “Optical sectioning using a digital Fresnel incoherent-holography-based confocal imaging system,” Optica 1(2), 70–74 (2014).
    [Crossref] [PubMed]
  14. R. Kelner and J. Rosen, “Parallel-mode scanning optical sectioning using digital Fresnel holography with three-wave interference phase-shifting,” Opt. Express 24(3), 2200–2214 (2016).
    [Crossref] [PubMed]
  15. B. Javidi and T. Nomura, “Securing information by use of digital holography,” Opt. Lett. 25(1), 28–30 (2000).
    [Crossref] [PubMed]
  16. E. Tajahuerce and B. Javidi, “Encrypting three-dimensional information with digital holography,” Appl. Opt. 39(35), 6595–6601 (2000).
    [Crossref] [PubMed]
  17. 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(25), 25113–25124 (2011).
    [Crossref] [PubMed]
  18. A. Jesacher, W. Harm, S. Bernet, and M. Ritsch-Marte, “Quantitative single-shot imaging of complex objects using phase retrieval with a designed periphery,” Opt. Express 20(5), 5470–5480 (2012).
    [Crossref] [PubMed]
  19. T. Oshima, Y. Matsudo, T. Kakue, D. Arai, T. Shimobaba, and T. Ito, “Twin-image reduction method for in-line digital holography using periphery and random reference phase-shifting techniques,” Opt. Commun. 350, 270–275 (2015).
    [Crossref]
  20. D. J. Goldstein, Understanding the Light Microscope: A Computer Aided Introduction (Academic, 1999) Chap. 1.
  21. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35(2), 227–246 (1972).
  22. G. Z. Yang, B. Z. Dong, B. Y. Gu, J. Y. Zhuang, and O. K. Ersoy, “Gerchberg-Saxton and Yang-Gu algorithms for phase retrieval in a nonunitary transform system: a comparison,” Appl. Opt. 33(2), 209–218 (1994).
    [Crossref] [PubMed]
  23. G. Brooker, N. Siegel, V. Wang, and J. Rosen, “Optimal resolution in Fresnel incoherent correlation holographic fluorescence microscopy,” Opt. Express 19(6), 5047–5062 (2011).
    [Crossref] [PubMed]
  24. M. K. Kim, “Adaptive optics by incoherent digital holography,” Opt. Lett. 37(13), 2694–2696 (2012).
    [Crossref] [PubMed]
  25. P. Bouchal, J. Kapitán, R. Chmelík, and Z. Bouchal, “Point spread function and two-point resolution in Fresnel incoherent correlation holography,” Opt. Express 19(16), 15603–15620 (2011).
    [Crossref] [PubMed]
  26. X. Lai, S. Xiao, Y. Guo, X. Lv, and S. Zeng, “Experimentally exploiting the violation of the Lagrange invariant for resolution improvement,” Opt. Express 23(24), 31408–31418 (2015).
    [Crossref] [PubMed]
  27. M. K. Kim, “Full color natural light holographic camera,” Opt. Express 21(8), 9636–9642 (2013).
    [Crossref] [PubMed]
  28. R. Kelner, J. Rosen, and G. Brooker, “Enhanced resolution in Fourier incoherent single channel holography (FISCH) with reduced optical path difference,” Opt. Express 21(17), 20131–20144 (2013).
    [Crossref] [PubMed]

2016 (1)

2015 (2)

T. Oshima, Y. Matsudo, T. Kakue, D. Arai, T. Shimobaba, and T. Ito, “Twin-image reduction method for in-line digital holography using periphery and random reference phase-shifting techniques,” Opt. Commun. 350, 270–275 (2015).
[Crossref]

X. Lai, S. Xiao, Y. Guo, X. Lv, and S. Zeng, “Experimentally exploiting the violation of the Lagrange invariant for resolution improvement,” Opt. Express 23(24), 31408–31418 (2015).
[Crossref] [PubMed]

2014 (2)

2013 (2)

2012 (2)

2011 (4)

2010 (1)

O. Mudanyali, D. Tseng, C. Oh, S. O. Isikman, I. Sencan, W. Bishara, C. Oztoprak, S. Seo, B. Khademhosseini, and A. Ozcan, “Compact, light-weight and cost-effective microscope based on lensless incoherent holography for telemedicine applications,” Lab Chip 10(11), 1417–1428 (2010).
[Crossref] [PubMed]

2007 (2)

2000 (4)

1995 (1)

M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Sevenfold improvement of axial resolution in 3D wide-field microscopy using two objective lenses,” Proc. SPIE 2412, 147–156 (1995).
[Crossref]

1994 (1)

1986 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35(2), 227–246 (1972).

1966 (1)

Agard, D. A.

M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Sevenfold improvement of axial resolution in 3D wide-field microscopy using two objective lenses,” Proc. SPIE 2412, 147–156 (1995).
[Crossref]

Arai, D.

T. Oshima, Y. Matsudo, T. Kakue, D. Arai, T. Shimobaba, and T. Ito, “Twin-image reduction method for in-line digital holography using periphery and random reference phase-shifting techniques,” Opt. Commun. 350, 270–275 (2015).
[Crossref]

Bernet, S.

Bishara, W.

O. Mudanyali, D. Tseng, C. Oh, S. O. Isikman, I. Sencan, W. Bishara, C. Oztoprak, S. Seo, B. Khademhosseini, and A. Ozcan, “Compact, light-weight and cost-effective microscope based on lensless incoherent holography for telemedicine applications,” Lab Chip 10(11), 1417–1428 (2010).
[Crossref] [PubMed]

Bouchal, P.

Bouchal, Z.

Brooker, G.

Chmelík, R.

Considine, P. S.

Dong, B. Z.

Ersoy, O. K.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35(2), 227–246 (1972).

Gu, B. Y.

Guo, Y.

Gustafsson, M. G. L.

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198(2), 82–87 (2000).
[Crossref] [PubMed]

M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Sevenfold improvement of axial resolution in 3D wide-field microscopy using two objective lenses,” Proc. SPIE 2412, 147–156 (1995).
[Crossref]

Harm, W.

Indebetouw, G.

Isikman, S. O.

O. Mudanyali, D. Tseng, C. Oh, S. O. Isikman, I. Sencan, W. Bishara, C. Oztoprak, S. Seo, B. Khademhosseini, and A. Ozcan, “Compact, light-weight and cost-effective microscope based on lensless incoherent holography for telemedicine applications,” Lab Chip 10(11), 1417–1428 (2010).
[Crossref] [PubMed]

Ito, T.

T. Oshima, Y. Matsudo, T. Kakue, D. Arai, T. Shimobaba, and T. Ito, “Twin-image reduction method for in-line digital holography using periphery and random reference phase-shifting techniques,” Opt. Commun. 350, 270–275 (2015).
[Crossref]

Javidi, B.

Jesacher, A.

Kakue, T.

T. Oshima, Y. Matsudo, T. Kakue, D. Arai, T. Shimobaba, and T. Ito, “Twin-image reduction method for in-line digital holography using periphery and random reference phase-shifting techniques,” Opt. Commun. 350, 270–275 (2015).
[Crossref]

Kapitán, J.

Katz, B.

Kelner, R.

Khademhosseini, B.

O. Mudanyali, D. Tseng, C. Oh, S. O. Isikman, I. Sencan, W. Bishara, C. Oztoprak, S. Seo, B. Khademhosseini, and A. Ozcan, “Compact, light-weight and cost-effective microscope based on lensless incoherent holography for telemedicine applications,” Lab Chip 10(11), 1417–1428 (2010).
[Crossref] [PubMed]

Kim, M. K.

Klysubun, P.

Lai, X.

Lv, X.

Matsudo, Y.

T. Oshima, Y. Matsudo, T. Kakue, D. Arai, T. Shimobaba, and T. Ito, “Twin-image reduction method for in-line digital holography using periphery and random reference phase-shifting techniques,” Opt. Commun. 350, 270–275 (2015).
[Crossref]

Mills, J. P.

Mudanyali, O.

O. Mudanyali, D. Tseng, C. Oh, S. O. Isikman, I. Sencan, W. Bishara, C. Oztoprak, S. Seo, B. Khademhosseini, and A. Ozcan, “Compact, light-weight and cost-effective microscope based on lensless incoherent holography for telemedicine applications,” Lab Chip 10(11), 1417–1428 (2010).
[Crossref] [PubMed]

Nomura, T.

Oh, C.

O. Mudanyali, D. Tseng, C. Oh, S. O. Isikman, I. Sencan, W. Bishara, C. Oztoprak, S. Seo, B. Khademhosseini, and A. Ozcan, “Compact, light-weight and cost-effective microscope based on lensless incoherent holography for telemedicine applications,” Lab Chip 10(11), 1417–1428 (2010).
[Crossref] [PubMed]

Oliver, B. M.

B. M. Oliver, “Sparkling spots and random diffraction,” Proc. IEEE Lett. 51(1), 220–221 (1963).

Oshima, T.

T. Oshima, Y. Matsudo, T. Kakue, D. Arai, T. Shimobaba, and T. Ito, “Twin-image reduction method for in-line digital holography using periphery and random reference phase-shifting techniques,” Opt. Commun. 350, 270–275 (2015).
[Crossref]

Ozcan, A.

O. Mudanyali, D. Tseng, C. Oh, S. O. Isikman, I. Sencan, W. Bishara, C. Oztoprak, S. Seo, B. Khademhosseini, and A. Ozcan, “Compact, light-weight and cost-effective microscope based on lensless incoherent holography for telemedicine applications,” Lab Chip 10(11), 1417–1428 (2010).
[Crossref] [PubMed]

Oztoprak, C.

O. Mudanyali, D. Tseng, C. Oh, S. O. Isikman, I. Sencan, W. Bishara, C. Oztoprak, S. Seo, B. Khademhosseini, and A. Ozcan, “Compact, light-weight and cost-effective microscope based on lensless incoherent holography for telemedicine applications,” Lab Chip 10(11), 1417–1428 (2010).
[Crossref] [PubMed]

Ritsch-Marte, M.

Rosen, J.

R. Kelner and J. Rosen, “Parallel-mode scanning optical sectioning using digital Fresnel holography with three-wave interference phase-shifting,” Opt. Express 24(3), 2200–2214 (2016).
[Crossref] [PubMed]

R. Kelner, B. Katz, and J. Rosen, “Optical sectioning using a digital Fresnel incoherent-holography-based confocal imaging system,” Optica 1(2), 70–74 (2014).
[Crossref] [PubMed]

J. Rosen and R. Kelner, “Modified Lagrange invariants and their role in determining transverse and axial imaging resolutions of self-interference incoherent holographic systems,” Opt. Express 22(23), 29048–29066 (2014).
[Crossref] [PubMed]

R. Kelner, J. Rosen, and G. Brooker, “Enhanced resolution in Fourier incoherent single channel holography (FISCH) with reduced optical path difference,” Opt. Express 21(17), 20131–20144 (2013).
[Crossref] [PubMed]

J. Rosen, N. Siegel, and G. Brooker, “Theoretical and experimental demonstration of resolution beyond the Rayleigh limit by FINCH fluorescence microscopic imaging,” Opt. Express 19(27), 26249–26268 (2011).
[Crossref] [PubMed]

G. Brooker, N. Siegel, V. Wang, and J. Rosen, “Optimal resolution in Fresnel incoherent correlation holographic fluorescence microscopy,” Opt. Express 19(6), 5047–5062 (2011).
[Crossref] [PubMed]

J. Rosen and G. Brooker, “Fluorescence incoherent color holography,” Opt. Express 15(5), 2244–2250 (2007).
[Crossref] [PubMed]

J. Rosen and G. Brooker, “Digital spatially incoherent Fresnel holography,” Opt. Lett. 32(8), 912–914 (2007).
[Crossref] [PubMed]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35(2), 227–246 (1972).

Sedat, J. W.

M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Sevenfold improvement of axial resolution in 3D wide-field microscopy using two objective lenses,” Proc. SPIE 2412, 147–156 (1995).
[Crossref]

Sencan, I.

O. Mudanyali, D. Tseng, C. Oh, S. O. Isikman, I. Sencan, W. Bishara, C. Oztoprak, S. Seo, B. Khademhosseini, and A. Ozcan, “Compact, light-weight and cost-effective microscope based on lensless incoherent holography for telemedicine applications,” Lab Chip 10(11), 1417–1428 (2010).
[Crossref] [PubMed]

Seo, S.

O. Mudanyali, D. Tseng, C. Oh, S. O. Isikman, I. Sencan, W. Bishara, C. Oztoprak, S. Seo, B. Khademhosseini, and A. Ozcan, “Compact, light-weight and cost-effective microscope based on lensless incoherent holography for telemedicine applications,” Lab Chip 10(11), 1417–1428 (2010).
[Crossref] [PubMed]

Shimobaba, T.

T. Oshima, Y. Matsudo, T. Kakue, D. Arai, T. Shimobaba, and T. Ito, “Twin-image reduction method for in-line digital holography using periphery and random reference phase-shifting techniques,” Opt. Commun. 350, 270–275 (2015).
[Crossref]

Siegel, N.

Tajahuerce, E.

Thompson, B. J.

Tseng, D.

O. Mudanyali, D. Tseng, C. Oh, S. O. Isikman, I. Sencan, W. Bishara, C. Oztoprak, S. Seo, B. Khademhosseini, and A. Ozcan, “Compact, light-weight and cost-effective microscope based on lensless incoherent holography for telemedicine applications,” Lab Chip 10(11), 1417–1428 (2010).
[Crossref] [PubMed]

Wang, V.

Xiao, S.

Yang, G. Z.

Zeng, S.

Zhuang, J. Y.

Appl. Opt. (2)

J. Microsc. (1)

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198(2), 82–87 (2000).
[Crossref] [PubMed]

J. Opt. Soc. Am. (1)

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

Lab Chip (1)

O. Mudanyali, D. Tseng, C. Oh, S. O. Isikman, I. Sencan, W. Bishara, C. Oztoprak, S. Seo, B. Khademhosseini, and A. Ozcan, “Compact, light-weight and cost-effective microscope based on lensless incoherent holography for telemedicine applications,” Lab Chip 10(11), 1417–1428 (2010).
[Crossref] [PubMed]

Opt. Commun. (1)

T. Oshima, Y. Matsudo, T. Kakue, D. Arai, T. Shimobaba, and T. Ito, “Twin-image reduction method for in-line digital holography using periphery and random reference phase-shifting techniques,” Opt. Commun. 350, 270–275 (2015).
[Crossref]

Opt. Express (11)

G. Brooker, N. Siegel, V. Wang, and J. Rosen, “Optimal resolution in Fresnel incoherent correlation holographic fluorescence microscopy,” Opt. Express 19(6), 5047–5062 (2011).
[Crossref] [PubMed]

P. Bouchal, J. Kapitán, R. Chmelík, and Z. Bouchal, “Point spread function and two-point resolution in Fresnel incoherent correlation holography,” Opt. Express 19(16), 15603–15620 (2011).
[Crossref] [PubMed]

X. Lai, S. Xiao, Y. Guo, X. Lv, and S. Zeng, “Experimentally exploiting the violation of the Lagrange invariant for resolution improvement,” Opt. Express 23(24), 31408–31418 (2015).
[Crossref] [PubMed]

M. K. Kim, “Full color natural light holographic camera,” Opt. Express 21(8), 9636–9642 (2013).
[Crossref] [PubMed]

R. Kelner, J. Rosen, and G. Brooker, “Enhanced resolution in Fourier incoherent single channel holography (FISCH) with reduced optical path difference,” Opt. Express 21(17), 20131–20144 (2013).
[Crossref] [PubMed]

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(25), 25113–25124 (2011).
[Crossref] [PubMed]

A. Jesacher, W. Harm, S. Bernet, and M. Ritsch-Marte, “Quantitative single-shot imaging of complex objects using phase retrieval with a designed periphery,” Opt. Express 20(5), 5470–5480 (2012).
[Crossref] [PubMed]

R. Kelner and J. Rosen, “Parallel-mode scanning optical sectioning using digital Fresnel holography with three-wave interference phase-shifting,” Opt. Express 24(3), 2200–2214 (2016).
[Crossref] [PubMed]

J. Rosen and G. Brooker, “Fluorescence incoherent color holography,” Opt. Express 15(5), 2244–2250 (2007).
[Crossref] [PubMed]

J. Rosen, N. Siegel, and G. Brooker, “Theoretical and experimental demonstration of resolution beyond the Rayleigh limit by FINCH fluorescence microscopic imaging,” Opt. Express 19(27), 26249–26268 (2011).
[Crossref] [PubMed]

J. Rosen and R. Kelner, “Modified Lagrange invariants and their role in determining transverse and axial imaging resolutions of self-interference incoherent holographic systems,” Opt. Express 22(23), 29048–29066 (2014).
[Crossref] [PubMed]

Opt. Lett. (4)

Optica (1)

Optik (Stuttg.) (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35(2), 227–246 (1972).

Proc. SPIE (1)

M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Sevenfold improvement of axial resolution in 3D wide-field microscopy using two objective lenses,” Proc. SPIE 2412, 147–156 (1995).
[Crossref]

Other (3)

J. B. Pawley, Handbook of biological and confocal microscopy, (Plenum, 1990) Chap. 1.

B. M. Oliver, “Sparkling spots and random diffraction,” Proc. IEEE Lett. 51(1), 220–221 (1963).

D. J. Goldstein, Understanding the Light Microscope: A Computer Aided Introduction (Academic, 1999) Chap. 1.

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Figures (7)

Fig. 1
Fig. 1 COACH configuration for recording object and PSF holograms.
Fig. 2
Fig. 2 Experimental setup of COACH with two illumination channels.
Fig. 3
Fig. 3 (a)–(c) PSF holograms with phase shift values of 0°, 120° and 240°, respectively; (d)–(f) object holograms with phase shift values of 0°, 120° and 240°, respectively; (g) and (h) magnitude of the complex hologram of the pinhole and the NBS chart; (i) phase image of the CPM; (j) and (k) phase of the complex hologram of the pinhole and the NBS chart; and (l) result of the reconstruction by correlating the complex PSF with the object hologram.
Fig. 4
Fig. 4 Normalized intensity of reconstruction/imaging at (x = 0,y = 0) versus the axial distance of the pinhole from the front focal plane of lens L2.
Fig. 5
Fig. 5 Experimental comparison results of regular imaging and reconstruction of the FINCH and COACH holograms at plane 1 (NBS chart) and plane 2 (USAF chart) of channels 1 and 2 respectively, when the location (Δd) of the USAF chart relative to NBS chart was varied from 3 cm to 3 cm in steps of 1 cm.
Fig. 6
Fig. 6 Average cross-sections of the 7.1 lp/mm gratings calculated based on regular imaging and on the reconstructions of FINCH and COACH acquired holograms.
Fig. 7
Fig. 7 Experimental results presenting reconstructions of two COACH holograms each recorded with a different CPM. For each CPM, a corresponding PSF hologram was also recorded. The reconstructions were performed by correlating each hologram with each of the two PSFs holograms. Only the reconstruction of a hologram with its corresponding PSF, both originated from the same CPM, yields a proper image of the recorded object.

Equations (6)

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

I k ( u , v ) = | A + G ( u , v ) exp ( i θ k ) | 2 , k = 1 , 2 , 3 ,
t ( x , y ) = j a j δ ( x x j , y y j ) .
I k ( u , v ) = j | A j exp [ i 2 π ( x j u + y j v ) λ f o ] + exp ( i θ k ) B j exp [ i 2 π ( x j u + y j v ) λ f o ] G ( u u j , v v j ) | 2 .
H ( u , v ) = I 1 ( u , v ) [ exp ( i θ 3 ) exp ( i θ 2 ) ] + I 2 ( u , v ) [ exp ( i θ 1 ) exp ( i θ 3 ) ] + I 3 ( u , v ) [ exp ( i θ 2 ) exp ( i θ 1 ) ] .
H ( u , v ) = j A j * B j G ( u u j , v v j ) .
P ( u ' , v ' ) = { j A j * B j G ( u u j , v v j ) } G * ( u u ' , v v ' ) d u d v j A j * B j δ ( u ' u j , v ' v j ) t ( u ' / M T , v ' / M T ) ,

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