Abstract

Recording digital holograms without wave interference simplifies the optical systems, increases their power efficiency and avoids complicated aligning procedures. We propose and demonstrate a new technique of digital hologram acquisition without two-wave interference. Incoherent light emitted from an object propagates through a random-like coded phase mask and recorded directly without interference by a digital camera. In the training stage of the system, a point spread hologram (PSH) is first recorded by modulating the light diffracted from a point object by the coded phase masks. At least two different masks should be used to record two different intensity distributions at all possible axial locations. The various recorded patterns at every axial location are superposed in the computer to obtain a complex valued PSH library cataloged to its axial location. Following the training stage, an object is placed within the axial boundaries of the PSH library and the light diffracted from the object is once again modulated by the same phase masks. The intensity patterns are recorded and superposed exactly as the PSH to yield a complex hologram of the object. The object information at any particular plane is reconstructed by a cross-correlation between the complex valued hologram and the appropriate element of the PSH library. The characteristics and the performance of the proposed system were compared with an equivalent regular imaging system.

© 2017 Optical Society of America

Full Article  |  PDF Article
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References

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    [Crossref] [PubMed]
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    [Crossref]
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  8. T. Leportier and M. C. Park, “Generation of binary holograms for deep scenes captured with a camera and a depth sensor,” Opt. Eng. 56(1), 013107 (2017).
    [Crossref]
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    [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. X. Lai, S. Zeng, X. Lv, J. Yuan, and L. Fu, “Violation of the Lagrange invariant in an optical imaging system,” Opt. Lett. 38(11), 1896–1898 (2013).
    [Crossref] [PubMed]
  13. 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]
  14. 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]
  15. Y. Kashter, A. Vijayakumar, Y. Miyamoto, and J. Rosen, “Enhanced super resolution using Fresnel incoherent correlation holography with structured illumination,” Opt. Lett. 41(7), 1558–1561 (2016).
    [Crossref] [PubMed]
  16. 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]
  17. 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).
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  20. A. Vijayakumar, Y. Kashter, R. Kelner, and J. Rosen, “Coded aperture correlation holography system with improved performance [Invited],” Appl. Opt. 56(13), F67–F77 (2017).
    [Crossref] [PubMed]
  21. A. Vijayakumar and J. Rosen, “Spectrum and space resolved 4D imaging by coded aperture correlation holography (COACH) with diffractive objective lens,” Opt. Lett. 42(5), 947–950 (2017).
    [Crossref] [PubMed]
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  23. A. Levin, R. Fergus, F. Durand, and W. T. Freeman, “Image and depth from a conventional camera with a coded aperture,” ACM Trans. Graph. 26(3), 70 (2007).
    [Crossref]
  24. C. Zhou, S. Lin, and S. Nayar, “Coded aperture pairs for depth from defocus,” in 2009 IEEE 12th International Conference on Computer Vision (ICCV), 325–332 (2009).
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  30. S. Jia, J. C. Vaughan, and X. Zhuang, “Isotropic three-dimensional super-resolution imaging with a self-bending point spread function,” Nat. Photonics 8(4), 302–306 (2014).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  32. C. Liu, S. Knitter, Z. Cong, I. Sencan, H. Cao, and M. A. Choma, “High-speed line-field confocal holographic microscope for quantitative phase imaging,” Opt. Express 24(9), 9251–9265 (2016).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]

2017 (4)

2016 (4)

2015 (2)

2014 (6)

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]

S. Jia, J. C. Vaughan, and X. Zhuang, “Isotropic three-dimensional super-resolution imaging with a self-bending point spread function,” Nat. Photonics 8(4), 302–306 (2014).
[Crossref] [PubMed]

Y. Kashter and J. Rosen, “Enhanced-resolution using modified configuration of Fresnel incoherent holographic recorder with synthetic aperture,” Opt. Express 22(17), 20551–20565 (2014).
[Crossref] [PubMed]

Y. Shechtman, S. J. Sahl, A. S. Backer, and W. E. Moerner, “Optimal point spread function design for 3D imaging,” Phys. Rev. Lett. 113(13), 133902 (2014).
[Crossref] [PubMed]

A. S. Backer and W. E. Moerner, “Extending single-molecule microscopy using optical Fourier processing,” J. Phys. Chem. B 118(28), 8313–8329 (2014).
[Crossref] [PubMed]

2013 (2)

2012 (3)

2011 (3)

2009 (2)

J. Rosen, G. Brooker, G. Indebetouw, and N. T. Shaked, “A review of incoherent digital Fresnel holography,” J. Hologr. Speckle 5(2), 124–140 (2009).
[Crossref]

N. T. Shaked, B. Katz, and J. Rosen, “Review of three-dimensional holographic imaging by multiple-viewpoint-projection based methods,” Appl. Opt. 48(34), H120–H136 (2009).
[Crossref] [PubMed]

2007 (2)

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

A. Levin, R. Fergus, F. Durand, and W. T. Freeman, “Image and depth from a conventional camera with a coded aperture,” ACM Trans. Graph. 26(3), 70 (2007).
[Crossref]

2006 (1)

2000 (1)

1994 (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).

Abe, R.

Backer, A. S.

Y. Shechtman, S. J. Sahl, A. S. Backer, and W. E. Moerner, “Optimal point spread function design for 3D imaging,” Phys. Rev. Lett. 113(13), 133902 (2014).
[Crossref] [PubMed]

A. S. Backer and W. E. Moerner, “Extending single-molecule microscopy using optical Fourier processing,” J. Phys. Chem. B 118(28), 8313–8329 (2014).
[Crossref] [PubMed]

Bouchal, P.

Bouchal, Z.

Brooker, G.

Cao, H.

Chi, W.

Chmelík, R.

Choi, K.-H.

Choma, M. A.

Cong, Z.

Dong, B. Z.

Durand, F.

A. Levin, R. Fergus, F. Durand, and W. T. Freeman, “Image and depth from a conventional camera with a coded aperture,” ACM Trans. Graph. 26(3), 70 (2007).
[Crossref]

Ersoy, O. K.

Fergus, R.

A. Levin, R. Fergus, F. Durand, and W. T. Freeman, “Image and depth from a conventional camera with a coded aperture,” ACM Trans. Graph. 26(3), 70 (2007).
[Crossref]

Freeman, W. T.

A. Levin, R. Fergus, F. Durand, and W. T. Freeman, “Image and depth from a conventional camera with a coded aperture,” ACM Trans. Graph. 26(3), 70 (2007).
[Crossref]

Fu, L.

George, N.

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).

Greengard, A.

Gu, B. Y.

Guo, Y.

Hatzvi, M. R.

Hayasaki, Y.

Hong, J.

Indebetouw, G.

J. Rosen, G. Brooker, G. Indebetouw, and N. T. Shaked, “A review of incoherent digital Fresnel holography,” J. Hologr. Speckle 5(2), 124–140 (2009).
[Crossref]

G. Indebetouw, P. Klysubun, T. Kim, and T.-C. Poon, “Imaging properties of scanning holographic microscopy,” J. Opt. Soc. Am. A 17(3), 380–390 (2000).
[Crossref] [PubMed]

Jia, S.

S. Jia, J. C. Vaughan, and X. Zhuang, “Isotropic three-dimensional super-resolution imaging with a self-bending point spread function,” Nat. Photonics 8(4), 302–306 (2014).
[Crossref] [PubMed]

Kapitán, J.

Kashter, Y.

Katz, B.

Kelner, R.

Kim, M. K.

Kim, T.

Klysubun, P.

Knitter, S.

Lai, X.

Leportier, T.

T. Leportier and M. C. Park, “Generation of binary holograms for deep scenes captured with a camera and a depth sensor,” Opt. Eng. 56(1), 013107 (2017).
[Crossref]

Levin, A.

A. Levin, R. Fergus, F. Durand, and W. T. Freeman, “Image and depth from a conventional camera with a coded aperture,” ACM Trans. Graph. 26(3), 70 (2007).
[Crossref]

Liu, C.

Lv, X.

Min, S.-W.

Miyamoto, Y.

Moerner, W. E.

Y. Shechtman, S. J. Sahl, A. S. Backer, and W. E. Moerner, “Optimal point spread function design for 3D imaging,” Phys. Rev. Lett. 113(13), 133902 (2014).
[Crossref] [PubMed]

A. S. Backer and W. E. Moerner, “Extending single-molecule microscopy using optical Fourier processing,” J. Phys. Chem. B 118(28), 8313–8329 (2014).
[Crossref] [PubMed]

Park, M. C.

T. Leportier and M. C. Park, “Generation of binary holograms for deep scenes captured with a camera and a depth sensor,” Opt. Eng. 56(1), 013107 (2017).
[Crossref]

Piestun, R.

Poon, T.-C.

Rosen, J.

A. Vijayakumar, Y. Kashter, R. Kelner, and J. Rosen, “Coded aperture correlation holography system with improved performance [Invited],” Appl. Opt. 56(13), F67–F77 (2017).
[Crossref] [PubMed]

A. Vijayakumar and J. Rosen, “Spectrum and space resolved 4D imaging by coded aperture correlation holography (COACH) with diffractive objective lens,” Opt. Lett. 42(5), 947–950 (2017).
[Crossref] [PubMed]

Y. Kashter, A. Vijayakumar, Y. Miyamoto, and J. Rosen, “Enhanced super resolution using Fresnel incoherent correlation holography with structured illumination,” Opt. Lett. 41(7), 1558–1561 (2016).
[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]

A. Vijayakumar, Y. Kashter, R. Kelner, and J. Rosen, “Coded aperture correlation holography-a new type of incoherent digital holograms,” Opt. Express 24(11), 12430–12441 (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]

Y. Kashter and J. Rosen, “Enhanced-resolution using modified configuration of Fresnel incoherent holographic recorder with synthetic aperture,” Opt. Express 22(17), 20551–20565 (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]

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]

N. T. Shaked, B. Katz, and J. Rosen, “Review of three-dimensional holographic imaging by multiple-viewpoint-projection based methods,” Appl. Opt. 48(34), H120–H136 (2009).
[Crossref] [PubMed]

J. Rosen, G. Brooker, G. Indebetouw, and N. T. Shaked, “A review of incoherent digital Fresnel holography,” J. Hologr. Speckle 5(2), 124–140 (2009).
[Crossref]

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

Sahl, S. J.

Y. Shechtman, S. J. Sahl, A. S. Backer, and W. E. Moerner, “Optimal point spread function design for 3D imaging,” Phys. Rev. Lett. 113(13), 133902 (2014).
[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).

Schechner, Y. Y.

Sencan, I.

Shaked, N. T.

N. T. Shaked, B. Katz, and J. Rosen, “Review of three-dimensional holographic imaging by multiple-viewpoint-projection based methods,” Appl. Opt. 48(34), H120–H136 (2009).
[Crossref] [PubMed]

J. Rosen, G. Brooker, G. Indebetouw, and N. T. Shaked, “A review of incoherent digital Fresnel holography,” J. Hologr. Speckle 5(2), 124–140 (2009).
[Crossref]

Shechtman, Y.

Y. Shechtman, S. J. Sahl, A. S. Backer, and W. E. Moerner, “Optimal point spread function design for 3D imaging,” Phys. Rev. Lett. 113(13), 133902 (2014).
[Crossref] [PubMed]

Siegel, N.

Vaughan, J. C.

S. Jia, J. C. Vaughan, and X. Zhuang, “Isotropic three-dimensional super-resolution imaging with a self-bending point spread function,” Nat. Photonics 8(4), 302–306 (2014).
[Crossref] [PubMed]

Vijayakumar, A.

Xiao, S.

Yanagawa, T.

Yang, G. Z.

Yim, J.

Yuan, J.

Zeng, S.

Zhuang, J. Y.

Zhuang, X.

S. Jia, J. C. Vaughan, and X. Zhuang, “Isotropic three-dimensional super-resolution imaging with a self-bending point spread function,” Nat. Photonics 8(4), 302–306 (2014).
[Crossref] [PubMed]

ACM Trans. Graph. (1)

A. Levin, R. Fergus, F. Durand, and W. T. Freeman, “Image and depth from a conventional camera with a coded aperture,” ACM Trans. Graph. 26(3), 70 (2007).
[Crossref]

Appl. Opt. (4)

J. Hologr. Speckle (1)

J. Rosen, G. Brooker, G. Indebetouw, and N. T. Shaked, “A review of incoherent digital Fresnel holography,” J. Hologr. Speckle 5(2), 124–140 (2009).
[Crossref]

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

J. Phys. Chem. B (1)

A. S. Backer and W. E. Moerner, “Extending single-molecule microscopy using optical Fourier processing,” J. Phys. Chem. B 118(28), 8313–8329 (2014).
[Crossref] [PubMed]

Nat. Photonics (1)

S. Jia, J. C. Vaughan, and X. Zhuang, “Isotropic three-dimensional super-resolution imaging with a self-bending point spread function,” Nat. Photonics 8(4), 302–306 (2014).
[Crossref] [PubMed]

Opt. Eng. (1)

T. Leportier and M. C. Park, “Generation of binary holograms for deep scenes captured with a camera and a depth sensor,” Opt. Eng. 56(1), 013107 (2017).
[Crossref]

Opt. Express (9)

W. Chi and N. George, “Optical imaging with phase-coded aperture,” Opt. Express 19(5), 4294–4300 (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]

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]

Y. Kashter and J. Rosen, “Enhanced-resolution using modified configuration of Fresnel incoherent holographic recorder with synthetic aperture,” Opt. Express 22(17), 20551–20565 (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]

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]

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]

C. Liu, S. Knitter, Z. Cong, I. Sencan, H. Cao, and M. A. Choma, “High-speed line-field confocal holographic microscope for quantitative phase imaging,” Opt. Express 24(9), 9251–9265 (2016).
[Crossref] [PubMed]

A. Vijayakumar, Y. Kashter, R. Kelner, and J. Rosen, “Coded aperture correlation holography-a new type of incoherent digital holograms,” Opt. Express 24(11), 12430–12441 (2016).
[Crossref] [PubMed]

Opt. Lett. (10)

A. Vijayakumar and J. Rosen, “Spectrum and space resolved 4D imaging by coded aperture correlation holography (COACH) with diffractive objective lens,” Opt. Lett. 42(5), 947–950 (2017).
[Crossref] [PubMed]

Y. Kashter, A. Vijayakumar, Y. Miyamoto, and J. Rosen, “Enhanced super resolution using Fresnel incoherent correlation holography with structured illumination,” Opt. Lett. 41(7), 1558–1561 (2016).
[Crossref] [PubMed]

T. Yanagawa, R. Abe, and Y. Hayasaki, “Three-dimensional mapping of fluorescent nanoparticles using incoherent digital holography,” Opt. Lett. 40(14), 3312–3315 (2015).
[Crossref] [PubMed]

A. Greengard, Y. Y. Schechner, and R. Piestun, “Depth from diffracted rotation,” Opt. Lett. 31(2), 181–183 (2006).
[Crossref] [PubMed]

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

M. K. Kim, “Adaptive optics by incoherent digital holography,” Opt. Lett. 37(13), 2694–2696 (2012).
[Crossref] [PubMed]

P. Bouchal and Z. Bouchal, “Selective edge enhancement in three-dimensional vortex imaging with incoherent light,” Opt. Lett. 37(14), 2949–2951 (2012).
[Crossref] [PubMed]

M. R. Hatzvi and Y. Y. Schechner, “Three-dimensional optical transfer of rotating beams,” Opt. Lett. 37(15), 3207–3209 (2012).
[Crossref] [PubMed]

X. Lai, S. Zeng, X. Lv, J. Yuan, and L. Fu, “Violation of the Lagrange invariant in an optical imaging system,” Opt. Lett. 38(11), 1896–1898 (2013).
[Crossref] [PubMed]

J. Hong and M. K. Kim, “Single-shot self-interference incoherent digital holography using off-axis configuration,” Opt. Lett. 38(23), 5196–5199 (2013).
[Crossref] [PubMed]

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).

Phys. Rev. Lett. (1)

Y. Shechtman, S. J. Sahl, A. S. Backer, and W. E. Moerner, “Optimal point spread function design for 3D imaging,” Phys. Rev. Lett. 113(13), 133902 (2014).
[Crossref] [PubMed]

Other (4)

D. J. Goldstein, Understanding the light microscope: A computer aided introduction (Academic, 1999) Chap.1.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968), Chap. 9, pp. 295–313.

C. Zhou, S. Lin, and S. Nayar, “Coded aperture pairs for depth from defocus,” in 2009 IEEE 12th International Conference on Computer Vision (ICCV), 325–332 (2009).

H. Nagahara, C. Zhou, T. Watanabe, H. Ishiguro, and S. K. Nayar, “Programmable aperture camera using LCoS,” in “Computer Vision–ECCV 2010,” (Springer, 2010), pp. 337–350.

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Figures (10)

Fig. 1
Fig. 1 Schemes of the three incoherent digital systems: (a) FINCH; (b) COACH; (c) I-COACH. The dashed blue and red lines are wave fronts. In FINCH and COACH the blue and the red waves interfere. In I-COACH there is only a single diffracted wave, and therefore there is no wave interference. CPM-coded phase mask, SLM-spatial light modulator, CCD-charge coupled device, SPC-superposition calculator, DH-digital hologram, RS-reconstruction system, X, Y, Z-Intensity pattern recorded by the CCD in three camera shots, a, b, c-complex constants, HOBJ-complex digital object hologram, HPSH-point spread hologram, - correlation sign, Q(1/ z r )=exp[ iπ ( z r λ ) 1 ( x 2 + y 2 ) ].
Fig. 2
Fig. 2 Optical configuration of I-COACH for recording object and PSHs.
Fig. 3
Fig. 3 Experimental setup of I-COACH with two illumination channels. There is no time overlap between the two types of modulation, and there is no spatial overlap between the two channels. Therefore, there is never two-wave interference in this experiment.
Fig. 4
Fig. 4 Coded phase masks (a) CPM1, (b) CPM2, and (c) CPM3; Intensity patterns recorded using the CPMs for the pinhole (d) IPSH(CPM1), (e) IPSH(CPM2), and (f) IPSH(CPM3); Intensity patterns recorded using the CPMs for the object (g) IOBJ(CPM1), (h) IOBJ(CPM2), and (i) IOBJ(CPM3); (j) Phase and (l) magnitude of the synthesized HPSH; (k) Phase and (m) magnitude of the synthesized HOBJ; Reconstruction results of the NBS object with (n) matched filter (SNR = 5), (o) phase-only filter (SNR = 16.7) and (p) regular imaging of the NBS object.
Fig. 5
Fig. 5 Normalized intensity of reconstruction/imaging at (x,y) = (0,0) versus the axial distance of the pinhole from the front focal plane of lens L2.
Fig. 6
Fig. 6 Experimental comparison results of regular imaging and reconstruction of the I-COACH complex patterns at plane 1 (NBS chart) and plane 2 (USAF chart) of channels 1 and 2 respectively, when the plane separation was varied from 1cm to 1 cm in steps of 0.5 cm.
Fig. 7
Fig. 7 Experimental setup of I-COACH with two illumination channels for studying reflective 3D objects.
Fig. 8
Fig. 8 Reconstruction and imaging results of I-COACH and regular imaging of the different planes of the LED.
Fig. 9
Fig. 9 Reconstruction and imaging results of I-COACH and regular imaging of the two planes of the two one-cent coins separated by a distance of 5 mm.
Fig. 10
Fig. 10 Reconstruction and imaging results of I-COACH and regular imaging of the different planes of the stapler pins.

Equations (7)

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I k ( r ¯ 0 ; r ¯ s , z s )=| I s C 1 Q( 1 z s )L( r ¯ s z s )Q( - 1 f 0 )exp[ i Φ k ( r ¯ ) ] *Q( 1 z h )| 2 ,
I k ( r ¯ 0 ; r ¯ s , z s )=| I s C 1 L( r ¯ s z s )Q( 1 z 1 )exp[ i Φ k ( r ¯ ) ] *Q( 1 z h )| 2 = I k ( r ¯ 0 z h z s r ¯ s ;0, z s ),
o( r ¯ s )= j N a j δ( r ¯ r ¯ s,j ) .
I OBJ,k ( r ¯ 0 ; z s )= j a j I k ( r ¯ 0 z h z s r ¯ s,j ;0, z s ) .
H PSH ( r ¯ 0 ; z s )= k=1 K I k ( r ¯ 0 ; z s )exp( i θ k ) .
H OBJ ( r ¯ 0 ; z s )= k=1 K I OBJ,k ( r ¯ 0 ; z s )exp( i θ k ) = k=1 K j a j I k ( r ¯ 0 z h z s r ¯ s,j ;0, z s ) exp( i θ k ) . = j a j H PSH ( r ¯ 0 z h z s r ¯ s,j ; z s ) ,
P( r ¯ R )= H OBJ ( r ¯ 0 ; z s ) H PSH * ( r ¯ 0 r ¯ R ; z s )d r ¯ 0 = j a j H PSH ( r ¯ 0 z h z s r ¯ s,j ; z s ) H PSH * ( r ¯ 0 r ¯ R ; z s )d r ¯ 0 . = j a j Λ( r ¯ R z h z s r ¯ s,j ) o( r ¯ s M T ).

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