Abstract

We propose a task-specific digital holographic capture system for three-dimensional scenes, which can reduce the amount of data sent from the camera system to the receiver and can effectively reconstruct partially occluded objects. The system requires knowledge of the object of interest, but it does not require a priori knowledge of either the occlusion or the distance the object is from the camera. Subwindows of the camera-plane Fresnel field are digitally propagated to reveal different perspectives of the scene, and these are combined to overcome the unknown foreground occlusions. The nature of the occlusions and the effect of subwindows are analyzed thoroughly by using the Wigner distribution function. We demonstrate that a careful combination of reconstructions from subwindows can reveal features that are not apparent in a reconstruction from the whole hologram. We provide results by using optically captured digital holograms of real-world objects and simulated occlusions.

© 2006 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  16. T. J. Naughton and B. Javidi, "Compression of encrypted three-dimensional objects using digital holography," Opt. Eng. 43, 2233-2238 (2004).
    [CrossRef]
  17. O. Matoba, T. J. Naughton, Y. Frauel, N. Bertaux, and B. Javidi, "Real-time three-dimensional object reconstruction by use of a phase-encoded digital hologram," Appl. Opt. 41, 6187-6192 (2002).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  30. B. M. Hennelly and J. T. Sheridan "Tracking the space bandwidth product in optical systems," in Opto-Ireland 2005: Photonic Engineering, B.W.Bowe, G.Byrne, A.J.Flanogan, T.G.Glynn, J.Magee, G.M.O'Connor, R.F.O'Dowd, G.D.O'Sullivan, and J.T.Sheridan, eds., 5827,334-345 (2005).
  31. C. J. Román-Moreno and R. Ortega-Martínez "The Wigner function in paraxial optics II. Optical diffraction pattern representation," Mex. Fis. 49, 290-295 (2003).
  32. A. Stern and B. Javidi, "General sampling theorem and application to digital holography," in Optical Information Systems II, B.Javidi and D.Psaltis, eds., Proc. SPIE 5557,110-123 (2004).
  33. T. Kreis, Handbook of Holographic Interferometry (Wiley-VCH, 2005).
  34. B. Javidi and D. Kim, "Three-dimensional-object recognition by use of singleexposure on-axis digital holography," Opt. Lett. 30, 236-238 (2005).
    [CrossRef] [PubMed]
  35. C. P. McElhinney, J. Maycock, T. J. Naughton, J. B. McDonald, and B. Javidi, "Extraction of three-dimensional shape information from a digital hologram," in Optical Information Systems III, B.Javidi and D.Psaltis, eds., Proc. SPIE 5908,30-41 (2005).

2005

H. Kim and Y. H. Lee, "Optimal watermarking of digital hologram of 3-D object," Opt. Express 13, 2881-2886 (2005).
[CrossRef] [PubMed]

T. Nomura, A. Okazaki, M. Kameda, Y. Morimoto, and B. Javidi, "Image reconstruction from compressed encrypted digital hologram," Opt. Eng. 44, 075801 (2005).
[CrossRef]

B. M. Hennelly and J. T. Sheridan "Generalizing, optimizing, and inventing numerical algorithms for the fractional Fourier, Fresnel, and linear canonical transforms," J. Opt. Soc. Am. A 21, 917-927 (2005).
[CrossRef]

B. Javidi and D. Kim, "Three-dimensional-object recognition by use of singleexposure on-axis digital holography," Opt. Lett. 30, 236-238 (2005).
[CrossRef] [PubMed]

2004

2003

T. J. Naughton, J. B. McDonald, and B. Javidi, "Efficient compression of Fresnel fields for Internet transmission of three-dimensional images," Appl. Opt. 42, 4758-4764 (2003).
[CrossRef] [PubMed]

C. J. Román-Moreno and R. Ortega-Martínez "The Wigner function in paraxial optics II. Optical diffraction pattern representation," Mex. Fis. 49, 290-295 (2003).

2002

2001

2000

1997

1996

1994

1993

1987

L. Onural and P. D. Scott, "Digital decoding of in-line holograms," Opt. Eng. 26, 1124-1132 (1987).

1974

1967

J. W. Goodman and R. W. Lawrence, "Digital image formation from electronically detected holograms," Appl. Phys. Lett. 11, 77-79 (1967).
[CrossRef]

1948

D. Gabor, "A new microscopic principle," Nature 161, 777-778 (1948).
[CrossRef] [PubMed]

Bastians, M. J.

M. J. Bastians, "Application of the Wigner distribution function in optics," in The Wigner Distribution—Theory and Applications in Signal Processing, W. Mecklenbrauker and F. Hlawatsch, eds. (Elsevier Science, 1997).

Bertaux, N.

Bimber, O.

O. Bimber, T. Zeidler, A. Grundhfer, G. Wetzstein, M. Mhring, S. Kndel, and U. Hahne, "Interacting with augmented holograms," in Practical Holography XIX: Materials and Applications, T.H.Jeong and H.I.Bjelkhagen, eds., Proc. SPIE 5742,41-54 (2005).

Brangaccio, D. J.

Bruning, J. H.

Castro, M. -A.

Caulfield, H. J.

H. J. Caulfield, ed., Handbook of Optical Holography (Academic, 1979).

Dorsch, R. G.

Ferreira, C.

Frauel, Y.

Gabor, D.

D. Gabor, "A new microscopic principle," Nature 161, 777-778 (1948).
[CrossRef] [PubMed]

Gallagher, J. E.

Goodman, J. W.

J. W. Goodman and R. W. Lawrence, "Digital image formation from electronically detected holograms," Appl. Phys. Lett. 11, 77-79 (1967).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics, 3rd ed., (Roberts & Company, 2005).

Grundhfer, A.

O. Bimber, T. Zeidler, A. Grundhfer, G. Wetzstein, M. Mhring, S. Kndel, and U. Hahne, "Interacting with augmented holograms," in Practical Holography XIX: Materials and Applications, T.H.Jeong and H.I.Bjelkhagen, eds., Proc. SPIE 5742,41-54 (2005).

Hahne, U.

O. Bimber, T. Zeidler, A. Grundhfer, G. Wetzstein, M. Mhring, S. Kndel, and U. Hahne, "Interacting with augmented holograms," in Practical Holography XIX: Materials and Applications, T.H.Jeong and H.I.Bjelkhagen, eds., Proc. SPIE 5742,41-54 (2005).

Hennelly, B. M.

B. M. Hennelly and J. T. Sheridan "Generalizing, optimizing, and inventing numerical algorithms for the fractional Fourier, Fresnel, and linear canonical transforms," J. Opt. Soc. Am. A 21, 917-927 (2005).
[CrossRef]

B. M. Hennelly and J. T. Sheridan "Tracking the space bandwidth product in optical systems," in Opto-Ireland 2005: Photonic Engineering, B.W.Bowe, G.Byrne, A.J.Flanogan, T.G.Glynn, J.Magee, G.M.O'Connor, R.F.O'Dowd, G.D.O'Sullivan, and J.T.Sheridan, eds., 5827,334-345 (2005).

Herriott, D. R.

Hong, S. -H.

S. -H. Hong and B. Javidi, "Three-dimensional visualization of partially occluded objects using integral imaging," IEEE J. Display Technol. 1,354-359 (2005).

Javidi, B.

T. Nomura, A. Okazaki, M. Kameda, Y. Morimoto, and B. Javidi, "Image reconstruction from compressed encrypted digital hologram," Opt. Eng. 44, 075801 (2005).
[CrossRef]

B. Javidi and D. Kim, "Three-dimensional-object recognition by use of singleexposure on-axis digital holography," Opt. Lett. 30, 236-238 (2005).
[CrossRef] [PubMed]

T. J. Naughton and B. Javidi, "Compression of encrypted three-dimensional objects using digital holography," Opt. Eng. 43, 2233-2238 (2004).
[CrossRef]

A. Stern and B. Javidi, "Sampling in the light of Wigner distribution: eratta," J. Opt. Soc. Am. A 21, 360-366 (2004).
[CrossRef]

A. Stern and B. Javidi, "Sampling in the light of Wigner distribution," J. Opt. Soc. Am. A 21, 2038-2038 (2004).
[CrossRef]

T. J. Naughton, J. B. McDonald, and B. Javidi, "Efficient compression of Fresnel fields for Internet transmission of three-dimensional images," Appl. Opt. 42, 4758-4764 (2003).
[CrossRef] [PubMed]

O. Matoba, T. J. Naughton, Y. Frauel, N. Bertaux, and B. Javidi, "Real-time three-dimensional object reconstruction by use of a phase-encoded digital hologram," Appl. Opt. 41, 6187-6192 (2002).
[CrossRef] [PubMed]

T. J. Naughton, Y. Frauel, B. Javidi, and E. Tajahuerce, "Compression of digital holograms for three-dimensional object reconstruction and recognition," Appl. Opt. 41, 4124-4132 (2002).
[CrossRef] [PubMed]

Y. Frauel, E. Tajahuerce, M. -A. Castro, and B. Javidi, "Distortion-tolerant three-dimensional object recognition with digital holography," Appl. Opt. 40, 3887-3893 (2001).
[CrossRef]

B. Javidi and E. Tajahuerce, "Three-dimensional object recognition by use of digital holography," Opt. Lett. 25, 610-612 (2000).
[CrossRef]

E. Tajahuerce and B. Javidi, "Encrypting three-dimensional information with digital holography," Appl. Opt. 39, 6595-6601 (2000).
[CrossRef]

A. Stern and B. Javidi, "General sampling theorem and application to digital holography," in Optical Information Systems II, B.Javidi and D.Psaltis, eds., Proc. SPIE 5557,110-123 (2004).

C. P. Mc Elhinney, J. Maycock, J. B. McDonald, T. J. Naughton, and B. Javidi, "Three-dimensional scene reconstruction using digital holography," in Opto-Ireland 2005: Imaging and Vision, F.D.Murtagh, ed., Proc. SPIE 5823, 48-57 (2005).

S. -H. Hong and B. Javidi, "Three-dimensional visualization of partially occluded objects using integral imaging," IEEE J. Display Technol. 1,354-359 (2005).

C. P. McElhinney, J. Maycock, T. J. Naughton, J. B. McDonald, and B. Javidi, "Extraction of three-dimensional shape information from a digital hologram," in Optical Information Systems III, B.Javidi and D.Psaltis, eds., Proc. SPIE 5908,30-41 (2005).

Jüptner, W. P. O.

Kameda, M.

T. Nomura, A. Okazaki, M. Kameda, Y. Morimoto, and B. Javidi, "Image reconstruction from compressed encrypted digital hologram," Opt. Eng. 44, 075801 (2005).
[CrossRef]

Kim, D.

Kim, H.

Kndel, S.

O. Bimber, T. Zeidler, A. Grundhfer, G. Wetzstein, M. Mhring, S. Kndel, and U. Hahne, "Interacting with augmented holograms," in Practical Holography XIX: Materials and Applications, T.H.Jeong and H.I.Bjelkhagen, eds., Proc. SPIE 5742,41-54 (2005).

Kreis, T.

T. Kreis, Handbook of Holographic Interferometry (Wiley-VCH, 2005).

Kutay, M. A.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Lawrence, R. W.

J. W. Goodman and R. W. Lawrence, "Digital image formation from electronically detected holograms," Appl. Phys. Lett. 11, 77-79 (1967).
[CrossRef]

Lee, Y. H.

Lohmann, A. W.

Matoba, O.

Maycock, J.

C. P. McElhinney, J. Maycock, T. J. Naughton, J. B. McDonald, and B. Javidi, "Extraction of three-dimensional shape information from a digital hologram," in Optical Information Systems III, B.Javidi and D.Psaltis, eds., Proc. SPIE 5908,30-41 (2005).

C. P. Mc Elhinney, J. Maycock, J. B. McDonald, T. J. Naughton, and B. Javidi, "Three-dimensional scene reconstruction using digital holography," in Opto-Ireland 2005: Imaging and Vision, F.D.Murtagh, ed., Proc. SPIE 5823, 48-57 (2005).

Mc Elhinney, C. P.

C. P. Mc Elhinney, J. Maycock, J. B. McDonald, T. J. Naughton, and B. Javidi, "Three-dimensional scene reconstruction using digital holography," in Opto-Ireland 2005: Imaging and Vision, F.D.Murtagh, ed., Proc. SPIE 5823, 48-57 (2005).

McDonald, J. B.

T. J. Naughton, J. B. McDonald, and B. Javidi, "Efficient compression of Fresnel fields for Internet transmission of three-dimensional images," Appl. Opt. 42, 4758-4764 (2003).
[CrossRef] [PubMed]

C. P. Mc Elhinney, J. Maycock, J. B. McDonald, T. J. Naughton, and B. Javidi, "Three-dimensional scene reconstruction using digital holography," in Opto-Ireland 2005: Imaging and Vision, F.D.Murtagh, ed., Proc. SPIE 5823, 48-57 (2005).

C. P. McElhinney, J. Maycock, T. J. Naughton, J. B. McDonald, and B. Javidi, "Extraction of three-dimensional shape information from a digital hologram," in Optical Information Systems III, B.Javidi and D.Psaltis, eds., Proc. SPIE 5908,30-41 (2005).

McElhinney, C. P.

C. P. McElhinney, J. Maycock, T. J. Naughton, J. B. McDonald, and B. Javidi, "Extraction of three-dimensional shape information from a digital hologram," in Optical Information Systems III, B.Javidi and D.Psaltis, eds., Proc. SPIE 5908,30-41 (2005).

Mendolovic, D.

Merzlyakov, N. S.

L. P. Yaroslavskii and N. S. Merzlyakov, Methods of Digital Holography (Consultants Bureau, Plenum, 1980).

Mhring, M.

O. Bimber, T. Zeidler, A. Grundhfer, G. Wetzstein, M. Mhring, S. Kndel, and U. Hahne, "Interacting with augmented holograms," in Practical Holography XIX: Materials and Applications, T.H.Jeong and H.I.Bjelkhagen, eds., Proc. SPIE 5742,41-54 (2005).

Morimoto, Y.

T. Nomura, A. Okazaki, M. Kameda, Y. Morimoto, and B. Javidi, "Image reconstruction from compressed encrypted digital hologram," Opt. Eng. 44, 075801 (2005).
[CrossRef]

Naughton, T. J.

T. J. Naughton and B. Javidi, "Compression of encrypted three-dimensional objects using digital holography," Opt. Eng. 43, 2233-2238 (2004).
[CrossRef]

T. J. Naughton, J. B. McDonald, and B. Javidi, "Efficient compression of Fresnel fields for Internet transmission of three-dimensional images," Appl. Opt. 42, 4758-4764 (2003).
[CrossRef] [PubMed]

O. Matoba, T. J. Naughton, Y. Frauel, N. Bertaux, and B. Javidi, "Real-time three-dimensional object reconstruction by use of a phase-encoded digital hologram," Appl. Opt. 41, 6187-6192 (2002).
[CrossRef] [PubMed]

T. J. Naughton, Y. Frauel, B. Javidi, and E. Tajahuerce, "Compression of digital holograms for three-dimensional object reconstruction and recognition," Appl. Opt. 41, 4124-4132 (2002).
[CrossRef] [PubMed]

C. P. McElhinney, J. Maycock, T. J. Naughton, J. B. McDonald, and B. Javidi, "Extraction of three-dimensional shape information from a digital hologram," in Optical Information Systems III, B.Javidi and D.Psaltis, eds., Proc. SPIE 5908,30-41 (2005).

C. P. Mc Elhinney, J. Maycock, J. B. McDonald, T. J. Naughton, and B. Javidi, "Three-dimensional scene reconstruction using digital holography," in Opto-Ireland 2005: Imaging and Vision, F.D.Murtagh, ed., Proc. SPIE 5823, 48-57 (2005).

Nomura, T.

T. Nomura, A. Okazaki, M. Kameda, Y. Morimoto, and B. Javidi, "Image reconstruction from compressed encrypted digital hologram," Opt. Eng. 44, 075801 (2005).
[CrossRef]

Okazaki, A.

T. Nomura, A. Okazaki, M. Kameda, Y. Morimoto, and B. Javidi, "Image reconstruction from compressed encrypted digital hologram," Opt. Eng. 44, 075801 (2005).
[CrossRef]

Onural, L.

L. Onural and P. D. Scott, "Digital decoding of in-line holograms," Opt. Eng. 26, 1124-1132 (1987).

Ortega-Martínez, R.

C. J. Román-Moreno and R. Ortega-Martínez "The Wigner function in paraxial optics II. Optical diffraction pattern representation," Mex. Fis. 49, 290-295 (2003).

Ozaktas, H. M.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Román-Moreno, C. J.

C. J. Román-Moreno and R. Ortega-Martínez "The Wigner function in paraxial optics II. Optical diffraction pattern representation," Mex. Fis. 49, 290-295 (2003).

Rosenfeld, D. P.

Schnars, U.

Scott, P. D.

L. Onural and P. D. Scott, "Digital decoding of in-line holograms," Opt. Eng. 26, 1124-1132 (1987).

Sheridan, J. T.

B. M. Hennelly and J. T. Sheridan "Generalizing, optimizing, and inventing numerical algorithms for the fractional Fourier, Fresnel, and linear canonical transforms," J. Opt. Soc. Am. A 21, 917-927 (2005).
[CrossRef]

B. M. Hennelly and J. T. Sheridan "Tracking the space bandwidth product in optical systems," in Opto-Ireland 2005: Photonic Engineering, B.W.Bowe, G.Byrne, A.J.Flanogan, T.G.Glynn, J.Magee, G.M.O'Connor, R.F.O'Dowd, G.D.O'Sullivan, and J.T.Sheridan, eds., 5827,334-345 (2005).

Stern, A.

A. Stern and B. Javidi, "Sampling in the light of Wigner distribution," J. Opt. Soc. Am. A 21, 2038-2038 (2004).
[CrossRef]

A. Stern and B. Javidi, "Sampling in the light of Wigner distribution: eratta," J. Opt. Soc. Am. A 21, 360-366 (2004).
[CrossRef]

A. Stern and B. Javidi, "General sampling theorem and application to digital holography," in Optical Information Systems II, B.Javidi and D.Psaltis, eds., Proc. SPIE 5557,110-123 (2004).

Tajahuerce, E.

Wetzstein, G.

O. Bimber, T. Zeidler, A. Grundhfer, G. Wetzstein, M. Mhring, S. Kndel, and U. Hahne, "Interacting with augmented holograms," in Practical Holography XIX: Materials and Applications, T.H.Jeong and H.I.Bjelkhagen, eds., Proc. SPIE 5742,41-54 (2005).

White, A. D.

Wigner, E.

E. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749-759 (1932).

Yamaguchi, I.

Yaroslavskii, L. P.

L. P. Yaroslavskii and N. S. Merzlyakov, Methods of Digital Holography (Consultants Bureau, Plenum, 1980).

Zalevsky, Z.

A. W. Lohmann, R. G. Dorsch, D. Mendolovic, Z. Zalevsky, and C. Ferreira, "Space-bandwidth product of optical signals and systems," J. Opt. Soc. Am. A 13, 470-473 (1996).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Zeidler, T.

O. Bimber, T. Zeidler, A. Grundhfer, G. Wetzstein, M. Mhring, S. Kndel, and U. Hahne, "Interacting with augmented holograms," in Practical Holography XIX: Materials and Applications, T.H.Jeong and H.I.Bjelkhagen, eds., Proc. SPIE 5742,41-54 (2005).

Zhang, T.

Appl. Opt.

Appl. Phys. Lett.

J. W. Goodman and R. W. Lawrence, "Digital image formation from electronically detected holograms," Appl. Phys. Lett. 11, 77-79 (1967).
[CrossRef]

J. Opt. Soc. Am. A

Mex. Fis.

C. J. Román-Moreno and R. Ortega-Martínez "The Wigner function in paraxial optics II. Optical diffraction pattern representation," Mex. Fis. 49, 290-295 (2003).

Nature

D. Gabor, "A new microscopic principle," Nature 161, 777-778 (1948).
[CrossRef] [PubMed]

Opt. Eng.

L. Onural and P. D. Scott, "Digital decoding of in-line holograms," Opt. Eng. 26, 1124-1132 (1987).

T. J. Naughton and B. Javidi, "Compression of encrypted three-dimensional objects using digital holography," Opt. Eng. 43, 2233-2238 (2004).
[CrossRef]

T. Nomura, A. Okazaki, M. Kameda, Y. Morimoto, and B. Javidi, "Image reconstruction from compressed encrypted digital hologram," Opt. Eng. 44, 075801 (2005).
[CrossRef]

Opt. Express

Opt. Lett.

Other

B. M. Hennelly and J. T. Sheridan "Tracking the space bandwidth product in optical systems," in Opto-Ireland 2005: Photonic Engineering, B.W.Bowe, G.Byrne, A.J.Flanogan, T.G.Glynn, J.Magee, G.M.O'Connor, R.F.O'Dowd, G.D.O'Sullivan, and J.T.Sheridan, eds., 5827,334-345 (2005).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed., (Roberts & Company, 2005).

H. J. Caulfield, ed., Handbook of Optical Holography (Academic, 1979).

E. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749-759 (1932).

M. J. Bastians, "Application of the Wigner distribution function in optics," in The Wigner Distribution—Theory and Applications in Signal Processing, W. Mecklenbrauker and F. Hlawatsch, eds. (Elsevier Science, 1997).

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

A. Stern and B. Javidi, "General sampling theorem and application to digital holography," in Optical Information Systems II, B.Javidi and D.Psaltis, eds., Proc. SPIE 5557,110-123 (2004).

T. Kreis, Handbook of Holographic Interferometry (Wiley-VCH, 2005).

C. P. McElhinney, J. Maycock, T. J. Naughton, J. B. McDonald, and B. Javidi, "Extraction of three-dimensional shape information from a digital hologram," in Optical Information Systems III, B.Javidi and D.Psaltis, eds., Proc. SPIE 5908,30-41 (2005).

L. P. Yaroslavskii and N. S. Merzlyakov, Methods of Digital Holography (Consultants Bureau, Plenum, 1980).

O. Bimber, T. Zeidler, A. Grundhfer, G. Wetzstein, M. Mhring, S. Kndel, and U. Hahne, "Interacting with augmented holograms," in Practical Holography XIX: Materials and Applications, T.H.Jeong and H.I.Bjelkhagen, eds., Proc. SPIE 5742,41-54 (2005).

S. -H. Hong and B. Javidi, "Three-dimensional visualization of partially occluded objects using integral imaging," IEEE J. Display Technol. 1,354-359 (2005).

C. P. Mc Elhinney, J. Maycock, J. B. McDonald, T. J. Naughton, and B. Javidi, "Three-dimensional scene reconstruction using digital holography," in Opto-Ireland 2005: Imaging and Vision, F.D.Murtagh, ed., Proc. SPIE 5823, 48-57 (2005).

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Figures (12)

Fig. 1
Fig. 1

Experimental setup for PSI: BE, beam expander; BS, beam splitter; RP, retardation plate; M, mirror.

Fig. 2
Fig. 2

Wigner chart of the propagating signal. (a) Wavefield immediately after the object, (b) Wigner chart after propagation of distance d 1, (c) Wigner chart after further propagation of distance d 2.

Fig. 3
Fig. 3

(a) Wigner chart of the object signal after propagating a distance d 1 to the occlusion, (b) occlusion Wigner chart, (c) Wigner chart of the occluded wavefront obtained by convolving (a) and (b), (d) Wigner chart after the occluded wave field propagation of a distance d 2 to the CCD plane, (e) Wigner chart of the CCD and the captured wave field, (f) Wigner chart of the reconstructed signal against the total reconstructed signal.

Fig. 4
Fig. 4

(a) Wigner chart of both the occluded wave field (having already propagated a distance d 1) and the second wave field, (b) Two Wigner charts after propagation to the CCD plane, (c) Wigner charts after propagation to the CCD and the result of convolution with the signal's WDF, (d) Wigner chart of the reconstructed signals in the plane of the first object if the CCD had been large enough to capture the entire wave fields during recording, (e) reconstruction using the signal in (c) plotted against total reconstruction.

Fig. 5
Fig. 5

Simulated experimental setup with an occlusion positioned at 260 mm from the object.

Fig. 6
Fig. 6

Occlusion positioned at 280 mm from 3D object: (a) a weak perspective view of the scene, (b) the occlusion plane, (c) the reconstruction along the optical axis of the die.

Fig. 7
Fig. 7

Simulated experimental setup in which one die acts as an occlusion to a second die.

Fig. 8
Fig. 8

Advantage of using a smaller window of pixels over the entire set of pixels when an occluded object is reconstructed: (a) reconstruction using the entire set of pixels and (b) reconstruction of the scene using the top right 1024 × 1024 window of pixels.

Fig. 9
Fig. 9

Comparison between (a) a traditional holographic system and (b) the proposed automated task-specific holographic system.

Fig. 10
Fig. 10

Illustration of the grid structure, the starting window containing the highest intensity pixel, and the directional vectors used by the algorithm.

Fig. 11
Fig. 11

Simulated experimental setup with an opaque occlusion positioned 60 mm from the object.

Fig. 12
Fig. 12

Reconstruction of the scene using (a) the pixels selected by the algorithm, (b) the entire set of pixels, (c) the combination of five 512 × 512 pixel region reconstructions.

Equations (88)

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( 514.5   nm )
d = 350   mm
U 0 ( x , y )
R P 1
R P 2
H 0 ( x , y )
H 0 ( x , y )
U ( x , y , d )  =  i λ d exp ( i 2 π λ d ) × H 0 ( x , y ) exp [ i π ( x 2 + y 2 ) λ d ] ,
z = d
U 0 ( x , y )
18.5   mm × 18.5   mm
9   mm
d = 350   mm
± 0.74 °
Ψ { f ( x ) } ( x , k )
Ψ { f ( x ) } ( x , k ) = f ( x + ξ 2 ) f * ( x ξ 2 ) × exp ( j 2 π k ξ ) d ξ ,
ψ { f ( x ξ ) } ( x , k ) = ψ { f ( x ) } ( x ξ , k ) .
ψ { f ( ξ ) g ( x ξ ) d ξ } ( x , k ) = ψ { f ( x ) } × ( x x , k ) ψ { g ( x ) } ( x , k ) d x ,
ψ { f ( x ) g ( x ) } ( x , k ) = ψ { f ( x ) } × ( x , k k ) ψ { g ( x ) } ( x , k ) d k .
ψ { f ( x ) + g ( x ) } ( x , k ) = ψ { f ( x ) } ( x , k ) + ψ { g ( x ) } ( x , k ) + 2 [ f ( x ξ / 2 ) g * ( x + ξ / 2 ) ] exp [ j 2 π k ξ ] d ξ .
| f ( x ) | 2 = ψ { f ( x ) } ( x , k ) d k ,
| F ( k ) | 2 = ψ { f ( x ) } ( x , k ) d x .
ψ { F λ d { f ( x ) } ( x ) } ( x , k ) = ψ { f ( x ) } ( x + λ d k , k )
[ x k ] = [ 1     λ d 0 1 ] [ x k ] .
d 2 > d 1
f ( x ) 0     | x | > W / 2 ,     { f ( x ) } ( k ) 0     | k | > B / 2 ,
δ T ( x ) = n = δ ( x n T )
δ ( x )
ψ { f ( x ) } ( x , k )
ψ { δ ( x ) } ( x , k )
ψ { δ T ( x ) } ( x , k ) = 1 2 T n q ( 1 ) q n δ ( x n T 2 ) δ ( x q 2 T ) .
S = [ W / 2     W / 2 W / 2     W / 2 B / 2 B / 2     B / 2 B / 2 ] .
S = [ 1     λ d 0 1 ] S .
r e c t ( x / w ) = 1 | x | < w / 2
= 0     | x | > w / 2 .
r e c t ( x / w )
ψ { r e c t ( x / w ) } ( x , k ) = sin ( k [ 2 | x | + w ] ) k , | x | < w / 2 ,
w λ
ψ { r e c t ( x / w ) } ( x , k ) = δ ( k ) , | x | < w / 2 .
o c c l u s i o n ( x ) = 1 r e c t ( x / w ) .
ψ { o c c l u s i o n ( x ) } ( x , k ) = ψ { 1 r e c t ( x / w ) } ( x , k ) = δ ( k ) ψ { r e c t ( x / w ) } ( x , k ) cos ( π x k ) sinc ( w k ) .
w λ
ψ { o c c l u s i o n ( x ) } ( x , k ) = δ ( k ) , | x | > w / 2 .
ψ { O ( x ) } ( x , k )
ψ { O ( x ) } ( x + λ d k , k )
w p
( w c )
( x 0 )
CCD ( x x c ) = [ δ T ( x ) r e ct ( x w p ) ] × r e c t ( x x c w c ) .
[ ψ { δ T ( x ) } ( x , k ) x ψ { r e c t ( x / w p ) } ( x , k ) ] k ψ { r e c t ( x / w c ) } ( x x c , k ) ,
ψ { O ( x ) } ( x λ d k ) k ψ { CCD ( x ) } ( x , k ) .
w o
d 1
d 2
x x x o
x 0
ψ { O ( x ) } ( x , k )
ψ { O ( x ) } ( x + λ d 1 k , k )
d = d 1
( x x o ) / w o
ψ { o c c l u s i o n ( x / w o ) } ( x , k )
ψ { O ( x ) } ( x , k ) k ψ { r e c t ( o c c l u s i o n x / w o ) } ( x x o , k ) ,
where    x = x + λ d .
d 2
ψ { O ( x ) } ( x , k ) k ψ { r e c t ( o c c l u s i o n x / w o ) } ( x x o , k ) ,
where    x = x + λ ( d 1 + d 2 ) k .
CCD ( x x c )
CCD ( x ) = r e c t ( x / w c )
ψ { hol ( x ) } ( x , k ) = [ ψ { O ( x ) } ( x , k ) k × ψ { r e c t ( o c c l u s i o n x / w o ) } ( x x o , k ) ] k × ψ { r e c t ( r e c t x / w o ) } × ( x x c , k ) .
( d 1 + d 2 )
ψ { hol ( x ) } ( x λ ( d 1 + d 2 ) , k )
x c
w c
d 2
d 2
[ 0 , 2 π )
260   mm
325   mm
z = d
260   mm
n log 2 n
n log 2 ( n / 4 )
D ( U ) = ( m = 0 N x 1 n = 0 N y 1 [ | U 0 ( m , n ) | 2 | U ( m , n ) | 2 ] 2 × { m = 0 N x 1 n = 0 N y 1 [ | U 0 ( m , n ) | 2 ] 2 } 1 ) 1 / 2 ,
U
N y
N x
72 %
12 %

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