Abstract

Digital holography is an imaging technique that enables recovery of topographic 3D information about an object under investigation. In digital holography, an interference pattern is recorded on a digital camera. Therefore, quantization of the recorded hologram is an integral part of the imaging process. We study the influence of quantization error in the recorded holograms on the fidelity of both the intensity and phase of the reconstructed image. We limit our analysis to the case of lensless Fourier off-axis digital holograms. We derive a theoretical model to predict the effect of quantization noise and we validate this model using experimental results. Based on this, we also show how the resultant noise in the reconstructed image, as well as the speckle that is inherent in digital holography, can be conveniently suppressed by standard speckle reduction techniques. We show that high-quality images can be obtained from binary holograms when speckle reduction is performed.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. Goodman and R. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77 (1967).
    [CrossRef]
  2. M. Kronrod, N. Merzlyakov, and L. Yaroslavskii, “Reconstruction of a hologram with a computer,” in SPIE Milestone Series 144 (SPIE Press, 1998), pp. 645–646.
  3. L. Yaroslavskii and N. Merzlyakov, Methods of Digital Holography (Consultants Bureau, 1980).
  4. U. Schnars and W. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179–181 (1994).
    [CrossRef] [PubMed]
  5. T. Colomb, J. Kühn, F. Charriere, C. Depeursinge, P. Marquet, and N. Aspert, “Total aberrations compensation in digital holographic microscopy with a reference conjugated hologram,” Opt. Express 14, 4300–4306 (2006).
    [CrossRef] [PubMed]
  6. P. Marquet, B. Rappaz, P. Magistretti, E. Cuche, Y. Emery, T. Colomb, and C. Depeursinge, “Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy,” Opt. Lett. 30, 468–470 (2005).
    [CrossRef] [PubMed]
  7. T. Baumbach, W. Osten, C. von Kopylow, and W. Jüptner, “Remote metrology by comparative digital holography,” Appl. Opt. 45, 925–934 (2006).
    [CrossRef] [PubMed]
  8. T. Poon, Digital Holography and Three-Dimensional Display: Principles and Applications (Springer, 2006).
    [CrossRef]
  9. N. Pandey and B. Hennelly, “Fixed-point numercial-reconstruction for digital holographic microscopy,” Opt. Lett. 35, 1076–1078 (2010).
    [CrossRef] [PubMed]
  10. T. 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]
  11. T. Naughton, J. McDonald, and B. Javidi, “Efficient compression of Fresnel fields for internet transmission of three-dimensional images,” Appl. Opt. 42, 4758–4764 (2003).
    [CrossRef] [PubMed]
  12. T. Naughton and B. Javidi, “Compression of encrypted three-dimensional objects using digital holography,” Opt. Eng. 43, 2233 (2004).
    [CrossRef]
  13. O. Matoba, T. 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]
  14. A. Shortt, T. Naughton, and B. Javidi, “A companding approach for nonuniform quantization of digital holograms of three-dimensional objects,” Opt. Express 14, 5129–5134(2006).
    [CrossRef] [PubMed]
  15. A. Shortt, T. Naughton, and B. Javidi, “Compression of optically encrypted digital holograms using artificial neural networks,” J. Display Technology 2, 401–410 (2006).
    [CrossRef]
  16. A. Shortt, T. J. Naughton, and B. Javidi, “Compression of digital holograms of three-dimensional objects using wavelets,” Opt. Express 14, 2625–2630 (2006).
    [CrossRef] [PubMed]
  17. G. Mills and I. Yamaguchi, “Effects of quantization in phase-shifting digital holography,” Appl. Opt. 44, 1216–1225(2005).
    [CrossRef] [PubMed]
  18. E. Darakis and J. Soraghan, “Reconstruction domain compression of phase-shifting digital holograms,” Appl. Opt. 46, 351–356 (2007).
    [CrossRef] [PubMed]
  19. E. Darakis, T. Naughton, and J. Soraghan, “Compression defects in different reconstructions from phase-shifting digital holographic data,” Appl. Opt. 46, 4579–4586 (2007).
    [CrossRef] [PubMed]
  20. A. Gotchev and L. Onural, “A survey on sampling and quantization in diffraction and holography,” in Workshop on Spectral Methods and Multirate Signal Processing, SMMSP (2006), pp. 179–190.
  21. D. Psaltis, E. Paek, and S. Venkatesh, “Optical image correlation with a binary spatial light modulator,” in SPIE Milestone Series, 156 (SPIE Press, 1999), pp. 482–488.
  22. B. Javidi and J. Horner, “Single spatial light modulator joint transform correlator,” Appl. Opt. 28, 1027–1032(1989).
    [CrossRef] [PubMed]
  23. A. Bourquard, F. Aguet, and M. Unser, “Optical imaging using binary sensors,” Opt. Express 18, 4876–4888 (2010).
    [CrossRef] [PubMed]
  24. W. Dallas and A. Lohmann, “Phase quantization in holograms-depth effects,” Appl. Opt. 11, 192–194 (1972).
    [CrossRef] [PubMed]
  25. A. W. Lohmann and D. P. Paris, “Binary Fraunhofer holograms, generated by computer,” Appl. Opt. 6, 1739–1748(1967).
    [CrossRef] [PubMed]
  26. F. Wyrowski, “Iterative quantization of digital amplitude holograms,” Appl. Opt. 28, 3864–3870 (1989).
    [CrossRef] [PubMed]
  27. R. Powers and J. Goodman, “Error rates in computer-generated holographic memories,” Appl. Opt. 14, 1690–1701(1975).
    [CrossRef] [PubMed]
  28. P. Naidu, “Quantization noise in binary holograms,” Opt. Commun. 15, 361–365 (1975).
    [CrossRef]
  29. M. Seldowitz, J. Allebach, and D. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 26, 2788–2798 (1987).
    [CrossRef] [PubMed]
  30. L. Schuchman, “Dither signals and their effect on quantization noise,” IEEE Trans. Communication Technology 12, 162–165 (1964).
    [CrossRef]
  31. J. Goodman, Introduction to Fourier Optics (Roberts & Co., 2005).
  32. D. Kelly, B. Hennelly, N. Pandey, T. Naughton, W. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
    [CrossRef]
  33. H. Jin, H. Wan, Y. Zhang, Y. Li, and P. Qiu, “The influence of structural parameters of CCD on the reconstruction image of digital holograms,” J. Mod. Opt. 55, 2989–3000(2008).
    [CrossRef]
  34. A. Oppenheim and R. Schafer, Discrete-Time Signal Processing (Prentice-Hall, 1999).
  35. A. Sripad and D. Snyder, “A necessary and sufficient condition for quantization errors to be uniform and white,” IEEE Trans. Acoust. Speech Signal Process. 25, 442–448 (1977).
    [CrossRef]
  36. J. Schoukens and J. Renneboog, “Modeling the noise influence on the Fourier coefficients after a discrete Fourier transform,” IEEE Trans. Instrum. Meas. 35, 278–286 (1986).
  37. R. Shiavi, Introduction to Applied Statistical Signal Analysis: Guide to Biomedical and Electrical Engineering Applications (Academic, 2007).
    [PubMed]
  38. J. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts & Co, 2007).
  39. O. A. Skydan, F. Lilley, M. J. Lalor, and D. R. Burton, “Quantization error of ccd cameras and their influence on phase calculation in fringe pattern analysis,” Appl. Opt. 42, 5302–5307 (2003).
    [CrossRef] [PubMed]
  40. H. Gudbjartsson and S. Patz, “The Rician distribution of noisy MRI data,” Magnetic Resonance Medicine 34, 910–914 (1995).
    [CrossRef]
  41. D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, 1960).
  42. J. Maycock, C. Elhinney, B. Hennelly, T. Naughton, J. McDonald, and B. Javidi, “Three-dimensional scene reconstruction of partially occluded objects using digital holograms,” Appl. Opt. 45, 2975–2985 (2006).
    [CrossRef] [PubMed]
  43. E. Buckley, “Holographic laser projection technology,” Information Display 24, 12 (2008).
  44. P. Hariharan and Z. Hegedus, “Reduction of speckle in coherent imaging by spatial frequency sampling,” J. Mod. Opt. 21, 345–356 (1974).
    [CrossRef]
  45. J. Dainty, “Laser speckle and related phenomena,” in Topics in Applied Physics (Springer-Verlag, 1975), Vol.  9, p. 298.
  46. J. W. Goodman, “Some fundamental properties of speckle,” J. Opt. Soc. Am. 66, 1145–1150 (1976).
    [CrossRef]
  47. J. Maycock, B. Hennelly, J. McDonald, Y. Frauel, A. Castro, B. Javidi, and T. Naughton, “Reduction of speckle in digital holography by discrete Fourier filtering,” J. Opt. Soc. Am. 24, 1617–1622 (2007).
    [CrossRef]
  48. J. Lim and H. Nawab, “Techniques for speckle noise removal,” Opt. Eng. 20, 472–480 (1981).
  49. T. Crimmins, “Geometric filter for speckle reduction,” Appl. Opt. 24, 1438–1443 (1985).
    [CrossRef] [PubMed]
  50. D. Monaghan, D. Kelly, N. Pandey, and B. Hennelly, “Twin removal in digital holography using diffuse illumination,” Opt. Lett. 34, 3610–3612 (2009).
    [CrossRef] [PubMed]
  51. F. Charriére, B. Rappaz, J. Kühn, T. Colomb, P. Marquet, and C. Depeursinge, “Influence of shot noise on phase measurement accuracy in digital holographic microscopy,” Opt. Express 15, 8818–8831 (2007).
    [CrossRef] [PubMed]
  52. M. Gross and M. Atlan, “Digital holography with ultimate sensitivity,” Opt. Lett. 32, 909–911 (2007).
    [CrossRef] [PubMed]
  53. J. Lukas, J. Fridrich, and M. Goljan, “Digital camera identification from sensor pattern noise,” IEEE Trans. Info. Foren. Sec. 1, 205–214 (2006).
    [CrossRef]

2010 (2)

2009 (2)

D. Kelly, B. Hennelly, N. Pandey, T. Naughton, W. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
[CrossRef]

D. Monaghan, D. Kelly, N. Pandey, and B. Hennelly, “Twin removal in digital holography using diffuse illumination,” Opt. Lett. 34, 3610–3612 (2009).
[CrossRef] [PubMed]

2008 (2)

H. Jin, H. Wan, Y. Zhang, Y. Li, and P. Qiu, “The influence of structural parameters of CCD on the reconstruction image of digital holograms,” J. Mod. Opt. 55, 2989–3000(2008).
[CrossRef]

E. Buckley, “Holographic laser projection technology,” Information Display 24, 12 (2008).

2007 (5)

2006 (7)

2005 (2)

2004 (1)

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

2003 (2)

2002 (2)

1995 (1)

H. Gudbjartsson and S. Patz, “The Rician distribution of noisy MRI data,” Magnetic Resonance Medicine 34, 910–914 (1995).
[CrossRef]

1994 (1)

1989 (2)

1987 (1)

1986 (1)

J. Schoukens and J. Renneboog, “Modeling the noise influence on the Fourier coefficients after a discrete Fourier transform,” IEEE Trans. Instrum. Meas. 35, 278–286 (1986).

1985 (1)

1981 (1)

J. Lim and H. Nawab, “Techniques for speckle noise removal,” Opt. Eng. 20, 472–480 (1981).

1977 (1)

A. Sripad and D. Snyder, “A necessary and sufficient condition for quantization errors to be uniform and white,” IEEE Trans. Acoust. Speech Signal Process. 25, 442–448 (1977).
[CrossRef]

1976 (1)

1975 (2)

1974 (1)

P. Hariharan and Z. Hegedus, “Reduction of speckle in coherent imaging by spatial frequency sampling,” J. Mod. Opt. 21, 345–356 (1974).
[CrossRef]

1972 (1)

1967 (2)

A. W. Lohmann and D. P. Paris, “Binary Fraunhofer holograms, generated by computer,” Appl. Opt. 6, 1739–1748(1967).
[CrossRef] [PubMed]

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

1964 (1)

L. Schuchman, “Dither signals and their effect on quantization noise,” IEEE Trans. Communication Technology 12, 162–165 (1964).
[CrossRef]

Aguet, F.

Allebach, J.

Aspert, N.

Atlan, M.

Baumbach, T.

Bertaux, N.

Bourquard, A.

Buckley, E.

E. Buckley, “Holographic laser projection technology,” Information Display 24, 12 (2008).

Burton, D. R.

Castro, A.

J. Maycock, B. Hennelly, J. McDonald, Y. Frauel, A. Castro, B. Javidi, and T. Naughton, “Reduction of speckle in digital holography by discrete Fourier filtering,” J. Opt. Soc. Am. 24, 1617–1622 (2007).
[CrossRef]

Charriere, F.

Charriére, F.

Colomb, T.

Crimmins, T.

Cuche, E.

Dainty, J.

J. Dainty, “Laser speckle and related phenomena,” in Topics in Applied Physics (Springer-Verlag, 1975), Vol.  9, p. 298.

Dallas, W.

Darakis, E.

Depeursinge, C.

Elhinney, C.

Emery, Y.

Frauel, Y.

Fridrich, J.

J. Lukas, J. Fridrich, and M. Goljan, “Digital camera identification from sensor pattern noise,” IEEE Trans. Info. Foren. Sec. 1, 205–214 (2006).
[CrossRef]

Goljan, M.

J. Lukas, J. Fridrich, and M. Goljan, “Digital camera identification from sensor pattern noise,” IEEE Trans. Info. Foren. Sec. 1, 205–214 (2006).
[CrossRef]

Goodman, J.

R. Powers and J. Goodman, “Error rates in computer-generated holographic memories,” Appl. Opt. 14, 1690–1701(1975).
[CrossRef] [PubMed]

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

J. Goodman, Introduction to Fourier Optics (Roberts & Co., 2005).

J. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts & Co, 2007).

Goodman, J. W.

Gotchev, A.

A. Gotchev and L. Onural, “A survey on sampling and quantization in diffraction and holography,” in Workshop on Spectral Methods and Multirate Signal Processing, SMMSP (2006), pp. 179–190.

Gross, M.

Gudbjartsson, H.

H. Gudbjartsson and S. Patz, “The Rician distribution of noisy MRI data,” Magnetic Resonance Medicine 34, 910–914 (1995).
[CrossRef]

Hariharan, P.

P. Hariharan and Z. Hegedus, “Reduction of speckle in coherent imaging by spatial frequency sampling,” J. Mod. Opt. 21, 345–356 (1974).
[CrossRef]

Hegedus, Z.

P. Hariharan and Z. Hegedus, “Reduction of speckle in coherent imaging by spatial frequency sampling,” J. Mod. Opt. 21, 345–356 (1974).
[CrossRef]

Hennelly, B.

Horner, J.

Javidi, B.

J. Maycock, B. Hennelly, J. McDonald, Y. Frauel, A. Castro, B. Javidi, and T. Naughton, “Reduction of speckle in digital holography by discrete Fourier filtering,” J. Opt. Soc. Am. 24, 1617–1622 (2007).
[CrossRef]

J. Maycock, C. Elhinney, B. Hennelly, T. Naughton, J. McDonald, and B. Javidi, “Three-dimensional scene reconstruction of partially occluded objects using digital holograms,” Appl. Opt. 45, 2975–2985 (2006).
[CrossRef] [PubMed]

A. Shortt, T. Naughton, and B. Javidi, “A companding approach for nonuniform quantization of digital holograms of three-dimensional objects,” Opt. Express 14, 5129–5134(2006).
[CrossRef] [PubMed]

A. Shortt, T. J. Naughton, and B. Javidi, “Compression of digital holograms of three-dimensional objects using wavelets,” Opt. Express 14, 2625–2630 (2006).
[CrossRef] [PubMed]

A. Shortt, T. Naughton, and B. Javidi, “Compression of optically encrypted digital holograms using artificial neural networks,” J. Display Technology 2, 401–410 (2006).
[CrossRef]

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

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

T. 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]

O. Matoba, T. 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]

B. Javidi and J. Horner, “Single spatial light modulator joint transform correlator,” Appl. Opt. 28, 1027–1032(1989).
[CrossRef] [PubMed]

Jin, H.

H. Jin, H. Wan, Y. Zhang, Y. Li, and P. Qiu, “The influence of structural parameters of CCD on the reconstruction image of digital holograms,” J. Mod. Opt. 55, 2989–3000(2008).
[CrossRef]

Jüptner, W.

Kelly, D.

D. Kelly, B. Hennelly, N. Pandey, T. Naughton, W. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
[CrossRef]

D. Monaghan, D. Kelly, N. Pandey, and B. Hennelly, “Twin removal in digital holography using diffuse illumination,” Opt. Lett. 34, 3610–3612 (2009).
[CrossRef] [PubMed]

Kronrod, M.

M. Kronrod, N. Merzlyakov, and L. Yaroslavskii, “Reconstruction of a hologram with a computer,” in SPIE Milestone Series 144 (SPIE Press, 1998), pp. 645–646.

Kühn, J.

Lalor, M. J.

Lawrence, R.

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

Li, Y.

H. Jin, H. Wan, Y. Zhang, Y. Li, and P. Qiu, “The influence of structural parameters of CCD on the reconstruction image of digital holograms,” J. Mod. Opt. 55, 2989–3000(2008).
[CrossRef]

Lilley, F.

Lim, J.

J. Lim and H. Nawab, “Techniques for speckle noise removal,” Opt. Eng. 20, 472–480 (1981).

Lohmann, A.

Lohmann, A. W.

Lukas, J.

J. Lukas, J. Fridrich, and M. Goljan, “Digital camera identification from sensor pattern noise,” IEEE Trans. Info. Foren. Sec. 1, 205–214 (2006).
[CrossRef]

Magistretti, P.

Marquet, P.

Matoba, O.

Maycock, J.

J. Maycock, B. Hennelly, J. McDonald, Y. Frauel, A. Castro, B. Javidi, and T. Naughton, “Reduction of speckle in digital holography by discrete Fourier filtering,” J. Opt. Soc. Am. 24, 1617–1622 (2007).
[CrossRef]

J. Maycock, C. Elhinney, B. Hennelly, T. Naughton, J. McDonald, and B. Javidi, “Three-dimensional scene reconstruction of partially occluded objects using digital holograms,” Appl. Opt. 45, 2975–2985 (2006).
[CrossRef] [PubMed]

McDonald, J.

Merzlyakov, N.

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

M. Kronrod, N. Merzlyakov, and L. Yaroslavskii, “Reconstruction of a hologram with a computer,” in SPIE Milestone Series 144 (SPIE Press, 1998), pp. 645–646.

Middleton, D.

D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, 1960).

Mills, G.

Monaghan, D.

Naidu, P.

P. Naidu, “Quantization noise in binary holograms,” Opt. Commun. 15, 361–365 (1975).
[CrossRef]

Naughton, T.

D. Kelly, B. Hennelly, N. Pandey, T. Naughton, W. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
[CrossRef]

J. Maycock, B. Hennelly, J. McDonald, Y. Frauel, A. Castro, B. Javidi, and T. Naughton, “Reduction of speckle in digital holography by discrete Fourier filtering,” J. Opt. Soc. Am. 24, 1617–1622 (2007).
[CrossRef]

E. Darakis, T. Naughton, and J. Soraghan, “Compression defects in different reconstructions from phase-shifting digital holographic data,” Appl. Opt. 46, 4579–4586 (2007).
[CrossRef] [PubMed]

A. Shortt, T. Naughton, and B. Javidi, “A companding approach for nonuniform quantization of digital holograms of three-dimensional objects,” Opt. Express 14, 5129–5134(2006).
[CrossRef] [PubMed]

A. Shortt, T. Naughton, and B. Javidi, “Compression of optically encrypted digital holograms using artificial neural networks,” J. Display Technology 2, 401–410 (2006).
[CrossRef]

J. Maycock, C. Elhinney, B. Hennelly, T. Naughton, J. McDonald, and B. Javidi, “Three-dimensional scene reconstruction of partially occluded objects using digital holograms,” Appl. Opt. 45, 2975–2985 (2006).
[CrossRef] [PubMed]

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

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

T. 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]

O. Matoba, T. 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]

Naughton, T. J.

Nawab, H.

J. Lim and H. Nawab, “Techniques for speckle noise removal,” Opt. Eng. 20, 472–480 (1981).

Onural, L.

A. Gotchev and L. Onural, “A survey on sampling and quantization in diffraction and holography,” in Workshop on Spectral Methods and Multirate Signal Processing, SMMSP (2006), pp. 179–190.

Oppenheim, A.

A. Oppenheim and R. Schafer, Discrete-Time Signal Processing (Prentice-Hall, 1999).

Osten, W.

Paek, E.

D. Psaltis, E. Paek, and S. Venkatesh, “Optical image correlation with a binary spatial light modulator,” in SPIE Milestone Series, 156 (SPIE Press, 1999), pp. 482–488.

Pandey, N.

Paris, D. P.

Patz, S.

H. Gudbjartsson and S. Patz, “The Rician distribution of noisy MRI data,” Magnetic Resonance Medicine 34, 910–914 (1995).
[CrossRef]

Poon, T.

T. Poon, Digital Holography and Three-Dimensional Display: Principles and Applications (Springer, 2006).
[CrossRef]

Powers, R.

Psaltis, D.

D. Psaltis, E. Paek, and S. Venkatesh, “Optical image correlation with a binary spatial light modulator,” in SPIE Milestone Series, 156 (SPIE Press, 1999), pp. 482–488.

Qiu, P.

H. Jin, H. Wan, Y. Zhang, Y. Li, and P. Qiu, “The influence of structural parameters of CCD on the reconstruction image of digital holograms,” J. Mod. Opt. 55, 2989–3000(2008).
[CrossRef]

Rappaz, B.

Renneboog, J.

J. Schoukens and J. Renneboog, “Modeling the noise influence on the Fourier coefficients after a discrete Fourier transform,” IEEE Trans. Instrum. Meas. 35, 278–286 (1986).

Rhodes, W.

D. Kelly, B. Hennelly, N. Pandey, T. Naughton, W. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
[CrossRef]

Schafer, R.

A. Oppenheim and R. Schafer, Discrete-Time Signal Processing (Prentice-Hall, 1999).

Schnars, U.

Schoukens, J.

J. Schoukens and J. Renneboog, “Modeling the noise influence on the Fourier coefficients after a discrete Fourier transform,” IEEE Trans. Instrum. Meas. 35, 278–286 (1986).

Schuchman, L.

L. Schuchman, “Dither signals and their effect on quantization noise,” IEEE Trans. Communication Technology 12, 162–165 (1964).
[CrossRef]

Seldowitz, M.

Shiavi, R.

R. Shiavi, Introduction to Applied Statistical Signal Analysis: Guide to Biomedical and Electrical Engineering Applications (Academic, 2007).
[PubMed]

Shortt, A.

Skydan, O. A.

Snyder, D.

A. Sripad and D. Snyder, “A necessary and sufficient condition for quantization errors to be uniform and white,” IEEE Trans. Acoust. Speech Signal Process. 25, 442–448 (1977).
[CrossRef]

Soraghan, J.

Sripad, A.

A. Sripad and D. Snyder, “A necessary and sufficient condition for quantization errors to be uniform and white,” IEEE Trans. Acoust. Speech Signal Process. 25, 442–448 (1977).
[CrossRef]

Sweeney, D.

Tajahuerce, E.

Unser, M.

Venkatesh, S.

D. Psaltis, E. Paek, and S. Venkatesh, “Optical image correlation with a binary spatial light modulator,” in SPIE Milestone Series, 156 (SPIE Press, 1999), pp. 482–488.

von Kopylow, C.

Wan, H.

H. Jin, H. Wan, Y. Zhang, Y. Li, and P. Qiu, “The influence of structural parameters of CCD on the reconstruction image of digital holograms,” J. Mod. Opt. 55, 2989–3000(2008).
[CrossRef]

Wyrowski, F.

Yamaguchi, I.

Yaroslavskii, L.

M. Kronrod, N. Merzlyakov, and L. Yaroslavskii, “Reconstruction of a hologram with a computer,” in SPIE Milestone Series 144 (SPIE Press, 1998), pp. 645–646.

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

Zhang, Y.

H. Jin, H. Wan, Y. Zhang, Y. Li, and P. Qiu, “The influence of structural parameters of CCD on the reconstruction image of digital holograms,” J. Mod. Opt. 55, 2989–3000(2008).
[CrossRef]

Appl. Opt. (17)

U. Schnars and W. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179–181 (1994).
[CrossRef] [PubMed]

T. Baumbach, W. Osten, C. von Kopylow, and W. Jüptner, “Remote metrology by comparative digital holography,” Appl. Opt. 45, 925–934 (2006).
[CrossRef] [PubMed]

T. 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]

T. Naughton, J. 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. 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]

G. Mills and I. Yamaguchi, “Effects of quantization in phase-shifting digital holography,” Appl. Opt. 44, 1216–1225(2005).
[CrossRef] [PubMed]

E. Darakis and J. Soraghan, “Reconstruction domain compression of phase-shifting digital holograms,” Appl. Opt. 46, 351–356 (2007).
[CrossRef] [PubMed]

E. Darakis, T. Naughton, and J. Soraghan, “Compression defects in different reconstructions from phase-shifting digital holographic data,” Appl. Opt. 46, 4579–4586 (2007).
[CrossRef] [PubMed]

W. Dallas and A. Lohmann, “Phase quantization in holograms-depth effects,” Appl. Opt. 11, 192–194 (1972).
[CrossRef] [PubMed]

A. W. Lohmann and D. P. Paris, “Binary Fraunhofer holograms, generated by computer,” Appl. Opt. 6, 1739–1748(1967).
[CrossRef] [PubMed]

F. Wyrowski, “Iterative quantization of digital amplitude holograms,” Appl. Opt. 28, 3864–3870 (1989).
[CrossRef] [PubMed]

R. Powers and J. Goodman, “Error rates in computer-generated holographic memories,” Appl. Opt. 14, 1690–1701(1975).
[CrossRef] [PubMed]

B. Javidi and J. Horner, “Single spatial light modulator joint transform correlator,” Appl. Opt. 28, 1027–1032(1989).
[CrossRef] [PubMed]

M. Seldowitz, J. Allebach, and D. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 26, 2788–2798 (1987).
[CrossRef] [PubMed]

J. Maycock, C. Elhinney, B. Hennelly, T. Naughton, J. McDonald, and B. Javidi, “Three-dimensional scene reconstruction of partially occluded objects using digital holograms,” Appl. Opt. 45, 2975–2985 (2006).
[CrossRef] [PubMed]

O. A. Skydan, F. Lilley, M. J. Lalor, and D. R. Burton, “Quantization error of ccd cameras and their influence on phase calculation in fringe pattern analysis,” Appl. Opt. 42, 5302–5307 (2003).
[CrossRef] [PubMed]

T. Crimmins, “Geometric filter for speckle reduction,” Appl. Opt. 24, 1438–1443 (1985).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

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

IEEE Trans. Acoust. Speech Signal Process. (1)

A. Sripad and D. Snyder, “A necessary and sufficient condition for quantization errors to be uniform and white,” IEEE Trans. Acoust. Speech Signal Process. 25, 442–448 (1977).
[CrossRef]

IEEE Trans. Communication Technology (1)

L. Schuchman, “Dither signals and their effect on quantization noise,” IEEE Trans. Communication Technology 12, 162–165 (1964).
[CrossRef]

IEEE Trans. Info. Foren. Sec. (1)

J. Lukas, J. Fridrich, and M. Goljan, “Digital camera identification from sensor pattern noise,” IEEE Trans. Info. Foren. Sec. 1, 205–214 (2006).
[CrossRef]

IEEE Trans. Instrum. Meas. (1)

J. Schoukens and J. Renneboog, “Modeling the noise influence on the Fourier coefficients after a discrete Fourier transform,” IEEE Trans. Instrum. Meas. 35, 278–286 (1986).

Information Display (1)

E. Buckley, “Holographic laser projection technology,” Information Display 24, 12 (2008).

J. Display Technology (1)

A. Shortt, T. Naughton, and B. Javidi, “Compression of optically encrypted digital holograms using artificial neural networks,” J. Display Technology 2, 401–410 (2006).
[CrossRef]

J. Mod. Opt. (2)

H. Jin, H. Wan, Y. Zhang, Y. Li, and P. Qiu, “The influence of structural parameters of CCD on the reconstruction image of digital holograms,” J. Mod. Opt. 55, 2989–3000(2008).
[CrossRef]

P. Hariharan and Z. Hegedus, “Reduction of speckle in coherent imaging by spatial frequency sampling,” J. Mod. Opt. 21, 345–356 (1974).
[CrossRef]

J. Opt. Soc. Am. (2)

J. W. Goodman, “Some fundamental properties of speckle,” J. Opt. Soc. Am. 66, 1145–1150 (1976).
[CrossRef]

J. Maycock, B. Hennelly, J. McDonald, Y. Frauel, A. Castro, B. Javidi, and T. Naughton, “Reduction of speckle in digital holography by discrete Fourier filtering,” J. Opt. Soc. Am. 24, 1617–1622 (2007).
[CrossRef]

Magnetic Resonance Medicine (1)

H. Gudbjartsson and S. Patz, “The Rician distribution of noisy MRI data,” Magnetic Resonance Medicine 34, 910–914 (1995).
[CrossRef]

Opt. Commun. (1)

P. Naidu, “Quantization noise in binary holograms,” Opt. Commun. 15, 361–365 (1975).
[CrossRef]

Opt. Eng. (3)

D. Kelly, B. Hennelly, N. Pandey, T. Naughton, W. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. 48, 095801 (2009).
[CrossRef]

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

J. Lim and H. Nawab, “Techniques for speckle noise removal,” Opt. Eng. 20, 472–480 (1981).

Opt. Express (5)

Opt. Lett. (4)

Other (11)

D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, 1960).

J. Dainty, “Laser speckle and related phenomena,” in Topics in Applied Physics (Springer-Verlag, 1975), Vol.  9, p. 298.

M. Kronrod, N. Merzlyakov, and L. Yaroslavskii, “Reconstruction of a hologram with a computer,” in SPIE Milestone Series 144 (SPIE Press, 1998), pp. 645–646.

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

T. Poon, Digital Holography and Three-Dimensional Display: Principles and Applications (Springer, 2006).
[CrossRef]

A. Gotchev and L. Onural, “A survey on sampling and quantization in diffraction and holography,” in Workshop on Spectral Methods and Multirate Signal Processing, SMMSP (2006), pp. 179–190.

D. Psaltis, E. Paek, and S. Venkatesh, “Optical image correlation with a binary spatial light modulator,” in SPIE Milestone Series, 156 (SPIE Press, 1999), pp. 482–488.

A. Oppenheim and R. Schafer, Discrete-Time Signal Processing (Prentice-Hall, 1999).

R. Shiavi, Introduction to Applied Statistical Signal Analysis: Guide to Biomedical and Electrical Engineering Applications (Academic, 2007).
[PubMed]

J. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts & Co, 2007).

J. Goodman, Introduction to Fourier Optics (Roberts & Co., 2005).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (16)

Fig. 1
Fig. 1

Setup for Fourier digital holography.

Fig. 2
Fig. 2

(a) Original hologram. (b) Histogram showing the grayscale value and the corresponding number of pixels. (c), (d) Reconstructed image.

Fig. 3
Fig. 3

Illustration of the notation used between the hologram plane and the reconstruction (Fourier) plane. Q l stands for the quantization noise in the hologram plane and q l stands for the quantization noise in the Fourier domain. q l is complex and the reconstruction is the complex sum of the ideal reconstruction h and the noise q l .

Fig. 4
Fig. 4

Histograms for various quantization levels and the corresponding reconstructed images. (a) and (b)  12   bits , (c) and (d)  8   bits , (e) and (f)  4   bits , (g) and (h)  1   bit (binary) cases.

Fig. 5
Fig. 5

Standard deviation of the quantization noise in the hologram plane versus the theoretical prediction.

Fig. 6
Fig. 6

Quantization of a sinusoidal signal and the corresponding quantization noise for three different cases. (a) and (b)  5   bits (b) and (c)  2   bits and (d) and (e)  1   bit . It can be seen that for 5   bits the quantization noise is almost randomly varying as in (b), and for the 1   bit case in (e), it has the particular characteristic frequency of the original signal.

Fig. 7
Fig. 7

Normalized histograms of the quanti zation noise in the hologram plane. (a)  8   bits . (b)  6   bits . (c)  4   bits . (d)  1   bit .

Fig. 8
Fig. 8

For low bit quantization like 1   bit , the quantization noise in the hologram plane is not uniform and white. This can be seen in the reconstruction from the quantization noise for four different cases. (a)  8   bits , (b)  6   bits , (c)  4   bits , (d)  1   bit . The reconstruction of the noise for 1   bit is not random and the object can be seen in it.

Fig. 9
Fig. 9

Theoretical and experimental probability distributions for various parts of the complex quantization noise in the Fourier plane for 4   bit quantization. (a) histogram of quantization noise in hologram plane. (b) distribution of real part of complex noise. (c) distribution of imaginary part of complex noise. (d) distribution of phase (e) distribution of amplitude. (f) distribution of intensity.

Fig. 10
Fig. 10

Complex quantization noise, q l , is distributed with a Rayleigh distribution and adds on to the original amplitude vector. The net phase detected is the angle subtended by the complex vector sum of the original signal and the complex noise vector.

Fig. 11
Fig. 11

(a) Recorded phase hologram of the a lens, (b) Fourier reconstruction, (c) amplitude in a 200 × 250 section of the reconstructed object, (d) phase distribution in the same area as (c).

Fig. 12
Fig. 12

Error distribution in phase π to + π for (a)  8   bits , (b)  6   bits , (c)  4   bits , (d)  1   bit .

Fig. 13
Fig. 13

Standard deviation of the error in reconstructed phase, θ err for different levels of quantization of the recorded phase hologram.

Fig. 14
Fig. 14

Standard deviation in a uniform section of the intensity of the reconstruction plane. We can see that the simple model above can be used to predict accurately the quantization error upto 2   bits (4 levels) of quantization. The deviation at 1   bit is due to the nonwhite nature of quantization noise in the hologram plane at binary quantization. In this case, the noise term has a pattern and structure similar to the original signal.

Fig. 15
Fig. 15

Results for speckle reduction on the 1   bit holograms (a) four diffuse holograms added together, (b) 16 patterns, (c) 36 patterns, (d) speckle index in the image.

Fig. 16
Fig. 16

Comparison of reconstructions from (a) a single 12   bit hologram and (b) reconstructions obtained from 12 1   bit holograms. The two reconstructions involve the same amount of memory usage (in bits).

Tables (1)

Tables Icon

Table 1 Probability Densities of Different Parts of Quantization Noise in the Fourier Domain

Equations (25)

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

H ( x ) = | O ( x ) | 2 + | R ( x ) | 2 + O ( x ) R ( x ) * + R ( x ) O ( x ) *
O ( x ) = exp ( j π x 2 λ d ) F { o ( x ) ( exp j π x 2 λ d ) } ( x ) .
F { H ( x ) } ( x ) = F { | O ( x ) | 2 } ( x ) + δ ( x ) + o * ( ξ x ) exp ( j π ( ξ x ) 2 λ d ) + o ( x ξ ) exp ( j π ( x ξ ) 2 λ d ) .
Δ l = M x M n 2 l ,
H l = H + Q l .
σ Q l 2 = Δ l / 2 Δ l / 2 x 2 ( 1 Δ l ) d x = Δ l 2 12 .
σ Q l = Δ l 12 = M x M n 2 l 12 .
h l = DFT ( H l ) = DFT ( H + Q l ) = h + q l .
q l ( k ) = R l ( k ) + i I l ( k ) ,
R l ( k ) = 1 N n = 0 N 1 Q l ( n ) cos ( 2 π k n N ) ,
I l ( k ) = 1 N n = 0 N 1 Q l ( n ) sin ( 2 π k n N ) .
E [ R l ] = 1 N n = 0 N 1 E [ Q l ( n ) cos ( 2 π k n N ) ] = 1 N n = 0 N 1 E [ Q l ( n ) ] E [ cos ( 2 π k n N ) ] = 0 ,
E [ I l ] = 1 N n = 0 N 1 E [ Q l ( n ) sin ( 2 π k n N ) ] = 1 N n = 0 N 1 E [ Q l ( n ) ] E [ sin ( 2 π k n N ) ] = 0 ,
σ R l 2 = E [ R l 2 ] = 1 N n = 0 N 1 m = 0 N 1 E [ Q l ( n ) Q l ( m ) ] E [ cos ( 2 π k n N ) cos ( 2 π k m N ) ] .
σ R l 2 = n = 0 N 1 1 N E [ Q l ( n ) 2 ] E [ cos 2 ϕ n ] = 1 N n = 0 N 1 E [ Q l ( n ) 2 ] E [ 1 2 + 1 2 cos 2 ϕ n ] = 1 N n = 0 N 1 E [ Q l ( n ) 2 ] 2 = σ Q l 2 2 .
q l = h 12 h l .
θ err = S n S o .
θ err = S n S o = [ S n exp ( i S o ) ] = [ A + N exp ( i S o ) ] .
p ( θ ) = 1 2 π exp ( A 2 2 σ R 2 ) { 1 + A σ R 2 π cos θ exp ( A 2 cos 2 θ 2 σ R 2 ) [ 1 Φ ( A cos θ σ R ) ] } ,
Φ ( x ) = 1 2 π x exp ( x 2 / 2 ) d x ,
p ( θ err ) = A 2 π σ R exp ( θ err 2 A 2 2 σ R 2 ) .
σ θ err = M x M n 2 l 24 A = σ Q l 2 A ,
| h l | 2 = | h + q l | 2 = | h | 2 + | q l | 2 + h q l * + h * q l ,
| h l | 2 = | h | 2 + | q l | 2 + 2 ( Real [ h ] Real [ q l ] Imag [ h ] Imag [ q l ] ) .
σ | h l | 2 2 = σ | h | 2 2 + σ | q l | 2 2 + 4 σ Real ( | h | ) 2 σ Imag ( | q l | ) 2 + 4 σ Imag ( | h | ) 2 σ Real ( | q l | ) 2 .

Metrics