Abstract

We present the results of what we believe is the first application of wavelet analysis to the compression of complex-valued digital holograms of three-dimensional real-world objects. We achieve compression through thresholding and quantization of the wavelet coefficients, followed by lossless encoding of the quantized data.

© 2006 Optical Society of America

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  1. J. W. Goodman and R. W. Lawrence, "Digital image formation from electronically detected holograms," Appl. Phys. Lett. 11, 77-79 (1967).
    [CrossRef]
  2. M. A. Kronod, N. S. Merzlyakov, and L. P. Yaroslavskii, "Reconstruction of a hologram with a computer," Sov. Phys. Tech. Phys. 17, 333-334 (1972).
  3. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, "Digital wavefront measuring interferometer for testing optical surfaces and lenses," Appl. Opt. 13, 2693-2703 (1974).
    [CrossRef] [PubMed]
  4. L. Onural and P. D. Scott, "Digital decoding of in-line holograms," Opt. Eng. 26, 1124-1132 (1987).
  5. U . Schnars and W. P. O. Jüptner, "Direct recording of holograms by a CCD target and numerical reconstruction," Appl. Opt. 33, 179-181 (1994).
    [CrossRef] [PubMed]
  6. I. Yamaguchi and T. Zhang, "Phase-shifting digital holography," Opt. Lett. 22, 1268-1270 (1997).
    [CrossRef] [PubMed]
  7. E. Cuche, F. Bevilacqua, and C. Depeursinge, "Digital holography for quantitative phase-contrast imaging," Opt. Lett. 24, 291-293 (1999).
    [CrossRef]
  8. T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley-VCH, Berlin, 2004).
    [CrossRef]
  9. B. Javidi and E. Tajahuerce, "Three-dimensional object recognition by use of digital holography," Opt. Lett. 25, 610-612 (2000).
    [CrossRef]
  10. 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]
  11. 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]
  12. B. Javidi and F. Okano, Three Dimensional Television, Video, and Display Technologies (Springer, Berlin, 2002).
  13. 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]
  14. H. J. Caulfield, ed., Handbook of Optical Holography (Academic Press, New York, 1979).
  15. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, Englewood, Colorado, 2005).
  16. M. Burrows and D. J. Wheeler, "A block-sorting lossless data compression algorithm," Tech. Rep. 124, Digital Systems Research Center, Palo Alto, California (1994).
  17. W. J. Dallas and A. W. Lohmann, "Phase quantization in holograms," Appl. Opt. 11, 192-194 (1972).
    [CrossRef] [PubMed]
  18. T. J. Naughton, J. B. Mc Donald, and B. Javidi, "Efficient compression of Fresnel fields for Internet transmission of three-dimensional images," Appl. Opt. 23, 4758-4764 (2003).
    [CrossRef]
  19. T. J. Naughton and B. Javidi, "Compression of encrypted three-dimensional objects using digital holography," Opt. Eng. 43, 2233-2238 (2004).
    [CrossRef]
  20. T. Nomura, A. Okazaki, M. Kameda, Y. Morimoto, and B. Javidi, "Image reconstruction from compressed encrypted digital hologram," Opt. Eng. 44, 075801 (2005).
    [CrossRef]
  21. D. Kayser, T. Kreis, and W. J ptner, "Compression of digital holographic data using its electromagnetic field properties," Optical Information Systems III, B. Javidi, and D. Psaltis, Eds, Proc. SPIE 5908, 97-105, (2005).
  22. I. Daubechies, Ten Lectures on Wavelets (Capital City Press, Vermont, 1992).
    [CrossRef]
  23. S. Mallat, "A theory for multiresolution signal decomposition: the wavelet representation," IEEE Trans. Pattern Anal. Mach. Intell. 11, 674-693 (1989).
    [CrossRef]
  24. R. DeVore, B. Jawerth, and B. Lucier, "Image compression through wavelet transform coding," IEEE Trans. Inform. Theory 38, 719-746 (1992).
    [CrossRef]
  25. L. Onural, "Diffraction from a wavelet point of view," Opt. Lett. 18, 846-848 (1993).
    [CrossRef] [PubMed]
  26. Y. Sheng, S. Desch enes, and H. J. Caulfield, "Monochromatic electromagnetic wavelets and the Huygens principle," Appl. Opt. 37, 828-833 (1998).
    [CrossRef]
  27. M. Liebling, T. Blu, and M. Unser, "Fresnelets: new multiresolution wavelet bases for digital holography," IEEE Trans. Image Process. 12, 29-43 (2003).
    [CrossRef]
  28. M. Liebling, T. Blu, and M. Unser, "Complex-wave retrieval from a single off-axis hologram," J. Opt. Soc. Am. A 21, 367-377 (2004).
    [CrossRef]
  29. L. Zeng, C. P. Jansen, S. Marsch, M. Unser, and P. R. Hunziker, "Wavelet compression of four-dimensional arbitrarily size echocardiographic data," IEEE Trans. Med. Imaging 21, 1179-1187 (2002).
    [CrossRef]
  30. Z. Zeng and I. G. Cumming, "SAR image data compression using a tree-structured wavelet transform," IEEE Trans. Geoscience and Remote Sensing 39, 546-552 (2001).
    [CrossRef]

2005

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

2004

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

M. Liebling, T. Blu, and M. Unser, "Complex-wave retrieval from a single off-axis hologram," J. Opt. Soc. Am. A 21, 367-377 (2004).
[CrossRef]

2003

T. J. Naughton, J. B. Mc Donald, and B. Javidi, "Efficient compression of Fresnel fields for Internet transmission of three-dimensional images," Appl. Opt. 23, 4758-4764 (2003).
[CrossRef]

M. Liebling, T. Blu, and M. Unser, "Fresnelets: new multiresolution wavelet bases for digital holography," IEEE Trans. Image Process. 12, 29-43 (2003).
[CrossRef]

2002

2001

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]

Z. Zeng and I. G. Cumming, "SAR image data compression using a tree-structured wavelet transform," IEEE Trans. Geoscience and Remote Sensing 39, 546-552 (2001).
[CrossRef]

2000

1999

1998

1997

1994

1993

1992

R. DeVore, B. Jawerth, and B. Lucier, "Image compression through wavelet transform coding," IEEE Trans. Inform. Theory 38, 719-746 (1992).
[CrossRef]

1989

S. Mallat, "A theory for multiresolution signal decomposition: the wavelet representation," IEEE Trans. Pattern Anal. Mach. Intell. 11, 674-693 (1989).
[CrossRef]

1987

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

1974

1972

W. J. Dallas and A. W. Lohmann, "Phase quantization in holograms," Appl. Opt. 11, 192-194 (1972).
[CrossRef] [PubMed]

M. A. Kronod, N. S. Merzlyakov, and L. P. Yaroslavskii, "Reconstruction of a hologram with a computer," Sov. Phys. Tech. Phys. 17, 333-334 (1972).

1967

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

Bertaux, N.

Bevilacqua, F.

Blu, T.

M. Liebling, T. Blu, and M. Unser, "Complex-wave retrieval from a single off-axis hologram," J. Opt. Soc. Am. A 21, 367-377 (2004).
[CrossRef]

M. Liebling, T. Blu, and M. Unser, "Fresnelets: new multiresolution wavelet bases for digital holography," IEEE Trans. Image Process. 12, 29-43 (2003).
[CrossRef]

Brangaccio, D. J.

Bruning, J. H.

Castro, M. -A.

Cuche, E.

Cumming, I. G.

Z. Zeng and I. G. Cumming, "SAR image data compression using a tree-structured wavelet transform," IEEE Trans. Geoscience and Remote Sensing 39, 546-552 (2001).
[CrossRef]

Dallas, W. J.

Depeursinge, C.

DeVore, R.

R. DeVore, B. Jawerth, and B. Lucier, "Image compression through wavelet transform coding," IEEE Trans. Inform. Theory 38, 719-746 (1992).
[CrossRef]

Frauel, Y.

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]

Herriott, D. R.

Hunziker, P. R.

L. Zeng, C. P. Jansen, S. Marsch, M. Unser, and P. R. Hunziker, "Wavelet compression of four-dimensional arbitrarily size echocardiographic data," IEEE Trans. Med. Imaging 21, 1179-1187 (2002).
[CrossRef]

Jansen, C. P.

L. Zeng, C. P. Jansen, S. Marsch, M. Unser, and P. R. Hunziker, "Wavelet compression of four-dimensional arbitrarily size echocardiographic data," IEEE Trans. Med. Imaging 21, 1179-1187 (2002).
[CrossRef]

Javidi, B.

Jawerth, B.

R. DeVore, B. Jawerth, and B. Lucier, "Image compression through wavelet transform coding," IEEE Trans. Inform. Theory 38, 719-746 (1992).
[CrossRef]

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]

Kronod, M. A.

M. A. Kronod, N. S. Merzlyakov, and L. P. Yaroslavskii, "Reconstruction of a hologram with a computer," Sov. Phys. Tech. Phys. 17, 333-334 (1972).

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]

Liebling, M.

M. Liebling, T. Blu, and M. Unser, "Complex-wave retrieval from a single off-axis hologram," J. Opt. Soc. Am. A 21, 367-377 (2004).
[CrossRef]

M. Liebling, T. Blu, and M. Unser, "Fresnelets: new multiresolution wavelet bases for digital holography," IEEE Trans. Image Process. 12, 29-43 (2003).
[CrossRef]

Lohmann, A. W.

Lucier, B.

R. DeVore, B. Jawerth, and B. Lucier, "Image compression through wavelet transform coding," IEEE Trans. Inform. Theory 38, 719-746 (1992).
[CrossRef]

Mallat, S.

S. Mallat, "A theory for multiresolution signal decomposition: the wavelet representation," IEEE Trans. Pattern Anal. Mach. Intell. 11, 674-693 (1989).
[CrossRef]

Marsch, S.

L. Zeng, C. P. Jansen, S. Marsch, M. Unser, and P. R. Hunziker, "Wavelet compression of four-dimensional arbitrarily size echocardiographic data," IEEE Trans. Med. Imaging 21, 1179-1187 (2002).
[CrossRef]

Matoba, O.

Mc Donald, J. B.

T. J. Naughton, J. B. Mc Donald, and B. Javidi, "Efficient compression of Fresnel fields for Internet transmission of three-dimensional images," Appl. Opt. 23, 4758-4764 (2003).
[CrossRef]

Merzlyakov, N. S.

M. A. Kronod, N. S. Merzlyakov, and L. P. Yaroslavskii, "Reconstruction of a hologram with a computer," Sov. Phys. Tech. Phys. 17, 333-334 (1972).

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. Mc Donald, and B. Javidi, "Efficient compression of Fresnel fields for Internet transmission of three-dimensional images," Appl. Opt. 23, 4758-4764 (2003).
[CrossRef]

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]

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]

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, "Diffraction from a wavelet point of view," Opt. Lett. 18, 846-848 (1993).
[CrossRef] [PubMed]

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

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).

Sheng, Y.

Tajahuerce, E.

Unser, M.

M. Liebling, T. Blu, and M. Unser, "Complex-wave retrieval from a single off-axis hologram," J. Opt. Soc. Am. A 21, 367-377 (2004).
[CrossRef]

M. Liebling, T. Blu, and M. Unser, "Fresnelets: new multiresolution wavelet bases for digital holography," IEEE Trans. Image Process. 12, 29-43 (2003).
[CrossRef]

L. Zeng, C. P. Jansen, S. Marsch, M. Unser, and P. R. Hunziker, "Wavelet compression of four-dimensional arbitrarily size echocardiographic data," IEEE Trans. Med. Imaging 21, 1179-1187 (2002).
[CrossRef]

White, A. D.

Yamaguchi, I.

Yaroslavskii, L. P.

M. A. Kronod, N. S. Merzlyakov, and L. P. Yaroslavskii, "Reconstruction of a hologram with a computer," Sov. Phys. Tech. Phys. 17, 333-334 (1972).

Zeng, L.

L. Zeng, C. P. Jansen, S. Marsch, M. Unser, and P. R. Hunziker, "Wavelet compression of four-dimensional arbitrarily size echocardiographic data," IEEE Trans. Med. Imaging 21, 1179-1187 (2002).
[CrossRef]

Zeng, Z.

Z. Zeng and I. G. Cumming, "SAR image data compression using a tree-structured wavelet transform," IEEE Trans. Geoscience and Remote Sensing 39, 546-552 (2001).
[CrossRef]

Zhang, T.

Appl. Opt.

T. J. Naughton, J. B. Mc Donald, and B. Javidi, "Efficient compression of Fresnel fields for Internet transmission of three-dimensional images," Appl. Opt. 23, 4758-4764 (2003).
[CrossRef]

W. J. Dallas and A. W. Lohmann, "Phase quantization in holograms," Appl. Opt. 11, 192-194 (1972).
[CrossRef] [PubMed]

J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, "Digital wavefront measuring interferometer for testing optical surfaces and lenses," Appl. Opt. 13, 2693-2703 (1974).
[CrossRef] [PubMed]

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

Y. Sheng, S. Desch enes, and H. J. Caulfield, "Monochromatic electromagnetic wavelets and the Huygens principle," Appl. Opt. 37, 828-833 (1998).
[CrossRef]

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]

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]

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]

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]

IEEE Trans. Geoscience and Remote Sensing

Z. Zeng and I. G. Cumming, "SAR image data compression using a tree-structured wavelet transform," IEEE Trans. Geoscience and Remote Sensing 39, 546-552 (2001).
[CrossRef]

IEEE Trans. Image Process.

M. Liebling, T. Blu, and M. Unser, "Fresnelets: new multiresolution wavelet bases for digital holography," IEEE Trans. Image Process. 12, 29-43 (2003).
[CrossRef]

IEEE Trans. Inform. Theory

R. DeVore, B. Jawerth, and B. Lucier, "Image compression through wavelet transform coding," IEEE Trans. Inform. Theory 38, 719-746 (1992).
[CrossRef]

IEEE Trans. Med. Imaging

L. Zeng, C. P. Jansen, S. Marsch, M. Unser, and P. R. Hunziker, "Wavelet compression of four-dimensional arbitrarily size echocardiographic data," IEEE Trans. Med. Imaging 21, 1179-1187 (2002).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell.

S. Mallat, "A theory for multiresolution signal decomposition: the wavelet representation," IEEE Trans. Pattern Anal. Mach. Intell. 11, 674-693 (1989).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Eng.

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]

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

Opt. Lett.

Sov. Phys. Tech. Phys.

M. A. Kronod, N. S. Merzlyakov, and L. P. Yaroslavskii, "Reconstruction of a hologram with a computer," Sov. Phys. Tech. Phys. 17, 333-334 (1972).

Other

T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley-VCH, Berlin, 2004).
[CrossRef]

B. Javidi and F. Okano, Three Dimensional Television, Video, and Display Technologies (Springer, Berlin, 2002).

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

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

M. Burrows and D. J. Wheeler, "A block-sorting lossless data compression algorithm," Tech. Rep. 124, Digital Systems Research Center, Palo Alto, California (1994).

D. Kayser, T. Kreis, and W. J ptner, "Compression of digital holographic data using its electromagnetic field properties," Optical Information Systems III, B. Javidi, and D. Psaltis, Eds, Proc. SPIE 5908, 97-105, (2005).

I. Daubechies, Ten Lectures on Wavelets (Capital City Press, Vermont, 1992).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

Vertical bars illustrating minimum and maximum values at each level of approximation for 20 resolution levels using the Haar wavelet for (a) die and (b) bolt, and each level of detail for (c) die and (d) bolt. x-axis: resolution level, y-axis: value at each level.

Fig. 2.
Fig. 2.

NRMS error of the reconstructed object plotted against compression ratio for the die object for wavelet levels 1, 3, 10 and 20 for (a) bior3.7, (b) Haar, and (c) db4.

Fig. 3.
Fig. 3.

NRMS error of the reconstructed object plotted against compression ratio for (a) die, and (b) bolt, for uniform quantization and 7 mother wavelets with 3 resolution levels.

Fig. 4.
Fig. 4.

Reconstructions of the die and bolt objects, respectively, with (a),(d) 2 bits, (b),(e) 3 bits, and (c),(f) 4 bits uniform quantization of the Haar wavelet coefficients.

Fig. 5.
Fig. 5.

Scatter plots of the complex-valued Haar wavelet coefficients for the die object. Each point in each complex plane represents the value of a thresholded wavelet coefficient. Plots show the distribution of values before and after quantization: (a) approximation, (b) level 3 details, (c) level 2 details, and (d) level 1 details, for a complex threshold of ±0.35±0.35i.

Fig. 6.
Fig. 6.

NRMS difference of the reconstructed object plotted against compression ratio for hologram (a) die, and (b) bolt, for thresholding and 4, 3 and 2 bit uniform quantization applied directly to the hologram data and seven mother wavelets all with 3 resolution levels. ● uniform., □ bior3.7., ◇ coif5., △ db4., x dmey., ∇ Haar., ▷ rbio3.7., ◁ sym5.

Fig. 7.
Fig. 7.

NRMS difference of the reconstructed object plotted against compression ratio for die hologram for thresholding and (a) 4 bits, (b) 3 bits, and (c) 2 bits uniform quantization applied directly to hologram data and seven mother wavelets all with 3 resolution levels.

Equations (5)

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

H′ x y = round { H x y × σ 1 × [ 2 ( b 1 ) 1 ] }
σ = max { min [ Im ( H ) ] , max [ Im ( H ) ] , min [ Re ( H ) ] , max [ Re ( H ) ] } .
D = [ m = 0 N x 1 n = 0 N y 1 { U m n U′ m n } 2 × ( m = 0 N x 1 n = 0 N y 1 U m n 2 ) 1 ] 1 / 2 ,
g t d = f ( x ) ψ t , d * ( x ) dx ,
ψ t , d ( x ) = d 0.5 ψ [ ( x t ) / d ] ,

Metrics