Abstract

The cosine function is a heavy computational operation in computer-generated hologram (CGH) calculation; therefore, it is implemented by substitution methods such as a look-up table. However, the computational load and required memory space of such methods are still large. In this study, we propose a simple and fast cosine function approximation method for CGH calculation. As a result, we succeeded in creating CGH with sufficient quality and made the calculation time 1.6 times as fast at maximum compared to using the look-up table of the cosine function on CPU implementation.

© 2015 Optical Society of America

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References

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  1. H. Sasaki, K. Yamamoto, Y. Ichihashi, and T. Senoh, “Image size scalable full-parallax coloured three-dimensional video by electronic holography,” Sci. Rep. 4, 4000 (2014).
    [PubMed]
  2. K. Matsushima and M. Takai, “Recurrence formulas for fast creation of synthetic three-dimensional holograms,” Appl. Opt. 39, 6587–6594 (2000).
    [Crossref]
  3. T. Shimobaba and T. Ito, “An efficient computational method suitable for hardware of computer-generated hologram with phase computation by addition,” Comput. Phys. Commun. 138, 44–52 (2001).
    [Crossref]
  4. H. Yoshikawa, “Fast computation of Fresnel holograms employing difference,” Opt. Rev. 8, 331–335 (2001).
    [Crossref]
  5. T. Ito, N. Masuda, K. Yoshimura, A. Shiraki, T. Shimobaba, and T. Sugie, “Special-purpose computer HORN-5 for a real-time electroholography,” Opt. Express 13, 1923–1932 (2005).
    [Crossref] [PubMed]
  6. Y. Ichihashi, H. Nakayama, T. Ito, N. Masuda, T. Shimobaba, A. Shiraki, and T. Sugie, “HORN-6 special-purpose clustered computing system for electroholography,” Opt. Express 17, 13895–13903 (2009).
    [Crossref] [PubMed]
  7. Y. Kimura, R. Kawaguchi, T. Sugie, T. Kakue, T. Shimobaba, and T. Ito, “Circuit design of special-purpose computer for holography HORN-8 using eight virtex-5 FPGAs,” in Proceedings of 3D Systems and Applicatons (3DSA, 2015), S3–2.
  8. J. Weng, T. Shimobaba, M. Oikawa, N. Masuda, and T. Ito, “Fast recurrence relation for computer-generated-hologram,” Comput. Phys. Commun. 183, 46–59 (2012).
    [Crossref]
  9. M. Lucente, “Interactive computation of holograms using a look-up table,” J. Electron. Imaging 2, 28–34 (1993).
    [Crossref]
  10. S. Kim, J. Kim, and E. Kim, “Effective memory reduction of the novel look-up table with one-dimensional sub-principle fringe patterns in computer-generated holograms,” Opt. Express 20, 12021–12034 (2012).
    [Crossref] [PubMed]
  11. S. Kim and E. Kim, “Fast one-step calculation of holographic videos of three-dimensional scenes by combined use of baseline and depth-compensating principal fringe patterns,” Opt. Express 22, 22513–22527 (2014).
    [Crossref] [PubMed]
  12. X. Dong, S.-C. Kim, and E.-S. Kim, “Three-directional motion compensation-based novel-look-up-table for video hologram generation of three-dimensional objects freely maneuvering in space,” Opt. Express 22, 16925–16944 (2014)
    [Crossref] [PubMed]
  13. T. Nishitsuji, T. Shimobaba, T. Kakue, N. Masuda, and T. Ito, “Fast calculation of computer-generated hologram using the circular symmetry of zone plates,” Opt. Express 20, 27496–27502 (2012).
    [Crossref] [PubMed]
  14. T. Nishitsuji, T. Shimobaba, T. Kakue, and T. Ito, “Fast calculation of computer-generated hologram using run-length encoding based recurrence relation,” Opt. Express 23, 9852–9857 (2015).
    [Crossref] [PubMed]
  15. E. Seevinc, “Simple, wide-range approximations to trigonometric and inverse trigonometric functions useful in real-time signal processing,” in IEE Proceedings G (Electronic Circuits and Systems) (IEE, 1981), 1, pp. 41–45
    [Crossref]
  16. J. A. McIntosh and E. E. Swartzlander, “High-Speed Cosine Generator,” in Proceedings of 1994 28th Asilomar Conference on Signals, Systems and Computers (IEEE, 1994), pp. 273–277.
    [Crossref]
  17. T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
    [Crossref]
  18. R. Gomes, W. Junior, E. Cerqueira, and A. Abelem, “A QoE Fuzzy Routing Protocol for Wireless Mesh Networks,” in Proceedings of the International Workshop, S. Zeadally, E. Cerqueira, M. Curado, and M. Leszczuk, eds. (Springer, 2010).

2015 (1)

2014 (3)

2012 (4)

J. Weng, T. Shimobaba, M. Oikawa, N. Masuda, and T. Ito, “Fast recurrence relation for computer-generated-hologram,” Comput. Phys. Commun. 183, 46–59 (2012).
[Crossref]

T. Nishitsuji, T. Shimobaba, T. Kakue, N. Masuda, and T. Ito, “Fast calculation of computer-generated hologram using the circular symmetry of zone plates,” Opt. Express 20, 27496–27502 (2012).
[Crossref] [PubMed]

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

S. Kim, J. Kim, and E. Kim, “Effective memory reduction of the novel look-up table with one-dimensional sub-principle fringe patterns in computer-generated holograms,” Opt. Express 20, 12021–12034 (2012).
[Crossref] [PubMed]

2009 (1)

2005 (1)

2001 (2)

T. Shimobaba and T. Ito, “An efficient computational method suitable for hardware of computer-generated hologram with phase computation by addition,” Comput. Phys. Commun. 138, 44–52 (2001).
[Crossref]

H. Yoshikawa, “Fast computation of Fresnel holograms employing difference,” Opt. Rev. 8, 331–335 (2001).
[Crossref]

2000 (1)

1993 (1)

M. Lucente, “Interactive computation of holograms using a look-up table,” J. Electron. Imaging 2, 28–34 (1993).
[Crossref]

Abelem, A.

R. Gomes, W. Junior, E. Cerqueira, and A. Abelem, “A QoE Fuzzy Routing Protocol for Wireless Mesh Networks,” in Proceedings of the International Workshop, S. Zeadally, E. Cerqueira, M. Curado, and M. Leszczuk, eds. (Springer, 2010).

Cerqueira, E.

R. Gomes, W. Junior, E. Cerqueira, and A. Abelem, “A QoE Fuzzy Routing Protocol for Wireless Mesh Networks,” in Proceedings of the International Workshop, S. Zeadally, E. Cerqueira, M. Curado, and M. Leszczuk, eds. (Springer, 2010).

Dong, X.

Gomes, R.

R. Gomes, W. Junior, E. Cerqueira, and A. Abelem, “A QoE Fuzzy Routing Protocol for Wireless Mesh Networks,” in Proceedings of the International Workshop, S. Zeadally, E. Cerqueira, M. Curado, and M. Leszczuk, eds. (Springer, 2010).

Ichihashi, Y.

H. Sasaki, K. Yamamoto, Y. Ichihashi, and T. Senoh, “Image size scalable full-parallax coloured three-dimensional video by electronic holography,” Sci. Rep. 4, 4000 (2014).
[PubMed]

Y. Ichihashi, H. Nakayama, T. Ito, N. Masuda, T. Shimobaba, A. Shiraki, and T. Sugie, “HORN-6 special-purpose clustered computing system for electroholography,” Opt. Express 17, 13895–13903 (2009).
[Crossref] [PubMed]

Ito, T.

T. Nishitsuji, T. Shimobaba, T. Kakue, and T. Ito, “Fast calculation of computer-generated hologram using run-length encoding based recurrence relation,” Opt. Express 23, 9852–9857 (2015).
[Crossref] [PubMed]

T. Nishitsuji, T. Shimobaba, T. Kakue, N. Masuda, and T. Ito, “Fast calculation of computer-generated hologram using the circular symmetry of zone plates,” Opt. Express 20, 27496–27502 (2012).
[Crossref] [PubMed]

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

J. Weng, T. Shimobaba, M. Oikawa, N. Masuda, and T. Ito, “Fast recurrence relation for computer-generated-hologram,” Comput. Phys. Commun. 183, 46–59 (2012).
[Crossref]

Y. Ichihashi, H. Nakayama, T. Ito, N. Masuda, T. Shimobaba, A. Shiraki, and T. Sugie, “HORN-6 special-purpose clustered computing system for electroholography,” Opt. Express 17, 13895–13903 (2009).
[Crossref] [PubMed]

T. Ito, N. Masuda, K. Yoshimura, A. Shiraki, T. Shimobaba, and T. Sugie, “Special-purpose computer HORN-5 for a real-time electroholography,” Opt. Express 13, 1923–1932 (2005).
[Crossref] [PubMed]

T. Shimobaba and T. Ito, “An efficient computational method suitable for hardware of computer-generated hologram with phase computation by addition,” Comput. Phys. Commun. 138, 44–52 (2001).
[Crossref]

Y. Kimura, R. Kawaguchi, T. Sugie, T. Kakue, T. Shimobaba, and T. Ito, “Circuit design of special-purpose computer for holography HORN-8 using eight virtex-5 FPGAs,” in Proceedings of 3D Systems and Applicatons (3DSA, 2015), S3–2.

Junior, W.

R. Gomes, W. Junior, E. Cerqueira, and A. Abelem, “A QoE Fuzzy Routing Protocol for Wireless Mesh Networks,” in Proceedings of the International Workshop, S. Zeadally, E. Cerqueira, M. Curado, and M. Leszczuk, eds. (Springer, 2010).

Kakue, T.

Kawaguchi, R.

Y. Kimura, R. Kawaguchi, T. Sugie, T. Kakue, T. Shimobaba, and T. Ito, “Circuit design of special-purpose computer for holography HORN-8 using eight virtex-5 FPGAs,” in Proceedings of 3D Systems and Applicatons (3DSA, 2015), S3–2.

Kim, E.

Kim, E.-S.

Kim, J.

Kim, S.

Kim, S.-C.

Kimura, Y.

Y. Kimura, R. Kawaguchi, T. Sugie, T. Kakue, T. Shimobaba, and T. Ito, “Circuit design of special-purpose computer for holography HORN-8 using eight virtex-5 FPGAs,” in Proceedings of 3D Systems and Applicatons (3DSA, 2015), S3–2.

Lucente, M.

M. Lucente, “Interactive computation of holograms using a look-up table,” J. Electron. Imaging 2, 28–34 (1993).
[Crossref]

Masuda, N.

Matsushima, K.

McIntosh, J. A.

J. A. McIntosh and E. E. Swartzlander, “High-Speed Cosine Generator,” in Proceedings of 1994 28th Asilomar Conference on Signals, Systems and Computers (IEEE, 1994), pp. 273–277.
[Crossref]

Nakayama, H.

Nishitsuji, T.

Oikawa, M.

J. Weng, T. Shimobaba, M. Oikawa, N. Masuda, and T. Ito, “Fast recurrence relation for computer-generated-hologram,” Comput. Phys. Commun. 183, 46–59 (2012).
[Crossref]

Okada, N.

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

Sakurai, T.

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

Sasaki, H.

H. Sasaki, K. Yamamoto, Y. Ichihashi, and T. Senoh, “Image size scalable full-parallax coloured three-dimensional video by electronic holography,” Sci. Rep. 4, 4000 (2014).
[PubMed]

Seevinc, E.

E. Seevinc, “Simple, wide-range approximations to trigonometric and inverse trigonometric functions useful in real-time signal processing,” in IEE Proceedings G (Electronic Circuits and Systems) (IEE, 1981), 1, pp. 41–45
[Crossref]

Senoh, T.

H. Sasaki, K. Yamamoto, Y. Ichihashi, and T. Senoh, “Image size scalable full-parallax coloured three-dimensional video by electronic holography,” Sci. Rep. 4, 4000 (2014).
[PubMed]

Shimobaba, T.

T. Nishitsuji, T. Shimobaba, T. Kakue, and T. Ito, “Fast calculation of computer-generated hologram using run-length encoding based recurrence relation,” Opt. Express 23, 9852–9857 (2015).
[Crossref] [PubMed]

T. Nishitsuji, T. Shimobaba, T. Kakue, N. Masuda, and T. Ito, “Fast calculation of computer-generated hologram using the circular symmetry of zone plates,” Opt. Express 20, 27496–27502 (2012).
[Crossref] [PubMed]

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

J. Weng, T. Shimobaba, M. Oikawa, N. Masuda, and T. Ito, “Fast recurrence relation for computer-generated-hologram,” Comput. Phys. Commun. 183, 46–59 (2012).
[Crossref]

Y. Ichihashi, H. Nakayama, T. Ito, N. Masuda, T. Shimobaba, A. Shiraki, and T. Sugie, “HORN-6 special-purpose clustered computing system for electroholography,” Opt. Express 17, 13895–13903 (2009).
[Crossref] [PubMed]

T. Ito, N. Masuda, K. Yoshimura, A. Shiraki, T. Shimobaba, and T. Sugie, “Special-purpose computer HORN-5 for a real-time electroholography,” Opt. Express 13, 1923–1932 (2005).
[Crossref] [PubMed]

T. Shimobaba and T. Ito, “An efficient computational method suitable for hardware of computer-generated hologram with phase computation by addition,” Comput. Phys. Commun. 138, 44–52 (2001).
[Crossref]

Y. Kimura, R. Kawaguchi, T. Sugie, T. Kakue, T. Shimobaba, and T. Ito, “Circuit design of special-purpose computer for holography HORN-8 using eight virtex-5 FPGAs,” in Proceedings of 3D Systems and Applicatons (3DSA, 2015), S3–2.

Shiraki, A.

Sugie, T.

Swartzlander, E. E.

J. A. McIntosh and E. E. Swartzlander, “High-Speed Cosine Generator,” in Proceedings of 1994 28th Asilomar Conference on Signals, Systems and Computers (IEEE, 1994), pp. 273–277.
[Crossref]

Takada, N.

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

Takai, M.

Weng, J.

J. Weng, T. Shimobaba, M. Oikawa, N. Masuda, and T. Ito, “Fast recurrence relation for computer-generated-hologram,” Comput. Phys. Commun. 183, 46–59 (2012).
[Crossref]

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

Yamamoto, K.

H. Sasaki, K. Yamamoto, Y. Ichihashi, and T. Senoh, “Image size scalable full-parallax coloured three-dimensional video by electronic holography,” Sci. Rep. 4, 4000 (2014).
[PubMed]

Yoshikawa, H.

H. Yoshikawa, “Fast computation of Fresnel holograms employing difference,” Opt. Rev. 8, 331–335 (2001).
[Crossref]

Yoshimura, K.

Appl. Opt. (1)

Comput. Phys. Commun. (3)

T. Shimobaba and T. Ito, “An efficient computational method suitable for hardware of computer-generated hologram with phase computation by addition,” Comput. Phys. Commun. 138, 44–52 (2001).
[Crossref]

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

J. Weng, T. Shimobaba, M. Oikawa, N. Masuda, and T. Ito, “Fast recurrence relation for computer-generated-hologram,” Comput. Phys. Commun. 183, 46–59 (2012).
[Crossref]

J. Electron. Imaging (1)

M. Lucente, “Interactive computation of holograms using a look-up table,” J. Electron. Imaging 2, 28–34 (1993).
[Crossref]

Opt. Express (7)

S. Kim, J. Kim, and E. Kim, “Effective memory reduction of the novel look-up table with one-dimensional sub-principle fringe patterns in computer-generated holograms,” Opt. Express 20, 12021–12034 (2012).
[Crossref] [PubMed]

S. Kim and E. Kim, “Fast one-step calculation of holographic videos of three-dimensional scenes by combined use of baseline and depth-compensating principal fringe patterns,” Opt. Express 22, 22513–22527 (2014).
[Crossref] [PubMed]

X. Dong, S.-C. Kim, and E.-S. Kim, “Three-directional motion compensation-based novel-look-up-table for video hologram generation of three-dimensional objects freely maneuvering in space,” Opt. Express 22, 16925–16944 (2014)
[Crossref] [PubMed]

T. Nishitsuji, T. Shimobaba, T. Kakue, N. Masuda, and T. Ito, “Fast calculation of computer-generated hologram using the circular symmetry of zone plates,” Opt. Express 20, 27496–27502 (2012).
[Crossref] [PubMed]

T. Nishitsuji, T. Shimobaba, T. Kakue, and T. Ito, “Fast calculation of computer-generated hologram using run-length encoding based recurrence relation,” Opt. Express 23, 9852–9857 (2015).
[Crossref] [PubMed]

T. Ito, N. Masuda, K. Yoshimura, A. Shiraki, T. Shimobaba, and T. Sugie, “Special-purpose computer HORN-5 for a real-time electroholography,” Opt. Express 13, 1923–1932 (2005).
[Crossref] [PubMed]

Y. Ichihashi, H. Nakayama, T. Ito, N. Masuda, T. Shimobaba, A. Shiraki, and T. Sugie, “HORN-6 special-purpose clustered computing system for electroholography,” Opt. Express 17, 13895–13903 (2009).
[Crossref] [PubMed]

Opt. Rev. (1)

H. Yoshikawa, “Fast computation of Fresnel holograms employing difference,” Opt. Rev. 8, 331–335 (2001).
[Crossref]

Sci. Rep. (1)

H. Sasaki, K. Yamamoto, Y. Ichihashi, and T. Senoh, “Image size scalable full-parallax coloured three-dimensional video by electronic holography,” Sci. Rep. 4, 4000 (2014).
[PubMed]

Other (4)

R. Gomes, W. Junior, E. Cerqueira, and A. Abelem, “A QoE Fuzzy Routing Protocol for Wireless Mesh Networks,” in Proceedings of the International Workshop, S. Zeadally, E. Cerqueira, M. Curado, and M. Leszczuk, eds. (Springer, 2010).

Y. Kimura, R. Kawaguchi, T. Sugie, T. Kakue, T. Shimobaba, and T. Ito, “Circuit design of special-purpose computer for holography HORN-8 using eight virtex-5 FPGAs,” in Proceedings of 3D Systems and Applicatons (3DSA, 2015), S3–2.

E. Seevinc, “Simple, wide-range approximations to trigonometric and inverse trigonometric functions useful in real-time signal processing,” in IEE Proceedings G (Electronic Circuits and Systems) (IEE, 1981), 1, pp. 41–45
[Crossref]

J. A. McIntosh and E. E. Swartzlander, “High-Speed Cosine Generator,” in Proceedings of 1994 28th Asilomar Conference on Signals, Systems and Computers (IEEE, 1994), pp. 273–277.
[Crossref]

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Figures (4)
Fig. 1
Fig. 1 Block diagram of CGH calculation system

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Fig. 2
Fig. 2 Overview of the proposed method: (a) Process of creating chirp wave from the output of the phase calculation part (b) Comparison between the ideal output using the conventional cosine function and the output of the proposed method.

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Fig. 3
Fig. 3 Focus property of each zoneplate: (a) Focus : 0.3 m (b) Focus : 1.0 m

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Fig. 4
Fig. 4 Reconstructed images: (a) Using the conventional cosine function method by numerical simulation; (b) Using the look-up table of the cosine function (1,024 samplings) by numerical simulation; (c) The proposed method by numerical simulation; (d) The proposed method by optical reconstruction

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Tables (1)

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Table 1 Calculation time and image quality

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Equations (2)

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ϕ ( x α , y α ) = arg [ ∑ j = 0 N − 1 A j exp ( 2 π i p λ { ( x α − x j ) 2 + ( y α − y j ) 2 + z j 2 } 1 2 ) ] ,
= arg [ ∑ j = 0 N − 1 A j { cos ( 2 π pr α j λ ) + i sin ( 2 π pr α j λ ) } ] ,

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