Abstract

In electroholography, a real-time reconstruction is one of the grand challenges. To realize it, we developed a parallelized high-performance computing board for computer-generated hologram, named HORN-5 board, where four large-scale field programmable gate array chips were mounted. The number of circuits for hologram calculation implemented to the board was 1,408. The board calculated a hologram at higher speed by 360 times than a personal computer with Pentium4 processor. A personal computer connected with four HORN-5 boards calculated a hologram of 1,408×1,050 made from a three-dimensional object consisting of 10,000 points at 0.0023 s. In other words, beyond at video rate (30 frames/s), it realized a real-time reconstruction.

© 2005 Optical Society of America

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References

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  1. P. S. Hilaire, S. A. Benton, M. Lucente, M. L. Jepsen, J. Kollin, H. Yoshikawa, and J. Underkoffler, “Electronic display system for computational holography,” Proc. SPIE 1212–20, 174–182 (1990).
  2. P. S. Hilaire, S. A. Benton, M. Lucente, J. D. Sutter, and W. J. Plesniak, “Advances in holographic video,” Proc. SPIE 1914–27, 188–196 (1993).
  3. P. S. Hilaire, “Scalable optical architecture for electronic holography,” Opt. Eng. 34, 2900–2911 (1995).
  4. G. Tricoles, “Computer generated holograms: an historical review,” Appl. Opt. 26, 4351–4360 (1987).
  5. M. Lucente, “Interactive three-dimensional holographic displays: seeing the future in depth,” Comp. Graphics 31, 63–67 (1997).
  6. T. Ito, T. Shimobaba, H. Godo, and M. Horiuchi, “Holographic reconstruction with a 10-µm pixel-pitch reflective liquid-crystal display by use of a light-emitting diode reference light,” Opt. Lett. 27, 1406–1408 (2002).
  7. T. Ito and K. Okano, “Color electroholography by three colored reference lights simultaneously incident upon one hologram panel,” Opt, Express 12, 4320–4325 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-18-4320.
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  12. T. Shimobaba and T. Ito, “An efficient computational method suitable for hardware of computer-generated hologram with phase computation by addition,” Comp. Phys. Commun. 138, 44–52 (2001).
  13. J. A. Watlington, M. Lucente, C. J. Sparrell, V. M. Bove, and I. Tamitani. “A hardware architecture for rapid generation of electro-holographic fringe patterns,” Proc SPIE 2406–23, 172–183 (1995).
  14. M. Lucente and T. A. Galyean, “Rendering interactive holographic images”, Proc. ACM SIGGRAPH 95, 387–394 (1995).
  15. T. Yabe, T. Ito, and M. Okazaki, “Holography machine HORN-1 for computer-aided retrieval of virtual three-dimensional image,” Jpn. J. Appl. Phys. 32, L1359–L1361 (1993).
  16. T. Ito, T. Yabe, M. Okazaki, and M. Yanagi, “Special-purpose computer HORN-1 for reconstruction of virtual image in three dimensions,” Comp. Phys. Commun. 82, 104–110 (1994).
  17. T. Ito, H. Eldeib, K. Yoshida, S. Takahashi, T. Yabe, and T. Kunugi, “Special-purpose computer for holography HORN-2,” Comp. Phys. Commun. 93, 13–20 (1996).
  18. T. Shimobaba, N. Masuda, T. Sugie, S. Hosono, S. Tsukui, and T. Ito, “Special-purpose computer for holography HORN-3 with PLD technology,” Comp. Phys. Commun. 130, 75–82 (2000).
  19. T. Shimobaba and T. Ito, “Special-purpose computer for holography HORN-4 with recurrence algorithm,” Comp. Phys. Commun. 148, 160–170 (2002).
  20. http://www.jvcdig.com/technology.htm.
  21. T. Ito and T. Shimobaba, “One-unit system for electroholography by use of a special-purpose computational chip with a high-resolution liquid-crystal display toward a three-dimensional television”, Opt. Express,  12, 1788–1793 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1788.

2004 (2)

T. Ito and K. Okano, “Color electroholography by three colored reference lights simultaneously incident upon one hologram panel,” Opt, Express 12, 4320–4325 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-18-4320.

T. Ito and T. Shimobaba, “One-unit system for electroholography by use of a special-purpose computational chip with a high-resolution liquid-crystal display toward a three-dimensional television”, Opt. Express,  12, 1788–1793 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1788.

2003 (1)

2002 (2)

2001 (1)

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

2000 (3)

H. Yoshikawa, S. Iwase, and T. Oneda, “Fast computation of Fresnel holograms employing difference,” Proc. SPIE 3956, 48–55 (2000).

K. Matsushima and M. Takai, “Fast computation of Fresnel holograms employing difference,” Appl. Opt. 39, 6587–6594 (2000).

T. Shimobaba, N. Masuda, T. Sugie, S. Hosono, S. Tsukui, and T. Ito, “Special-purpose computer for holography HORN-3 with PLD technology,” Comp. Phys. Commun. 130, 75–82 (2000).

1997 (1)

M. Lucente, “Interactive three-dimensional holographic displays: seeing the future in depth,” Comp. Graphics 31, 63–67 (1997).

1996 (1)

T. Ito, H. Eldeib, K. Yoshida, S. Takahashi, T. Yabe, and T. Kunugi, “Special-purpose computer for holography HORN-2,” Comp. Phys. Commun. 93, 13–20 (1996).

1995 (3)

J. A. Watlington, M. Lucente, C. J. Sparrell, V. M. Bove, and I. Tamitani. “A hardware architecture for rapid generation of electro-holographic fringe patterns,” Proc SPIE 2406–23, 172–183 (1995).

M. Lucente and T. A. Galyean, “Rendering interactive holographic images”, Proc. ACM SIGGRAPH 95, 387–394 (1995).

P. S. Hilaire, “Scalable optical architecture for electronic holography,” Opt. Eng. 34, 2900–2911 (1995).

1994 (1)

T. Ito, T. Yabe, M. Okazaki, and M. Yanagi, “Special-purpose computer HORN-1 for reconstruction of virtual image in three dimensions,” Comp. Phys. Commun. 82, 104–110 (1994).

1993 (3)

T. Yabe, T. Ito, and M. Okazaki, “Holography machine HORN-1 for computer-aided retrieval of virtual three-dimensional image,” Jpn. J. Appl. Phys. 32, L1359–L1361 (1993).

P. S. Hilaire, S. A. Benton, M. Lucente, J. D. Sutter, and W. J. Plesniak, “Advances in holographic video,” Proc. SPIE 1914–27, 188–196 (1993).

M. Lucente, “Interactive Computation of Holograms Using a Look-Up Table,” J. Electron. Imaging 2, pp. 28–34 (1993).

1990 (1)

P. S. Hilaire, S. A. Benton, M. Lucente, M. L. Jepsen, J. Kollin, H. Yoshikawa, and J. Underkoffler, “Electronic display system for computational holography,” Proc. SPIE 1212–20, 174–182 (1990).

1987 (1)

Benton, S. A.

P. S. Hilaire, S. A. Benton, M. Lucente, J. D. Sutter, and W. J. Plesniak, “Advances in holographic video,” Proc. SPIE 1914–27, 188–196 (1993).

P. S. Hilaire, S. A. Benton, M. Lucente, M. L. Jepsen, J. Kollin, H. Yoshikawa, and J. Underkoffler, “Electronic display system for computational holography,” Proc. SPIE 1212–20, 174–182 (1990).

Bove, V. M.

J. A. Watlington, M. Lucente, C. J. Sparrell, V. M. Bove, and I. Tamitani. “A hardware architecture for rapid generation of electro-holographic fringe patterns,” Proc SPIE 2406–23, 172–183 (1995).

Eldeib, H.

T. Ito, H. Eldeib, K. Yoshida, S. Takahashi, T. Yabe, and T. Kunugi, “Special-purpose computer for holography HORN-2,” Comp. Phys. Commun. 93, 13–20 (1996).

Galyean, T. A.

M. Lucente and T. A. Galyean, “Rendering interactive holographic images”, Proc. ACM SIGGRAPH 95, 387–394 (1995).

Garner, R. G.

Godo, H.

Hilaire, P. S.

P. S. Hilaire, “Scalable optical architecture for electronic holography,” Opt. Eng. 34, 2900–2911 (1995).

P. S. Hilaire, S. A. Benton, M. Lucente, J. D. Sutter, and W. J. Plesniak, “Advances in holographic video,” Proc. SPIE 1914–27, 188–196 (1993).

P. S. Hilaire, S. A. Benton, M. Lucente, M. L. Jepsen, J. Kollin, H. Yoshikawa, and J. Underkoffler, “Electronic display system for computational holography,” Proc. SPIE 1212–20, 174–182 (1990).

Horiuchi, M.

Hosono, S.

T. Shimobaba, N. Masuda, T. Sugie, S. Hosono, S. Tsukui, and T. Ito, “Special-purpose computer for holography HORN-3 with PLD technology,” Comp. Phys. Commun. 130, 75–82 (2000).

Huebschman, M.

Ito, T.

T. Ito and T. Shimobaba, “One-unit system for electroholography by use of a special-purpose computational chip with a high-resolution liquid-crystal display toward a three-dimensional television”, Opt. Express,  12, 1788–1793 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1788.

T. Ito and K. Okano, “Color electroholography by three colored reference lights simultaneously incident upon one hologram panel,” Opt, Express 12, 4320–4325 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-18-4320.

T. Ito, T. Shimobaba, H. Godo, and M. Horiuchi, “Holographic reconstruction with a 10-µm pixel-pitch reflective liquid-crystal display by use of a light-emitting diode reference light,” Opt. Lett. 27, 1406–1408 (2002).

T. Shimobaba and T. Ito, “Special-purpose computer for holography HORN-4 with recurrence algorithm,” Comp. Phys. Commun. 148, 160–170 (2002).

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

T. Shimobaba, N. Masuda, T. Sugie, S. Hosono, S. Tsukui, and T. Ito, “Special-purpose computer for holography HORN-3 with PLD technology,” Comp. Phys. Commun. 130, 75–82 (2000).

T. Ito, H. Eldeib, K. Yoshida, S. Takahashi, T. Yabe, and T. Kunugi, “Special-purpose computer for holography HORN-2,” Comp. Phys. Commun. 93, 13–20 (1996).

T. Ito, T. Yabe, M. Okazaki, and M. Yanagi, “Special-purpose computer HORN-1 for reconstruction of virtual image in three dimensions,” Comp. Phys. Commun. 82, 104–110 (1994).

T. Yabe, T. Ito, and M. Okazaki, “Holography machine HORN-1 for computer-aided retrieval of virtual three-dimensional image,” Jpn. J. Appl. Phys. 32, L1359–L1361 (1993).

Iwase, S.

H. Yoshikawa, S. Iwase, and T. Oneda, “Fast computation of Fresnel holograms employing difference,” Proc. SPIE 3956, 48–55 (2000).

Jepsen, M. L.

P. S. Hilaire, S. A. Benton, M. Lucente, M. L. Jepsen, J. Kollin, H. Yoshikawa, and J. Underkoffler, “Electronic display system for computational holography,” Proc. SPIE 1212–20, 174–182 (1990).

Kollin, J.

P. S. Hilaire, S. A. Benton, M. Lucente, M. L. Jepsen, J. Kollin, H. Yoshikawa, and J. Underkoffler, “Electronic display system for computational holography,” Proc. SPIE 1212–20, 174–182 (1990).

Kunugi, T.

T. Ito, H. Eldeib, K. Yoshida, S. Takahashi, T. Yabe, and T. Kunugi, “Special-purpose computer for holography HORN-2,” Comp. Phys. Commun. 93, 13–20 (1996).

Lucente, M.

M. Lucente, “Interactive three-dimensional holographic displays: seeing the future in depth,” Comp. Graphics 31, 63–67 (1997).

J. A. Watlington, M. Lucente, C. J. Sparrell, V. M. Bove, and I. Tamitani. “A hardware architecture for rapid generation of electro-holographic fringe patterns,” Proc SPIE 2406–23, 172–183 (1995).

M. Lucente and T. A. Galyean, “Rendering interactive holographic images”, Proc. ACM SIGGRAPH 95, 387–394 (1995).

M. Lucente, “Interactive Computation of Holograms Using a Look-Up Table,” J. Electron. Imaging 2, pp. 28–34 (1993).

P. S. Hilaire, S. A. Benton, M. Lucente, J. D. Sutter, and W. J. Plesniak, “Advances in holographic video,” Proc. SPIE 1914–27, 188–196 (1993).

P. S. Hilaire, S. A. Benton, M. Lucente, M. L. Jepsen, J. Kollin, H. Yoshikawa, and J. Underkoffler, “Electronic display system for computational holography,” Proc. SPIE 1212–20, 174–182 (1990).

Masuda, N.

T. Shimobaba, N. Masuda, T. Sugie, S. Hosono, S. Tsukui, and T. Ito, “Special-purpose computer for holography HORN-3 with PLD technology,” Comp. Phys. Commun. 130, 75–82 (2000).

Matsushima, K.

Munjuluri, B.

Okano, K.

T. Ito and K. Okano, “Color electroholography by three colored reference lights simultaneously incident upon one hologram panel,” Opt, Express 12, 4320–4325 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-18-4320.

Okazaki, M.

T. Ito, T. Yabe, M. Okazaki, and M. Yanagi, “Special-purpose computer HORN-1 for reconstruction of virtual image in three dimensions,” Comp. Phys. Commun. 82, 104–110 (1994).

T. Yabe, T. Ito, and M. Okazaki, “Holography machine HORN-1 for computer-aided retrieval of virtual three-dimensional image,” Jpn. J. Appl. Phys. 32, L1359–L1361 (1993).

Oneda, T.

H. Yoshikawa, S. Iwase, and T. Oneda, “Fast computation of Fresnel holograms employing difference,” Proc. SPIE 3956, 48–55 (2000).

Plesniak, W. J.

P. S. Hilaire, S. A. Benton, M. Lucente, J. D. Sutter, and W. J. Plesniak, “Advances in holographic video,” Proc. SPIE 1914–27, 188–196 (1993).

Shimobaba, T.

T. Ito and T. Shimobaba, “One-unit system for electroholography by use of a special-purpose computational chip with a high-resolution liquid-crystal display toward a three-dimensional television”, Opt. Express,  12, 1788–1793 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1788.

T. Ito, T. Shimobaba, H. Godo, and M. Horiuchi, “Holographic reconstruction with a 10-µm pixel-pitch reflective liquid-crystal display by use of a light-emitting diode reference light,” Opt. Lett. 27, 1406–1408 (2002).

T. Shimobaba and T. Ito, “Special-purpose computer for holography HORN-4 with recurrence algorithm,” Comp. Phys. Commun. 148, 160–170 (2002).

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

T. Shimobaba, N. Masuda, T. Sugie, S. Hosono, S. Tsukui, and T. Ito, “Special-purpose computer for holography HORN-3 with PLD technology,” Comp. Phys. Commun. 130, 75–82 (2000).

Sparrell, C. J.

J. A. Watlington, M. Lucente, C. J. Sparrell, V. M. Bove, and I. Tamitani. “A hardware architecture for rapid generation of electro-holographic fringe patterns,” Proc SPIE 2406–23, 172–183 (1995).

Sugie, T.

T. Shimobaba, N. Masuda, T. Sugie, S. Hosono, S. Tsukui, and T. Ito, “Special-purpose computer for holography HORN-3 with PLD technology,” Comp. Phys. Commun. 130, 75–82 (2000).

Sutter, J. D.

P. S. Hilaire, S. A. Benton, M. Lucente, J. D. Sutter, and W. J. Plesniak, “Advances in holographic video,” Proc. SPIE 1914–27, 188–196 (1993).

Takahashi, S.

T. Ito, H. Eldeib, K. Yoshida, S. Takahashi, T. Yabe, and T. Kunugi, “Special-purpose computer for holography HORN-2,” Comp. Phys. Commun. 93, 13–20 (1996).

Takai, M.

Tamitani, I.

J. A. Watlington, M. Lucente, C. J. Sparrell, V. M. Bove, and I. Tamitani. “A hardware architecture for rapid generation of electro-holographic fringe patterns,” Proc SPIE 2406–23, 172–183 (1995).

Tricoles, G.

Tsukui, S.

T. Shimobaba, N. Masuda, T. Sugie, S. Hosono, S. Tsukui, and T. Ito, “Special-purpose computer for holography HORN-3 with PLD technology,” Comp. Phys. Commun. 130, 75–82 (2000).

Underkoffler, J.

P. S. Hilaire, S. A. Benton, M. Lucente, M. L. Jepsen, J. Kollin, H. Yoshikawa, and J. Underkoffler, “Electronic display system for computational holography,” Proc. SPIE 1212–20, 174–182 (1990).

Watlington, J. A.

J. A. Watlington, M. Lucente, C. J. Sparrell, V. M. Bove, and I. Tamitani. “A hardware architecture for rapid generation of electro-holographic fringe patterns,” Proc SPIE 2406–23, 172–183 (1995).

Yabe, T.

T. Ito, H. Eldeib, K. Yoshida, S. Takahashi, T. Yabe, and T. Kunugi, “Special-purpose computer for holography HORN-2,” Comp. Phys. Commun. 93, 13–20 (1996).

T. Ito, T. Yabe, M. Okazaki, and M. Yanagi, “Special-purpose computer HORN-1 for reconstruction of virtual image in three dimensions,” Comp. Phys. Commun. 82, 104–110 (1994).

T. Yabe, T. Ito, and M. Okazaki, “Holography machine HORN-1 for computer-aided retrieval of virtual three-dimensional image,” Jpn. J. Appl. Phys. 32, L1359–L1361 (1993).

Yanagi, M.

T. Ito, T. Yabe, M. Okazaki, and M. Yanagi, “Special-purpose computer HORN-1 for reconstruction of virtual image in three dimensions,” Comp. Phys. Commun. 82, 104–110 (1994).

Yoshida, K.

T. Ito, H. Eldeib, K. Yoshida, S. Takahashi, T. Yabe, and T. Kunugi, “Special-purpose computer for holography HORN-2,” Comp. Phys. Commun. 93, 13–20 (1996).

Yoshikawa, H.

H. Yoshikawa, S. Iwase, and T. Oneda, “Fast computation of Fresnel holograms employing difference,” Proc. SPIE 3956, 48–55 (2000).

P. S. Hilaire, S. A. Benton, M. Lucente, M. L. Jepsen, J. Kollin, H. Yoshikawa, and J. Underkoffler, “Electronic display system for computational holography,” Proc. SPIE 1212–20, 174–182 (1990).

Appl. Opt. (2)

Comp. Graphics (1)

M. Lucente, “Interactive three-dimensional holographic displays: seeing the future in depth,” Comp. Graphics 31, 63–67 (1997).

Comp. Phys. Commun. (5)

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

T. Ito, T. Yabe, M. Okazaki, and M. Yanagi, “Special-purpose computer HORN-1 for reconstruction of virtual image in three dimensions,” Comp. Phys. Commun. 82, 104–110 (1994).

T. Ito, H. Eldeib, K. Yoshida, S. Takahashi, T. Yabe, and T. Kunugi, “Special-purpose computer for holography HORN-2,” Comp. Phys. Commun. 93, 13–20 (1996).

T. Shimobaba, N. Masuda, T. Sugie, S. Hosono, S. Tsukui, and T. Ito, “Special-purpose computer for holography HORN-3 with PLD technology,” Comp. Phys. Commun. 130, 75–82 (2000).

T. Shimobaba and T. Ito, “Special-purpose computer for holography HORN-4 with recurrence algorithm,” Comp. Phys. Commun. 148, 160–170 (2002).

J. Electron. Imaging (1)

M. Lucente, “Interactive Computation of Holograms Using a Look-Up Table,” J. Electron. Imaging 2, pp. 28–34 (1993).

Jpn. J. Appl. Phys. (1)

T. Yabe, T. Ito, and M. Okazaki, “Holography machine HORN-1 for computer-aided retrieval of virtual three-dimensional image,” Jpn. J. Appl. Phys. 32, L1359–L1361 (1993).

Opt, Express (1)

T. Ito and K. Okano, “Color electroholography by three colored reference lights simultaneously incident upon one hologram panel,” Opt, Express 12, 4320–4325 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-18-4320.

Opt. Eng. (1)

P. S. Hilaire, “Scalable optical architecture for electronic holography,” Opt. Eng. 34, 2900–2911 (1995).

Opt. Express (2)

Opt. Lett. (1)

Proc SPIE (1)

J. A. Watlington, M. Lucente, C. J. Sparrell, V. M. Bove, and I. Tamitani. “A hardware architecture for rapid generation of electro-holographic fringe patterns,” Proc SPIE 2406–23, 172–183 (1995).

Proc. ACM SIGGRAPH (1)

M. Lucente and T. A. Galyean, “Rendering interactive holographic images”, Proc. ACM SIGGRAPH 95, 387–394 (1995).

Proc. SPIE (3)

P. S. Hilaire, S. A. Benton, M. Lucente, M. L. Jepsen, J. Kollin, H. Yoshikawa, and J. Underkoffler, “Electronic display system for computational holography,” Proc. SPIE 1212–20, 174–182 (1990).

P. S. Hilaire, S. A. Benton, M. Lucente, J. D. Sutter, and W. J. Plesniak, “Advances in holographic video,” Proc. SPIE 1914–27, 188–196 (1993).

H. Yoshikawa, S. Iwase, and T. Oneda, “Fast computation of Fresnel holograms employing difference,” Proc. SPIE 3956, 48–55 (2000).

Other (1)

http://www.jvcdig.com/technology.htm.

Supplementary Material (2)

» Media 1: MOV (781 KB)     
» Media 2: MOV (353 KB)     

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Figures (10)

Fig. 1.
Fig. 1.

Basic structure of HORN system.

Fig. 2.
Fig. 2.

Real-time reconstruction system by HORN-5 including the optical setup.

Fig. 3.
Fig. 3.

Schematic drawing of our recurrence formulas algorithm.

Fig. 4.
Fig. 4.

Block diagram of BPU (Basic Processing Unit).

Fig. 5.
Fig. 5.

Block diagram of APU (Additional Processing Unit).

Fig. 6.
Fig. 6.

Block diagram of the HORN-5 pipeline.

Fig. 7.
Fig. 7.

Top view of the HORN-5 board.

Fig. 8.
Fig. 8.

Snapshot of a real-time electroholography by HORN-5; (a) the original graphics [Media 1], (b) the CGH and (c) the constructed image [Media 2].

Fig. 9.
Fig. 9.

Basic structure of HORN system without delay by communications.

Fig. 10.
Fig. 10.

Parallel system by HORN.

Tables (1)

Tables Icon

Table 1. Performance of the HORN-5 system compared with a PC

Equations (7)

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

I ( x α , y α ) = j = 1 M A j cos [ 2 π λ ( x α x j ) 2 + ( y α y j ) 2 + z j 2 ] .
I ( x α , y α ) = j = 1 M A j cos [ 2 π λ ( z j + x α j 2 + y α j 2 2 z j ) ] .
I ( X α , Y α ) = j = 1 M A j cos [ 2 π ( p Z j λ + p 2 λ Z j ( X α j 2 + Y α j 2 ) ) ] .
I ( X α + k , Y α ) = j = 1 M A j cos ( 2 π Θ k ) .
Θ 0 = p Z j λ + p 2 λ Z j ( X α j 2 + Y α j 2 ) , Γ 0 = p 2 λ Z j ( 2 X α j + 1 ) , Δ = p λ Z j .
Θ k + 1 = Θ k + Γ k , Γ k + 1 = Γ k + Δ .
Z ( 1 ) j = p 2 λ Z j , Z ( 2 ) j = p Z j λ .

Metrics