Abstract

A multichannel holographic recording method is presented for three-dimensional (3D) displays, utilizing pixel-based recording instead of image-based recording in order to realize parallel processing. The proposed approach is composed of two main stages. In the first stage, each two-dimensional (2D) image acquired from multiple viewpoints is partitioned by holographic recording channels (HRC) into nonoverlapping subimages. In the second stage, the corresponding pixels of the subimages are rearranged to constitute an encoding image. The encoding images are recorded simultaneously by each HRC, respectively, so the recording speed is improved significantly. The experimental results have demonstrated that the three-channel system is feasible and the full-parallax hologram reconstructed with white light is acceptable in quality. The three-channel system saves approximately 60% of the recording time in comparison with the single-channel system. More importantly, the proposed method can accomplish a large-scale final hologram composed of multichannel holograms without sacrificing the hologram quality. Several 3D imaging applications such as medical diagnosis and advertisements could benefit from this research.

© 2011 Optical Society of America

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References

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  1. P. Hariharan, Optical Holography, Principles, Techniques and Applications (Cambridge University Press, 1996).
  2. R. J. Collier, C. B. Burckhardt, and L. H. Lin, Optical Holography (Academic, 1971).
  3. N. T. Shaked, B. Katz, and J. Rosen, “Review of three-dimensional holographic imaging by multiple-viewpoint-projection based methods,” Appl. Opt. 48, H120–H136 (2009).
    [CrossRef]
  4. M. Agus, F. Bettio, A. Giachetti, E. Gobbetti, J. A. I. Guitián, F. Marton, J. Nilsson, and G. Pintore, “An interactive 3D medical visualization system based on a light field display,” Visual Computer 25, 883–893 (2009).
    [CrossRef]
  5. M. L. Huebschman, B. Munjuluri, and H. R. Garner, “Dynamic holographic 3-D image projection,” Opt. Express 11, 437–445(2003).
    [CrossRef]
  6. P. St. Hilaire, M. E. Lucente, J. D. Sutter, R. Pappu, C. J. Sparrell, and S. A. Benton, “Scaling up the MIT holographic video system,” Proc. SPIE 2333, 374–380 (1995).
    [CrossRef]
  7. M. Stanley, P. B. Conway, S. Coomber, J. C. Jones, D. C. Scattergood, C. W. Slinger, B. W. Bannister, C. V. Brown, W. A. Crossland, and A. R. L. Travis, “Novel electro-optic modulator system for the production of dynamic images from giga-pixel computer-generated holograms,” Proc. SPIE 3956, 13–22 (2000).
    [CrossRef]
  8. L. E. Helseth and I. Singstad, “Diffusers for holographic stereography,” Opt. Commun. 193, 81–86 (2001).
    [CrossRef]
  9. T. Honda, K. Okada, and J. Tsujiuchi, “3-D distortion of observed images reconstructed from a cylindrical holographic stereogram. (1) laser light reconstruction type,” Opt. Commun. 36, 11–16 (1981).
    [CrossRef]
  10. K. Okada, T. Honda, and J. Tsujiuchi, “3-D distortion of observed images reconstructed from a cylindrical holographic stereogram. (2) white light reconstruction type,” Opt. Commun. 36, 17–21 (1981).
    [CrossRef]
  11. M. A. Klug, M. W. Halle, M. Lucente, and W. J. Plesniak, “A compact prototype one-step ultragram printer,” Proc. SPIE 1914, 15–23 (1993).
    [CrossRef]
  12. A. Ravishankar Rao and Alejandro Jaimes, “Digital stereoscopic imaging,” Proc. SPIE 3639, 144–154 (1999).
    [CrossRef]
  13. P. Benzie, J. Watson, P. Surman, I. Rakkolainen, K. Hopf, H. Urey, V. Sainov, and C. von Kopylow, “A survey of 3DTV displays: techniques and technologies,” IEEE Trans. Circuits Syst. Video Technol. 17, 1647–1658 (2007).
    [CrossRef]

2009 (2)

M. Agus, F. Bettio, A. Giachetti, E. Gobbetti, J. A. I. Guitián, F. Marton, J. Nilsson, and G. Pintore, “An interactive 3D medical visualization system based on a light field display,” Visual Computer 25, 883–893 (2009).
[CrossRef]

N. T. Shaked, B. Katz, and J. Rosen, “Review of three-dimensional holographic imaging by multiple-viewpoint-projection based methods,” Appl. Opt. 48, H120–H136 (2009).
[CrossRef]

2007 (1)

P. Benzie, J. Watson, P. Surman, I. Rakkolainen, K. Hopf, H. Urey, V. Sainov, and C. von Kopylow, “A survey of 3DTV displays: techniques and technologies,” IEEE Trans. Circuits Syst. Video Technol. 17, 1647–1658 (2007).
[CrossRef]

2003 (1)

2001 (1)

L. E. Helseth and I. Singstad, “Diffusers for holographic stereography,” Opt. Commun. 193, 81–86 (2001).
[CrossRef]

2000 (1)

M. Stanley, P. B. Conway, S. Coomber, J. C. Jones, D. C. Scattergood, C. W. Slinger, B. W. Bannister, C. V. Brown, W. A. Crossland, and A. R. L. Travis, “Novel electro-optic modulator system for the production of dynamic images from giga-pixel computer-generated holograms,” Proc. SPIE 3956, 13–22 (2000).
[CrossRef]

1999 (1)

A. Ravishankar Rao and Alejandro Jaimes, “Digital stereoscopic imaging,” Proc. SPIE 3639, 144–154 (1999).
[CrossRef]

1995 (1)

P. St. Hilaire, M. E. Lucente, J. D. Sutter, R. Pappu, C. J. Sparrell, and S. A. Benton, “Scaling up the MIT holographic video system,” Proc. SPIE 2333, 374–380 (1995).
[CrossRef]

1993 (1)

M. A. Klug, M. W. Halle, M. Lucente, and W. J. Plesniak, “A compact prototype one-step ultragram printer,” Proc. SPIE 1914, 15–23 (1993).
[CrossRef]

1981 (2)

T. Honda, K. Okada, and J. Tsujiuchi, “3-D distortion of observed images reconstructed from a cylindrical holographic stereogram. (1) laser light reconstruction type,” Opt. Commun. 36, 11–16 (1981).
[CrossRef]

K. Okada, T. Honda, and J. Tsujiuchi, “3-D distortion of observed images reconstructed from a cylindrical holographic stereogram. (2) white light reconstruction type,” Opt. Commun. 36, 17–21 (1981).
[CrossRef]

Agus, M.

M. Agus, F. Bettio, A. Giachetti, E. Gobbetti, J. A. I. Guitián, F. Marton, J. Nilsson, and G. Pintore, “An interactive 3D medical visualization system based on a light field display,” Visual Computer 25, 883–893 (2009).
[CrossRef]

Bannister, B. W.

M. Stanley, P. B. Conway, S. Coomber, J. C. Jones, D. C. Scattergood, C. W. Slinger, B. W. Bannister, C. V. Brown, W. A. Crossland, and A. R. L. Travis, “Novel electro-optic modulator system for the production of dynamic images from giga-pixel computer-generated holograms,” Proc. SPIE 3956, 13–22 (2000).
[CrossRef]

Benton, S. A.

P. St. Hilaire, M. E. Lucente, J. D. Sutter, R. Pappu, C. J. Sparrell, and S. A. Benton, “Scaling up the MIT holographic video system,” Proc. SPIE 2333, 374–380 (1995).
[CrossRef]

Benzie, P.

P. Benzie, J. Watson, P. Surman, I. Rakkolainen, K. Hopf, H. Urey, V. Sainov, and C. von Kopylow, “A survey of 3DTV displays: techniques and technologies,” IEEE Trans. Circuits Syst. Video Technol. 17, 1647–1658 (2007).
[CrossRef]

Bettio, F.

M. Agus, F. Bettio, A. Giachetti, E. Gobbetti, J. A. I. Guitián, F. Marton, J. Nilsson, and G. Pintore, “An interactive 3D medical visualization system based on a light field display,” Visual Computer 25, 883–893 (2009).
[CrossRef]

Brown, C. V.

M. Stanley, P. B. Conway, S. Coomber, J. C. Jones, D. C. Scattergood, C. W. Slinger, B. W. Bannister, C. V. Brown, W. A. Crossland, and A. R. L. Travis, “Novel electro-optic modulator system for the production of dynamic images from giga-pixel computer-generated holograms,” Proc. SPIE 3956, 13–22 (2000).
[CrossRef]

Burckhardt, C. B.

R. J. Collier, C. B. Burckhardt, and L. H. Lin, Optical Holography (Academic, 1971).

Collier, R. J.

R. J. Collier, C. B. Burckhardt, and L. H. Lin, Optical Holography (Academic, 1971).

Conway, P. B.

M. Stanley, P. B. Conway, S. Coomber, J. C. Jones, D. C. Scattergood, C. W. Slinger, B. W. Bannister, C. V. Brown, W. A. Crossland, and A. R. L. Travis, “Novel electro-optic modulator system for the production of dynamic images from giga-pixel computer-generated holograms,” Proc. SPIE 3956, 13–22 (2000).
[CrossRef]

Coomber, S.

M. Stanley, P. B. Conway, S. Coomber, J. C. Jones, D. C. Scattergood, C. W. Slinger, B. W. Bannister, C. V. Brown, W. A. Crossland, and A. R. L. Travis, “Novel electro-optic modulator system for the production of dynamic images from giga-pixel computer-generated holograms,” Proc. SPIE 3956, 13–22 (2000).
[CrossRef]

Crossland, W. A.

M. Stanley, P. B. Conway, S. Coomber, J. C. Jones, D. C. Scattergood, C. W. Slinger, B. W. Bannister, C. V. Brown, W. A. Crossland, and A. R. L. Travis, “Novel electro-optic modulator system for the production of dynamic images from giga-pixel computer-generated holograms,” Proc. SPIE 3956, 13–22 (2000).
[CrossRef]

Garner, H. R.

Giachetti, A.

M. Agus, F. Bettio, A. Giachetti, E. Gobbetti, J. A. I. Guitián, F. Marton, J. Nilsson, and G. Pintore, “An interactive 3D medical visualization system based on a light field display,” Visual Computer 25, 883–893 (2009).
[CrossRef]

Gobbetti, E.

M. Agus, F. Bettio, A. Giachetti, E. Gobbetti, J. A. I. Guitián, F. Marton, J. Nilsson, and G. Pintore, “An interactive 3D medical visualization system based on a light field display,” Visual Computer 25, 883–893 (2009).
[CrossRef]

Guitián, J. A. I.

M. Agus, F. Bettio, A. Giachetti, E. Gobbetti, J. A. I. Guitián, F. Marton, J. Nilsson, and G. Pintore, “An interactive 3D medical visualization system based on a light field display,” Visual Computer 25, 883–893 (2009).
[CrossRef]

Halle, M. W.

M. A. Klug, M. W. Halle, M. Lucente, and W. J. Plesniak, “A compact prototype one-step ultragram printer,” Proc. SPIE 1914, 15–23 (1993).
[CrossRef]

Hariharan, P.

P. Hariharan, Optical Holography, Principles, Techniques and Applications (Cambridge University Press, 1996).

Helseth, L. E.

L. E. Helseth and I. Singstad, “Diffusers for holographic stereography,” Opt. Commun. 193, 81–86 (2001).
[CrossRef]

Honda, T.

K. Okada, T. Honda, and J. Tsujiuchi, “3-D distortion of observed images reconstructed from a cylindrical holographic stereogram. (2) white light reconstruction type,” Opt. Commun. 36, 17–21 (1981).
[CrossRef]

T. Honda, K. Okada, and J. Tsujiuchi, “3-D distortion of observed images reconstructed from a cylindrical holographic stereogram. (1) laser light reconstruction type,” Opt. Commun. 36, 11–16 (1981).
[CrossRef]

Hopf, K.

P. Benzie, J. Watson, P. Surman, I. Rakkolainen, K. Hopf, H. Urey, V. Sainov, and C. von Kopylow, “A survey of 3DTV displays: techniques and technologies,” IEEE Trans. Circuits Syst. Video Technol. 17, 1647–1658 (2007).
[CrossRef]

Huebschman, M. L.

Jaimes, Alejandro

A. Ravishankar Rao and Alejandro Jaimes, “Digital stereoscopic imaging,” Proc. SPIE 3639, 144–154 (1999).
[CrossRef]

Jones, J. C.

M. Stanley, P. B. Conway, S. Coomber, J. C. Jones, D. C. Scattergood, C. W. Slinger, B. W. Bannister, C. V. Brown, W. A. Crossland, and A. R. L. Travis, “Novel electro-optic modulator system for the production of dynamic images from giga-pixel computer-generated holograms,” Proc. SPIE 3956, 13–22 (2000).
[CrossRef]

Katz, B.

Klug, M. A.

M. A. Klug, M. W. Halle, M. Lucente, and W. J. Plesniak, “A compact prototype one-step ultragram printer,” Proc. SPIE 1914, 15–23 (1993).
[CrossRef]

Lin, L. H.

R. J. Collier, C. B. Burckhardt, and L. H. Lin, Optical Holography (Academic, 1971).

Lucente, M.

M. A. Klug, M. W. Halle, M. Lucente, and W. J. Plesniak, “A compact prototype one-step ultragram printer,” Proc. SPIE 1914, 15–23 (1993).
[CrossRef]

Lucente, M. E.

P. St. Hilaire, M. E. Lucente, J. D. Sutter, R. Pappu, C. J. Sparrell, and S. A. Benton, “Scaling up the MIT holographic video system,” Proc. SPIE 2333, 374–380 (1995).
[CrossRef]

Marton, F.

M. Agus, F. Bettio, A. Giachetti, E. Gobbetti, J. A. I. Guitián, F. Marton, J. Nilsson, and G. Pintore, “An interactive 3D medical visualization system based on a light field display,” Visual Computer 25, 883–893 (2009).
[CrossRef]

Munjuluri, B.

Nilsson, J.

M. Agus, F. Bettio, A. Giachetti, E. Gobbetti, J. A. I. Guitián, F. Marton, J. Nilsson, and G. Pintore, “An interactive 3D medical visualization system based on a light field display,” Visual Computer 25, 883–893 (2009).
[CrossRef]

Okada, K.

K. Okada, T. Honda, and J. Tsujiuchi, “3-D distortion of observed images reconstructed from a cylindrical holographic stereogram. (2) white light reconstruction type,” Opt. Commun. 36, 17–21 (1981).
[CrossRef]

T. Honda, K. Okada, and J. Tsujiuchi, “3-D distortion of observed images reconstructed from a cylindrical holographic stereogram. (1) laser light reconstruction type,” Opt. Commun. 36, 11–16 (1981).
[CrossRef]

Pappu, R.

P. St. Hilaire, M. E. Lucente, J. D. Sutter, R. Pappu, C. J. Sparrell, and S. A. Benton, “Scaling up the MIT holographic video system,” Proc. SPIE 2333, 374–380 (1995).
[CrossRef]

Pintore, G.

M. Agus, F. Bettio, A. Giachetti, E. Gobbetti, J. A. I. Guitián, F. Marton, J. Nilsson, and G. Pintore, “An interactive 3D medical visualization system based on a light field display,” Visual Computer 25, 883–893 (2009).
[CrossRef]

Plesniak, W. J.

M. A. Klug, M. W. Halle, M. Lucente, and W. J. Plesniak, “A compact prototype one-step ultragram printer,” Proc. SPIE 1914, 15–23 (1993).
[CrossRef]

Rakkolainen, I.

P. Benzie, J. Watson, P. Surman, I. Rakkolainen, K. Hopf, H. Urey, V. Sainov, and C. von Kopylow, “A survey of 3DTV displays: techniques and technologies,” IEEE Trans. Circuits Syst. Video Technol. 17, 1647–1658 (2007).
[CrossRef]

Rao, A. Ravishankar

A. Ravishankar Rao and Alejandro Jaimes, “Digital stereoscopic imaging,” Proc. SPIE 3639, 144–154 (1999).
[CrossRef]

Rosen, J.

Sainov, V.

P. Benzie, J. Watson, P. Surman, I. Rakkolainen, K. Hopf, H. Urey, V. Sainov, and C. von Kopylow, “A survey of 3DTV displays: techniques and technologies,” IEEE Trans. Circuits Syst. Video Technol. 17, 1647–1658 (2007).
[CrossRef]

Scattergood, D. C.

M. Stanley, P. B. Conway, S. Coomber, J. C. Jones, D. C. Scattergood, C. W. Slinger, B. W. Bannister, C. V. Brown, W. A. Crossland, and A. R. L. Travis, “Novel electro-optic modulator system for the production of dynamic images from giga-pixel computer-generated holograms,” Proc. SPIE 3956, 13–22 (2000).
[CrossRef]

Shaked, N. T.

Singstad, I.

L. E. Helseth and I. Singstad, “Diffusers for holographic stereography,” Opt. Commun. 193, 81–86 (2001).
[CrossRef]

Slinger, C. W.

M. Stanley, P. B. Conway, S. Coomber, J. C. Jones, D. C. Scattergood, C. W. Slinger, B. W. Bannister, C. V. Brown, W. A. Crossland, and A. R. L. Travis, “Novel electro-optic modulator system for the production of dynamic images from giga-pixel computer-generated holograms,” Proc. SPIE 3956, 13–22 (2000).
[CrossRef]

Sparrell, C. J.

P. St. Hilaire, M. E. Lucente, J. D. Sutter, R. Pappu, C. J. Sparrell, and S. A. Benton, “Scaling up the MIT holographic video system,” Proc. SPIE 2333, 374–380 (1995).
[CrossRef]

St. Hilaire, P.

P. St. Hilaire, M. E. Lucente, J. D. Sutter, R. Pappu, C. J. Sparrell, and S. A. Benton, “Scaling up the MIT holographic video system,” Proc. SPIE 2333, 374–380 (1995).
[CrossRef]

Stanley, M.

M. Stanley, P. B. Conway, S. Coomber, J. C. Jones, D. C. Scattergood, C. W. Slinger, B. W. Bannister, C. V. Brown, W. A. Crossland, and A. R. L. Travis, “Novel electro-optic modulator system for the production of dynamic images from giga-pixel computer-generated holograms,” Proc. SPIE 3956, 13–22 (2000).
[CrossRef]

Surman, P.

P. Benzie, J. Watson, P. Surman, I. Rakkolainen, K. Hopf, H. Urey, V. Sainov, and C. von Kopylow, “A survey of 3DTV displays: techniques and technologies,” IEEE Trans. Circuits Syst. Video Technol. 17, 1647–1658 (2007).
[CrossRef]

Sutter, J. D.

P. St. Hilaire, M. E. Lucente, J. D. Sutter, R. Pappu, C. J. Sparrell, and S. A. Benton, “Scaling up the MIT holographic video system,” Proc. SPIE 2333, 374–380 (1995).
[CrossRef]

Travis, A. R. L.

M. Stanley, P. B. Conway, S. Coomber, J. C. Jones, D. C. Scattergood, C. W. Slinger, B. W. Bannister, C. V. Brown, W. A. Crossland, and A. R. L. Travis, “Novel electro-optic modulator system for the production of dynamic images from giga-pixel computer-generated holograms,” Proc. SPIE 3956, 13–22 (2000).
[CrossRef]

Tsujiuchi, J.

K. Okada, T. Honda, and J. Tsujiuchi, “3-D distortion of observed images reconstructed from a cylindrical holographic stereogram. (2) white light reconstruction type,” Opt. Commun. 36, 17–21 (1981).
[CrossRef]

T. Honda, K. Okada, and J. Tsujiuchi, “3-D distortion of observed images reconstructed from a cylindrical holographic stereogram. (1) laser light reconstruction type,” Opt. Commun. 36, 11–16 (1981).
[CrossRef]

Urey, H.

P. Benzie, J. Watson, P. Surman, I. Rakkolainen, K. Hopf, H. Urey, V. Sainov, and C. von Kopylow, “A survey of 3DTV displays: techniques and technologies,” IEEE Trans. Circuits Syst. Video Technol. 17, 1647–1658 (2007).
[CrossRef]

von Kopylow, C.

P. Benzie, J. Watson, P. Surman, I. Rakkolainen, K. Hopf, H. Urey, V. Sainov, and C. von Kopylow, “A survey of 3DTV displays: techniques and technologies,” IEEE Trans. Circuits Syst. Video Technol. 17, 1647–1658 (2007).
[CrossRef]

Watson, J.

P. Benzie, J. Watson, P. Surman, I. Rakkolainen, K. Hopf, H. Urey, V. Sainov, and C. von Kopylow, “A survey of 3DTV displays: techniques and technologies,” IEEE Trans. Circuits Syst. Video Technol. 17, 1647–1658 (2007).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Circuits Syst. Video Technol. (1)

P. Benzie, J. Watson, P. Surman, I. Rakkolainen, K. Hopf, H. Urey, V. Sainov, and C. von Kopylow, “A survey of 3DTV displays: techniques and technologies,” IEEE Trans. Circuits Syst. Video Technol. 17, 1647–1658 (2007).
[CrossRef]

Opt. Commun. (3)

L. E. Helseth and I. Singstad, “Diffusers for holographic stereography,” Opt. Commun. 193, 81–86 (2001).
[CrossRef]

T. Honda, K. Okada, and J. Tsujiuchi, “3-D distortion of observed images reconstructed from a cylindrical holographic stereogram. (1) laser light reconstruction type,” Opt. Commun. 36, 11–16 (1981).
[CrossRef]

K. Okada, T. Honda, and J. Tsujiuchi, “3-D distortion of observed images reconstructed from a cylindrical holographic stereogram. (2) white light reconstruction type,” Opt. Commun. 36, 17–21 (1981).
[CrossRef]

Opt. Express (1)

Proc. SPIE (4)

M. A. Klug, M. W. Halle, M. Lucente, and W. J. Plesniak, “A compact prototype one-step ultragram printer,” Proc. SPIE 1914, 15–23 (1993).
[CrossRef]

A. Ravishankar Rao and Alejandro Jaimes, “Digital stereoscopic imaging,” Proc. SPIE 3639, 144–154 (1999).
[CrossRef]

P. St. Hilaire, M. E. Lucente, J. D. Sutter, R. Pappu, C. J. Sparrell, and S. A. Benton, “Scaling up the MIT holographic video system,” Proc. SPIE 2333, 374–380 (1995).
[CrossRef]

M. Stanley, P. B. Conway, S. Coomber, J. C. Jones, D. C. Scattergood, C. W. Slinger, B. W. Bannister, C. V. Brown, W. A. Crossland, and A. R. L. Travis, “Novel electro-optic modulator system for the production of dynamic images from giga-pixel computer-generated holograms,” Proc. SPIE 3956, 13–22 (2000).
[CrossRef]

Visual Computer (1)

M. Agus, F. Bettio, A. Giachetti, E. Gobbetti, J. A. I. Guitián, F. Marton, J. Nilsson, and G. Pintore, “An interactive 3D medical visualization system based on a light field display,” Visual Computer 25, 883–893 (2009).
[CrossRef]

Other (2)

P. Hariharan, Optical Holography, Principles, Techniques and Applications (Cambridge University Press, 1996).

R. J. Collier, C. B. Burckhardt, and L. H. Lin, Optical Holography (Academic, 1971).

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Figures (5)

Fig. 1
Fig. 1

Schematic of relationship between V ( a , b ) , S ( e , f ) and p ( m , n ) . (a) image array from A × B viewpoints; (b) segmentation of V ( a , b ) with E × F optical channels; (c) M × N pixel array of a sectional image S ( e , f ) .

Fig. 2
Fig. 2

Three of the original images.

Fig. 3
Fig. 3

Recording setup: (S) shutter, (SP) beam splitter, (SPO) object beam splitter, (M) mirror, (EBL) beam expanding lens, (SLM) spatial light modulator, (LO) object lens, (SPR) reference beam splitter, (H) hologram.

Fig. 4
Fig. 4

Three encoding images on the SLM.

Fig. 5
Fig. 5

Reconstructed images from three different viewpoints.

Equations (6)

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

V = [ V ( 1 , 1 ) , V ( 1 , 2 ) , , V ( 1 , B ) ; V ( 2 , 1 ) , V ( 2 , 2 ) , , V ( 2 , B ) ; ; V ( A , 1 ) , V ( A , 2 ) , , V ( A , B ) ] .
V ( a , b ) = [ S ( 1 , 1 ) , S ( 1 , 2 ) , , S ( 1 , F ) ; S ( 2 , 1 ) , S ( 2 , 2 ) , , S ( 2 , F ) ; ; S ( E , 1 ) , S ( E , 2 ) , , S ( E , F ) ] .
S ( e , f ) = [ p ( 1 , 1 ) , p ( 1 , 2 ) , , p ( 1 , N ) ; p ( 2 , 1 ) , p ( 2 , 2 ) , , p ( 2 , N ) ; ; p ( M , 1 ) , p ( M , 2 ) , , p ( M , N ) ] .
H = [ H ( 1 , 1 ) , H ( 1 , 2 ) , , H ( 1 , N ) ; H ( 2 , 1 ) , H ( 2 , 2 ) , , H ( 2 , N ) ; ; H ( M , 1 ) , H ( M , 2 ) , , H ( M , N ) ] .
H ( m , n ) = [ h ( 1 , 1 ) , h ( 1 , 2 ) , , h ( 1 , B ) ; h ( 2 , 1 ) , h ( 2 , 2 ) , , h ( 2 , B ) ; ; h ( A , 1 ) , h ( A , 2 ) , , h ( A , B ) ] .
H ( m , n ) = [ p ( m , n ) | V ( 1 , 1 ) S ( e , f ) , p ( m , n ) | V ( 1 , 2 ) S ( e , f ) , , p ( m , n ) | V ( 1 , B ) S ( e , f ) ; p ( m , n ) | V ( 2 , 1 ) S ( e , f ) , p ( m , n ) | V ( 2 , 2 ) S ( e , f ) , , p ( m , n ) | V ( 2 , B ) S ( e , f ) ; ; p ( m , n ) | V ( A , 1 ) S ( e , f ) , p ( m , n ) | V ( A , 2 ) S ( e , f ) , , p ( m , n ) | V ( A , B ) S ( e , f ) ] .

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