Abstract

Abstract: We propose a hogel overlapping method for the holographic printer to enhance the lateral resolution of holographic stereograms. The hogel size is directly related to the lateral resolution of the holographic stereogram. Our analysis by computer simulation shows that there is a limit to decreasing the hogel size while printing holographic stereograms. Instead of reducing the size of hogel, the lateral resolution of holographic stereograms can be enhanced by printing overlapped hogels, which makes it possible to take advantage of multiplexing property of the volume hologram. We built a holographic printer, and recorded two holographic stereograms using the conventional and proposed overlapping methods. The images and movies of the holographic stereograms experimentally captured were compared between the conventional and proposed methods. The experimental results confirm that the proposed hogel overlapping method improves the lateral resolution of holographic stereograms compared to the conventional holographic printing method.

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S. J. Daly, R. T. Held, and D. M. Hoffman, “Perceptual issues in stereoscopic signal processing,” IEEE Trans. Broadcast57(2), 347–361 (2011).
[CrossRef]

D. Brotherton-Ratcliffe, S. J. Zacharovas, R. J. Bakanas, J. Pileckas, A. Nikoskij, and J. Kuchin, “Digital holographic printing using pulsed RGB lasers,” Opt. Eng.50(9), 091307 (2011).
[CrossRef]

2010

P.-A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W.-Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature468(7320), 80–83 (2010).
[CrossRef] [PubMed]

2008

N. Peyghambarian, S. Tay, P.-A. Blanche, R. Norwood, and M. Yamamoto, “Rewritable holographic 3D displays,” Opt. Photon. News19(7), 22–27 (2008).
[CrossRef]

S. Tay, P.-A. Blanche, R. Voorakaranam, A. V. Tunç, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. St Hilaire, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic three-dimensional display,” Nature451(7179), 694–698 (2008).
[CrossRef] [PubMed]

1998

1997

1996

1995

M. Yamaguchi, T. Koyama, H. Endoh, N. Ohyama, S. Takahashi, and F. Iwata, “Development of a prototype full-parallax holoprinter,” Proc. SPIE2406, 50–56 (1995).
[CrossRef]

1992

1983

S. A. Benton, “Survey of holographic stereograms,” Proc. SPIE367, 15–19 (1983).
[CrossRef]

Bablumian, A.

P.-A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W.-Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature468(7320), 80–83 (2010).
[CrossRef] [PubMed]

Bakanas, R. J.

D. Brotherton-Ratcliffe, S. J. Zacharovas, R. J. Bakanas, J. Pileckas, A. Nikoskij, and J. Kuchin, “Digital holographic printing using pulsed RGB lasers,” Opt. Eng.50(9), 091307 (2011).
[CrossRef]

Benton, S. A.

S. A. Benton, “Survey of holographic stereograms,” Proc. SPIE367, 15–19 (1983).
[CrossRef]

Blanche, P.-A.

P.-A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W.-Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature468(7320), 80–83 (2010).
[CrossRef] [PubMed]

S. Tay, P.-A. Blanche, R. Voorakaranam, A. V. Tunç, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. St Hilaire, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic three-dimensional display,” Nature451(7179), 694–698 (2008).
[CrossRef] [PubMed]

N. Peyghambarian, S. Tay, P.-A. Blanche, R. Norwood, and M. Yamamoto, “Rewritable holographic 3D displays,” Opt. Photon. News19(7), 22–27 (2008).
[CrossRef]

Brotherton-Ratcliffe, D.

D. Brotherton-Ratcliffe, S. J. Zacharovas, R. J. Bakanas, J. Pileckas, A. Nikoskij, and J. Kuchin, “Digital holographic printing using pulsed RGB lasers,” Opt. Eng.50(9), 091307 (2011).
[CrossRef]

Burr, G. W.

Christenson, C.

P.-A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W.-Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature468(7320), 80–83 (2010).
[CrossRef] [PubMed]

Daly, S. J.

S. J. Daly, R. T. Held, and D. M. Hoffman, “Perceptual issues in stereoscopic signal processing,” IEEE Trans. Broadcast57(2), 347–361 (2011).
[CrossRef]

Endoh, H.

M. Yamaguchi, T. Koyama, H. Endoh, N. Ohyama, S. Takahashi, and F. Iwata, “Development of a prototype full-parallax holoprinter,” Proc. SPIE2406, 50–56 (1995).
[CrossRef]

Flores, D.

P.-A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W.-Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature468(7320), 80–83 (2010).
[CrossRef] [PubMed]

S. Tay, P.-A. Blanche, R. Voorakaranam, A. V. Tunç, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. St Hilaire, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic three-dimensional display,” Nature451(7179), 694–698 (2008).
[CrossRef] [PubMed]

Gu, T.

P.-A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W.-Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature468(7320), 80–83 (2010).
[CrossRef] [PubMed]

S. Tay, P.-A. Blanche, R. Voorakaranam, A. V. Tunç, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. St Hilaire, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic three-dimensional display,” Nature451(7179), 694–698 (2008).
[CrossRef] [PubMed]

Held, R. T.

S. J. Daly, R. T. Held, and D. M. Hoffman, “Perceptual issues in stereoscopic signal processing,” IEEE Trans. Broadcast57(2), 347–361 (2011).
[CrossRef]

Hoffman, D. M.

S. J. Daly, R. T. Held, and D. M. Hoffman, “Perceptual issues in stereoscopic signal processing,” IEEE Trans. Broadcast57(2), 347–361 (2011).
[CrossRef]

Honda, T.

Hsieh, W.-Y.

P.-A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W.-Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature468(7320), 80–83 (2010).
[CrossRef] [PubMed]

Iwata, F.

M. Yamaguchi, T. Koyama, H. Endoh, N. Ohyama, S. Takahashi, and F. Iwata, “Development of a prototype full-parallax holoprinter,” Proc. SPIE2406, 50–56 (1995).
[CrossRef]

Kang, Y. H.

Kathaperumal, M.

P.-A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W.-Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature468(7320), 80–83 (2010).
[CrossRef] [PubMed]

Kim, K. H.

Koyama, T.

M. Yamaguchi, T. Koyama, H. Endoh, N. Ohyama, S. Takahashi, and F. Iwata, “Development of a prototype full-parallax holoprinter,” Proc. SPIE2406, 50–56 (1995).
[CrossRef]

Kuchin, J.

D. Brotherton-Ratcliffe, S. J. Zacharovas, R. J. Bakanas, J. Pileckas, A. Nikoskij, and J. Kuchin, “Digital holographic printing using pulsed RGB lasers,” Opt. Eng.50(9), 091307 (2011).
[CrossRef]

Lee, B.

Li, G.

S. Tay, P.-A. Blanche, R. Voorakaranam, A. V. Tunç, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. St Hilaire, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic three-dimensional display,” Nature451(7179), 694–698 (2008).
[CrossRef] [PubMed]

Lin, W.

P.-A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W.-Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature468(7320), 80–83 (2010).
[CrossRef] [PubMed]

S. Tay, P.-A. Blanche, R. Voorakaranam, A. V. Tunç, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. St Hilaire, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic three-dimensional display,” Nature451(7179), 694–698 (2008).
[CrossRef] [PubMed]

Mok, F. H.

Nikoskij, A.

D. Brotherton-Ratcliffe, S. J. Zacharovas, R. J. Bakanas, J. Pileckas, A. Nikoskij, and J. Kuchin, “Digital holographic printing using pulsed RGB lasers,” Opt. Eng.50(9), 091307 (2011).
[CrossRef]

Norwood, R.

N. Peyghambarian, S. Tay, P.-A. Blanche, R. Norwood, and M. Yamamoto, “Rewritable holographic 3D displays,” Opt. Photon. News19(7), 22–27 (2008).
[CrossRef]

Norwood, R. A.

P.-A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W.-Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature468(7320), 80–83 (2010).
[CrossRef] [PubMed]

S. Tay, P.-A. Blanche, R. Voorakaranam, A. V. Tunç, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. St Hilaire, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic three-dimensional display,” Nature451(7179), 694–698 (2008).
[CrossRef] [PubMed]

Ohyama, N.

Peyghambarian, N.

P.-A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W.-Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature468(7320), 80–83 (2010).
[CrossRef] [PubMed]

S. Tay, P.-A. Blanche, R. Voorakaranam, A. V. Tunç, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. St Hilaire, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic three-dimensional display,” Nature451(7179), 694–698 (2008).
[CrossRef] [PubMed]

N. Peyghambarian, S. Tay, P.-A. Blanche, R. Norwood, and M. Yamamoto, “Rewritable holographic 3D displays,” Opt. Photon. News19(7), 22–27 (2008).
[CrossRef]

Pileckas, J.

D. Brotherton-Ratcliffe, S. J. Zacharovas, R. J. Bakanas, J. Pileckas, A. Nikoskij, and J. Kuchin, “Digital holographic printing using pulsed RGB lasers,” Opt. Eng.50(9), 091307 (2011).
[CrossRef]

Psaltis, D.

Rachwal, B.

P.-A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W.-Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature468(7320), 80–83 (2010).
[CrossRef] [PubMed]

Rokutanda, S.

S. Tay, P.-A. Blanche, R. Voorakaranam, A. V. Tunç, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. St Hilaire, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic three-dimensional display,” Nature451(7179), 694–698 (2008).
[CrossRef] [PubMed]

Siddiqui, O.

P.-A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W.-Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature468(7320), 80–83 (2010).
[CrossRef] [PubMed]

St Hilaire, P.

S. Tay, P.-A. Blanche, R. Voorakaranam, A. V. Tunç, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. St Hilaire, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic three-dimensional display,” Nature451(7179), 694–698 (2008).
[CrossRef] [PubMed]

Sugiura, H.

Takahashi, S.

M. Yamaguchi, T. Koyama, H. Endoh, N. Ohyama, S. Takahashi, and F. Iwata, “Development of a prototype full-parallax holoprinter,” Proc. SPIE2406, 50–56 (1995).
[CrossRef]

Tay, S.

N. Peyghambarian, S. Tay, P.-A. Blanche, R. Norwood, and M. Yamamoto, “Rewritable holographic 3D displays,” Opt. Photon. News19(7), 22–27 (2008).
[CrossRef]

S. Tay, P.-A. Blanche, R. Voorakaranam, A. V. Tunç, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. St Hilaire, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic three-dimensional display,” Nature451(7179), 694–698 (2008).
[CrossRef] [PubMed]

Thomas, J.

P.-A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W.-Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature468(7320), 80–83 (2010).
[CrossRef] [PubMed]

S. Tay, P.-A. Blanche, R. Voorakaranam, A. V. Tunç, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. St Hilaire, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic three-dimensional display,” Nature451(7179), 694–698 (2008).
[CrossRef] [PubMed]

Tunç, A. V.

S. Tay, P.-A. Blanche, R. Voorakaranam, A. V. Tunç, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. St Hilaire, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic three-dimensional display,” Nature451(7179), 694–698 (2008).
[CrossRef] [PubMed]

Voorakaranam, R.

P.-A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W.-Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature468(7320), 80–83 (2010).
[CrossRef] [PubMed]

S. Tay, P.-A. Blanche, R. Voorakaranam, A. V. Tunç, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. St Hilaire, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic three-dimensional display,” Nature451(7179), 694–698 (2008).
[CrossRef] [PubMed]

Wang, P.

P.-A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W.-Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature468(7320), 80–83 (2010).
[CrossRef] [PubMed]

S. Tay, P.-A. Blanche, R. Voorakaranam, A. V. Tunç, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. St Hilaire, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic three-dimensional display,” Nature451(7179), 694–698 (2008).
[CrossRef] [PubMed]

Yamaguchi, M.

Yamamoto, M.

P.-A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W.-Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature468(7320), 80–83 (2010).
[CrossRef] [PubMed]

N. Peyghambarian, S. Tay, P.-A. Blanche, R. Norwood, and M. Yamamoto, “Rewritable holographic 3D displays,” Opt. Photon. News19(7), 22–27 (2008).
[CrossRef]

S. Tay, P.-A. Blanche, R. Voorakaranam, A. V. Tunç, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. St Hilaire, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic three-dimensional display,” Nature451(7179), 694–698 (2008).
[CrossRef] [PubMed]

Zacharovas, S. J.

D. Brotherton-Ratcliffe, S. J. Zacharovas, R. J. Bakanas, J. Pileckas, A. Nikoskij, and J. Kuchin, “Digital holographic printing using pulsed RGB lasers,” Opt. Eng.50(9), 091307 (2011).
[CrossRef]

Appl. Opt.

IEEE Trans. Broadcast

S. J. Daly, R. T. Held, and D. M. Hoffman, “Perceptual issues in stereoscopic signal processing,” IEEE Trans. Broadcast57(2), 347–361 (2011).
[CrossRef]

J. Opt. Soc. Am. A

Nature

S. Tay, P.-A. Blanche, R. Voorakaranam, A. V. Tunç, W. Lin, S. Rokutanda, T. Gu, D. Flores, P. Wang, G. Li, P. St Hilaire, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “An updatable holographic three-dimensional display,” Nature451(7179), 694–698 (2008).
[CrossRef] [PubMed]

P.-A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W.-Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature468(7320), 80–83 (2010).
[CrossRef] [PubMed]

Opt. Eng.

D. Brotherton-Ratcliffe, S. J. Zacharovas, R. J. Bakanas, J. Pileckas, A. Nikoskij, and J. Kuchin, “Digital holographic printing using pulsed RGB lasers,” Opt. Eng.50(9), 091307 (2011).
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Opt. Photon. News

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Supplementary Material (2)

» Media 1: MOV (3302 KB)     
» Media 2: MOV (3247 KB)     

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram of a holographic printer.

Fig. 2
Fig. 2

Optical arrangement on the signal beam path of a holographic printer.

Fig. 3
Fig. 3

Degradation of the image quality for the reconstructed hogel images by decresing the size of the recorded hogel: (a) input image on the SLM, and reconstructed hogel images for the recorded hogel sizes of (b) 100 μm, (c) 200 μm, and (d) 400 μm, respectively. (e) The dependency of PSNR on the hogel size in a range of 0 μm to 427 μm.

Fig. 4
Fig. 4

Graphical descriptions for recorded hogels (a) in a conventional scheme (M = 1 and α = 1) and (b) in a proposed hogel overlapping scheme (M = 4 and α = 2).

Fig. 5
Fig. 5

Schematic diagram of the experimental setup for the holographic printer system.

Fig. 6
Fig. 6

Photograph of the holographic printer system used for the experiments.

Fig. 7
Fig. 7

Perspective images captured from different viewing points: (a) conventional method and (b) proposed method.

Fig. 8
Fig. 8

Single-frame excerpts from video recordings of printed holographic stereograms: (a) conventional method (Media 1) and (b) proposed method (Media 2).

Tables (1)

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Table 1 Parameter values used in the experiments for the conventional and proposed holographic printing methods

Equations (3)

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H(u,v)=rect( u W )rect( v W ).
I re (x,y)= F 1 { F[ I in (x,y) ]×H( f X λf, f Y λf) },
M=2 p δ ,

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