Abstract

Here, a new method for calculating the computer-generated holograms of three-dimensional (3D) objects is presented along with a review of current techniques. The method, the planar layers method (PLM), is established on the idea of representing 3D objects in discrete planar layers perpendicular to the observation plane, then calculating the total far field pattern by summing up the far field patterns of each layer. Simulation results, computational complexity, and error comparisons reveal that this new method can be used to calculate far field patterns—hence, the holograms—of computer-synthesized objects very efficiently.

© 2010 Optical Society of America

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B. Katz, N. T. Shaked, and J. Rosen, “Synthesizing computer-generated holograms with reduced number of perspective projections,” Opt. Express 15, 13250–13255(2007).
[CrossRef] [PubMed]

R. Ziegler, P. Kaufmann, and M. Gross, “A framework for holographic scene representation and image synthesis,” IEEE Trans. Vis. Comput. Graph. 13, 403–415 (2007).
[CrossRef] [PubMed]

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[CrossRef]

2006 (5)

2005 (2)

K. Matsushima, “Computer-generated holograms for three-dimensional surface objects with shade and texture,” Appl. Opt. 44, 4607–4614 (2005).
[CrossRef] [PubMed]

K. Matsushima, “Exact hidden-surface removal in digitally synthetic full-parallax holograms,” Proc. SPIE 5742, 25–32 (2005) .
[CrossRef]

2004 (3)

2003 (3)

2002 (1)

2001 (4)

2000 (4)

C. B. Lefebvre, S. Coëtmellec, D. Lebrun, and C. Özkul, “Application of wavelet transform to hologram analysis: Three-dimensional location of particles,” Opt. Lasers Eng. 33, 409–421 (2000).
[CrossRef]

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

Y. Takaki and H. Ohzu, “Hybrid holographic microscopy: visualization of three-dimensional object information by use of viewing angles,” Appl. Opt. 39, 5302–5308 (2000).
[CrossRef]

K. Matsushima and M. Takai, “Recurrence formulas for fast creation of synthetic three-dimensional holograms,” Appl. Opt. 39, 6587–6594 (2000).
[CrossRef]

1999 (2)

1998 (1)

1997 (2)

J. L. Juárez-Pérez, A. Olivares-Pérez, and L. R. Berriel-Valdos, “Nonredundant calculations for creating digital Fresnel holograms,” Appl. Opt. 36, 7437–7443 (1997).
[CrossRef]

J. S. Underkoffler, “Occlusion processing and smooth surface shading for fully computed synthetic holography,” Proc. SPIE 3011, 19–30 (1997).
[CrossRef]

1994 (2)

1993 (1)

M. Lucente, “Interactive computation of holograms using a look-up table,” J. Electron. Imaging 02, 28–34(1993).
[CrossRef]

1992 (1)

1991 (1)

M. W. Halle, S. A. Benton, M. A. Klug, and J. S. Underkoffler, “Ultragram: a generalized holographic stereogram,” Proc. SPIE 1461, 142–155 (1991).
[CrossRef]

1988 (1)

1985 (1)

H. J. Rabal, N. Bolognini, and E. E. Sicre, “Diffraction by a tilted aperture,” Opt. Acta 32, 1309–1311 (1985).
[CrossRef]

1983 (1)

K. Patorski, “Fraunhofer diffraction patterns of titled planar objects,” Opt. Acta 30, 673–679 (1983).
[CrossRef]

1981 (1)

S. Ganci, “Fourier diffraction through a tilted slit,” Eur. J. Phys. 2, 158–160 (1981).
[CrossRef]

1976 (1)

1972 (1)

M. A. Kronrod, N. S. Merzlyakov, and L. P. Yaroslavsky, “Reconstruction of a hologram with a computer,” Sov. Phys. Tech. Phys. 17, 419–420 (1972).

1971 (2)

T. Huang, “Digital holography,” Proc. IEEE 59, 1335–1346(1971).
[CrossRef]

J. Goodman, “An introduction to the principles and applications of holography,” Proc. IEEE 59, 1292–1304 (1971).
[CrossRef]

1970 (1)

R. Dändliker and K. Weiss, “Reconstruction of the three-dimensional refractive index from scattered waves,” Opt. Commun. 1, 323–328 (1970).
[CrossRef]

1948 (1)

D. Gabor, “A new microscopic principle,” Nature 161, 777–778(1948).
[CrossRef] [PubMed]

Abookasis, D.

Ahrenberg, L.

Aoki, Y.

Y. Aoki, “Watermarking technique using computer-generated holograms,” Electron. Commun. Jpn. 84, 21–31 (2001).
[CrossRef]

Bayraktar, M.

M. Özcan and M. Bayraktar, “Digital holography image reconstruction methods,” Proc. SPIE 7233, 72330B (2009).
[CrossRef]

M. Bayraktar and M. Özcan, “A new method for computer-generated holography of 3D objects,” in 24th International Symposium on Computer and Information Sciences (ISCIS) (IEEE, 2009), pp. 66–69.
[CrossRef]

Benton, S. A.

M. W. Halle, S. A. Benton, M. A. Klug, and J. S. Underkoffler, “Ultragram: a generalized holographic stereogram,” Proc. SPIE 1461, 142–155 (1991).
[CrossRef]

Benzie, P.

Berriel-Valdos, L. R.

Bianco, B.

Bolognini, N.

H. J. Rabal, N. Bolognini, and E. E. Sicre, “Diffraction by a tilted aperture,” Opt. Acta 32, 1309–1311 (1985).
[CrossRef]

Böttger, J.

Castro, M.-A.

Chong, T.-C.

Coëtmellec, S.

C. B. Lefebvre, S. Coëtmellec, D. Lebrun, and C. Özkul, “Application of wavelet transform to hologram analysis: Three-dimensional location of particles,” Opt. Lasers Eng. 33, 409–421 (2000).
[CrossRef]

Dändliker, R.

R. Dändliker and K. Weiss, “Reconstruction of the three-dimensional refractive index from scattered waves,” Opt. Commun. 1, 323–328 (1970).
[CrossRef]

Delen, N.

Deussen, O.

Frauel, Y.

Frere, C.

Gabor, D.

D. Gabor, “A new microscopic principle,” Nature 161, 777–778(1948).
[CrossRef] [PubMed]

Ganci, S.

S. Ganci, “Fourier diffraction through a tilted slit,” Eur. J. Phys. 2, 158–160 (1981).
[CrossRef]

Goodman, J.

J. Goodman, “An introduction to the principles and applications of holography,” Proc. IEEE 59, 1292–1304 (1971).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Gopinathan, U.

A. Nelleri, U. Gopinathan, J. Joseph, and K. Singh, “Three-dimensional object recognition from digital Fresnel hologram by wavelet matched filtering,” Opt. Commun. 259, 499–506(2006).
[CrossRef]

Gotchev, A.

L. Onural, A. Gotchev, H. M. Özaktaş, and E. Stoykova, “A survey of signal processing problems and tools in holographic three-dimensional television,” IEEE Trans. Circuits Syst. Video Technol. 17, 1631–1646 (2007).
[CrossRef]

Gross, M.

R. Ziegler, P. Kaufmann, and M. Gross, “A framework for holographic scene representation and image synthesis,” IEEE Trans. Vis. Comput. Graph. 13, 403–415 (2007).
[CrossRef] [PubMed]

Hahn, J.

Halle, M. W.

M. W. Halle, S. A. Benton, M. A. Klug, and J. S. Underkoffler, “Ultragram: a generalized holographic stereogram,” Proc. SPIE 1461, 142–155 (1991).
[CrossRef]

Hooker, B.

Huang, T.

T. Huang, “Digital holography,” Proc. IEEE 59, 1335–1346(1971).
[CrossRef]

Ichihashi, Y.

Ito, T.

Itoh, M.

Iwase, S.

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

Javidi, B.

Joseph, J.

A. Nelleri, U. Gopinathan, J. Joseph, and K. Singh, “Three-dimensional object recognition from digital Fresnel hologram by wavelet matched filtering,” Opt. Commun. 259, 499–506(2006).
[CrossRef]

Juárez-Pérez, J. L.

Jüptner, W.

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

U. Schnars and W. Jüptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer-Verlag, 2005).

Jüptner, W. P. O.

Kato, J.

Katz, B.

Kaufmann, P.

R. Ziegler, P. Kaufmann, and M. Gross, “A framework for holographic scene representation and image synthesis,” IEEE Trans. Vis. Comput. Graph. 13, 403–415 (2007).
[CrossRef] [PubMed]

Kawai, H.

Kim, E.-S.

Kim, H.

Kim, S.-C.

Klug, M. A.

M. W. Halle, S. A. Benton, M. A. Klug, and J. S. Underkoffler, “Ultragram: a generalized holographic stereogram,” Proc. SPIE 1461, 142–155 (1991).
[CrossRef]

Kondoh, A.

K. Matsushima and A. Kondoh, “A wave-optical algorithm for hidden-surface removal in digitally synthetic full-parallax holograms for three-dimensional objects,” Proc. SPIE 5290, 90–97 (2004).
[CrossRef]

K. Matsushima and A. Kondoh, “Wave optical algorithm for creating digitally snythetic holograms of three-dimensional surface objects,” Proc. SPIE 5005, 190–197 (2003).
[CrossRef]

König, M.

Kronrod, M. A.

M. A. Kronrod, N. S. Merzlyakov, and L. P. Yaroslavsky, “Reconstruction of a hologram with a computer,” Sov. Phys. Tech. Phys. 17, 419–420 (1972).

Lebrun, D.

C. B. Lefebvre, S. Coëtmellec, D. Lebrun, and C. Özkul, “Application of wavelet transform to hologram analysis: Three-dimensional location of particles,” Opt. Lasers Eng. 33, 409–421 (2000).
[CrossRef]

Lee, B.

Lefebvre, C. B.

C. B. Lefebvre, S. Coëtmellec, D. Lebrun, and C. Özkul, “Application of wavelet transform to hologram analysis: Three-dimensional location of particles,” Opt. Lasers Eng. 33, 409–421 (2000).
[CrossRef]

Leseberg, D.

Li, Y.

Liang, X.

Lucente, M.

M. Lucente, “Interactive computation of holograms using a look-up table,” J. Electron. Imaging 02, 28–34(1993).
[CrossRef]

Magnor, M.

Masuda, N.

Matoba, O.

Y. Frauel, T. Naughton, O. Matoba, E. Tajahuerce, and B. Javidi, “Three-dimensional imaging and processing using computational holographic imaging,” Proc. IEEE 94, 636–653 (2006).
[CrossRef]

Matsushima, K.

K. Matsushima and S. Nakahara, “Extremely high-definition full-parallax computer-generated hologram created by the polygon-based method,” Appl. Opt. 48, H54–H63 (2009).
[CrossRef] [PubMed]

K. Matsushima, “Exact hidden-surface removal in digitally synthetic full-parallax holograms,” Proc. SPIE 5742, 25–32 (2005) .
[CrossRef]

K. Matsushima, “Computer-generated holograms for three-dimensional surface objects with shade and texture,” Appl. Opt. 44, 4607–4614 (2005).
[CrossRef] [PubMed]

K. Matsushima and A. Kondoh, “A wave-optical algorithm for hidden-surface removal in digitally synthetic full-parallax holograms for three-dimensional objects,” Proc. SPIE 5290, 90–97 (2004).
[CrossRef]

K. Matsushima and A. Kondoh, “Wave optical algorithm for creating digitally snythetic holograms of three-dimensional surface objects,” Proc. SPIE 5005, 190–197 (2003).
[CrossRef]

K. Matsushima and M. Takai, “Recurrence formulas for fast creation of synthetic three-dimensional holograms,” Appl. Opt. 39, 6587–6594 (2000).
[CrossRef]

Merzlyakov, N. S.

M. A. Kronrod, N. S. Merzlyakov, and L. P. Yaroslavsky, “Reconstruction of a hologram with a computer,” Sov. Phys. Tech. Phys. 17, 419–420 (1972).

Mizuno, J.

Nakahara, S.

Nakayama, H.

Naughton, T.

Y. Frauel, T. Naughton, O. Matoba, E. Tajahuerce, and B. Javidi, “Three-dimensional imaging and processing using computational holographic imaging,” Proc. IEEE 94, 636–653 (2006).
[CrossRef]

Naughton, T. J.

Nelleri, A.

A. Nelleri, U. Gopinathan, J. Joseph, and K. Singh, “Three-dimensional object recognition from digital Fresnel hologram by wavelet matched filtering,” Opt. Commun. 259, 499–506(2006).
[CrossRef]

Ohta, S.

Ohzu, H.

Olivares-Pérez, A.

Oneda, T.

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

Onural, L.

L. Onural, A. Gotchev, H. M. Özaktaş, and E. Stoykova, “A survey of signal processing problems and tools in holographic three-dimensional television,” IEEE Trans. Circuits Syst. Video Technol. 17, 1631–1646 (2007).
[CrossRef]

Özaktas, H. M.

L. Onural, A. Gotchev, H. M. Özaktaş, and E. Stoykova, “A survey of signal processing problems and tools in holographic three-dimensional television,” IEEE Trans. Circuits Syst. Video Technol. 17, 1631–1646 (2007).
[CrossRef]

Özcan, M.

M. Özcan and M. Bayraktar, “Digital holography image reconstruction methods,” Proc. SPIE 7233, 72330B (2009).
[CrossRef]

M. Bayraktar and M. Özcan, “A new method for computer-generated holography of 3D objects,” in 24th International Symposium on Computer and Information Sciences (ISCIS) (IEEE, 2009), pp. 66–69.
[CrossRef]

Özkul, C.

C. B. Lefebvre, S. Coëtmellec, D. Lebrun, and C. Özkul, “Application of wavelet transform to hologram analysis: Three-dimensional location of particles,” Opt. Lasers Eng. 33, 409–421 (2000).
[CrossRef]

Pan, Y.

Patorski, K.

K. Patorski, “Fraunhofer diffraction patterns of titled planar objects,” Opt. Acta 30, 673–679 (1983).
[CrossRef]

Rabal, H. J.

H. J. Rabal, N. Bolognini, and E. E. Sicre, “Diffraction by a tilted aperture,” Opt. Acta 32, 1309–1311 (1985).
[CrossRef]

Ritter, A.

Rosen, J.

Sando, Y.

Schnars, U.

Shaked, N. T.

Shen, F.

Shimobaba, T.

Shiraki, A.

Sicre, E. E.

H. J. Rabal, N. Bolognini, and E. E. Sicre, “Diffraction by a tilted aperture,” Opt. Acta 32, 1309–1311 (1985).
[CrossRef]

Singh, K.

A. Nelleri, U. Gopinathan, J. Joseph, and K. Singh, “Three-dimensional object recognition from digital Fresnel hologram by wavelet matched filtering,” Opt. Commun. 259, 499–506(2006).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

(a) Perspective view of a simple pyramid, where its visible surfaces are painted with different gray levels. For a simple illustration of the idea, we assume the pyramid is located head-on with respect to the recording plane ( x y plane) as shown with the coordinate axes. (b) Pyramid is represented by 14 × 14 pixels in the intensity map and (c) it is represented by four different layers in the depth map such that the distances (z = 0.70, 0.72, 0.74, 0.76 m) are the distances of the individual layers from the recording plane. (Note that the coloring on the depth map is just for illustration.) The bottom row of the figure shows (d) the perspective views of the layers with their associated intensity distributions, and (e) each layer is propagated via the Fresnel transform, and the results are superposed (fields are added) and the resultant intensity distribution in the recording plane is shown.

Fig. 2
Fig. 2

Discretization of the object in depth is done according to the desired maximum angular separation between the two layers, which depends on the distance between the layer and the hologram, the hologram size and the lateral resolution of the object (please see text for explanation).

Fig. 3
Fig. 3

(a) Complexity comparison of the pointwise method and the PLM with respect to the object plane dimensions ( N o × N o ) and the recording plane dimensions ( N × N ). Dashed lines are for the pointwise method, and solid lines are for the PLM. The complexity of the pointwise method increases quadratically and much higher than the complexity of the PLM. (b) Ratio of the simulation times of the pointwise method and the PLM for M = 2 N o 1 and N = 512 . The time ratio is calculated by dividing the simulation times required to calculate the wave field of randomly generated intensity maps using the pointwise method and the PLM. There is an exponential speed gain over the pointwise method, and as N o increases, the speedup becomes more significant. For example, for the 256 × 256 input image (where log 2 N o = 8 ), the speedup is 102.

Fig. 4
Fig. 4

(a) Intensity map of a cube that is covered with a U.S. Air Force resolution chart with dimensions of 5.9 mm × 5.9 mm × 5.9 mm is shown here. The cube is rotated by 45 ° in the x and y axes in the counterclockwise direction and viewed from the + z direction. (b) Depth map for the cube. There are 202 unique depth values where the nearest point is at d = 0.7 m and the farthest point is at d = 0.7071 m . The magnitude of the object wave was calculated using (c) the pointwise method and (d) the PLM. For this example, the MSE is calculated as 0.037 (see text).

Fig. 5
Fig. 5

Reconstruction results at d = 0.7 m for the holograms generated by (a) the pointwise method and (b) the PLM.

Equations (8)

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U ( ξ , η , z ) = i λ U 0 ( x 0 , y 0 , z 0 ) exp ( i 2 π λ ρ ) ρ cos θ d x 0 d y 0 ,
U ( ξ , η ) = U 0 ( x 0 , y 0 ) g ( ξ x 0 , η y 0 , z ) d x 0 d y 0 ,
T convolution = T a + T b + T c + T d + T e = τ 2 ( N 2 log 2 N + N 2 + N 2 log 2 N + N 2 + N 2 log 2 N ) ,
U ( ξ , η ) = i λ d exp [ i 2 π λ d ] exp [ i π λ d ( ξ 2 + η 2 ) ] × F 1 { U 0 ( x 0 , y 0 ) exp [ i π λ d ( x 0 2 + y 0 2 ) ] } .
T fresnel = T a + T b + T c + T d = τ 3 ( N 2 + N 2 + N 2 log 2 N + N 2 ) ,
d θ = arctan ( z + Δ z L h ) arctan ( z L h ) ,
MSE = m = 1 N n = 1 N ( | U PLM ( m , n ) | | U pointwise ( m , n ) | ) 2 N 2 .
I h = | U o + U r | 2 = U o 2 + U r 2 + U o * U r + U o U r * .

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