Abstract

In the holographic three-dimensional (3D) display, the numerical synthesis of the computer-generated holograms needs tremendous calculation. To solve the problem, a fast polygon-based method based on two-dimensional Fourier analysis of 3D affine transformation is proposed. From one primitive polygon, the proposed method calculates the diffracted optical field of each arbitrary polygon in the 3D model, where the pseudo-inverse matrix, the interpolation, and the compensation of the power spectral density are employed. The proposed method could save the computation time in the hologram synthesis since it does not need the fast Fourier transform for each polygonal surface and the additional diffusion computation. The numerical simulation and the optical experimental results are presented to demonstrate the effectiveness of the method. The results reveal the proposed method could reconstruct the 3D scene with the solid effect and without the depth limitation. The factors that influence the image quality are discussed, and the thresholds are proposed to ensure the reconstruction quality.

© 2012 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  8. C. Frére and D. Leseberg, “Large objects reconstructed from computer-generated holograms,” Appl. Opt. 28, 2422–2425 (1989).
    [CrossRef]
  9. T. Tommasi and B. Bianco, “Frequency analysis of light diffraction between rotated planes,” Opt. Lett. 17, 556–558 (1992).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  12. K. Matsushima, “Computer-generated holograms for three-dimensional surface objects with shade and texture,” Appl. Opt. 44, 4607–4614 (2005).
    [CrossRef]
  13. K. Matsushima, “Formulation of the rotational transformation of wave fields and their application to digital holography,” Appl. Opt. 47, D110–D116 (2008).
    [CrossRef]
  14. 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]
  15. H. Nishi, K. Matsushima, and S. Nakahara, “Rendering of specular surfaces in polygon-based computer-generated holograms,” Appl. Opt. 50, H245–H252 (2011).
    [CrossRef]
  16. K. Matsushima, H. Nishi, and S. Nakahara, “Simple wave-field rendering for photorealistic reconstruction in polygon-based high-definition computer holography,” J. Electron. Imaging 21, 023002–23003 (2012).
    [CrossRef]
  17. L. Onural, “Exact solution for scalar diffraction between tilted and translated planes using impulse functions over a surface,” J. Opt. Soc. Am. A 28, 290–295 (2011).
    [CrossRef]
  18. L. Ahrenberg, P. Benzie, M. Magnor, and J. Watson, “Computer generated holograms from three dimensional meshes using an analytic light transport model,” Appl. Opt. 47, 1567–1574 (2008).
    [CrossRef]
  19. H. Kim, J. Hahn, and B. Lee, “Mathematical modeling of triangle-mesh-modeled three-dimensional surface objects for digital holography,” Appl. Opt. 47, D117–D127(2008).
    [CrossRef]
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    [CrossRef]
  21. R. N. Bracewell, K. Y. Chang, A. K. Jha, and Y.-H. Wang, “Affine theorem for two-dimensional Fourier transform,” Electron. Lett. 29, 304–306 (1993).
    [CrossRef]
  22. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  23. R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision, 2nd ed. (Cambridge University, 2003).
  24. A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, 2nd ed. (Springer-Verlag, 2003).
  25. T. Greville, “Some applications of the pseudoinverse of a matrix,” SIAM Rev. 2, 15–22 (1960).
    [CrossRef]
  26. W. M. Newman and R. F. Sproull, Principles of Interactive Computer Graphics, 2nd ed. (McGraw-Hill, 1979).
  27. H. Zhang, J. Xie, J. Liu, and Y. Wang, “Elimination of a zero-order beam induced by a pixelated spatial light modulator for holographic projection,” Appl. Opt. 48, 5834–5841 (2009).
    [CrossRef]
  28. K. Matsushima, “Exact hidden-surface removal in digitally synthetic full-parallax holograms,” Proc. SPIE 5742, 25–32 (2005).
    [CrossRef]

2012

K. Matsushima, H. Nishi, and S. Nakahara, “Simple wave-field rendering for photorealistic reconstruction in polygon-based high-definition computer holography,” J. Electron. Imaging 21, 023002–23003 (2012).
[CrossRef]

2011

2009

2008

2005

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]

C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38, 46–53 (2005).
[CrossRef]

2003

1993

T. Tommasi and B. Bianco, “Computer-generated holograms of tilted planes by a spatial frequency approach,” J. Opt. Soc. Am. A 10, 299–305 (1993).
[CrossRef]

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

R. N. Bracewell, K. Y. Chang, A. K. Jha, and Y.-H. Wang, “Affine theorem for two-dimensional Fourier transform,” Electron. Lett. 29, 304–306 (1993).
[CrossRef]

1992

1989

1988

1985

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

1981

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

1960

T. Greville, “Some applications of the pseudoinverse of a matrix,” SIAM Rev. 2, 15–22 (1960).
[CrossRef]

Ahrenberg, L.

Ben-Israel, A.

A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, 2nd ed. (Springer-Verlag, 2003).

Benzie, P.

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]

Bracewell, R. N.

R. N. Bracewell, K. Y. Chang, A. K. Jha, and Y.-H. Wang, “Affine theorem for two-dimensional Fourier transform,” Electron. Lett. 29, 304–306 (1993).
[CrossRef]

Cameron, C.

C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38, 46–53 (2005).
[CrossRef]

Chang, K. Y.

R. N. Bracewell, K. Y. Chang, A. K. Jha, and Y.-H. Wang, “Affine theorem for two-dimensional Fourier transform,” Electron. Lett. 29, 304–306 (1993).
[CrossRef]

Chong, T.-C.

Frére, C.

Ganci, S.

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

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Greville, T.

T. Greville, “Some applications of the pseudoinverse of a matrix,” SIAM Rev. 2, 15–22 (1960).
[CrossRef]

Greville, T. N. E.

A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, 2nd ed. (Springer-Verlag, 2003).

Hahn, J.

Hartley, R.

R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision, 2nd ed. (Cambridge University, 2003).

Jha, A. K.

R. N. Bracewell, K. Y. Chang, A. K. Jha, and Y.-H. Wang, “Affine theorem for two-dimensional Fourier transform,” Electron. Lett. 29, 304–306 (1993).
[CrossRef]

Kim, E.-S.

Kim, H.

Kim, S.-C.

Lee, B.

Leseberg, D.

Liang, X.

Liu, J.

Lucente, M.

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

Magnor, M.

Matsushima, K.

Nakahara, S.

Newman, W. M.

W. M. Newman and R. F. Sproull, Principles of Interactive Computer Graphics, 2nd ed. (McGraw-Hill, 1979).

Nishi, H.

K. Matsushima, H. Nishi, and S. Nakahara, “Simple wave-field rendering for photorealistic reconstruction in polygon-based high-definition computer holography,” J. Electron. Imaging 21, 023002–23003 (2012).
[CrossRef]

H. Nishi, K. Matsushima, and S. Nakahara, “Rendering of specular surfaces in polygon-based computer-generated holograms,” Appl. Opt. 50, H245–H252 (2011).
[CrossRef]

Onural, L.

Pan, Y.

Rabal, H. J.

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

Sakamoto, Y.

Sakata, H.

Schimmel, H.

Sicre, E. E.

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

Slinger, C.

C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38, 46–53 (2005).
[CrossRef]

Solanki, S.

Sproull, R. F.

W. M. Newman and R. F. Sproull, Principles of Interactive Computer Graphics, 2nd ed. (McGraw-Hill, 1979).

Stanley, M.

C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38, 46–53 (2005).
[CrossRef]

Tan, C.

Tanjung, R.

Tommasi, T.

Wang, Y.

Wang, Y.-H.

R. N. Bracewell, K. Y. Chang, A. K. Jha, and Y.-H. Wang, “Affine theorem for two-dimensional Fourier transform,” Electron. Lett. 29, 304–306 (1993).
[CrossRef]

Watson, J.

Wyrowski, F.

Xie, J.

Xu, X.

Zhang, H.

Zisserman, A.

R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision, 2nd ed. (Cambridge University, 2003).

Appl. Opt.

S.-C. Kim and E.-S. Kim, “Effective generation of digital holograms of three-dimensional objects using a novel look-up table method,” Appl. Opt. 47, D55–D62 (2008).
[CrossRef]

D. Leseberg and C. Frére, “Computer-generated holograms of 3-D objects composed of tilted planar segments,” Appl. Opt. 27, 3020–3024 (1988).
[CrossRef]

C. Frére and D. Leseberg, “Large objects reconstructed from computer-generated holograms,” Appl. Opt. 28, 2422–2425 (1989).
[CrossRef]

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

K. Matsushima, “Formulation of the rotational transformation of wave fields and their application to digital holography,” Appl. Opt. 47, D110–D116 (2008).
[CrossRef]

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]

H. Nishi, K. Matsushima, and S. Nakahara, “Rendering of specular surfaces in polygon-based computer-generated holograms,” Appl. Opt. 50, H245–H252 (2011).
[CrossRef]

L. Ahrenberg, P. Benzie, M. Magnor, and J. Watson, “Computer generated holograms from three dimensional meshes using an analytic light transport model,” Appl. Opt. 47, 1567–1574 (2008).
[CrossRef]

H. Kim, J. Hahn, and B. Lee, “Mathematical modeling of triangle-mesh-modeled three-dimensional surface objects for digital holography,” Appl. Opt. 47, D117–D127(2008).
[CrossRef]

H. Sakata and Y. Sakamoto, “Fast computation method for a Fresnel hologram using three-dimensional affine transformations in real space,” Appl. Opt. 48, H212–H221 (2009).
[CrossRef]

H. Zhang, J. Xie, J. Liu, and Y. Wang, “Elimination of a zero-order beam induced by a pixelated spatial light modulator for holographic projection,” Appl. Opt. 48, 5834–5841 (2009).
[CrossRef]

Computer

C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38, 46–53 (2005).
[CrossRef]

Electron. Lett.

R. N. Bracewell, K. Y. Chang, A. K. Jha, and Y.-H. Wang, “Affine theorem for two-dimensional Fourier transform,” Electron. Lett. 29, 304–306 (1993).
[CrossRef]

Eur. J. Phys.

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

J. Electron. Imaging

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

K. Matsushima, H. Nishi, and S. Nakahara, “Simple wave-field rendering for photorealistic reconstruction in polygon-based high-definition computer holography,” J. Electron. Imaging 21, 023002–23003 (2012).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Acta

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

Opt. Express

Opt. Lett.

Proc. SPIE

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

SIAM Rev.

T. Greville, “Some applications of the pseudoinverse of a matrix,” SIAM Rev. 2, 15–22 (1960).
[CrossRef]

Other

W. M. Newman and R. F. Sproull, Principles of Interactive Computer Graphics, 2nd ed. (McGraw-Hill, 1979).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision, 2nd ed. (Cambridge University, 2003).

A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, 2nd ed. (Springer-Verlag, 2003).

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Figures (9)

Fig. 1.
Fig. 1.

(a) Primitive triangle and local coordinates. (b) Arbitrary triangle, global coordinates, and local coordinates.

Fig. 2.
Fig. 2.

Isosceles right primitive triangle in the local coordinates.

Fig. 3.
Fig. 3.

Flowchart of the implementation.

Fig. 4.
Fig. 4.

Experimental setup: SF, spatial filter; CL, collimating lens; HWP, half-wave plate; SLM, spatial light modulator; FL, Fourier lens; HPF, high-pass filter.

Fig. 5.
Fig. 5.

Optical experimental results. (a) Left square with side length 11 mm and 45 deg rotation about x and y axes, right small square with side length 1.1 mm and no rotation. Image recorded at the screen (b) 131 mm and (c) 262 mm off the rear focus for a 3D scene with large depth. (d) Reconstructed complicated 3D teapot.

Fig. 6.
Fig. 6.

Computation time ratio as a function of the number of the effective facets.

Fig. 7.
Fig. 7.

Numerical results of experiments using primitive signal with different side length to reconstruct 1D and 2D signals: reconstructed (a), (d) unit phase 1D signal, (b), (e) random phase 1D signal, and (c), (f) random phase 2D signal, all with 2.4 mm (300 pixel) feature length, from 1D and 2D primitive signal with 12.8 mm (1600 pixel) (a)–(c) and 5.6 mm (700 pixel) (d)–(f) side length, respectively. Dashed–dotted line, expected reconstructed signal; solid line, actual reconstructed signal; dashed line, corresponding sinc2 function.

Fig. 8.
Fig. 8.

Four typical (but not all) complex relationships between the new frequency domain and the original frequency domain. The square frame shows the original frequency domain, and the parallelogram frame represents the new frequency domain.

Fig. 9.
Fig. 9.

Numerical results of the experiments that have different scaling factors in the reconstructions: (a) spectrum of the primitive isosceles right triangle with 0.8 mm (100 pixel) side length and the random phase and (b) interpolated spectrum in the reconstruction of the isosceles right triangle with 2.4 mm (300 pixel) right side length and 60°, 60°, 0° rotational angle about the x, y, z axes from the spectrum of (a) (scaling factors kx, ky equal 3.33). Dark area, zero padding area of the interpolation method. (c) Spectrum of a primitive isosceles right triangle with 4.8 mm (600 pixel) side length and the random phase. (d) Interpolated spectrum in the same reconstruction of (b) from the spectrum of (c) (scaling factors equal 0.5). (e), (f) Numerical reconstructed hexagonal prism by flat shading illumination approach with (e) all the scaling factors exceeding the threshold and (f) no scaling factors exceeding the threshold.

Equations (23)

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

FΔpri(μ,ν;z=0)=fΔpri(x,y,z=0)ei2π(μx+νy)dxdy,
fΔpri(x,y,z=0)={a(x,y)Ψ(x,y)if(x,y)lies insideΔpri0else,
GΔarb(μ,ν;z=0)=exp(i2πϖz)gΔarb(x,y,z)ei2π(μx+νy)dxdy=gΔarb(x,y,z)ei2π(μx+νy+ϖz)dxdy,
GΔarb(μ,ν;z=0)=fΔpri(x,y,z=0)ei2π(μx+νy+ϖz)dxdy.
[a1,a2,a3,1]T=T[a1,a2,a3,1]T,
T=[a11a12a13t1a21a22a23t2a31a32a33t30001].
x=α(x,y,z)=a11x+a12y+a13z+t1,y=β(x,y,z)=a21x+a22y+a23z+t2,z=γ(x,y,z)=a31x+a32y+a33z+t3.
GΔarb(μ,ν;z=0)=E1(μ,ν)E2(μ,ν)fΔpri(x,y,z=0)ei2π(μx+νy)dxdy,
μ=a11μ+a21ν+a31ϖ(μ,ν),ν=a12μ+a22ν+a32ϖ(μ,ν),E1(μ,ν)=ei2πz(a13μ+a23ν+a33ϖ(μ,ν)),E2(μ,ν)=ei2π(t1μ+t2ν+t3ϖ(μ,ν)).
μ0=a31/λ,ν0=a32/λ.
μ=μμ0=a11μ+a21ν+a31o(μ,ν),ν=νν0=a12μ+a22ν+a33o(μ,ν).
J=αxβyαyβx=a11a22a12a21.
E1(μ,ν)=1.
GΔarb(μ,ν;z=0)=|J|E2(μ,ν)FΔpri(μ,ν;z=0).
P=[a11a120t1a21a220t2a31a320t30001].
FΔpri(μ,ν)=[(m+1)Λμ,μ][FΔpri(mΛ,nΛ)FΔpri(mΛ,(n+1)Λ)FΔpri((m+1)Λ,nΛ)FΔpri((m+1)Λ,(n+1)Λ)][(n+1)Λνν].
Computational time ratio=Computation time of the proposed methodComputation time of the traditonal method.
Π(xk·L1)*comb(xk·L2)sinc2(xk·L2),
c=AMPmaxAMPminAMPmax+AMPmin=1sinc2(L12L2)1+sinc2(L12L2),
T=[At0T1],
A=[kx000ky000kz]R(ϕ,θ,ψ),
R(ϕ,θ,ψ)=[cosθcosψcosϕsinψ+sinϕsinθcosψsinϕsinψ+cosϕsinθcosψcosθsinψcosϕcosψ+sinϕsinθsinψsinϕcosψ+cosϕsinθsinψsinθsinϕcosθcosϕcosθ],
μa11μ+a21ν,νa12μ+a22ν.

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