Abstract

In this paper, a fast algorithm is proposed for accurate calculation of the scalar optical diffraction on a pixelated optical device used in the reconstruction process from a three-dimensional object that is formed by scattered sample points over the space. In computer-generated holography, fast and accurate calculation of the diffraction field is an important and a challenging problem. Therefore, several fast algorithms can be found in the literature. The accuracy of the calculations can be determined by the signal processing techniques and the numerical methods used in the calculation of diffraction fields. Furthermore, the quality of reconstructed objects can be affected by the properties of optical devices employed in the reconstruction process. For instance, the pixelated structure of those devices has a significant effect on the reconstruction process. Therefore, the pixelated structure of the display device has to be taken into account. Furthermore, fast calculation of the diffraction pattern can be a bottleneck in dynamic holographic content generation. As a solution to the problems, we propose a fast and accurate algorithm based on a precomputed one-dimensional kernel and scaling of that kernel for the computation of the diffraction pattern for a pixelated display.

© 2013 Optical Society of America

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2010 (2)

2009 (3)

2008 (6)

2007 (2)

2005 (1)

2003 (1)

2001 (1)

1998 (1)

1997 (1)

T. Haist, M. Schönleber, and H. J. Tiziani, “Computer-generated holograms from 3D-objects written on twisted-nematic liquid crystal displays,” Opt. Commun. 140, 299–308(1997).
[CrossRef]

1993 (1)

1992 (1)

1989 (1)

1988 (1)

1980 (1)

F. N. Fritsch and R. E. Carlton, “Monotone piecewise cubic interpolation,” SIAM J. Numer. Anal. 17, 238–246 (1980).
[CrossRef]

Abookasis, D.

Ahrenberg, L.

Astola, J.

Benzie, P.

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]

M. Kovachev, R. Ilieva, P. Benzie, G. B. Esmer, L. Onural, J. Watson, and T. Reyhan, “Holographic 3DTV displays using spatial light modulators,” in Three-Dimensional Television: Capture, Transmission, Display (Springer-Verlag, 2008), pp. 529–556.

Bianco, B.

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1980).

Brooker, G.

Cai, L.

Carlton, R. E.

F. N. Fritsch and R. E. Carlton, “Monotone piecewise cubic interpolation,” SIAM J. Numer. Anal. 17, 238–246 (1980).
[CrossRef]

Chen, B.-C.

Delen, N.

Dong, J.-W.

Egiazarian, K.

Esmer, G. B.

G. B. Esmer, “Computation of holographic patterns between tilted planes,” M.S. thesis (Bilkent University, 2004).

M. Kovachev, R. Ilieva, P. Benzie, G. B. Esmer, L. Onural, J. Watson, and T. Reyhan, “Holographic 3DTV displays using spatial light modulators,” in Three-Dimensional Television: Capture, Transmission, Display (Springer-Verlag, 2008), pp. 529–556.

Frére, C.

Fritsch, F. N.

F. N. Fritsch and R. E. Carlton, “Monotone piecewise cubic interpolation,” SIAM J. Numer. Anal. 17, 238–246 (1980).
[CrossRef]

Goodman, J. W.

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

Hahn, J.

Haist, T.

T. Haist, M. Schönleber, and H. J. Tiziani, “Computer-generated holograms from 3D-objects written on twisted-nematic liquid crystal displays,” Opt. Commun. 140, 299–308(1997).
[CrossRef]

He, H.-X.

Heath, M. T.

M. T. Heath, Scientific Computing: An Introductory Survey, 2nd ed. (McGraw-Hill, 2002).

Hooker, B.

Ilieva, R.

M. Kovachev, R. Ilieva, P. Benzie, G. B. Esmer, L. Onural, J. Watson, and T. Reyhan, “Holographic 3DTV displays using spatial light modulators,” in Three-Dimensional Television: Capture, Transmission, Display (Springer-Verlag, 2008), pp. 529–556.

Ito, T.

Kahaner, D.

D. Kahaner, C. Maler, and S. Nash, Numerical Methods and Software (Prentice-Hall, 1988).

Kang, H.

Katkovnik, V.

Kim, E.-S.

Kim, H.

Kim, S.-C.

Kovachev, M.

M. Kovachev, R. Ilieva, P. Benzie, G. B. Esmer, L. Onural, J. Watson, and T. Reyhan, “Holographic 3DTV displays using spatial light modulators,” in Three-Dimensional Television: Capture, Transmission, Display (Springer-Verlag, 2008), pp. 529–556.

Kurita, T.

K. Yamamoto, T. Senoh, R. Oi, and T. Kurita, “8K4K-size computer generated hologram for 3-D visual system using rendering technology,” in Proceedings of 4th International Universal Communication Symposium (IUCS) (IEEE, 2010), pp. 193–196.

Lee, B.

Leseberg, D.

Liu, Y.-Z.

Lucente, M.

M. Lucente, “Diffraction-specific fringe computation for electro-holography,” Ph.D. dissertation (Massachusetts Institute of Technology, 1994).

Magnor, M.

Maler, C.

D. Kahaner, C. Maler, and S. Nash, Numerical Methods and Software (Prentice-Hall, 1988).

Masuda, N.

Matsushima, K.

Migukin, A.

Miura, J.

Muffoletto, R. P.

Nakayama, H.

Nash, S.

D. Kahaner, C. Maler, and S. Nash, Numerical Methods and Software (Prentice-Hall, 1988).

Oi, R.

K. Yamamoto, T. Senoh, R. Oi, and T. Kurita, “8K4K-size computer generated hologram for 3-D visual system using rendering technology,” in Proceedings of 4th International Universal Communication Symposium (IUCS) (IEEE, 2010), pp. 193–196.

Onural, L.

H. Kang, F. Yaras, and L. Onural, “Graphics processing unit accelerated computation of digital holograms,” Appl. Opt. 48, H137–H143 (2009).
[CrossRef]

M. Kovachev, R. Ilieva, P. Benzie, G. B. Esmer, L. Onural, J. Watson, and T. Reyhan, “Holographic 3DTV displays using spatial light modulators,” in Three-Dimensional Television: Capture, Transmission, Display (Springer-Verlag, 2008), pp. 529–556.

Reyhan, T.

M. Kovachev, R. Ilieva, P. Benzie, G. B. Esmer, L. Onural, J. Watson, and T. Reyhan, “Holographic 3DTV displays using spatial light modulators,” in Three-Dimensional Television: Capture, Transmission, Display (Springer-Verlag, 2008), pp. 529–556.

Rosen, J.

Sato, Y.

Schönleber, M.

T. Haist, M. Schönleber, and H. J. Tiziani, “Computer-generated holograms from 3D-objects written on twisted-nematic liquid crystal displays,” Opt. Commun. 140, 299–308(1997).
[CrossRef]

Senoh, T.

K. Yamamoto, T. Senoh, R. Oi, and T. Kurita, “8K4K-size computer generated hologram for 3-D visual system using rendering technology,” in Proceedings of 4th International Universal Communication Symposium (IUCS) (IEEE, 2010), pp. 193–196.

Shimobaba, T.

Takenouchi, M.

Tiziani, H. J.

T. Haist, M. Schönleber, and H. J. Tiziani, “Computer-generated holograms from 3D-objects written on twisted-nematic liquid crystal displays,” Opt. Commun. 140, 299–308(1997).
[CrossRef]

Tohline, J. E.

Tommasi, T.

Tyler, J. M.

Wang, H.-Z.

Watson, J.

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]

M. Kovachev, R. Ilieva, P. Benzie, G. B. Esmer, L. Onural, J. Watson, and T. Reyhan, “Holographic 3DTV displays using spatial light modulators,” in Three-Dimensional Television: Capture, Transmission, Display (Springer-Verlag, 2008), pp. 529–556.

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1980).

Yamaguchi, T.

Yamamoto, K.

K. Yamamoto, T. Senoh, R. Oi, and T. Kurita, “8K4K-size computer generated hologram for 3-D visual system using rendering technology,” in Proceedings of 4th International Universal Communication Symposium (IUCS) (IEEE, 2010), pp. 193–196.

Yaras, F.

Yoshikawa, H.

Yu, L.

Appl. Opt. (11)

D. Leseberg and C. Frére, “Computer generated holograms of 3D 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]

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]

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]

V. Katkovnik, J. Astola, and K. Egiazarian, “Discrete diffraction transform for propagation, reconstruction, and design of wavefield distributions,” Appl. Opt. 47, 3481–3493 (2008).
[CrossRef]

H. Kang, T. Yamaguchi, H. Yoshikawa, S.-C. Kim, and E.-S. Kim, “Acceleration method of computing a compensated phase-added stereogram on a graphic processing unit,” Appl. Opt. 47, 5784–5789 (2008).
[CrossRef]

S.-C. Kim and E.-S. Kim, “Fast computation of hologram patterns of a 3D object using run-length encoding and novel look-up table methods,” Appl. Opt. 48, 1030–1041 (2009).
[CrossRef]

V. Katkovnik, A. Migukin, and J. Astola, “Backward discrete wave field propagation modelling as an inverse problem: toward reconstruction of wave field distributions,” Appl. Opt. 48, 3407–3423 (2009).
[CrossRef]

H. Kang, F. Yaras, and L. Onural, “Graphics processing unit accelerated computation of digital holograms,” Appl. Opt. 48, H137–H143 (2009).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Commun. (1)

T. Haist, M. Schönleber, and H. J. Tiziani, “Computer-generated holograms from 3D-objects written on twisted-nematic liquid crystal displays,” Opt. Commun. 140, 299–308(1997).
[CrossRef]

Opt. Express (4)

Opt. Lett. (2)

SIAM J. Numer. Anal. (1)

F. N. Fritsch and R. E. Carlton, “Monotone piecewise cubic interpolation,” SIAM J. Numer. Anal. 17, 238–246 (1980).
[CrossRef]

Other (9)

D. Kahaner, C. Maler, and S. Nash, Numerical Methods and Software (Prentice-Hall, 1988).

M. T. Heath, Scientific Computing: An Introductory Survey, 2nd ed. (McGraw-Hill, 2002).

M. Lucente, “Diffraction-specific fringe computation for electro-holography,” Ph.D. dissertation (Massachusetts Institute of Technology, 1994).

K. Yamamoto, T. Senoh, R. Oi, and T. Kurita, “8K4K-size computer generated hologram for 3-D visual system using rendering technology,” in Proceedings of 4th International Universal Communication Symposium (IUCS) (IEEE, 2010), pp. 193–196.

G. B. Esmer, “Computation of holographic patterns between tilted planes,” M.S. thesis (Bilkent University, 2004).

M. Kovachev, R. Ilieva, P. Benzie, G. B. Esmer, L. Onural, J. Watson, and T. Reyhan, “Holographic 3DTV displays using spatial light modulators,” in Three-Dimensional Television: Capture, Transmission, Display (Springer-Verlag, 2008), pp. 529–556.

HoloEye, “LC-R 720 Spatial Light Modulators”.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1980).

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

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Figures (5)

Fig. 1.
Fig. 1.

Structure of a pixelated SLM used in reconstruction process.

Fig. 2.
Fig. 2.

Pixel structure of employed ideal SLM in the reconstruction process.

Fig. 3.
Fig. 3.

Illustration of a 3D object formed by 4090 sample points scattered over the space.

Fig. 4.
Fig. 4.

Illustration of the simulated optical setup.

Fig. 5.
Fig. 5.

(a) Real part of the calculated diffraction pattern on the SLM from the object given in Fig.  3 by Eq. (15). (b) Reconstruction of the object on the plane at z = z 0 from the diffraction pattern that is shown in (a). The gray level distribution is changed to improve the visibility of the reconstructed object. (c) Real part of the calculated diffraction pattern on the SLM from the object given in Fig. 3 by Eq. (19). (d) Reconstruction of the object on the plane at z = z 0 from the diffraction pattern that is shown in Fig. 5(c). The gray level distribution is changed to improve the visibility of the reconstructed object.

Equations (20)

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ψ ( r o ) = S ψ ( r s ) h RS ( r o r s ) d S ,
h RS ( r ) = 1 j λ exp ( j k | r | ) | r | cos ( θ ) ,
h F ( r ) = exp ( j k z ) j λ z exp [ j k 2 z ( x 2 + y 2 ) ] .
ψ ( r o ) = S ψ ( r s ) h F ( r o r s ) d S .
ψ ( r 0 ) = l = 1 L ψ ( r l ) h F ( r 0 r l ) ,
h F ( r ) = exp ( j k z ) exp ( j k 2 z x 2 ) j λ z exp ( j k 2 z y 2 ) j λ z .
ψ 1 D ( x , 0 ) = l = 1 L ψ ( x l , z l ) exp ( j k z l ) j λ z l exp [ j k 2 z l ( x l x ) 2 ] .
ψ SLM , 1 D ( i ) = x i x i + 1 ψ 1 D ( x , 0 ) d x = x i x i + 1 l = 1 L ψ ( x l , z l ) exp ( j k z l ) j λ z l exp ( j k 2 z l [ ( x l x ) 2 ] ) d x ,
ψ SLM , 1 D ( i ) = l = 1 L P ( x l ) x i x i + 1 1 λ z l exp ( j k 2 z l [ ( x l x ) 2 ] ) d x ,
ψ SLM , 1 D ( i ) = l = 1 L P ( x l ) ξ l , i ξ l , i + 1 exp ( j π ξ l 2 ) d ξ l = l = 1 L P ( x l ) [ C ( ξ l , i + 1 ) + j S ( ξ l , i + 1 ) C ( ξ l , i ) j S ( ξ l , i ) ] ,
ψ SLM , 1 D ( i ) = l = 1 L P ( x l ) K α l ( i ) ,
ψ SLM , 1 D = l = 1 L P ( x l ) K α l , 1 D ,
K α l , 1 D = [ K α l ( 1 ) K α l ( 2 ) K α l ( N ) ] ,
K α l , 2 D = K α l , 1 D x l K α l , 1 D y l T ,
ψ ^ SLM , 2 D = l = 1 L P ( x l , y l ) K α l , 2 D ,
K α 0 , 1 D interpolation K ˜ α l , 1 D ,
K α 0 , 1 D = [ K α 0 ( 1 ) K α 0 ( 2 ) K α 0 ( N ) ] ,
K ˜ α l , 2 D = ( K ˜ α l , 1 D x l ) T ( K ˜ α l , 1 D y l ) ,
ψ ~ SLM , 2 D = l = 1 L P ( x l , y l ) K ˜ α l , 2 D .
NMSE = n = 1 N m = 1 M | ψ ˜ SLM , 2 D ( n , m ) ψ ^ SLM , 2 D ( n , m ) | n = 1 N m = 1 M | ψ ^ SLM , 2 D ( n , m ) | ,

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