Abstract

Fast two-step layer-based and sub-sparse two-dimensional Fast Fourier transform (SS-2DFFT) algorithms are proposed to speed up the calculation of computer-generated holograms. In a layer-based method, each layer image may contain large areas in which the pixel values are zero considering the occlusion effect among the different depth layers. By taking advantage of this feature, the two-step layer-based algorithm only calculates the non-zero image areas of each layer. In addition, the SS-2DFFT method implements two one-dimensional fast Fourier transforms (1DFFT) to compute a 2DFFT without calculating the rows or columns in which the image pixels are all zero. Since the size of the active calculation is reduced, the computational speed is considerably improved. Numerical simulations and optical experiments are performed to confirm these methods. The results show that the total computational time can be reduced by 5 times for a three-dimensional (3D) object of a train, 3.4 times for a 3D object of a castle and 10 times for a 3D object of a statue head when compared with a conventional layer-based method if combining the two proposed methods together.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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Accurate calculation of computer-generated holograms using angular-spectrum layer-oriented method

Yan Zhao, Liangcai Cao, Hao Zhang, Dezhao Kong, and Guofan Jin
Opt. Express 23(20) 25440-25449 (2015)

References

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2017 (1)

2016 (4)

2015 (4)

2014 (2)

J.-S. Chen, D. Chu, and Q. Smithwick, “Rapid hologram generation utilizing layer-based approach and graphic rendering for realistic three-dimensional image reconstruction by angular tiling,” J. Electron. Imaging 23(2), 023016 (2014).
[Crossref]

X.-B. Dong, S.-C. Kim, and E.-S. Kim, “MPEG-based novel look-up table for rapid generation of video holograms of fast-moving three-dimensional objects,” Opt. Express 22(7), 8047–8067 (2014).
[Crossref] [PubMed]

2013 (1)

2012 (2)

2009 (1)

2000 (1)

Arai, D.

Barada, D.

Cao, L.

Chang, C.

Chen, J.-S.

J.-S. Chen and D. P. Chu, “Improved layer-based method for rapid hologram generation and real-time interactive holographic display applications,” Opt. Express 23(14), 18143–18155 (2015).
[Crossref] [PubMed]

J.-S. Chen, D. Chu, and Q. Smithwick, “Rapid hologram generation utilizing layer-based approach and graphic rendering for realistic three-dimensional image reconstruction by angular tiling,” J. Electron. Imaging 23(2), 023016 (2014).
[Crossref]

Chu, D.

J.-S. Chen, D. Chu, and Q. Smithwick, “Rapid hologram generation utilizing layer-based approach and graphic rendering for realistic three-dimensional image reconstruction by angular tiling,” J. Electron. Imaging 23(2), 023016 (2014).
[Crossref]

Chu, D. P.

Dong, X.-B.

Gao, C.

Hassanieh, H.

H. Hassanieh, P. Indyk, D. Katabi, and E. Price, “Simple and Practical Algorithm for Sparse Fourier Transform,” in Proceedings Twenty-third Annual ACM-SIAM Symposium Discrete Algorithms, SODA’12 (Society for Industrial and Applied Mathematics, 2012), pp. 1183–1194.
[Crossref]

Indyk, P.

H. Hassanieh, P. Indyk, D. Katabi, and E. Price, “Simple and Practical Algorithm for Sparse Fourier Transform,” in Proceedings Twenty-third Annual ACM-SIAM Symposium Discrete Algorithms, SODA’12 (Society for Industrial and Applied Mathematics, 2012), pp. 1183–1194.
[Crossref]

Ito, T.

Jeong, H.

Jia, J.

Jiang, W.

Jin, G.

Kakue, T.

Kang, H.

Katabi, D.

H. Hassanieh, P. Indyk, D. Katabi, and E. Price, “Simple and Practical Algorithm for Sparse Fourier Transform,” in Proceedings Twenty-third Annual ACM-SIAM Symposium Discrete Algorithms, SODA’12 (Society for Industrial and Applied Mathematics, 2012), pp. 1183–1194.
[Crossref]

Kim, E.-S.

Kim, H. G.

Kim, S.-C.

Kong, D.

Kwon, M.-W.

Li, X.

Liu, J.

Liu, J.-P.

Man Ro, Y.

Matsushima, K.

Nie, S.

Nishitsuji, T.

Onural, L.

Pan, Y.

Price, E.

H. Hassanieh, P. Indyk, D. Katabi, and E. Price, “Simple and Practical Algorithm for Sparse Fourier Transform,” in Proceedings Twenty-third Annual ACM-SIAM Symposium Discrete Algorithms, SODA’12 (Society for Industrial and Applied Mathematics, 2012), pp. 1183–1194.
[Crossref]

Qi, Y.

Sando, Y.

Shimobaba, T.

Smithwick, Q.

J.-S. Chen, D. Chu, and Q. Smithwick, “Rapid hologram generation utilizing layer-based approach and graphic rendering for realistic three-dimensional image reconstruction by angular tiling,” J. Electron. Imaging 23(2), 023016 (2014).
[Crossref]

Stoykova, E.

Sun, Z.

Wang, Y.

Wu, J.

Xia, J.

Xue, G.

Yatagai, T.

Yoshikawa, H.

Yuan, C.

Zhang, B.

Zhang, H.

Zhao, Q.

Zhao, Y.

Appl. Opt. (5)

J. Electron. Imaging (1)

J.-S. Chen, D. Chu, and Q. Smithwick, “Rapid hologram generation utilizing layer-based approach and graphic rendering for realistic three-dimensional image reconstruction by angular tiling,” J. Electron. Imaging 23(2), 023016 (2014).
[Crossref]

J. Opt. Soc. Am. A (1)

Opt. Express (9)

K. Matsushima and T. Shimobaba, “Band-limited angular spectrum method for numerical simulation of free-space propagation in far and near fields,” Opt. Express 17(22), 19662–19673 (2009).
[Crossref] [PubMed]

H. G. Kim, H. Jeong, and Y. Man Ro, “Acceleration of the calculation speed of computer-generated holograms using the sparsity of the holographic fringe pattern for a 3D object,” Opt. Express 24(22), 25317–25328 (2016).
[Crossref] [PubMed]

H. G. Kim and Y. Man Ro, “Ultrafast layer based computer-generated hologram calculation with sparse template holographic fringe pattern for 3-D object,” Opt. Express 25(24), 30418–30427 (2017).
[Crossref] [PubMed]

X.-B. Dong, S.-C. Kim, and E.-S. Kim, “MPEG-based novel look-up table for rapid generation of video holograms of fast-moving three-dimensional objects,” Opt. Express 22(7), 8047–8067 (2014).
[Crossref] [PubMed]

C. Gao, J. Liu, X. Li, G. Xue, J. Jia, and Y. Wang, “Accurate compressed look up table method for CGH in 3D holographic display,” Opt. Express 23(26), 33194–33204 (2015).
[Crossref] [PubMed]

J.-S. Chen and D. P. Chu, “Improved layer-based method for rapid hologram generation and real-time interactive holographic display applications,” Opt. Express 23(14), 18143–18155 (2015).
[Crossref] [PubMed]

Y. Zhao, L. Cao, H. Zhang, D. Kong, and G. Jin, “Accurate calculation of computer-generated holograms using angular-spectrum layer-oriented method,” Opt. Express 23(20), 25440–25449 (2015).
[Crossref] [PubMed]

T. Nishitsuji, T. Shimobaba, T. Kakue, D. Arai, and T. Ito, “Simple and fast cosine approximation method for computer-generated hologram calculation,” Opt. Express 23(25), 32465–32470 (2015).
[Crossref] [PubMed]

Y. Sando, D. Barada, and T. Yatagai, “Fast calculation of computer-generated holograms based on 3-D Fourier spectrum for omnidirectional diffraction from a 3-D voxel-based object,” Opt. Express 20(19), 20962–20969 (2012).
[Crossref] [PubMed]

Other (4)

H. Hassanieh, P. Indyk, D. Katabi, and E. Price, “Nearly Optimal Sparse Fourier Transform,” in Proceedings Forty-fourth Annual ACM SymposiumTheory Computing, STOC’12 (ACM, 2012), pp. 563–578.

H. Hassanieh, P. Indyk, D. Katabi, and E. Price, “Simple and Practical Algorithm for Sparse Fourier Transform,” in Proceedings Twenty-third Annual ACM-SIAM Symposium Discrete Algorithms, SODA’12 (Society for Industrial and Applied Mathematics, 2012), pp. 1183–1194.
[Crossref]

J. Hu, Z. Wang, Q. Qiu, W. Xiao, and D. Lilja, “Sparse Fast Fourier Transform on GPUs and Multi-core CPUs,” in Computer Architecture and High Performance Computing (SBAC-PAD), 2012 IEEE 24th International Symposium on 2012, pp. 83–91.
[Crossref]

http://www.3d66.com/

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Figures (7)

Fig. 1
Fig. 1 A traditional layer based method using 2DFFT for each depth layer.
Fig. 2
Fig. 2 Sub-sparse 2DFFT.
Fig. 3
Fig. 3 Schedule of principle, (a) Step 1: Calculation the hologram of the non-zero image areas of each layer to Reference plane (Ref); (b) Step 2: Calculation the final hologram from Ref plane to the hologram plane.
Fig. 4
Fig. 4 Aliasing error, (a) forward angular spectrum transform (AST), (b) inverse angular spectrum transform (IAST) with the same spatial size, (c) forward angular spectrum transform with small spatial size, (d) inverse angular spectrum transform with the extend spatial size.
Fig. 5
Fig. 5 The Numerical simulation reconstruction images, (a) the target image, (b) reduced the Fourier transform area without using dynamical zero padding, (c) reduced the Fourier transform the area with dynamical zero padding.
Fig. 6
Fig. 6 Comparison of calculation time for total Fourier transform, non-zero Fourier transform with/without zero padding, non-zero Fourier transform with zero padding and SS-2DFFT.
Fig. 7
Fig. 7 (a) Target 3D objects, (b) simulation and (c) optical experimental results with different focus points. [The 3D objects are from 3D66 [20], a 3D model open-resource website.]

Tables (2)

Tables Icon

Table 1 The comparison of computational complexity

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Table 2 Comparison of computational complexity and total time (Unit: seconds)

Equations (9)

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H(u,v)= l=0 L1 2DFF T x,y (f(x,y, z l ))×lens( z l )
F(u,v)= x=0 M1 y=0 N1 f(x,y) e 2πj( u M x+ v N y)
2DFFT{f(x,y)}=1DFF T y {1DFF T x {f(x,y)}}=1DFF T x {1DFF T y {f(x,y)}}
F(u,y)= x=0 M1 f(x,y) e 2πjux M ,
F(u,v)= y=0 N1 F(u,y) e 2πjvy N ,
H r (u,v)= l=0 L1 2DFF T 1 [2DFFT(f( x l , y l , z l ))×T( z l ) ],
T( z l )=exp(j2π z l 1/ λ 2 f r 2 ),
w 2 +| 1 2π d d f r [2πz 1/ λ 2 f r 2 ] max | 1 2 Δ f ,
w 2z Δ 0 4 ( Δ 0 /λ) 2 1 +w.

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