Abstract

Computer-generated holograms (CGHs) are generated by superimposing complex amplitudes emitted from a number of object points. However, this superposition process remains very time-consuming even when using the latest computers. We propose a fast calculation algorithm for CGHs that uses a wavelet shrinkage method, eliminating small wavelet coefficient values to express approximated complex amplitudes using only a few representative wavelet coefficients.

© 2017 Optical Society of America

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References

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  1. T. C. Poon, ed., Digital Holography and Three-Dimensional Display - Principles and Applications (Springer, 2006).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
  11. T. Shimobaba, H. Nakayama, N. Masuda, and T. Ito, “Rapid calculation algorithm of Fresnel computer-generated-hologram using look-up table and wavefront-recording plane methods for three-dimensional display,” Opt. Express 18, 19504–19509 (2010).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  16. L. Onural, “Diffraction from a wavelet point of view,” Opt. Lett. 18, 846–848 (1993).
    [Crossref] [PubMed]
  17. M. Liebling, T. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43, (2003).
    [Crossref]
  18. M. Liebling and M. Unser, “Autofocus for digital Fresnel holograms by use of a Fresnelet-sparsity criterion,” J. Opt. Soc. Am. A 21, 2424–2430 (2004).
    [Crossref]
  19. J. Weng, J. Zhong, and C. Hu, “Digital reconstruction based on angular spectrum diffraction with the ridge of wavelet transform in holographic phase-contrast microscopy,” Opt. Express 16, 21971–21981 (2008).
    [Crossref] [PubMed]
  20. S. G. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989).
    [Crossref]
  21. T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
    [Crossref]

2016 (1)

T. Shimobaba, T. Kakue, and T. Ito, “Review of fast algorithms and hardware implementations on computer holography,” IEEE Trans. Ind. Informat. 12, 1611–1622 (2016).
[Crossref]

2015 (2)

2012 (1)

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

2011 (2)

2010 (1)

2009 (1)

2008 (2)

2004 (1)

2003 (1)

M. Liebling, T. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43, (2003).
[Crossref]

2001 (2)

H. Yoshikawa, “Fast computation of Fresnel holograms employing difference,” Opt. Rev. 8, 331–335, (2001).
[Crossref]

T. Shimobaba and T. Ito, “An efficient computational method suitable for hardware of computer-generated hologram with phase computation by addition,” Comput. Phys. Commun. 138, 44–52 (2001).
[Crossref]

2000 (1)

1995 (1)

D. L. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inf. Theory 41, 613–627 (1995).
[Crossref]

1993 (2)

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

L. Onural, “Diffraction from a wavelet point of view,” Opt. Lett. 18, 846–848 (1993).
[Crossref] [PubMed]

1989 (1)

S. G. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989).
[Crossref]

Blinder, D.

Blu, T.

M. Liebling, T. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43, (2003).
[Crossref]

Cheung, W.-K.

Donoho, D. L.

D. L. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inf. Theory 41, 613–627 (1995).
[Crossref]

Hu, C.

Ito, T.

T. Shimobaba, T. Kakue, and T. Ito, “Review of fast algorithms and hardware implementations on computer holography,” IEEE Trans. Ind. Informat. 12, 1611–1622 (2016).
[Crossref]

T. Nishitsuji, T. Shimobaba, T. Kakue, and T. Ito, “Fast calculation of computer-generated hologram using run-length encoding based recurrence relation,” Opt. Express 23, 9852–9857 (2015).
[Crossref] [PubMed]

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

T. Shimobaba, H. Nakayama, N. Masuda, and T. Ito, “Rapid calculation algorithm of Fresnel computer-generated-hologram using look-up table and wavefront-recording plane methods for three-dimensional display,” Opt. Express 18, 19504–19509 (2010).
[Crossref] [PubMed]

T. Shimobaba and T. Ito, “An efficient computational method suitable for hardware of computer-generated hologram with phase computation by addition,” Comput. Phys. Commun. 138, 44–52 (2001).
[Crossref]

Ito., T.

Kakue, T.

T. Shimobaba, T. Kakue, and T. Ito, “Review of fast algorithms and hardware implementations on computer holography,” IEEE Trans. Ind. Informat. 12, 1611–1622 (2016).
[Crossref]

T. Nishitsuji, T. Shimobaba, T. Kakue, and T. Ito, “Fast calculation of computer-generated hologram using run-length encoding based recurrence relation,” Opt. Express 23, 9852–9857 (2015).
[Crossref] [PubMed]

Kim, E.-S.

Kim, S.-C.

Liebling, M.

M. Liebling and M. Unser, “Autofocus for digital Fresnel holograms by use of a Fresnelet-sparsity criterion,” J. Opt. Soc. Am. A 21, 2424–2430 (2004).
[Crossref]

M. Liebling, T. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43, (2003).
[Crossref]

Lucente, M.

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

Mallat, S. G.

S. G. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989).
[Crossref]

Masuda, N.

Matsushima, K.

Munteanu, A.

Nakayama, H.

Nishitsuji, T.

T. Nishitsuji, T. Shimobaba, T. Kakue, and T. Ito, “Fast calculation of computer-generated hologram using run-length encoding based recurrence relation,” Opt. Express 23, 9852–9857 (2015).
[Crossref] [PubMed]

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

Okada, N.

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

Onural, L.

Poon, T.-C.

Sakurai, T.

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

Schelkens, P.

Shimobaba, T.

T. Shimobaba, T. Kakue, and T. Ito, “Review of fast algorithms and hardware implementations on computer holography,” IEEE Trans. Ind. Informat. 12, 1611–1622 (2016).
[Crossref]

T. Nishitsuji, T. Shimobaba, T. Kakue, and T. Ito, “Fast calculation of computer-generated hologram using run-length encoding based recurrence relation,” Opt. Express 23, 9852–9857 (2015).
[Crossref] [PubMed]

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

T. Shimobaba, H. Nakayama, N. Masuda, and T. Ito, “Rapid calculation algorithm of Fresnel computer-generated-hologram using look-up table and wavefront-recording plane methods for three-dimensional display,” Opt. Express 18, 19504–19509 (2010).
[Crossref] [PubMed]

T. Shimobaba, N. Masuda, and T. Ito., “Simple and fast calculation algorithm for computer-generated hologram with wavefront recording plane,” Opt. Lett. 343133–3135 (2009).
[Crossref] [PubMed]

T. Shimobaba and T. Ito, “An efficient computational method suitable for hardware of computer-generated hologram with phase computation by addition,” Comput. Phys. Commun. 138, 44–52 (2001).
[Crossref]

Shiraki, A.

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

Symeonidou, A.

Takada, N.

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

Takai, M.

Tsang, P.

Unser, M.

M. Liebling and M. Unser, “Autofocus for digital Fresnel holograms by use of a Fresnelet-sparsity criterion,” J. Opt. Soc. Am. A 21, 2424–2430 (2004).
[Crossref]

M. Liebling, T. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43, (2003).
[Crossref]

Weng, J.

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

J. Weng, J. Zhong, and C. Hu, “Digital reconstruction based on angular spectrum diffraction with the ridge of wavelet transform in holographic phase-contrast microscopy,” Opt. Express 16, 21971–21981 (2008).
[Crossref] [PubMed]

Yamaguchi, T.

Yoshikawa, H.

T. Yamaguchi and H. Yoshikawa, “Computer-generated image hologram,” Chin. Opt. Lett. 9, 120006 (2011).
[Crossref]

H. Yoshikawa, “Fast computation of Fresnel holograms employing difference,” Opt. Rev. 8, 331–335, (2001).
[Crossref]

Zhong, J.

Zhou, C.

Appl. Opt. (2)

Chin. Opt. Lett. (1)

Comput. Phys. Commun. (2)

T. Shimobaba and T. Ito, “An efficient computational method suitable for hardware of computer-generated hologram with phase computation by addition,” Comput. Phys. Commun. 138, 44–52 (2001).
[Crossref]

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

IEEE Trans. Image Process. (1)

M. Liebling, T. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43, (2003).
[Crossref]

IEEE Trans. Ind. Informat. (1)

T. Shimobaba, T. Kakue, and T. Ito, “Review of fast algorithms and hardware implementations on computer holography,” IEEE Trans. Ind. Informat. 12, 1611–1622 (2016).
[Crossref]

IEEE Trans. Inf. Theory (1)

D. L. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inf. Theory 41, 613–627 (1995).
[Crossref]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

S. G. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989).
[Crossref]

J. Electron. Imaging (1)

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

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

Opt. Express (5)

Opt. Lett. (2)

Opt. Rev. (1)

H. Yoshikawa, “Fast computation of Fresnel holograms employing difference,” Opt. Rev. 8, 331–335, (2001).
[Crossref]

Other (2)

T. C. Poon, ed., Digital Holography and Three-Dimensional Display - Principles and Applications (Springer, 2006).
[Crossref]

C. K. Chui, ed., An Introduction to Wavelets (Academic Press, 2014).

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Figures (4)

Fig. 1
Fig. 1 Distribution of the scaling and wavelet coefficients : (a) Distribution of the scaling and wavelet coefficients, (b) a PSF whose calculation conditions are zj = 100mm, a pixel pitch of 10μm and a wavelength of 633nm and (c) the wavelet decomposition of Fig. 1(b).
Fig. 2
Fig. 2 Point spread functions: (a) the original PSF and (b) approximated PSFs with r = 20%, (c) r = 10%, and (d) r = 5%. Interference patters constructed from two object points using (e) the original PSF and (f) approximated PSFs with r = 20%, (g) r = 10% and (h) r = 5%.
Fig. 3
Fig. 3 Intensity profiles of Fig. 2(a) and Fig. 2(d) reconstructed from the PSFs. Intensity profiles reconstructed from the interference pattern shown in Fig. 2(e) and the approximated interference pattern shown in Fig. 2(h) with r = 5%.
Fig. 4
Fig. 4 Reconstructed images at two different distances of 0.5 m and 0.05 m : (a) the dinosaur composed of 11,646 object points (b) the merry-go-round composed of 95,949 points and (c) the merry-go-round composed of 978,416 points.

Tables (3)

Tables Icon

Table 1 Calculation times for the dinosaur image (11,646 object points) using the conventional method [4] and WASABI with r = 5% and r = 1%.

Tables Icon

Table 2 Calculation time for the merry-go-round image (95,949 object points) using the conventional method [4] and WASABI with r = 5% and r = 1%.

Tables Icon

Table 3 Calculation time for the fountain image (978,416 object points) using the conventional method [4] and WASABI with r = 5% and r = 1%.

Equations (15)

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u ( x h , y h ) = j N a j exp ( i 2 π λ r h j ) = j N a j u z j ( x h x j , y h y j ) ,
r h j = ( x h x j ) 2 + ( y h y j ) 2 + z j 2 , z j + ( ( x h x j ) 2 + ( y h y j ) 2 ) / ( 2 z j ) .
W ¯ = 1 N j N W j = 1 N j N z j tan θ ,
u z j ( x h , y h ) = m = n = s m , n ( 0 ) ϕ ( x h m ) ϕ ( y h n ) ,
u z j ( x h , y h ) s m , n ( 0 ) .
s m , n ( + 1 ) = k 2 k 1 p k 1 2 m p k 2 2 n s m , n ( ) , w L H , m , n ( + 1 ) = k 2 k 1 p k 1 2 m q k 2 2 n s m , n ( ) , w H L , m , n ( + 1 ) = k 2 k 1 q k 1 2 m p k 2 2 n s m , n ( ) , w H H , m , n ( + 1 ) = k 2 k 1 q k 1 2 m q k 2 2 n s m , n ( ) ,
N r = 2 π W j 2 × r .
v z = [ c z , 0 , α z , 0 , , c z , N r 1 , α z , N r 1 ] ,
c z , k { s m , n ( ) , w L H , m , n ( ) , w H L , m , n ( ) , w H H , m , n ( ) } ,
α z , k = 2 .
ψ ( m , n ) = j = 0 N a j k = 0 N r 1 c z j , k δ ( m α z j , k x j , n α z j , k y j ) ,
s m , n ( ) = k 2 k 1 p m 2 k 2 p n 2 k 1 s m , n ( + 1 ) + p m 2 k 2 q n 2 k 1 w L H , m , n ( + 1 ) + q m 2 k 2 p n 2 k 1 w H L , m , n ( + 1 ) + q m 2 k 2 q n 2 k 1 w H H , m , n ( + 1 ) .
u ( x h , y j ) = j N a j u z j ( x h x j , y h y j , z j ) s m , n ( 0 ) .
O ( r W ¯ 2 N ) + O ( N h 2 ) .
O ( r W ¯ 2 N ) + O ( N h 2 ) O ( r W ¯ 2 N ) .

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