Abstract

Computer-generated holography is computationally intensive, making it especially challenging for holographic displays where high-resolutions and video rates are needed. To this end, we propose a technique for directly calculating short-time Fourier transform coefficients without the need for a look-up table. Because point spread functions are sparse in this transform domain, only a small fraction of the coefficients need to be updated, enabling significant speed gains. Twenty-fold accelerations are reported over the reference implementation. This approach generalizes the notion of the phase-added stereogram, allowing for the calculatiion of an arbitrary number of Fourier coefficients per block, enabling high calculation speed with holograms of good visual quality, targeting minimal memory requirements.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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Accelerated computer generated holography using sparse bases in the STFT domain

David Blinder and Peter Schelkens
Opt. Express 26(2) 1461-1473 (2018)

References

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  1. D. Blinder, A. Ahar, S. Bettens, T. Birnbaum, A. Symeonidou, H. Ottevaere, C. Schretter, and P. Schelkens, “Signal processing challenges for digital holographic video display systems,” Signal Process. Image 70, 114–130 (2019).
    [Crossref]
  2. J.-H. Park, “Recent progress in computer-generated holography for three-dimensional scenes,” J. Inf. Displ. 18, 1–12 (2017).
    [Crossref]
  3. M. E. Lucente, “Interactive computation of holograms using a look-up table,” J. Electron. Imaging 2, 28–34 (1993).
    [Crossref]
  4. 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] [PubMed]
  5. T. Shimobaba, N. Masuda, and T. Ito, “Simple and fast calculation algorithm for computer-generated hologram with wavefront recording plane,” Opt. Lett. 34, 3133–3135 (2009).
    [Crossref] [PubMed]
  6. P. Tsang, W.-K. Cheung, T.-C. Poon, and C. Zhou, “Holographic video at 40 frames per second for 4-million object points,” Opt. Express 19, 15205–15211 (2011).
    [Crossref] [PubMed]
  7. S. Jiao, Z. Zhuang, and W. Zou, “Fast computer generated hologram calculation with a mini look-up table incorporated with radial symmetric interpolation,” Opt. Express 25, 112–123 (2017).
    [Crossref] [PubMed]
  8. Y. Yamamoto, H. Nakayama, N. Takada, T. Nishitsuji, T. Sugie, T. Kakue, T. Shimobaba, and T. Ito, “Large-scale electroholography by horn-8 from a point-cloud model with 400,000 points,” Opt. Express 26, 34259–34265 (2018).
    [Crossref]
  9. A. Symeonidou, D. Blinder, A. Munteanu, and P. Schelkens, “Computer-generated holograms by multiple wavefront recording plane method with occlusion culling,” Opt. Express 23, 22149–22161 (2015).
    [Crossref] [PubMed]
  10. 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, 25440–25449 (2015).
    [Crossref] [PubMed]
  11. A. Symeonidou, D. Blinder, and P. Schelkens, “Colour computer-generated holography for point clouds utilizing the phong illumination model,” Opt. Express 26, 10282–10298 (2018).
    [Crossref] [PubMed]
  12. A. Gilles and P. Gioia, “Real-time layer-based computer-generated hologram calculation for the fourier transform optical system,” Appl. Opt. 57, 8508–8517 (2018).
    [Crossref] [PubMed]
  13. 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] [PubMed]
  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] [PubMed]
  15. K. Matsushima, M. Nakamura, and S. Nakahara, “Silhouette method for hidden surface removal in computer holography and its acceleration using the switch-back technique,” Opt. Express 22, 24450–24465 (2014).
    [Crossref] [PubMed]
  16. H. Nishi and K. Matsushima, “Rendering of specular curved objects in polygon-based computer holography,” Appl. Opt. 56, F37–F44 (2017).
    [Crossref] [PubMed]
  17. N. Okada, T. Shimobaba, Y. Ichihashi, R. Oi, K. Yamamoto, M. Oikawa, T. Kakue, N. Masuda, and T. Ito, “Band-limited double-step fresnel diffraction and its application to computer-generated holograms,” Opt. Express 21, 9192–9197 (2013).
    [Crossref] [PubMed]
  18. T. Shimobaba, T. Kakue, and T. Ito, “Acceleration of color computer-generated hologram from three-dimensional scenes with texture and depth information,” Proc.SPIE 9117, 91170B (2014).
  19. T. Yatagai, “Stereoscopic approach to 3-d display using computer-generated holograms,” Appl. Opt. 15, 2722–2729 (1976).
    [Crossref] [PubMed]
  20. M. Yamaguchi, H. Hoshino, T. Honda, and N. Ohyama, “Phase-added stereogram: calculation of hologram using computer graphics technique,” Proc. SPIE 1914, 25–31 (1993).
    [Crossref]
  21. R. H.-Y. Chen and T. D. Wilkinson, “Computer generated hologram with geometric occlusion using gpu-accelerated depth buffer rasterization for three-dimensional display,” Appl. Opt. 48, 4246–4255 (2009).
    [Crossref] [PubMed]
  22. K. Wakunami, H. Yamashita, and M. Yamaguchi, “Occlusion culling for computer generated hologram based on ray-wavefront conversion,” Opt. Express 21, 21811–21822 (2013).
    [Crossref] [PubMed]
  23. H. Zhang, Y. Zhao, L. Cao, and G. Jin, “Fully computed holographic stereogram based algorithm for computer-generated holograms with accurate depth cues,” Opt. Express 23, 3901–3913 (2015).
    [Crossref] [PubMed]
  24. A. Gilles, P. Gioia, R. Cozot, and L. Morin, “Hybrid approach for fast occlusion processing in computer-generated hologram calculation,” Appl. Opt. 55, 5459–5470 (2016).
    [Crossref] [PubMed]
  25. P. Schelkens, A. Skodras, and T. Ebrahimi, The JPEG 2000 Suite (Wiley Publishing, 2009).
    [Crossref]
  26. J. W. Goodman, Introduction to Fourier Optics (W. H. Freeman and Company, 2017).
  27. T. Shimobaba and T. Ito, “Fast generation of computer-generated holograms using wavelet shrinkage,” Opt. Express 25, 77–87 (2017).
    [Crossref] [PubMed]
  28. D. Blinder and P. Schelkens, “Accelerated computer generated holography using sparse bases in the STFT domain,” Opt. Express 26, 1461–1473 (2018).
    [Crossref] [PubMed]
  29. D. Arai, T. Shimobaba, T. Nishitsuji, T. Kakue, N. Masuda, and T. Ito, “An accelerated hologram calculation using the wavefront recording plane method and wavelet transform,” Opt. Commun. 393, 107–112 (2017).
    [Crossref]
  30. H. Kang, T. Fujii, T. Yamaguchi, and H. Yoshikawa, “Compensated phase-added stereogram for real-time holographic display,” Opt. Eng. 46, 095802 (2007).
    [Crossref]
  31. H. Kang, E. Stoykova, and H. Yoshikawa, “Fast phase-added stereogram algorithm for generation of photorealistic 3d content,” Appl. Opt. 55, A135–A143 (2016).
    [Crossref] [PubMed]

2019 (1)

D. Blinder, A. Ahar, S. Bettens, T. Birnbaum, A. Symeonidou, H. Ottevaere, C. Schretter, and P. Schelkens, “Signal processing challenges for digital holographic video display systems,” Signal Process. Image 70, 114–130 (2019).
[Crossref]

2018 (4)

2017 (5)

2016 (2)

2015 (3)

2014 (2)

T. Shimobaba, T. Kakue, and T. Ito, “Acceleration of color computer-generated hologram from three-dimensional scenes with texture and depth information,” Proc.SPIE 9117, 91170B (2014).

K. Matsushima, M. Nakamura, and S. Nakahara, “Silhouette method for hidden surface removal in computer holography and its acceleration using the switch-back technique,” Opt. Express 22, 24450–24465 (2014).
[Crossref] [PubMed]

2013 (2)

2011 (1)

2009 (3)

2008 (2)

2007 (1)

H. Kang, T. Fujii, T. Yamaguchi, and H. Yoshikawa, “Compensated phase-added stereogram for real-time holographic display,” Opt. Eng. 46, 095802 (2007).
[Crossref]

1993 (2)

M. Yamaguchi, H. Hoshino, T. Honda, and N. Ohyama, “Phase-added stereogram: calculation of hologram using computer graphics technique,” Proc. SPIE 1914, 25–31 (1993).
[Crossref]

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

1976 (1)

Ahar, A.

D. Blinder, A. Ahar, S. Bettens, T. Birnbaum, A. Symeonidou, H. Ottevaere, C. Schretter, and P. Schelkens, “Signal processing challenges for digital holographic video display systems,” Signal Process. Image 70, 114–130 (2019).
[Crossref]

Arai, D.

D. Arai, T. Shimobaba, T. Nishitsuji, T. Kakue, N. Masuda, and T. Ito, “An accelerated hologram calculation using the wavefront recording plane method and wavelet transform,” Opt. Commun. 393, 107–112 (2017).
[Crossref]

Bettens, S.

D. Blinder, A. Ahar, S. Bettens, T. Birnbaum, A. Symeonidou, H. Ottevaere, C. Schretter, and P. Schelkens, “Signal processing challenges for digital holographic video display systems,” Signal Process. Image 70, 114–130 (2019).
[Crossref]

Birnbaum, T.

D. Blinder, A. Ahar, S. Bettens, T. Birnbaum, A. Symeonidou, H. Ottevaere, C. Schretter, and P. Schelkens, “Signal processing challenges for digital holographic video display systems,” Signal Process. Image 70, 114–130 (2019).
[Crossref]

Blinder, D.

Cao, L.

Chen, R. H.-Y.

Cheung, W.-K.

Cozot, R.

Ebrahimi, T.

P. Schelkens, A. Skodras, and T. Ebrahimi, The JPEG 2000 Suite (Wiley Publishing, 2009).
[Crossref]

Fujii, T.

H. Kang, T. Fujii, T. Yamaguchi, and H. Yoshikawa, “Compensated phase-added stereogram for real-time holographic display,” Opt. Eng. 46, 095802 (2007).
[Crossref]

Gilles, A.

Gioia, P.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (W. H. Freeman and Company, 2017).

Hahn, J.

Honda, T.

M. Yamaguchi, H. Hoshino, T. Honda, and N. Ohyama, “Phase-added stereogram: calculation of hologram using computer graphics technique,” Proc. SPIE 1914, 25–31 (1993).
[Crossref]

Hoshino, H.

M. Yamaguchi, H. Hoshino, T. Honda, and N. Ohyama, “Phase-added stereogram: calculation of hologram using computer graphics technique,” Proc. SPIE 1914, 25–31 (1993).
[Crossref]

Ichihashi, Y.

Ito, T.

Jiao, S.

Jin, G.

Kakue, T.

Y. Yamamoto, H. Nakayama, N. Takada, T. Nishitsuji, T. Sugie, T. Kakue, T. Shimobaba, and T. Ito, “Large-scale electroholography by horn-8 from a point-cloud model with 400,000 points,” Opt. Express 26, 34259–34265 (2018).
[Crossref]

D. Arai, T. Shimobaba, T. Nishitsuji, T. Kakue, N. Masuda, and T. Ito, “An accelerated hologram calculation using the wavefront recording plane method and wavelet transform,” Opt. Commun. 393, 107–112 (2017).
[Crossref]

T. Shimobaba, T. Kakue, and T. Ito, “Acceleration of color computer-generated hologram from three-dimensional scenes with texture and depth information,” Proc.SPIE 9117, 91170B (2014).

N. Okada, T. Shimobaba, Y. Ichihashi, R. Oi, K. Yamamoto, M. Oikawa, T. Kakue, N. Masuda, and T. Ito, “Band-limited double-step fresnel diffraction and its application to computer-generated holograms,” Opt. Express 21, 9192–9197 (2013).
[Crossref] [PubMed]

Kang, H.

H. Kang, E. Stoykova, and H. Yoshikawa, “Fast phase-added stereogram algorithm for generation of photorealistic 3d content,” Appl. Opt. 55, A135–A143 (2016).
[Crossref] [PubMed]

H. Kang, T. Fujii, T. Yamaguchi, and H. Yoshikawa, “Compensated phase-added stereogram for real-time holographic display,” Opt. Eng. 46, 095802 (2007).
[Crossref]

Kim, E.-S.

Kim, H.

Kim, S.-C.

Kong, D.

Lee, B.

Lucente, M. E.

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

Masuda, N.

Matsushima, K.

Morin, L.

Munteanu, A.

Nakahara, S.

Nakamura, M.

Nakayama, H.

Nishi, H.

Nishitsuji, T.

Y. Yamamoto, H. Nakayama, N. Takada, T. Nishitsuji, T. Sugie, T. Kakue, T. Shimobaba, and T. Ito, “Large-scale electroholography by horn-8 from a point-cloud model with 400,000 points,” Opt. Express 26, 34259–34265 (2018).
[Crossref]

D. Arai, T. Shimobaba, T. Nishitsuji, T. Kakue, N. Masuda, and T. Ito, “An accelerated hologram calculation using the wavefront recording plane method and wavelet transform,” Opt. Commun. 393, 107–112 (2017).
[Crossref]

Ohyama, N.

M. Yamaguchi, H. Hoshino, T. Honda, and N. Ohyama, “Phase-added stereogram: calculation of hologram using computer graphics technique,” Proc. SPIE 1914, 25–31 (1993).
[Crossref]

Oi, R.

Oikawa, M.

Okada, N.

Ottevaere, H.

D. Blinder, A. Ahar, S. Bettens, T. Birnbaum, A. Symeonidou, H. Ottevaere, C. Schretter, and P. Schelkens, “Signal processing challenges for digital holographic video display systems,” Signal Process. Image 70, 114–130 (2019).
[Crossref]

Park, J.-H.

J.-H. Park, “Recent progress in computer-generated holography for three-dimensional scenes,” J. Inf. Displ. 18, 1–12 (2017).
[Crossref]

Poon, T.-C.

Schelkens, P.

Schretter, C.

D. Blinder, A. Ahar, S. Bettens, T. Birnbaum, A. Symeonidou, H. Ottevaere, C. Schretter, and P. Schelkens, “Signal processing challenges for digital holographic video display systems,” Signal Process. Image 70, 114–130 (2019).
[Crossref]

Shimobaba, T.

Skodras, A.

P. Schelkens, A. Skodras, and T. Ebrahimi, The JPEG 2000 Suite (Wiley Publishing, 2009).
[Crossref]

Stoykova, E.

Sugie, T.

Symeonidou, A.

Takada, N.

Tsang, P.

Wakunami, K.

Wilkinson, T. D.

Yamaguchi, M.

K. Wakunami, H. Yamashita, and M. Yamaguchi, “Occlusion culling for computer generated hologram based on ray-wavefront conversion,” Opt. Express 21, 21811–21822 (2013).
[Crossref] [PubMed]

M. Yamaguchi, H. Hoshino, T. Honda, and N. Ohyama, “Phase-added stereogram: calculation of hologram using computer graphics technique,” Proc. SPIE 1914, 25–31 (1993).
[Crossref]

Yamaguchi, T.

H. Kang, T. Fujii, T. Yamaguchi, and H. Yoshikawa, “Compensated phase-added stereogram for real-time holographic display,” Opt. Eng. 46, 095802 (2007).
[Crossref]

Yamamoto, K.

Yamamoto, Y.

Yamashita, H.

Yatagai, T.

Yoshikawa, H.

H. Kang, E. Stoykova, and H. Yoshikawa, “Fast phase-added stereogram algorithm for generation of photorealistic 3d content,” Appl. Opt. 55, A135–A143 (2016).
[Crossref] [PubMed]

H. Kang, T. Fujii, T. Yamaguchi, and H. Yoshikawa, “Compensated phase-added stereogram for real-time holographic display,” Opt. Eng. 46, 095802 (2007).
[Crossref]

Zhang, H.

Zhao, Y.

Zhou, C.

Zhuang, Z.

Zou, W.

Appl. Opt. (9)

A. Gilles, P. Gioia, R. Cozot, and L. Morin, “Hybrid approach for fast occlusion processing in computer-generated hologram calculation,” Appl. Opt. 55, 5459–5470 (2016).
[Crossref] [PubMed]

R. H.-Y. Chen and T. D. Wilkinson, “Computer generated hologram with geometric occlusion using gpu-accelerated depth buffer rasterization for three-dimensional display,” Appl. Opt. 48, 4246–4255 (2009).
[Crossref] [PubMed]

H. Nishi and K. Matsushima, “Rendering of specular curved objects in polygon-based computer holography,” Appl. Opt. 56, F37–F44 (2017).
[Crossref] [PubMed]

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] [PubMed]

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] [PubMed]

A. Gilles and P. Gioia, “Real-time layer-based computer-generated hologram calculation for the fourier transform optical system,” Appl. Opt. 57, 8508–8517 (2018).
[Crossref] [PubMed]

T. Yatagai, “Stereoscopic approach to 3-d display using computer-generated holograms,” Appl. Opt. 15, 2722–2729 (1976).
[Crossref] [PubMed]

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]

H. Kang, E. Stoykova, and H. Yoshikawa, “Fast phase-added stereogram algorithm for generation of photorealistic 3d content,” Appl. Opt. 55, A135–A143 (2016).
[Crossref] [PubMed]

J. Electron. Imaging (1)

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

J. Inf. Displ. (1)

J.-H. Park, “Recent progress in computer-generated holography for three-dimensional scenes,” J. Inf. Displ. 18, 1–12 (2017).
[Crossref]

Opt. Commun. (1)

D. Arai, T. Shimobaba, T. Nishitsuji, T. Kakue, N. Masuda, and T. Ito, “An accelerated hologram calculation using the wavefront recording plane method and wavelet transform,” Opt. Commun. 393, 107–112 (2017).
[Crossref]

Opt. Eng. (1)

H. Kang, T. Fujii, T. Yamaguchi, and H. Yoshikawa, “Compensated phase-added stereogram for real-time holographic display,” Opt. Eng. 46, 095802 (2007).
[Crossref]

Opt. Express (12)

K. Matsushima, M. Nakamura, and S. Nakahara, “Silhouette method for hidden surface removal in computer holography and its acceleration using the switch-back technique,” Opt. Express 22, 24450–24465 (2014).
[Crossref] [PubMed]

P. Tsang, W.-K. Cheung, T.-C. Poon, and C. Zhou, “Holographic video at 40 frames per second for 4-million object points,” Opt. Express 19, 15205–15211 (2011).
[Crossref] [PubMed]

N. Okada, T. Shimobaba, Y. Ichihashi, R. Oi, K. Yamamoto, M. Oikawa, T. Kakue, N. Masuda, and T. Ito, “Band-limited double-step fresnel diffraction and its application to computer-generated holograms,” Opt. Express 21, 9192–9197 (2013).
[Crossref] [PubMed]

T. Shimobaba and T. Ito, “Fast generation of computer-generated holograms using wavelet shrinkage,” Opt. Express 25, 77–87 (2017).
[Crossref] [PubMed]

K. Wakunami, H. Yamashita, and M. Yamaguchi, “Occlusion culling for computer generated hologram based on ray-wavefront conversion,” Opt. Express 21, 21811–21822 (2013).
[Crossref] [PubMed]

D. Blinder and P. Schelkens, “Accelerated computer generated holography using sparse bases in the STFT domain,” Opt. Express 26, 1461–1473 (2018).
[Crossref] [PubMed]

A. Symeonidou, D. Blinder, and P. Schelkens, “Colour computer-generated holography for point clouds utilizing the phong illumination model,” Opt. Express 26, 10282–10298 (2018).
[Crossref] [PubMed]

Y. Yamamoto, H. Nakayama, N. Takada, T. Nishitsuji, T. Sugie, T. Kakue, T. Shimobaba, and T. Ito, “Large-scale electroholography by horn-8 from a point-cloud model with 400,000 points,” Opt. Express 26, 34259–34265 (2018).
[Crossref]

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, 25440–25449 (2015).
[Crossref] [PubMed]

A. Symeonidou, D. Blinder, A. Munteanu, and P. Schelkens, “Computer-generated holograms by multiple wavefront recording plane method with occlusion culling,” Opt. Express 23, 22149–22161 (2015).
[Crossref] [PubMed]

S. Jiao, Z. Zhuang, and W. Zou, “Fast computer generated hologram calculation with a mini look-up table incorporated with radial symmetric interpolation,” Opt. Express 25, 112–123 (2017).
[Crossref] [PubMed]

H. Zhang, Y. Zhao, L. Cao, and G. Jin, “Fully computed holographic stereogram based algorithm for computer-generated holograms with accurate depth cues,” Opt. Express 23, 3901–3913 (2015).
[Crossref] [PubMed]

Opt. Lett. (1)

Proc. SPIE (1)

M. Yamaguchi, H. Hoshino, T. Honda, and N. Ohyama, “Phase-added stereogram: calculation of hologram using computer graphics technique,” Proc. SPIE 1914, 25–31 (1993).
[Crossref]

Proc.SPIE (1)

T. Shimobaba, T. Kakue, and T. Ito, “Acceleration of color computer-generated hologram from three-dimensional scenes with texture and depth information,” Proc.SPIE 9117, 91170B (2014).

Signal Process. Image (1)

D. Blinder, A. Ahar, S. Bettens, T. Birnbaum, A. Symeonidou, H. Ottevaere, C. Schretter, and P. Schelkens, “Signal processing challenges for digital holographic video display systems,” Signal Process. Image 70, 114–130 (2019).
[Crossref]

Other (2)

P. Schelkens, A. Skodras, and T. Ebrahimi, The JPEG 2000 Suite (Wiley Publishing, 2009).
[Crossref]

J. W. Goodman, Introduction to Fourier Optics (W. H. Freeman and Company, 2017).

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Figures (12)

Fig. 1
Fig. 1 MSE of the approximations for small values (ES(x)) and large values (EL(x)) compared to the reference E(x), on a logarithmic scale. The crossing point of both curves (at x ≈ 1.609) is chosen to select the approximation with the smallest error depending on x. Given the symmetry of E(x), the curves look exactly the same (but mirrored) for negative values of x.
Fig. 2
Fig. 2 Comparison of the (imaginary parts of) E(x) and (x). The curves show good agreement (see Fig. 1 for the error analysis).
Fig. 3
Fig. 3 Time-frequency charts of PSF curves superimposed on STFT coefficients. The red curves represent the instantaneous frequency of the two PSF instances, whose slope will depend on z and lateral displacement on δ. The (Heisenberg) boxes correspond to individual coefficients, indicating where their energy is concentrated in space (or time) and frequency. The blue boxes are the coefficients that will be computed for this PSF. Both (a) and (b) have the same sparsity rates s, but (b) has a larger block size N trading off more frequency resolution for less spatial resolution, which is better for points with large |z| values.
Fig. 4
Fig. 4 Absolute value of STFT coefficients for a single PSF. The reference PSF is on the left, the proposed sparse PSF is on the right. A single block is magnified for both algorithms in the middle. Note how the highest-magnitude coefficients are concentrated in a single spot for every N × N block. Only the coefficients located in the n × n subblock at position (jx, jy) are calculated.
Fig. 5
Fig. 5 Various renderings of a PSF placed at distance z = 3 cm using different algorithm settings. The subfigures denoted with “reference” are rendered using the standard PSF expression P(x, y) in the spatial domain, while the others are generated with the proposed approximate STFT algorithm. n denotes the subblock size and s the corresponding sparsity. ℜ means the real part is shown, φ indicates that the argument or (wrapped) phase is shown. Note how the PSF progressively resembles a phase-added stereogram more as n decreases.
Fig. 6
Fig. 6 Example sparse PSF renderings at various distances z ranging from 2 to 8 cm, showing the wrapped phase for a constant sparsity of s ≈ 1.56% (n = 4).
Fig. 7
Fig. 7 NMSE for various subblock dimensions n compared to a reference PSF (N = 32).
Fig. 8
Fig. 8 Calculation time for computing the hologram of the “Ship” point cloud, for various subblock dimensions n (for N = 32) compared to the reference method, shown by the full horizontal line on the graph.
Fig. 9
Fig. 9 Diagram of the “Ship” scene geometry. (a) front view, showing the lateral dimensions of the model. (b) top view, showing the distance to the hologram plane and the depth extent.
Fig. 10
Fig. 10 Three views taken from the holograms generated by the reference algorithm (top row) and the proposed algorithm (bottom row). The views were rendered by taking a vertically centered 1024 × 1024 subhologram crop at the left side, middle and right side of the hologram, followed by a backpropagation with z = −10.5 cm (using the ASM) and taking the absolute value.
Fig. 11
Fig. 11 Diagram of the “Train” scene geometry. (a) front view of the scene, including lateral dimensions. (b) top view, showing how the train is tilted w.r.t. the z-axis.
Fig. 12
Fig. 12 Reconstructed images of the “Train” hologram, for various values of subblock size n. The front focus is shown by taking the absolute value after backpropagating at z = −10.3 cm with the ASM. The back focus uses z = −11.2 cm.

Tables (1)

Tables Icon

Table 1 “Train” hologram calculation times for different values of n, and PSNR w.r.t. the unsparsified representation for views focused at the front and back of the scene.

Equations (17)

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P ( x , y ) = exp ( i γ 2 [ ( x δ ) 2 + ( y ) 2 ] )
H ( x , y ) = j = 1 M P j ( x , y ) = j = 1 M exp ( i γ j 2 [ ( x δ j ) 2 + ( y j ) 2 ] )
H ( x , y ) = 𝒯 1 { 𝒯 { H ( x , y ) } } = 𝒯 1 { j = 1 M 𝒯 { P j ( x , y ) } }
f ( ω , η ) = B + B B + B P ( x , y ) exp ( i ( x ω + y η ) ) d x d y
f x ( ω ) = B + B exp ( i [ γ 2 ( x δ ) 2 + x ω ] ) d x and f y ( η ) = B + B exp ( i [ γ 2 ( y ) 2 + y η ] ) d y
erf ( x ) = 2 π 0 x e t 2 d t
1 + i 4 γ 2 π exp ( i ( ω δ + ω 2 4 γ 2 ) ) [ erf ( 1 i 2 ( ( B + δ ) γ + ω 2 γ ) ) + erf ( 1 i 2 ( ( B δ ) γ ω 2 γ ) ) ] .
f x ( ω ) 1 γ exp ( i ( ω δ + α 2 ) ) [ E ( B γ + β ) + E ( B γ β ) ]
E ( x ) = c + exp ( i x 2 ) x i exp ( i x 2 ) 2 x 3 3 exp ( i x 2 ) 4 x 5 + 𝒪 ( 1 x 7 ) . ( for x + )
E L ( x ) = exp ( i x 2 ) x + c sgn ( x )
{ E ( x ) } { E S ( x ) } = 2 3 x 3 + 1 21 x 7 1 660 x 11
{ E ( x ) } { E S ( x ) } = 2 x 1 5 x 5 + 1 108 x 9
E ˜ ( x ) = { E S ( x ) , if | x | < 1.609 E L ( x ) , otherwise .
ν ( x ) = 1 2 π x P ( x ) = x δ λ z
j x = max ( 0 , min ( N n , N n + 1 2 N p ( b x δ ) λ z ) )
NMSE ( R , S ) = j R j S j 2 j R j 2
PSNR ( R , S ) = 10 log 10 ( C R max 2 j C ( R j S j ) 2 )

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