Abstract

Holographic video requires impractical bitrates for storage and transmission without data compression. We introduce an end-to-end compression pipeline for compressing holographic sequences with known ground truth motion. The compression strategy employs a motion compensation algorithm based on the rotational transformation of an angular spectrum. Residuals arising from the compensation step are represented using short-time Fourier transforms and quantized with uniform mid-rise quantizers whose bit depth is determined by a Lagrangian rate-distortion optimization criterion where the distortion metric is the mean squared error. Experiments use computer-generated holographic videos, and we report Bjøntegaard delta peak signal-to-noise ratio gains of around 20 dB when compared to traditional image/video codecs.

© 2019 Optical Society of America

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Efficient holographic video generation based on rotational transformation of wavefields

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References

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  1. D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
    [Crossref]
  2. J. Goodman, Introduction to Fourier Optics, McGraw-Hill Physical and Quantum Electronics Series (W. H. Freeman, 2005).
  3. S. A. Benton, J. Bove, and V. Michael, Holographic Imaging (Wiley-Interscience, 2008).
  4. J. Barabas and V. M. Bove, “Visual perception and holographic displays,” J. Phys.: Conf. Ser. 415, 012056 (2013).
    [Crossref]
  5. 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 Commun. 70, 114–130 (2019).
    [Crossref]
  6. P. Schelkens, D. Blinder, T. Birnbaum, A. Symeonidou, A. Ahar, and C. Schretter, “Source coding of holographic data: challenges, algorithms and standardization efforts,” Proc. SPIE 10834, 257–262 (2018).
    [Crossref]
  7. D. Blinder, C. Schretter, and P. Schelkens, “Global motion compensation for compressing holographic videos,” Opt. Express 26, 25524–25533 (2018).
    [Crossref]
  8. L. T. Bang, Z. Ali, P. D. Quang, J.-H. Park, and N. Kim, “Compression of digital hologram for three-dimensional object using wavelet-bandelets transform,” Opt. Express 19, 8019–8031 (2011).
    [Crossref]
  9. E. Darakis and J. J. Soraghan, “Use of Fresnelets for phase-shifting digital hologram compression,” IEEE Trans. Image Process. 15, 3804–3811 (2006).
    [Crossref]
  10. A. E. Rhammad, P. Gioia, A. Gilles, M. Cagnazzo, and B. Pesquet-Popescu, “Color digital hologram compression based on matching pursuit,” Appl. Opt. 57, 4930–4942 (2018).
    [Crossref]
  11. K. Viswanathan, P. Gioia, and L. Morin, “Wavelet compression of digital holograms: towards a view-dependent framework,” Proc. SPIE 8856, 539–548 (2013).
    [Crossref]
  12. Y. Xing, B. Pesquet-Popescu, and F. Dufaux, “Comparative study of scalar and vector quantization on different phase-shifting digital holographic data representations,” in 3DTV-Conference: The True Vision—Capture, Transmission and Display of 3D Video (3DTV-CON) (2014), pp. 1–4.
  13. E. Darakis and T. J. Naughton, “Compression of digital hologram sequences using MPEG-4,” Proc. SPIE 7358, 269–276 (2009).
    [Crossref]
  14. Y. Xing, B. Pesquet-Popescu, and F. Dufaux, “Compression of computer generated phase-shifting hologram sequence using AVC and HEVC,” Proc. SPIE 8856, 531–538 (2013).
    [Crossref]
  15. H. Takahashi, K. Tanaka, H. Okamoto, H. Ueda, and E. Shimizu, “Direct volume access by an improved electro-holography image generator,” Proc. SPIE 2406, 220–225 (1995).
    [Crossref]
  16. K. Matsushima, “Formulation of the rotational transformation of wave fields and their application to digital holography,” Appl. Opt. 47, D110–D116 (2008).
    [Crossref]
  17. S. Widnall and J. Peraire, Lecture L8—Relative Motion Using Rotating Axes (MIT Lect., 2009).
  18. Y. You, Audio Coding: Theory and Applications (Springer US, 2010).
  19. Y. Shoham and A. Gersho, “Efficient bit allocation for an arbitrary set of quantizers (speech coding),” IEEE Trans. Acoust. Speech Signal Process. 36, 1445–1453 (1988).
    [Crossref]
  20. D. Taubman and M. Marcellin, JPEG2000 Image Compression Fundamentals, Standards and Practice (Springer Publishing Company, Incorporated, 2013).
  21. A. Ahar, T. Birnbaum, D. Blinder, A. Symeonidou, and P. Schelkens, “Performance evaluation of sparseness significance ranking measure (SSRM) on holographic content,” in Imaging and Applied Optics 2018 (3D, AO, AIO, COSI, DH, IS, LACSEA, LS&C, MATH, pcAOP) (Optical Society of America, 2018), p. JTu4A.10.
  22. G. Bjontegaard, “Calculation of average PSNR differences between RD-curves,” , 2001.

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 Commun. 70, 114–130 (2019).
[Crossref]

2018 (3)

2013 (3)

K. Viswanathan, P. Gioia, and L. Morin, “Wavelet compression of digital holograms: towards a view-dependent framework,” Proc. SPIE 8856, 539–548 (2013).
[Crossref]

J. Barabas and V. M. Bove, “Visual perception and holographic displays,” J. Phys.: Conf. Ser. 415, 012056 (2013).
[Crossref]

Y. Xing, B. Pesquet-Popescu, and F. Dufaux, “Compression of computer generated phase-shifting hologram sequence using AVC and HEVC,” Proc. SPIE 8856, 531–538 (2013).
[Crossref]

2011 (1)

2009 (1)

E. Darakis and T. J. Naughton, “Compression of digital hologram sequences using MPEG-4,” Proc. SPIE 7358, 269–276 (2009).
[Crossref]

2008 (1)

2006 (1)

E. Darakis and J. J. Soraghan, “Use of Fresnelets for phase-shifting digital hologram compression,” IEEE Trans. Image Process. 15, 3804–3811 (2006).
[Crossref]

1995 (1)

H. Takahashi, K. Tanaka, H. Okamoto, H. Ueda, and E. Shimizu, “Direct volume access by an improved electro-holography image generator,” Proc. SPIE 2406, 220–225 (1995).
[Crossref]

1988 (1)

Y. Shoham and A. Gersho, “Efficient bit allocation for an arbitrary set of quantizers (speech coding),” IEEE Trans. Acoust. Speech Signal Process. 36, 1445–1453 (1988).
[Crossref]

1948 (1)

D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
[Crossref]

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 Commun. 70, 114–130 (2019).
[Crossref]

P. Schelkens, D. Blinder, T. Birnbaum, A. Symeonidou, A. Ahar, and C. Schretter, “Source coding of holographic data: challenges, algorithms and standardization efforts,” Proc. SPIE 10834, 257–262 (2018).
[Crossref]

A. Ahar, T. Birnbaum, D. Blinder, A. Symeonidou, and P. Schelkens, “Performance evaluation of sparseness significance ranking measure (SSRM) on holographic content,” in Imaging and Applied Optics 2018 (3D, AO, AIO, COSI, DH, IS, LACSEA, LS&C, MATH, pcAOP) (Optical Society of America, 2018), p. JTu4A.10.

Ali, Z.

Bang, L. T.

Barabas, J.

J. Barabas and V. M. Bove, “Visual perception and holographic displays,” J. Phys.: Conf. Ser. 415, 012056 (2013).
[Crossref]

Benton, S. A.

S. A. Benton, J. Bove, and V. Michael, Holographic Imaging (Wiley-Interscience, 2008).

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 Commun. 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 Commun. 70, 114–130 (2019).
[Crossref]

P. Schelkens, D. Blinder, T. Birnbaum, A. Symeonidou, A. Ahar, and C. Schretter, “Source coding of holographic data: challenges, algorithms and standardization efforts,” Proc. SPIE 10834, 257–262 (2018).
[Crossref]

A. Ahar, T. Birnbaum, D. Blinder, A. Symeonidou, and P. Schelkens, “Performance evaluation of sparseness significance ranking measure (SSRM) on holographic content,” in Imaging and Applied Optics 2018 (3D, AO, AIO, COSI, DH, IS, LACSEA, LS&C, MATH, pcAOP) (Optical Society of America, 2018), p. JTu4A.10.

Bjontegaard, G.

G. Bjontegaard, “Calculation of average PSNR differences between RD-curves,” , 2001.

Blinder, D.

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 Commun. 70, 114–130 (2019).
[Crossref]

P. Schelkens, D. Blinder, T. Birnbaum, A. Symeonidou, A. Ahar, and C. Schretter, “Source coding of holographic data: challenges, algorithms and standardization efforts,” Proc. SPIE 10834, 257–262 (2018).
[Crossref]

D. Blinder, C. Schretter, and P. Schelkens, “Global motion compensation for compressing holographic videos,” Opt. Express 26, 25524–25533 (2018).
[Crossref]

A. Ahar, T. Birnbaum, D. Blinder, A. Symeonidou, and P. Schelkens, “Performance evaluation of sparseness significance ranking measure (SSRM) on holographic content,” in Imaging and Applied Optics 2018 (3D, AO, AIO, COSI, DH, IS, LACSEA, LS&C, MATH, pcAOP) (Optical Society of America, 2018), p. JTu4A.10.

Bove, J.

S. A. Benton, J. Bove, and V. Michael, Holographic Imaging (Wiley-Interscience, 2008).

Bove, V. M.

J. Barabas and V. M. Bove, “Visual perception and holographic displays,” J. Phys.: Conf. Ser. 415, 012056 (2013).
[Crossref]

Cagnazzo, M.

Darakis, E.

E. Darakis and T. J. Naughton, “Compression of digital hologram sequences using MPEG-4,” Proc. SPIE 7358, 269–276 (2009).
[Crossref]

E. Darakis and J. J. Soraghan, “Use of Fresnelets for phase-shifting digital hologram compression,” IEEE Trans. Image Process. 15, 3804–3811 (2006).
[Crossref]

Dufaux, F.

Y. Xing, B. Pesquet-Popescu, and F. Dufaux, “Compression of computer generated phase-shifting hologram sequence using AVC and HEVC,” Proc. SPIE 8856, 531–538 (2013).
[Crossref]

Y. Xing, B. Pesquet-Popescu, and F. Dufaux, “Comparative study of scalar and vector quantization on different phase-shifting digital holographic data representations,” in 3DTV-Conference: The True Vision—Capture, Transmission and Display of 3D Video (3DTV-CON) (2014), pp. 1–4.

Gabor, D.

D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
[Crossref]

Gersho, A.

Y. Shoham and A. Gersho, “Efficient bit allocation for an arbitrary set of quantizers (speech coding),” IEEE Trans. Acoust. Speech Signal Process. 36, 1445–1453 (1988).
[Crossref]

Gilles, A.

Gioia, P.

A. E. Rhammad, P. Gioia, A. Gilles, M. Cagnazzo, and B. Pesquet-Popescu, “Color digital hologram compression based on matching pursuit,” Appl. Opt. 57, 4930–4942 (2018).
[Crossref]

K. Viswanathan, P. Gioia, and L. Morin, “Wavelet compression of digital holograms: towards a view-dependent framework,” Proc. SPIE 8856, 539–548 (2013).
[Crossref]

Goodman, J.

J. Goodman, Introduction to Fourier Optics, McGraw-Hill Physical and Quantum Electronics Series (W. H. Freeman, 2005).

Kim, N.

Marcellin, M.

D. Taubman and M. Marcellin, JPEG2000 Image Compression Fundamentals, Standards and Practice (Springer Publishing Company, Incorporated, 2013).

Matsushima, K.

Michael, V.

S. A. Benton, J. Bove, and V. Michael, Holographic Imaging (Wiley-Interscience, 2008).

Morin, L.

K. Viswanathan, P. Gioia, and L. Morin, “Wavelet compression of digital holograms: towards a view-dependent framework,” Proc. SPIE 8856, 539–548 (2013).
[Crossref]

Naughton, T. J.

E. Darakis and T. J. Naughton, “Compression of digital hologram sequences using MPEG-4,” Proc. SPIE 7358, 269–276 (2009).
[Crossref]

Okamoto, H.

H. Takahashi, K. Tanaka, H. Okamoto, H. Ueda, and E. Shimizu, “Direct volume access by an improved electro-holography image generator,” Proc. SPIE 2406, 220–225 (1995).
[Crossref]

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 Commun. 70, 114–130 (2019).
[Crossref]

Park, J.-H.

Peraire, J.

S. Widnall and J. Peraire, Lecture L8—Relative Motion Using Rotating Axes (MIT Lect., 2009).

Pesquet-Popescu, B.

A. E. Rhammad, P. Gioia, A. Gilles, M. Cagnazzo, and B. Pesquet-Popescu, “Color digital hologram compression based on matching pursuit,” Appl. Opt. 57, 4930–4942 (2018).
[Crossref]

Y. Xing, B. Pesquet-Popescu, and F. Dufaux, “Compression of computer generated phase-shifting hologram sequence using AVC and HEVC,” Proc. SPIE 8856, 531–538 (2013).
[Crossref]

Y. Xing, B. Pesquet-Popescu, and F. Dufaux, “Comparative study of scalar and vector quantization on different phase-shifting digital holographic data representations,” in 3DTV-Conference: The True Vision—Capture, Transmission and Display of 3D Video (3DTV-CON) (2014), pp. 1–4.

Quang, P. D.

Rhammad, A. E.

Schelkens, P.

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 Commun. 70, 114–130 (2019).
[Crossref]

P. Schelkens, D. Blinder, T. Birnbaum, A. Symeonidou, A. Ahar, and C. Schretter, “Source coding of holographic data: challenges, algorithms and standardization efforts,” Proc. SPIE 10834, 257–262 (2018).
[Crossref]

D. Blinder, C. Schretter, and P. Schelkens, “Global motion compensation for compressing holographic videos,” Opt. Express 26, 25524–25533 (2018).
[Crossref]

A. Ahar, T. Birnbaum, D. Blinder, A. Symeonidou, and P. Schelkens, “Performance evaluation of sparseness significance ranking measure (SSRM) on holographic content,” in Imaging and Applied Optics 2018 (3D, AO, AIO, COSI, DH, IS, LACSEA, LS&C, MATH, pcAOP) (Optical Society of America, 2018), p. JTu4A.10.

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 Commun. 70, 114–130 (2019).
[Crossref]

D. Blinder, C. Schretter, and P. Schelkens, “Global motion compensation for compressing holographic videos,” Opt. Express 26, 25524–25533 (2018).
[Crossref]

P. Schelkens, D. Blinder, T. Birnbaum, A. Symeonidou, A. Ahar, and C. Schretter, “Source coding of holographic data: challenges, algorithms and standardization efforts,” Proc. SPIE 10834, 257–262 (2018).
[Crossref]

Shimizu, E.

H. Takahashi, K. Tanaka, H. Okamoto, H. Ueda, and E. Shimizu, “Direct volume access by an improved electro-holography image generator,” Proc. SPIE 2406, 220–225 (1995).
[Crossref]

Shoham, Y.

Y. Shoham and A. Gersho, “Efficient bit allocation for an arbitrary set of quantizers (speech coding),” IEEE Trans. Acoust. Speech Signal Process. 36, 1445–1453 (1988).
[Crossref]

Soraghan, J. J.

E. Darakis and J. J. Soraghan, “Use of Fresnelets for phase-shifting digital hologram compression,” IEEE Trans. Image Process. 15, 3804–3811 (2006).
[Crossref]

Symeonidou, 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 Commun. 70, 114–130 (2019).
[Crossref]

P. Schelkens, D. Blinder, T. Birnbaum, A. Symeonidou, A. Ahar, and C. Schretter, “Source coding of holographic data: challenges, algorithms and standardization efforts,” Proc. SPIE 10834, 257–262 (2018).
[Crossref]

A. Ahar, T. Birnbaum, D. Blinder, A. Symeonidou, and P. Schelkens, “Performance evaluation of sparseness significance ranking measure (SSRM) on holographic content,” in Imaging and Applied Optics 2018 (3D, AO, AIO, COSI, DH, IS, LACSEA, LS&C, MATH, pcAOP) (Optical Society of America, 2018), p. JTu4A.10.

Takahashi, H.

H. Takahashi, K. Tanaka, H. Okamoto, H. Ueda, and E. Shimizu, “Direct volume access by an improved electro-holography image generator,” Proc. SPIE 2406, 220–225 (1995).
[Crossref]

Tanaka, K.

H. Takahashi, K. Tanaka, H. Okamoto, H. Ueda, and E. Shimizu, “Direct volume access by an improved electro-holography image generator,” Proc. SPIE 2406, 220–225 (1995).
[Crossref]

Taubman, D.

D. Taubman and M. Marcellin, JPEG2000 Image Compression Fundamentals, Standards and Practice (Springer Publishing Company, Incorporated, 2013).

Ueda, H.

H. Takahashi, K. Tanaka, H. Okamoto, H. Ueda, and E. Shimizu, “Direct volume access by an improved electro-holography image generator,” Proc. SPIE 2406, 220–225 (1995).
[Crossref]

Viswanathan, K.

K. Viswanathan, P. Gioia, and L. Morin, “Wavelet compression of digital holograms: towards a view-dependent framework,” Proc. SPIE 8856, 539–548 (2013).
[Crossref]

Widnall, S.

S. Widnall and J. Peraire, Lecture L8—Relative Motion Using Rotating Axes (MIT Lect., 2009).

Xing, Y.

Y. Xing, B. Pesquet-Popescu, and F. Dufaux, “Compression of computer generated phase-shifting hologram sequence using AVC and HEVC,” Proc. SPIE 8856, 531–538 (2013).
[Crossref]

Y. Xing, B. Pesquet-Popescu, and F. Dufaux, “Comparative study of scalar and vector quantization on different phase-shifting digital holographic data representations,” in 3DTV-Conference: The True Vision—Capture, Transmission and Display of 3D Video (3DTV-CON) (2014), pp. 1–4.

You, Y.

Y. You, Audio Coding: Theory and Applications (Springer US, 2010).

Appl. Opt. (2)

IEEE Trans. Acoust. Speech Signal Process. (1)

Y. Shoham and A. Gersho, “Efficient bit allocation for an arbitrary set of quantizers (speech coding),” IEEE Trans. Acoust. Speech Signal Process. 36, 1445–1453 (1988).
[Crossref]

IEEE Trans. Image Process. (1)

E. Darakis and J. J. Soraghan, “Use of Fresnelets for phase-shifting digital hologram compression,” IEEE Trans. Image Process. 15, 3804–3811 (2006).
[Crossref]

J. Phys.: Conf. Ser. (1)

J. Barabas and V. M. Bove, “Visual perception and holographic displays,” J. Phys.: Conf. Ser. 415, 012056 (2013).
[Crossref]

Nature (1)

D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
[Crossref]

Opt. Express (2)

Proc. SPIE (5)

P. Schelkens, D. Blinder, T. Birnbaum, A. Symeonidou, A. Ahar, and C. Schretter, “Source coding of holographic data: challenges, algorithms and standardization efforts,” Proc. SPIE 10834, 257–262 (2018).
[Crossref]

K. Viswanathan, P. Gioia, and L. Morin, “Wavelet compression of digital holograms: towards a view-dependent framework,” Proc. SPIE 8856, 539–548 (2013).
[Crossref]

E. Darakis and T. J. Naughton, “Compression of digital hologram sequences using MPEG-4,” Proc. SPIE 7358, 269–276 (2009).
[Crossref]

Y. Xing, B. Pesquet-Popescu, and F. Dufaux, “Compression of computer generated phase-shifting hologram sequence using AVC and HEVC,” Proc. SPIE 8856, 531–538 (2013).
[Crossref]

H. Takahashi, K. Tanaka, H. Okamoto, H. Ueda, and E. Shimizu, “Direct volume access by an improved electro-holography image generator,” Proc. SPIE 2406, 220–225 (1995).
[Crossref]

Signal Process. Image Commun. (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 Commun. 70, 114–130 (2019).
[Crossref]

Other (8)

J. Goodman, Introduction to Fourier Optics, McGraw-Hill Physical and Quantum Electronics Series (W. H. Freeman, 2005).

S. A. Benton, J. Bove, and V. Michael, Holographic Imaging (Wiley-Interscience, 2008).

S. Widnall and J. Peraire, Lecture L8—Relative Motion Using Rotating Axes (MIT Lect., 2009).

Y. You, Audio Coding: Theory and Applications (Springer US, 2010).

Y. Xing, B. Pesquet-Popescu, and F. Dufaux, “Comparative study of scalar and vector quantization on different phase-shifting digital holographic data representations,” in 3DTV-Conference: The True Vision—Capture, Transmission and Display of 3D Video (3DTV-CON) (2014), pp. 1–4.

D. Taubman and M. Marcellin, JPEG2000 Image Compression Fundamentals, Standards and Practice (Springer Publishing Company, Incorporated, 2013).

A. Ahar, T. Birnbaum, D. Blinder, A. Symeonidou, and P. Schelkens, “Performance evaluation of sparseness significance ranking measure (SSRM) on holographic content,” in Imaging and Applied Optics 2018 (3D, AO, AIO, COSI, DH, IS, LACSEA, LS&C, MATH, pcAOP) (Optical Society of America, 2018), p. JTu4A.10.

G. Bjontegaard, “Calculation of average PSNR differences between RD-curves,” , 2001.

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Figures (11)

Fig. 1.
Fig. 1. Block diagram of the proposed holographic video coder. (a) The encoding procedure initially uses the previous decoded frame to predict the current frame using motion compensation. The residual between the predicted frame and the ground truth frame is compressed. (b) The decoding procedure performs the motion compensation step and decodes the compressed residual to obtain the decoded current frame.
Fig. 2.
Fig. 2. Rotation of an object (cube) around a pivot point $ P({x_{\text{o}}},{y_{\text{o}}},{z_{\text{o}}}) $ for $ {\theta _{\text{x}}},{\theta _{\text{y}}} $ , and $ {\theta _{\text{z}}} $ radians about three orthogonal axes. These orthogonal axes are parallel to the $ X $ , $ Y $ , and $ Z $ axes where the origin $ O(0,0,0) $ is the center of the hologram, which is indicated by the plane.
Fig. 3.
Fig. 3. Equivalent relative rotation between object and virtual hologram. (a) Object (cube) and virtual hologram (plane) before rotational motion around the pivot point $ (x_{\text{o}},y_{\text{o}},z_{\text{o}}) $ indicated by the dot $ P $ . (b) Rotation of 45° for object about the vertical axis parallel to $ Y $ axis while keeping the virtual hologram plane stationary. (c) Rotation of $ -45^\circ $ for virtual hologram plane about the vertical axis parallel to $ Y $ axis while keeping the object stationary.
Fig. 4.
Fig. 4. Amplitude (black indicates absence of signal) of a zero-padded motion-compensated hologram in the spatial domain is shown here for different types of rotations and translations. The rectangular border in the center indicates the spatial boundary of the ground truth hologram. (a) Rotation about axis parallel to $ X $ axis. (b) Rotation about axis parallel to $ Y $ axis. (c) Rotation about axis parallel to $ Z $ axis. (d) Translation along $ X $ axis. (e) Translation along $ Y $ axis. (f) Translation along $ Z $ axis (towards hologram plane).
Fig. 5.
Fig. 5. Illustration of the procedure described by Eqs. (22)–(25) to obtain space–frequency blocks by using the STFT.
Fig. 6.
Fig. 6. Uniform scalar mid-rise quantizer. (a) Input-output relationship for a quantizer with quantization range [ $ -4,4 $ ] and 8 levels. (b) Quantization error dependence on quantization range for an examplary SFB.
Fig. 7.
Fig. 7. Bitstream produced by the encoder for an SFB chosen for storage contains side information that is included along with the quantization levels.
Fig. 8.
Fig. 8. Reconstructed central view in object plane. (a) Ground truth for Cat; (b) proposed codec at 0.1 bpp for Cat; (c) HEVC Inter at 0.1 bpp for Cat; (d) JPEG 2000 at 0.1 bpp for Cat; (e) ground truth for Venus; (f) proposed codec at 0.1 bpp for Venus; (g) HEVC Inter at 0.1 bpp for Venus; (h) JPEG 2000 at 0.1 bpp for Venus; (i) ground truth for Dragon; (j) proposed codec at 0.1 bpp for Dragon; (k) HEVC Inter at 0.1 bpp for Dragon; (l) JPEG 2000 at 0.1 bpp for Dragon.
Fig. 9.
Fig. 9. Reconstructed central view in object plane for Cat (close-up). (a) Ground truth; (b) proposed codec at 0.1 bpp; (c) HEVC Inter at 0.1 bpp; (d) JPEG 2000 at 0.1 bpp.
Fig. 10.
Fig. 10. Effect of different types of rotations and translations on rate-distortion performance. (a) Rotation about axis ∥ to $ Y $ axis (b) Rotation about axis ∥ to $ Z $ axis (In-plane) (c) Translation along $ Y $ axis (In-plane) (d) Translation along $ Z $ axis (towards hologram plane).
Fig. 11.
Fig. 11. Rate-distortion performance for different codecs.

Tables (6)

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Algorithm 1. Golden-section Search for Determining X maxtab k , l , m , n ( b ) and D tab k , l , m , n ( b )

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Algorithm 2. Lagrangian Rate-Distortion Optimization for Determining Bit Depth Allocation

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Table 1. Hologram and Codec Parameters

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Table 2. Parameters of Objects Tested

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Table 3. BD-PSNR Gain of the Proposed Codec Over Traditional Codecs for Different Objects

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Table 4. BD-PSNR Gain of the Proposed Codec Over HEVC Inter for Different Motions (0.1–1.0 bpp)

Equations (37)

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sin ( θ FOV 2 ) = λ 2 p .
g ( x , y ; z ) = F 1 { G ( u , v ; 0 ) H ( u , v ; z ) } .
H ( u , v ; z ) = { e i 2 π z λ 2 u 2 v 2 u 2 + v 2 λ 2 0 otherwise } .
p = R ( θ x , θ y , θ z ) p ,
R ( θ x , θ y , θ z ) = [ cos ( θ x ) cos ( θ y ) cos ( θ x ) sin ( θ y ) sin ( θ z ) sin ( θ x ) cos ( θ z ) cos ( θ x ) sin ( θ y ) cos ( θ z ) + sin ( θ x ) sin ( θ z ) sin ( θ x ) cos ( θ y ) sin ( θ x ) sin ( θ y ) sin ( θ z ) + cos ( θ x ) cos θ z ) sin ( θ x ) sin ( θ y ) cos ( θ z ) cos ( θ x ) cos ( θ z ) sin ( θ y ) cos ( θ y ) sin ( θ z ) cos ( θ y ) cos ( θ z ) ] .
g 1 ( x , y ; 0 ) = g ref ( x + x o , y + y o ; z o ) .
G 1 ( u , v ; 0 ) = G ref ( u , v ; z o ) e j 2 π ( u x o + v y o ) .
G ref ( u , v ; z o ) = G ref ( u , v ; 0 ) H ( u , v ; z o ) .
G 1 ( u , v ; 0 ) = G ref ( u , v ; 0 ) H ( u , v ; z o ) e j 2 π ( u x o + v y o ) .
v 1 = Ω 1 × r ,
v 2 = Ω 2 × r .
R ( θ x , θ y , θ z ) = [ a 1 a 4 a 7 a 2 a 5 a 8 a 3 a 6 a 9 ] .
G 2 ( u , v ; 0 ) = G 1 ( α ( u , v ) , β ( u , v ) ; 0 ) | J ( u , v ) | ,
α ( u , v ) = a 1 u + a 2 v + a 3 w ( u , v ) ,
β ( u , v ) = a 4 u + a 5 v + a 6 w ( u , v ) ,
w ( u , v ) = λ 2 u 2 v 2 ,
J ( u , v ) = ( a 2 a 6 a 3 a 5 ) u w ( u , v ) + ( a 3 a 4 a 1 a 6 ) v w ( u , v ) + ( a 1 a 5 a 2 a 4 ) .
G 3 ( u , v ; 0 ) = G 2 ( u , v ; 0 ) H ( u , v ; z o ) e j 2 π ( u x o + v y o ) .
g mc ( x , y ; 0 ) = g 3 ( x x t , y y t ; z t ) .
G mc ( u , v ; 0 ) = G 3 ( u , v ; 0 ) H ( u , v ; z t ) e j 2 π ( u x t + v y t ) .
G mc ( u , v ; 0 ) = G ref ( α ( u , v ) , β ( u , v ) ) Resampled hologram H mc ( u , v ) transfer function ,
H mc ( u , v ) = H ( u , v ; z t z o ) H ( α ( u , v ) , β ( u , v ) ; z o ) × | J ( u , v ) | e j 2 π ( x t u + y t v x o ( α ( u , v ) u ) y o ( β ( u , v ) v ) ) .
g mc ( x , y ; 0 ) = F 1 { G mc ( u , v ; 0 ) } .
[ x , y ] = g gt [ x , y ] g mc [ x , y ] , where x { 1 , 2 , . . . , A } and y { 1 , 2 , . . . , B } .
g s k , l [ x , y ] = g [ ( k 1 ) N x + x , ( l 1 ) N y + y ] , where x { 1 , 2 , . . , N x } , y { 1 , 2 , . . , N y } , k { 1 , 2 , . . , A N x } and , l { 1 , 2 , . . , B N y } .
G s k , l [ u , v ] = 1 N x N x x = 1 N x y = 1 N y g s k , l [ x , y ] e 2 π j ( u x N x + v y N y ) , where u { 1 , 2 , . . , N x } and v { 1 , 2 , . . , N y } .
G sf k , l , m , n [ u , v ] = G s k , l [ ( m 1 ) N u + u , ( n 1 ) N v + v ] , where u { 1 , 2 , . . , N u } , v { 1 , 2 , . . , N v } , m { 1 , 2 , . . , N x N u } and , n { 1 , 2 , . . , N y N v } .
g ^ gt [ x , y ] = g ^ [ x , y ] + g mc [ x , y ] , where x { 1 , 2 , . . . , A } and y { 1 , 2 , . . . , B } .
x = 1 A y = 1 B | g gt [ x , y ] g ^ gt [ x , y ] | 2 = k = 1 A N x l = 1 B N y m = 1 N x N u n = 1 N y N v D k , l , m , n , where D k , l , m , n = u = 1 N u v = 1 N v | G sf k , l , m , n [ u , v ] G ^ sf k , l , m , n [ u , v ] | 2 .
Q ( x , L , X max ) = { 0 if L = 1 ( L 2 + 0.5 ) 2 X max L else if x < X max ( x L 2 X max + 0.5 ) 2 X max L else if X max x X max ( L 2 0.5 ) 2 X max L otherwise } .
D ( B sol ) = k = 1 A N x l = 1 B N y m = 1 N x N u n = 1 N y N v D tab k , l , m , n ( b sol k , l , m , n ) .
R ( b sol k , l , m , n ) = { 0 if b sol k , l , m , n = 0 log 2 ( A B N u N v ) + 32 + 3 Side   Information + 2 b sol k , l , m , n N u N v Quantized   values if 1 b sol k , l , m , n 8 } .
R ( B sol ) = k = 1 A N x l = 1 B N y m = 1 N y N u n = 1 N y N v R ( b sol k , l , m , n ) .
min B sol S ( D ( B sol ) + Λ R ( B sol ) ) for 0 Λ < .
min b sol k , l , m , n { 0 , 1 , . .8 } ( D tab k , l , m , n ( b sol k , l , m , n ) + Λ R ( b sol k , l , m , n ) ) .
g [ x , y ] = j P a j r j e i 2 π r j Λ , where x { 1 , 2... A } and y { 1 , 2... B } ,
SNR = 10 log 10 ( x = 1 A y = 1 B | g gt obj [ x , y ] | 2 x = 1 A y = 1 B ( | g gt obj [ x , y ] | | g com obj [ x , y ] | ) 2 ) .

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