Abstract

Large high-resolution digital holographic displays may become feasible in the near future, and they will need considerable amounts of data. Handling this bandwidth is particularly challenging for dynamic content operating at video rates. Conventional motion compensation algorithms from classical video coders are ineffective on holograms because, in contrast to natural imagery, each pixel contains partial information from the whole scene. We propose an accurate motion compensation model predicting how hologram content changes with respect to 3D rigid-body motion that arises in natural scenes. Using diffraction theory, we derive tractable closed form expressions for transforming 2D complex-valued holographic video frames. Our experiments use computer generated hologram videos with known ground truth motion. We integrated the proposed motion compensation model into the HEVC codec. We report Bjøntegaard delta-PSNR ratio gains of 8 dB over standard HEVC.

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

Full Article  |  PDF Article
OSA Recommended Articles
Compression of 3D color integral images

Sekwon Yeom, Adrian Stern, and Bahram Javidi
Opt. Express 12(8) 1632-1642 (2004)

MPEG-based novel look-up table for rapid generation of video holograms of fast-moving three-dimensional objects

Xiao-Bin Dong, Seung-Cheol Kim, and Eun-Soo Kim
Opt. Express 22(7) 8047-8067 (2014)

References

  • View by:
  • |
  • |
  • |

  1. V. M. Bove, “Display holography’s digital second act,” Proc. IEEE 100, 918–928 (2012).
    [Crossref]
  2. C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38, 46–53 (2005).
    [Crossref]
  3. J. Ostermann, J. Bormans, P. List, D. Marpe, M. Narroschke, F. Pereira, T. Stockhammer, and T. Wedi, “Video coding with H.264/AVC: tools, performance, and complexity,” IEEE Circuits Syst. Mag. 4, 7–28 (2004).
    [Crossref]
  4. G. J. Sullivan, J. R. Ohm, W. J. Han, and T. Wiegand, “Overview of the high efficiency video coding (HEVC) standard,” IEEE Transactions on Circuits Syst. for Video Technol. 22, 1649–1668 (2012).
    [Crossref]
  5. Y. Xing, B. Pesquet-Popescu, and F. Dufaux, “Compression of computer generated phase-shifting hologram sequence using AVC and HEVC,” Proc. SPIE8856, Applications of Digital Image Processing XXXVI (2009).
  6. F. Dufaux, Y. Xing, B. Pesquet-Popescu, and P. Schelkens, “Compression of digital holographic data: an overview,” Proc. SPIE9599, Applications of Digital Image Processing XXXVIII (2015).
  7. E. Darakis and T. J. Naughton, “Compression of digital hologram sequences using MPEG-4,” Proc. SPIE7358, Holography: Advances and Modern Trends (2009).
  8. H. Yoshikawa and J. Tamai, “Holographic image compression by motion picture coding,” Proc. SPIE2652, Practical Holography X (1996).
  9. Y.-H. Seo, H.-J. Choi, and D.-W. Kim, “3d scanning-based compression technique for digital hologram video,” Signal Process. Image Commun. 22, 144–156 (2007).
    [Crossref]
  10. Y.-H. Seo, Y.-H. Lee, J.-S. Yoo, and D.-W. Kim, “Scalable hologram video coding for adaptive transmitting service,” Appl. Opt. 52, A254–A268 (2013).
    [Crossref] [PubMed]
  11. Y.-Z. Liu, J.-W. Dong, Y.-Y. Pu, B.-C. Chen, H.-X. He, and H.-Z. Wang, “High-speed full analytical holographic computations for true-life scenes,” Opt. Express 18, 3345–3351 (2010).
    [Crossref] [PubMed]
  12. T. Senoh, K. Wakunami, Y. Ichihashi, H. Sasaki, R. Oi, and K. Yamamoto, “Multiview image and depth map coding for holographic tv system,” Opt. Eng. 53, 112302 (2014).
    [Crossref]
  13. A. Gilles, P. Gioia, R. Cozot, and L. Morin, “Fast generation of complex modulation video holograms using temporal redundancy compression and hybrid point-source/wave-field approaches,” Proc. SPIE2652, Applications of Digital Image Processing XXXVIII (2015).
  14. J. J. Healy, M. A. Kutay, H. M. Ozaktas, and J. T. Sheridan, Linear Canonical Transforms (Springer, 2016).
    [Crossref]
  15. M. de Gosson, The Wigner Transform, Advanced textbooks in mathematics(World Scientific Publishing Company Pte Limited, 2017).
    [Crossref]
  16. D. G. Abdelsalam, R. Magnusson, and D. Kim, “Single-shot, dual-wavelength digital holography based on polarizing separation,” Appl. Opt. 50, 3360–3368 (2011).
    [Crossref] [PubMed]
  17. L. Zhao, J. J. Healy, and J. T. Sheridan, “Two-dimensional nonseparable linear canonical transform: sampling theorem and unitary discretization,” J. Opt. Soc. Am. A 31, 2631–2641 (2014).
    [Crossref]
  18. “High Efficiency Video Coding (HEVC) reference software,” HM-15.0 (Revision 4879), https://hevc.hhi.fraunhofer.de/svn/svn_HEVCSoftware/branches/HM-dev/ . [retrieved 31 March 2017].
  19. G. Bjontegaard, Calculation of average PSNR differences between RD-curves, ITU-VCEG (2001).

2014 (2)

T. Senoh, K. Wakunami, Y. Ichihashi, H. Sasaki, R. Oi, and K. Yamamoto, “Multiview image and depth map coding for holographic tv system,” Opt. Eng. 53, 112302 (2014).
[Crossref]

L. Zhao, J. J. Healy, and J. T. Sheridan, “Two-dimensional nonseparable linear canonical transform: sampling theorem and unitary discretization,” J. Opt. Soc. Am. A 31, 2631–2641 (2014).
[Crossref]

2013 (1)

2012 (2)

V. M. Bove, “Display holography’s digital second act,” Proc. IEEE 100, 918–928 (2012).
[Crossref]

G. J. Sullivan, J. R. Ohm, W. J. Han, and T. Wiegand, “Overview of the high efficiency video coding (HEVC) standard,” IEEE Transactions on Circuits Syst. for Video Technol. 22, 1649–1668 (2012).
[Crossref]

2011 (1)

2010 (1)

2007 (1)

Y.-H. Seo, H.-J. Choi, and D.-W. Kim, “3d scanning-based compression technique for digital hologram video,” Signal Process. Image Commun. 22, 144–156 (2007).
[Crossref]

2005 (1)

C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38, 46–53 (2005).
[Crossref]

2004 (1)

J. Ostermann, J. Bormans, P. List, D. Marpe, M. Narroschke, F. Pereira, T. Stockhammer, and T. Wedi, “Video coding with H.264/AVC: tools, performance, and complexity,” IEEE Circuits Syst. Mag. 4, 7–28 (2004).
[Crossref]

Abdelsalam, D. G.

Bjontegaard, G.

G. Bjontegaard, Calculation of average PSNR differences between RD-curves, ITU-VCEG (2001).

Bormans, J.

J. Ostermann, J. Bormans, P. List, D. Marpe, M. Narroschke, F. Pereira, T. Stockhammer, and T. Wedi, “Video coding with H.264/AVC: tools, performance, and complexity,” IEEE Circuits Syst. Mag. 4, 7–28 (2004).
[Crossref]

Bove, V. M.

V. M. Bove, “Display holography’s digital second act,” Proc. IEEE 100, 918–928 (2012).
[Crossref]

Cameron, C.

C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38, 46–53 (2005).
[Crossref]

Chen, B.-C.

Choi, H.-J.

Y.-H. Seo, H.-J. Choi, and D.-W. Kim, “3d scanning-based compression technique for digital hologram video,” Signal Process. Image Commun. 22, 144–156 (2007).
[Crossref]

Cozot, R.

A. Gilles, P. Gioia, R. Cozot, and L. Morin, “Fast generation of complex modulation video holograms using temporal redundancy compression and hybrid point-source/wave-field approaches,” Proc. SPIE2652, Applications of Digital Image Processing XXXVIII (2015).

Darakis, E.

E. Darakis and T. J. Naughton, “Compression of digital hologram sequences using MPEG-4,” Proc. SPIE7358, Holography: Advances and Modern Trends (2009).

de Gosson, M.

M. de Gosson, The Wigner Transform, Advanced textbooks in mathematics(World Scientific Publishing Company Pte Limited, 2017).
[Crossref]

Dong, J.-W.

Dufaux, F.

Y. Xing, B. Pesquet-Popescu, and F. Dufaux, “Compression of computer generated phase-shifting hologram sequence using AVC and HEVC,” Proc. SPIE8856, Applications of Digital Image Processing XXXVI (2009).

F. Dufaux, Y. Xing, B. Pesquet-Popescu, and P. Schelkens, “Compression of digital holographic data: an overview,” Proc. SPIE9599, Applications of Digital Image Processing XXXVIII (2015).

Gilles, A.

A. Gilles, P. Gioia, R. Cozot, and L. Morin, “Fast generation of complex modulation video holograms using temporal redundancy compression and hybrid point-source/wave-field approaches,” Proc. SPIE2652, Applications of Digital Image Processing XXXVIII (2015).

Gioia, P.

A. Gilles, P. Gioia, R. Cozot, and L. Morin, “Fast generation of complex modulation video holograms using temporal redundancy compression and hybrid point-source/wave-field approaches,” Proc. SPIE2652, Applications of Digital Image Processing XXXVIII (2015).

Han, W. J.

G. J. Sullivan, J. R. Ohm, W. J. Han, and T. Wiegand, “Overview of the high efficiency video coding (HEVC) standard,” IEEE Transactions on Circuits Syst. for Video Technol. 22, 1649–1668 (2012).
[Crossref]

He, H.-X.

Healy, J. J.

Ichihashi, Y.

T. Senoh, K. Wakunami, Y. Ichihashi, H. Sasaki, R. Oi, and K. Yamamoto, “Multiview image and depth map coding for holographic tv system,” Opt. Eng. 53, 112302 (2014).
[Crossref]

Kim, D.

Kim, D.-W.

Y.-H. Seo, Y.-H. Lee, J.-S. Yoo, and D.-W. Kim, “Scalable hologram video coding for adaptive transmitting service,” Appl. Opt. 52, A254–A268 (2013).
[Crossref] [PubMed]

Y.-H. Seo, H.-J. Choi, and D.-W. Kim, “3d scanning-based compression technique for digital hologram video,” Signal Process. Image Commun. 22, 144–156 (2007).
[Crossref]

Kutay, M. A.

J. J. Healy, M. A. Kutay, H. M. Ozaktas, and J. T. Sheridan, Linear Canonical Transforms (Springer, 2016).
[Crossref]

Lee, Y.-H.

List, P.

J. Ostermann, J. Bormans, P. List, D. Marpe, M. Narroschke, F. Pereira, T. Stockhammer, and T. Wedi, “Video coding with H.264/AVC: tools, performance, and complexity,” IEEE Circuits Syst. Mag. 4, 7–28 (2004).
[Crossref]

Liu, Y.-Z.

Magnusson, R.

Marpe, D.

J. Ostermann, J. Bormans, P. List, D. Marpe, M. Narroschke, F. Pereira, T. Stockhammer, and T. Wedi, “Video coding with H.264/AVC: tools, performance, and complexity,” IEEE Circuits Syst. Mag. 4, 7–28 (2004).
[Crossref]

Morin, L.

A. Gilles, P. Gioia, R. Cozot, and L. Morin, “Fast generation of complex modulation video holograms using temporal redundancy compression and hybrid point-source/wave-field approaches,” Proc. SPIE2652, Applications of Digital Image Processing XXXVIII (2015).

Narroschke, M.

J. Ostermann, J. Bormans, P. List, D. Marpe, M. Narroschke, F. Pereira, T. Stockhammer, and T. Wedi, “Video coding with H.264/AVC: tools, performance, and complexity,” IEEE Circuits Syst. Mag. 4, 7–28 (2004).
[Crossref]

Naughton, T. J.

E. Darakis and T. J. Naughton, “Compression of digital hologram sequences using MPEG-4,” Proc. SPIE7358, Holography: Advances and Modern Trends (2009).

Ohm, J. R.

G. J. Sullivan, J. R. Ohm, W. J. Han, and T. Wiegand, “Overview of the high efficiency video coding (HEVC) standard,” IEEE Transactions on Circuits Syst. for Video Technol. 22, 1649–1668 (2012).
[Crossref]

Oi, R.

T. Senoh, K. Wakunami, Y. Ichihashi, H. Sasaki, R. Oi, and K. Yamamoto, “Multiview image and depth map coding for holographic tv system,” Opt. Eng. 53, 112302 (2014).
[Crossref]

Ostermann, J.

J. Ostermann, J. Bormans, P. List, D. Marpe, M. Narroschke, F. Pereira, T. Stockhammer, and T. Wedi, “Video coding with H.264/AVC: tools, performance, and complexity,” IEEE Circuits Syst. Mag. 4, 7–28 (2004).
[Crossref]

Ozaktas, H. M.

J. J. Healy, M. A. Kutay, H. M. Ozaktas, and J. T. Sheridan, Linear Canonical Transforms (Springer, 2016).
[Crossref]

Pereira, F.

J. Ostermann, J. Bormans, P. List, D. Marpe, M. Narroschke, F. Pereira, T. Stockhammer, and T. Wedi, “Video coding with H.264/AVC: tools, performance, and complexity,” IEEE Circuits Syst. Mag. 4, 7–28 (2004).
[Crossref]

Pesquet-Popescu, B.

Y. Xing, B. Pesquet-Popescu, and F. Dufaux, “Compression of computer generated phase-shifting hologram sequence using AVC and HEVC,” Proc. SPIE8856, Applications of Digital Image Processing XXXVI (2009).

F. Dufaux, Y. Xing, B. Pesquet-Popescu, and P. Schelkens, “Compression of digital holographic data: an overview,” Proc. SPIE9599, Applications of Digital Image Processing XXXVIII (2015).

Pu, Y.-Y.

Sasaki, H.

T. Senoh, K. Wakunami, Y. Ichihashi, H. Sasaki, R. Oi, and K. Yamamoto, “Multiview image and depth map coding for holographic tv system,” Opt. Eng. 53, 112302 (2014).
[Crossref]

Schelkens, P.

F. Dufaux, Y. Xing, B. Pesquet-Popescu, and P. Schelkens, “Compression of digital holographic data: an overview,” Proc. SPIE9599, Applications of Digital Image Processing XXXVIII (2015).

Senoh, T.

T. Senoh, K. Wakunami, Y. Ichihashi, H. Sasaki, R. Oi, and K. Yamamoto, “Multiview image and depth map coding for holographic tv system,” Opt. Eng. 53, 112302 (2014).
[Crossref]

Seo, Y.-H.

Y.-H. Seo, Y.-H. Lee, J.-S. Yoo, and D.-W. Kim, “Scalable hologram video coding for adaptive transmitting service,” Appl. Opt. 52, A254–A268 (2013).
[Crossref] [PubMed]

Y.-H. Seo, H.-J. Choi, and D.-W. Kim, “3d scanning-based compression technique for digital hologram video,” Signal Process. Image Commun. 22, 144–156 (2007).
[Crossref]

Sheridan, J. T.

Slinger, C.

C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38, 46–53 (2005).
[Crossref]

Stanley, M.

C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38, 46–53 (2005).
[Crossref]

Stockhammer, T.

J. Ostermann, J. Bormans, P. List, D. Marpe, M. Narroschke, F. Pereira, T. Stockhammer, and T. Wedi, “Video coding with H.264/AVC: tools, performance, and complexity,” IEEE Circuits Syst. Mag. 4, 7–28 (2004).
[Crossref]

Sullivan, G. J.

G. J. Sullivan, J. R. Ohm, W. J. Han, and T. Wiegand, “Overview of the high efficiency video coding (HEVC) standard,” IEEE Transactions on Circuits Syst. for Video Technol. 22, 1649–1668 (2012).
[Crossref]

Tamai, J.

H. Yoshikawa and J. Tamai, “Holographic image compression by motion picture coding,” Proc. SPIE2652, Practical Holography X (1996).

Wakunami, K.

T. Senoh, K. Wakunami, Y. Ichihashi, H. Sasaki, R. Oi, and K. Yamamoto, “Multiview image and depth map coding for holographic tv system,” Opt. Eng. 53, 112302 (2014).
[Crossref]

Wang, H.-Z.

Wedi, T.

J. Ostermann, J. Bormans, P. List, D. Marpe, M. Narroschke, F. Pereira, T. Stockhammer, and T. Wedi, “Video coding with H.264/AVC: tools, performance, and complexity,” IEEE Circuits Syst. Mag. 4, 7–28 (2004).
[Crossref]

Wiegand, T.

G. J. Sullivan, J. R. Ohm, W. J. Han, and T. Wiegand, “Overview of the high efficiency video coding (HEVC) standard,” IEEE Transactions on Circuits Syst. for Video Technol. 22, 1649–1668 (2012).
[Crossref]

Xing, Y.

Y. Xing, B. Pesquet-Popescu, and F. Dufaux, “Compression of computer generated phase-shifting hologram sequence using AVC and HEVC,” Proc. SPIE8856, Applications of Digital Image Processing XXXVI (2009).

F. Dufaux, Y. Xing, B. Pesquet-Popescu, and P. Schelkens, “Compression of digital holographic data: an overview,” Proc. SPIE9599, Applications of Digital Image Processing XXXVIII (2015).

Yamamoto, K.

T. Senoh, K. Wakunami, Y. Ichihashi, H. Sasaki, R. Oi, and K. Yamamoto, “Multiview image and depth map coding for holographic tv system,” Opt. Eng. 53, 112302 (2014).
[Crossref]

Yoo, J.-S.

Yoshikawa, H.

H. Yoshikawa and J. Tamai, “Holographic image compression by motion picture coding,” Proc. SPIE2652, Practical Holography X (1996).

Zhao, L.

Appl. Opt. (2)

Computer (1)

C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38, 46–53 (2005).
[Crossref]

IEEE Circuits Syst. Mag. (1)

J. Ostermann, J. Bormans, P. List, D. Marpe, M. Narroschke, F. Pereira, T. Stockhammer, and T. Wedi, “Video coding with H.264/AVC: tools, performance, and complexity,” IEEE Circuits Syst. Mag. 4, 7–28 (2004).
[Crossref]

IEEE Transactions on Circuits Syst. for Video Technol. (1)

G. J. Sullivan, J. R. Ohm, W. J. Han, and T. Wiegand, “Overview of the high efficiency video coding (HEVC) standard,” IEEE Transactions on Circuits Syst. for Video Technol. 22, 1649–1668 (2012).
[Crossref]

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

Opt. Eng. (1)

T. Senoh, K. Wakunami, Y. Ichihashi, H. Sasaki, R. Oi, and K. Yamamoto, “Multiview image and depth map coding for holographic tv system,” Opt. Eng. 53, 112302 (2014).
[Crossref]

Opt. Express (1)

Proc. IEEE (1)

V. M. Bove, “Display holography’s digital second act,” Proc. IEEE 100, 918–928 (2012).
[Crossref]

Signal Process. Image Commun. (1)

Y.-H. Seo, H.-J. Choi, and D.-W. Kim, “3d scanning-based compression technique for digital hologram video,” Signal Process. Image Commun. 22, 144–156 (2007).
[Crossref]

Other (9)

Y. Xing, B. Pesquet-Popescu, and F. Dufaux, “Compression of computer generated phase-shifting hologram sequence using AVC and HEVC,” Proc. SPIE8856, Applications of Digital Image Processing XXXVI (2009).

F. Dufaux, Y. Xing, B. Pesquet-Popescu, and P. Schelkens, “Compression of digital holographic data: an overview,” Proc. SPIE9599, Applications of Digital Image Processing XXXVIII (2015).

E. Darakis and T. J. Naughton, “Compression of digital hologram sequences using MPEG-4,” Proc. SPIE7358, Holography: Advances and Modern Trends (2009).

H. Yoshikawa and J. Tamai, “Holographic image compression by motion picture coding,” Proc. SPIE2652, Practical Holography X (1996).

A. Gilles, P. Gioia, R. Cozot, and L. Morin, “Fast generation of complex modulation video holograms using temporal redundancy compression and hybrid point-source/wave-field approaches,” Proc. SPIE2652, Applications of Digital Image Processing XXXVIII (2015).

J. J. Healy, M. A. Kutay, H. M. Ozaktas, and J. T. Sheridan, Linear Canonical Transforms (Springer, 2016).
[Crossref]

M. de Gosson, The Wigner Transform, Advanced textbooks in mathematics(World Scientific Publishing Company Pte Limited, 2017).
[Crossref]

“High Efficiency Video Coding (HEVC) reference software,” HM-15.0 (Revision 4879), https://hevc.hhi.fraunhofer.de/svn/svn_HEVCSoftware/branches/HM-dev/ . [retrieved 31 March 2017].

G. Bjontegaard, Calculation of average PSNR differences between RD-curves, ITU-VCEG (2001).

Supplementary Material (1)

NameDescription
» Visualization 1       Side-by-side comparison of a compressed holographic video of Venus, reconstructed and backpropagated to the center of the mesh. From left to right, we show the standard HEVC-intra, proposed motion compensation model, and the reference uncompressed vi

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1 Illustration of the virtual setup for generating the video of the moving Venus head object, intended for display with a planar reference wave.
Fig. 2
Fig. 2 Simplified diagram of the TF space intersection of a motion compensated 1D hologram (Frame 2) using an ACT w.r.t. another frame (Frame 1). The surface area of the intersection is a good indication of the mutual information between the frames, thereby predicting the efficacy of the proposed motion compensation algorithm.
Fig. 3
Fig. 3 (a) 3D model of the Venus head, (b) the associated point cloud, (c) amplitude of backpropagated hologram generated from the pointcloud with occlusion. The model of Venus is courtesy of Direct Dimensions Inc. (Head sculpture).
Fig. 4
Fig. 4 Diagram of a CGH contribution of a single occluded PSF using ray-tracing. This process is repeated for every PSF and summed over to obtain the hologram frame.
Fig. 5
Fig. 5 Amplitude of several reconstructed hologram frames from the tested reference video sequence, backpropagated to the center of the mesh.
Fig. 6
Fig. 6 PSNR as a function of the allocated rate for all six codec configurations. The proposed motion compensation outperforms local in-plane motion compensation.

Tables (2)

Tables Icon

Algorithm 1 Holographic video compression with global motion compensation.

Tables Icon

Table 1 Compression results with the BD-PSNR improvements (in dB) w.r.t. to the default HEVC configuration for video compression in the range of 0.125 to 2 bpp.

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

S Sp ( 2 n , ) SL ( 2 n , ) S T JS = J
J = ( 0 n I n I n 0 n ) a n d S = ( A B C D ) ,
S [ f ] ( x ˜ ) : = | i B | 1 2 n e i π ( x ˜ D B 1 x ˜ 2 x B 1 x ˜ + x B 1 A x ) f ( x ) d x
A ( 2 n , ) = Sp ( 2 n , ) 2 n
{ S , b } : 2 n 2 n by p Sp + b ,
{ S , b } [ f ] ( x ˜ ) : = e 2 π i b ω x ˜ | i B | 1 2 n e i π ( x ˜ D B 1 x ˜ 2 x B 1 x ˜ + x B 1 A x ) f ( x b x ) d x
{ S , b } [ f ] ( x ) : = | A | 1 2 e i π ( x C A 1 x + 2 b ω x ) f ( x b x ) .
u ^ ( ξ , η ) = 2 u ( x , y ) e 2 π i ( x ξ + y η ) d x d y
R x , y ( θ x , θ y ) : u ^ ( ξ , η ) u ^ ( ξ θ x , η θ y ) u ( x , y ) u ( x , y ) e 2 π i ( x θ x + y θ y )
R z ( θ z ) = ( R 2 0 2 0 2 R 2 ) , R 2 = ( cos ( θ z ) sin ( θ z ) sin ( θ z ) cos ( θ z ) ) .
T x , y ( t x , t y ) : u ( x , y ) u ( x t x , y t y ) u ^ ( ξ , η ) u ^ ( ξ , η ) e 2 π i ( t x ξ + t y η )
T z ( t z ) : u ^ ( ξ , η ) u ^ ( ξ , η ) e 2 π i λ t z ( ξ 2 + η 2 )
T z ( t z ) = ( I 2 2 t z λ I 2 0 2 I 2 ) .
x ˜ = R z ( ψ ) T z ( d ) x + b
I = Vol ( C T ( C ) ) Vol ( C ) [ 0 , 1 ] ,
U ( x , y ) = j = 1 N a j r j exp ( i 2 π λ r j )
PSNR = 1 N i = 1 N 10 log 10 ( max ( | R i | ) 2 mean ( | R i X i | 2 ) )

Metrics