Abstract

The interocular affine similarity of three-dimensional scenes is investigated and a novel accelerated reconfiguration algorithm for intermediate-view polygon computer-generated holograms based on interocular affine similarity is proposed. We demonstrate by using the numerical simulations of full-color polygon computer-generation holograms that the proposed intermediate view reconfiguration algorithm is particularly useful for the computation of wide-viewing angle polygon computer-generated holograms.

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

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    [Crossref]
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    [Crossref] [PubMed]
  4. H. Kang, E. Stoykova, and H. Yoshikawa, “Fast phase-added stereogram algorithm for generation of photorealistic 3D content,” Appl. Opt. 55(3), A135–A143 (2016).
    [Crossref] [PubMed]
  5. 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(19), 19504–19509 (2010).
    [Crossref] [PubMed]
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  10. T. Ichikawa, T. Yoneyama, and Y. Sakamoto, “CGH calculation with the ray tracing method for the Fourier transform optical system,” Opt. Express 21(26), 32019–32031 (2013).
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  13. J. Roh, K. Kim, E. Moon, S. Kim, B. Yang, J. Hahn, and H. Kim, “Full-color holographic projection display system featuring an achromatic Fourier filter,” Opt. Express 25(13), 14774–14782 (2017).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  29. J. Cho, J. Hahn, and H. Kim, “Fast reconfiguration algorithm of computer generated holograms for adaptive view direction change in holographic three-dimensional display,” Opt. Express 20(27), 28282–28291 (2012).
    [Crossref] [PubMed]
  30. D. Im, J. Cho, J. Hahn, B. Lee, and H. Kim, “Accelerated synthesis algorithm of polygon computer-generated holograms,” Opt. Express 23(3), 2863–2871 (2015).
    [Crossref] [PubMed]
  31. Y. Pan, Y. Wang, J. Liu, X. Li, and J. Jia, “Fast polygon-based method for calculating computer-generated holograms in three-dimensional display,” Appl. Opt. 52(1), A290–A299 (2013).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  34. H. Kim, J. Hahn, and B. Lee, “Image volume analysis of omnidirectional parallax regular-polyhedron three-dimensional displays,” Opt. Express 17(8), 6389–6396 (2009).
    [Crossref] [PubMed]
  35. Y. Lim, K. Hong, H. Kim, H.-E. Kim, E.-Y. Chang, S. Lee, T. Kim, J. Nam, H.-G. Choo, J. Kim, and J. Hahn, “360-degree tabletop electronic holographic display,” Opt. Express 24(22), 24999–25009 (2016).
    [Crossref] [PubMed]
  36. T. Inoue and Y. Takaki, “Table screen 360-degree holographic display using circular viewing-zone scanning,” Opt. Express 23(5), 6533–6542 (2015).
    [Crossref] [PubMed]
  37. Y. Takaki and S. Uchida, “Table screen 360-degree three-dimensional display using a small array of high-speed projectors,” Opt. Express 20(8), 8848–8861 (2012).
    [Crossref] [PubMed]
  38. T. Kakue, T. Nishitsuji, T. Kawashima, K. Suzuki, T. Shimobaba, and T. Ito, “Aerial projection of three-dimensional motion pictures by electro-holography and parabolic mirrors,” Sci. Rep. 5(1), 11750 (2015).
    [Crossref] [PubMed]

2018 (2)

2017 (6)

2016 (2)

2015 (5)

2014 (3)

2013 (4)

2012 (4)

2011 (3)

2010 (2)

2009 (2)

2008 (2)

2006 (1)

2005 (1)

1996 (1)

Ahrenberg, L.

Benzie, P.

Blinder, D.

Cao, L.

Chang, E.-Y.

Chen, N.

Cho, J.

Choi, H.-J.

Choo, H.-G.

Dorsch, R. G.

Ferreira, C.

Hahn, J.

J. Roh, K. Kim, E. Moon, S. Kim, B. Yang, J. Hahn, and H. Kim, “Full-color holographic projection display system featuring an achromatic Fourier filter,” Opt. Express 25(13), 14774–14782 (2017).
[Crossref] [PubMed]

Y. Lim, K. Hong, H. Kim, H.-E. Kim, E.-Y. Chang, S. Lee, T. Kim, J. Nam, H.-G. Choo, J. Kim, and J. Hahn, “360-degree tabletop electronic holographic display,” Opt. Express 24(22), 24999–25009 (2016).
[Crossref] [PubMed]

D. Im, J. Cho, J. Hahn, B. Lee, and H. Kim, “Accelerated synthesis algorithm of polygon computer-generated holograms,” Opt. Express 23(3), 2863–2871 (2015).
[Crossref] [PubMed]

D. Im, E. Moon, Y. Park, D. Lee, J. Hahn, and H. Kim, “Phase-regularized polygon computer-generated holograms,” Opt. Lett. 39(12), 3642–3645 (2014).
[Crossref] [PubMed]

W. Lee, D. Im, J. Paek, J. Hahn, and H. Kim, “Semi-analytic texturing algorithm for polygon computer-generated holograms,” Opt. Express 22(25), 31180–31191 (2014).
[Crossref] [PubMed]

J. Cho, J. Hahn, and H. Kim, “Fast reconfiguration algorithm of computer generated holograms for adaptive view direction change in holographic three-dimensional display,” Opt. Express 20(27), 28282–28291 (2012).
[Crossref] [PubMed]

J. Hong, Y. Kim, H.-J. Choi, J. Hahn, J.-H. Park, H. Kim, S.-W. Min, N. Chen, and B. Lee, “Three-dimensional display technologies of recent interest: principles, status, and issues,” Appl. Opt. 50(34), H87–H115 (2011).
[Crossref] [PubMed]

H. Kim, J. Hahn, and B. Lee, “Image volume analysis of omnidirectional parallax regular-polyhedron three-dimensional displays,” Opt. Express 17(8), 6389–6396 (2009).
[Crossref] [PubMed]

J. Hahn, H. Kim, Y. Lim, G. Park, and B. Lee, “Wide viewing angle dynamic holographic stereogram with a curved array of spatial light modulators,” Opt. Express 16(16), 12372–12386 (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(19), D117–D127 (2008).
[Crossref] [PubMed]

Hong, J.

Hong, K.

Ichihashi, Y.

T. Senoh, Y. Ichihashi, R. Oi, H. Sasaki, and K. Yamamoto, “Study on a holographic TV system based on multi-view images and depth maps,” Proc. SPIE 8644, 86440A (2013).

T. Shimobaba, T. Ito, N. Masuda, Y. Ichihashi, and N. Takada, “Fast calculation of computer-generated-hologram on AMD HD5000 series GPU and OpenCL,” Opt. Express 18(10), 9955–9960 (2010).
[Crossref] [PubMed]

Ichikawa, T.

Im, D.

Inoue, T.

Ito, T.

Ji, Y.-M.

Jia, J.

Jiao, S.

Jin, G.

Kakue, T.

T. Kakue, T. Nishitsuji, T. Kawashima, K. Suzuki, T. Shimobaba, and T. Ito, “Aerial projection of three-dimensional motion pictures by electro-holography and parabolic mirrors,” Sci. Rep. 5(1), 11750 (2015).
[Crossref] [PubMed]

Kang, H.

Kawashima, T.

T. Kakue, T. Nishitsuji, T. Kawashima, K. Suzuki, T. Shimobaba, and T. Ito, “Aerial projection of three-dimensional motion pictures by electro-holography and parabolic mirrors,” Sci. Rep. 5(1), 11750 (2015).
[Crossref] [PubMed]

Kim, D.-W.

Kim, E.-S.

Kim, H.

J. Roh, K. Kim, E. Moon, S. Kim, B. Yang, J. Hahn, and H. Kim, “Full-color holographic projection display system featuring an achromatic Fourier filter,” Opt. Express 25(13), 14774–14782 (2017).
[Crossref] [PubMed]

Y. Lim, K. Hong, H. Kim, H.-E. Kim, E.-Y. Chang, S. Lee, T. Kim, J. Nam, H.-G. Choo, J. Kim, and J. Hahn, “360-degree tabletop electronic holographic display,” Opt. Express 24(22), 24999–25009 (2016).
[Crossref] [PubMed]

D. Im, J. Cho, J. Hahn, B. Lee, and H. Kim, “Accelerated synthesis algorithm of polygon computer-generated holograms,” Opt. Express 23(3), 2863–2871 (2015).
[Crossref] [PubMed]

D. Im, E. Moon, Y. Park, D. Lee, J. Hahn, and H. Kim, “Phase-regularized polygon computer-generated holograms,” Opt. Lett. 39(12), 3642–3645 (2014).
[Crossref] [PubMed]

W. Lee, D. Im, J. Paek, J. Hahn, and H. Kim, “Semi-analytic texturing algorithm for polygon computer-generated holograms,” Opt. Express 22(25), 31180–31191 (2014).
[Crossref] [PubMed]

J. Cho, J. Hahn, and H. Kim, “Fast reconfiguration algorithm of computer generated holograms for adaptive view direction change in holographic three-dimensional display,” Opt. Express 20(27), 28282–28291 (2012).
[Crossref] [PubMed]

J. Hong, Y. Kim, H.-J. Choi, J. Hahn, J.-H. Park, H. Kim, S.-W. Min, N. Chen, and B. Lee, “Three-dimensional display technologies of recent interest: principles, status, and issues,” Appl. Opt. 50(34), H87–H115 (2011).
[Crossref] [PubMed]

H. Kim, J. Hahn, and B. Lee, “Image volume analysis of omnidirectional parallax regular-polyhedron three-dimensional displays,” Opt. Express 17(8), 6389–6396 (2009).
[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(19), D117–D127 (2008).
[Crossref] [PubMed]

J. Hahn, H. Kim, Y. Lim, G. Park, and B. Lee, “Wide viewing angle dynamic holographic stereogram with a curved array of spatial light modulators,” Opt. Express 16(16), 12372–12386 (2008).
[Crossref] [PubMed]

Kim, H.-E.

Kim, H.-J.

Kim, J.

Kim, J.-M.

Kim, K.

Kim, N.

Kim, S.

Kim, S.-B.

Kim, S.-C.

Kim, S.-H.

Kim, T.

Kim, Y.

Ko, S.-B.

Kong, D.

Lee, B.

Lee, D.

Lee, S.

Lee, W.

Li, B.

Li, G.

Li, X.

Lim, Y.

Liu, J.

Lohman, A. W.

Magnor, M.

Masuda, N.

Matsushima, K.

Mendlovic, D.

Min, S.-W.

Moon, E.

Nakahara, S.

Nakayama, H.

Nam, J.

Nishitsuji, T.

T. Kakue, T. Nishitsuji, T. Kawashima, K. Suzuki, T. Shimobaba, and T. Ito, “Aerial projection of three-dimensional motion pictures by electro-holography and parabolic mirrors,” Sci. Rep. 5(1), 11750 (2015).
[Crossref] [PubMed]

Oi, R.

T. Senoh, Y. Ichihashi, R. Oi, H. Sasaki, and K. Yamamoto, “Study on a holographic TV system based on multi-view images and depth maps,” Proc. SPIE 8644, 86440A (2013).

Oikawa, M.

Okada, N.

Paek, J.

Pan, Y.

Park, G.

Park, J.-H.

Park, Y.

Roh, J.

Sakamoto, Y.

Sasaki, H.

T. Senoh, Y. Ichihashi, R. Oi, H. Sasaki, and K. Yamamoto, “Study on a holographic TV system based on multi-view images and depth maps,” Proc. SPIE 8644, 86440A (2013).

Schelkens, P.

Senoh, T.

T. Senoh, Y. Ichihashi, R. Oi, H. Sasaki, and K. Yamamoto, “Study on a holographic TV system based on multi-view images and depth maps,” Proc. SPIE 8644, 86440A (2013).

Seo, Y.-H.

Shimobaba, T.

Shiraki, A.

Sonobe, N.

Stoykova, E.

Suzuki, K.

T. Kakue, T. Nishitsuji, T. Kawashima, K. Suzuki, T. Shimobaba, and T. Ito, “Aerial projection of three-dimensional motion pictures by electro-holography and parabolic mirrors,” Sci. Rep. 5(1), 11750 (2015).
[Crossref] [PubMed]

Symeonidou, A.

Takada, N.

Takaki, Y.

Tsuchiyama, Y.

Uchida, S.

Wakunami, K.

Wang, Y.

Watson, J.

Yamaguchi, K.

Yamaguchi, M.

Yamamoto, K.

T. Senoh, Y. Ichihashi, R. Oi, H. Sasaki, and K. Yamamoto, “Study on a holographic TV system based on multi-view images and depth maps,” Proc. SPIE 8644, 86440A (2013).

Yang, B.

Yeom, H.-J.

Yeom, J.

Yoneyama, T.

Yoo, J.-S.

Yoshikawa, H.

Zalevsky, Z.

Zhang, H.

Zhao, Y.

Zhuang, Z.

Zou, W.

Appl. Opt. (9)

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

T. Ichikawa, K. Yamaguchi, and Y. Sakamoto, “Realistic expression for full-parallax computer-generated holograms with the ray-tracing method,” Appl. Opt. 52(1), A201–A209 (2013).
[Crossref] [PubMed]

J. Hong, Y. Kim, H.-J. Choi, J. Hahn, J.-H. Park, H. Kim, S.-W. Min, N. Chen, and B. Lee, “Three-dimensional display technologies of recent interest: principles, status, and issues,” Appl. Opt. 50(34), H87–H115 (2011).
[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(19), D117–D127 (2008).
[Crossref] [PubMed]

K. Matsushima and N. Sonobe, “Full-color digitized holography for large-scale holographic 3D imaging of physical and nonphysical objects,” Appl. Opt. 57(1), A150–A156 (2018).
[Crossref] [PubMed]

Y. Pan, Y. Wang, J. Liu, X. Li, and J. Jia, “Fast polygon-based method for calculating computer-generated holograms in three-dimensional display,” Appl. Opt. 52(1), A290–A299 (2013).
[Crossref] [PubMed]

K. Matsushima, “Computer-generated holograms for electro-holography,” Appl. Opt. 44, 4607–4614 (2005).
[Crossref] [PubMed]

K. Matsushima and S. Nakahara, “Extremely high-definition full-parallax computer-generated hologram created by the polygon-based method,” Appl. Opt. 48(34), H54–H63 (2009).
[Crossref] [PubMed]

N. Takada, T. Shimobaba, H. Nakayama, A. Shiraki, N. Okada, M. Oikawa, N. Masuda, and T. Ito, “Fast high-resolution computer-generated hologram computation using multiple graphics processing unit cluster system,” Appl. Opt. 51(30), 7303–7307 (2012).
[Crossref] [PubMed]

Chin. Opt. Lett. (1)

J. Inform. Displ. (1)

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

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

Opt. Express (23)

J. Hahn, H. Kim, Y. Lim, G. Park, and B. Lee, “Wide viewing angle dynamic holographic stereogram with a curved array of spatial light modulators,” Opt. Express 16(16), 12372–12386 (2008).
[Crossref] [PubMed]

H. Kim, J. Hahn, and B. Lee, “Image volume analysis of omnidirectional parallax regular-polyhedron three-dimensional displays,” Opt. Express 17(8), 6389–6396 (2009).
[Crossref] [PubMed]

Y. Lim, K. Hong, H. Kim, H.-E. Kim, E.-Y. Chang, S. Lee, T. Kim, J. Nam, H.-G. Choo, J. Kim, and J. Hahn, “360-degree tabletop electronic holographic display,” Opt. Express 24(22), 24999–25009 (2016).
[Crossref] [PubMed]

T. Inoue and Y. Takaki, “Table screen 360-degree holographic display using circular viewing-zone scanning,” Opt. Express 23(5), 6533–6542 (2015).
[Crossref] [PubMed]

Y. Takaki and S. Uchida, “Table screen 360-degree three-dimensional display using a small array of high-speed projectors,” Opt. Express 20(8), 8848–8861 (2012).
[Crossref] [PubMed]

T. Shimobaba, T. Ito, N. Masuda, Y. Ichihashi, and N. Takada, “Fast calculation of computer-generated-hologram on AMD HD5000 series GPU and OpenCL,” Opt. Express 18(10), 9955–9960 (2010).
[Crossref] [PubMed]

Y.-H. Seo, H.-J. Choi, J.-S. Yoo, and D.-W. Kim, “Cell-based hardware architecture for full-parallel generation algorithm of digital holograms,” Opt. Express 19(9), 8750–8761 (2011).
[Crossref] [PubMed]

Y. Tsuchiyama and K. Matsushima, “Full-color large-scaled computer-generated holograms using RGB color filters,” Opt. Express 25(3), 2016–2030 (2017).
[Crossref] [PubMed]

J. Cho, J. Hahn, and H. Kim, “Fast reconfiguration algorithm of computer generated holograms for adaptive view direction change in holographic three-dimensional display,” Opt. Express 20(27), 28282–28291 (2012).
[Crossref] [PubMed]

D. Im, J. Cho, J. Hahn, B. Lee, and H. Kim, “Accelerated synthesis algorithm of polygon computer-generated holograms,” Opt. Express 23(3), 2863–2871 (2015).
[Crossref] [PubMed]

S.-B. Ko and J.-H. Park, “Speckle reduction using angular spectrum interleaving for triangular mesh based computer generated hologram,” Opt. Express 25(24), 29788–29797 (2017).
[Crossref] [PubMed]

S.-C. Kim, J.-M. Kim, and E.-S. Kim, “Effective memory reduction of the novel look-up table with one-dimensional sub-principle fringe patterns in computer-generated holograms,” Opt. Express 20(11), 12021–12034 (2012).
[Crossref] [PubMed]

J.-H. Park, S.-B. Kim, H.-J. Yeom, H.-J. Kim, H. Zhang, B. Li, Y.-M. Ji, S.-H. Kim, and S.-B. Ko, “Continuous shading and its fast update in fully analytic triangular-mesh-based computer generated hologram,” Opt. Express 23(26), 33893–33901 (2015).
[Crossref] [PubMed]

W. Lee, D. Im, J. Paek, J. Hahn, and H. Kim, “Semi-analytic texturing algorithm for polygon computer-generated holograms,” Opt. Express 22(25), 31180–31191 (2014).
[Crossref] [PubMed]

L. Ahrenberg, P. Benzie, M. Magnor, and J. Watson, “Computer generated holography using parallel commodity graphics hardware,” Opt. Express 14(17), 7636–7641 (2006).
[Crossref] [PubMed]

T. Ichikawa, T. Yoneyama, and Y. Sakamoto, “CGH calculation with the ray tracing method for the Fourier transform optical system,” Opt. Express 21(26), 32019–32031 (2013).
[Crossref] [PubMed]

K. Wakunami and M. Yamaguchi, “Calculation for computer generated hologram using ray-sampling plane,” Opt. Express 19(10), 9086–9101 (2011).
[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]

J. Roh, K. Kim, E. Moon, S. Kim, B. Yang, J. Hahn, and H. Kim, “Full-color holographic projection display system featuring an achromatic Fourier filter,” Opt. Express 25(13), 14774–14782 (2017).
[Crossref] [PubMed]

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(19), 19504–19509 (2010).
[Crossref] [PubMed]

T. Shimobaba and T. Ito, “Fast generation of computer-generated holograms using wavelet shrinkage,” Opt. Express 25(1), 77–87 (2017).
[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(1), 112–123 (2017).
[Crossref] [PubMed]

A. Symeonidou, D. Blinder, and P. Schelkens, “Colour computer-generated holography for point clouds utilizing the Phong illumination model,” Opt. Express 26(8), 10282–10298 (2018).
[Crossref] [PubMed]

Opt. Lett. (1)

Proc. SPIE (1)

T. Senoh, Y. Ichihashi, R. Oi, H. Sasaki, and K. Yamamoto, “Study on a holographic TV system based on multi-view images and depth maps,” Proc. SPIE 8644, 86440A (2013).

Sci. Rep. (1)

T. Kakue, T. Nishitsuji, T. Kawashima, K. Suzuki, T. Shimobaba, and T. Ito, “Aerial projection of three-dimensional motion pictures by electro-holography and parabolic mirrors,” Sci. Rep. 5(1), 11750 (2015).
[Crossref] [PubMed]

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Figures (9)

Fig. 1
Fig. 1 Convergence and accommodation in the binocular visual perception system (a) global and local coordinates and (b) adaptive global coordinate system.
Fig. 2
Fig. 2 Image transport of an eye system (a) collinear transform and (b) rotational transform in object space and retina space.
Fig. 3
Fig. 3 (a) Schematic design of the computational simulation to verify the non-linear conversion relationship between two eyes, and (b) the retina image and non-linear grid map for the left and right eyes.
Fig. 4
Fig. 4 (a) Disparity in observed texture patterns at different locations. (b) The approximately linear relationship among the adjacent observation points.
Fig. 5
Fig. 5 (a) Schematic diagram for analyzing interocular similarity and (b) the comparison of two grids calculated by the exact and approximate method.
Fig. 6
Fig. 6 Analysis results for interocular similarity: (a) two comparison grids calculated by the exact and approximate methods, (b) the RMS error graph for total area, and (c) the RMS error for the interior area of the triangle.
Fig. 7
Fig. 7 The observation image of a full-color CGH calculated using (a) the exact method and (b) the approximate method.
Fig. 8
Fig. 8 The elapsed time for calculating the CGHs using (a) the exact method and (b) the approximate method.
Fig. 9
Fig. 9 Verifying the properties of a 3D holographic image with the accommodation effect: (a) when focusing on the cube and (b) when focusing on the checker board.

Equations (55)

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N( x 0 y 0 z 0 )=( x c y c z c )+( cosϕsinθ sinϕsinθ cosθ )t,
t= ( x 0 x c ) 2 + ( y 0 y c ) 2 + ( z 0 z c ) 2 ,
( cosθ,sinθ )=( ( z 0 z c )/t , ( x 0 x c ) 2 + ( y 0 y c ) 2 /t ),
( cosϕ,sinϕ )=( ( x 0 x c )/ ( tsinθ ) , ( y 0 y c )/ ( tsinθ ) ).
v=( cosϕcosθsinτsinϕcosτ,sinϕcosθsinτ+cosϕcosτ,sinθsinτ ),
f= d e t/ ( d e +t ) ,
( x y z )=( r 11 r 12 r 13 r 21 r 22 r 23 r 31 r 32 r 33 )( x x 0 y y 0 z z 0 )+( x 0 y 0 z 0 ),
( r 11 r 12 r 13 r 21 r 22 r 23 r 31 r 32 r 33 )=( cosτ sinτ 0 sinτ cosτ 0 0 0 1 )( cosθ 0 sinθ 0 1 0 sinθ 0 cosθ )( cosϕ sinϕ 0 sinϕ cosϕ 0 0 0 1 ),
( x 0 , y 0 , z 0 )=( 0,0, x 0 2 + y 0 2 + z 0 2 ).
1/f =1/ ( d 1 z c ) +1/ d 2 .
( u,v,w )=( D 2 x/ D 1 , D 2 y/ D 1 , D 2 d 2 ).
cosϕsinθ( x x c )+sinϕsinθ( y y c )+cosθ( z z c )=0,
cos ϕ r sin θ r ( u u c )+sin ϕ r sin θ r ( v v c )+cos θ r ( w w c )=0,
( x y z )=( 0u 0v d 1 ( d 1 + d 2 +w ) )t+( u v d 1 + d 2 +w ),
t= ( x c u )cosϕsinθ+( y c v )sinϕsinθ+( z c w d 1 d 2 )cosθ x r cosϕsinθ y r sinϕsinθ+[ d 1 ( d 1 + d 2 + z r ) ]cosθ .
( u v w )= 1 ( 1t ) ( x y z d 1 )+( 0 0 d 2 )=s( x y z d 1 )+( 0 0 d 2 ),
s= u c cos ϕ r sin θ r + v c sin ϕ r sin θ r +( d 2 + w c )cos θ r cos ϕ r sin θ r x+sin ϕ r sin θ r y+( z d 1 )cos θ r .
( x' y' z' )=( cosθcosϕ cosθsinϕ sinθ sinϕ cosϕ 0 sinθcosϕ sinθsinϕ cosθ )( x x c y y c z z c ).
( u' v' w' )=( cos θ r cos ϕ r cos θ r sin ϕ r sin θ r sin ϕ r cos ϕ r 0 sin θ r cos ϕ r sin θ r sin ϕ r cos θ r )( u u c v v c w w c ).
( x' y' z' )=( cosθcosϕ cosθsinϕ sinθ sinϕ cosϕ 0 sinθcosϕ sinθsinϕ cosθ )[ ( 0u 0v d 1 ( d 1 + d 2 +w ) )t+( u x c v y c d 1 + d 2 +w z c ) ],
( u v w )=( cos ϕ r cos θ r sin θ r cos ϕ r sin θ r sin ϕ r cos θ r cos ϕ r sin ϕ r sin θ r sin θ r 0 cos θ r )( u v w )+( u c v c w c ).
( u v w )=( cos θ r cos ϕ r cos θ r sin ϕ r sin θ r sin ϕ r cos ϕ r 0 sin θ r cos ϕ r sin θ r sin ϕ r cos θ r )( u u c v v c w w c ),
( u v w )=s{ ( cos θ r cos ϕ r cos θ r sin ϕ r sin θ r sin ϕ r cos ϕ r 0 sin θ r cos ϕ r sin θ r sin ϕ r cos θ r ) 1 ( x' y' z' )+( x c y c z c ) }+( 0 0 d 2 s d 1 ).
( x n y n z n )=Gto L n RLtoG( x' y' z' ).
( a 11 a 12 a 21 a 22 )( u ' ref,1 v ' ref,1 )+( b 1 b 2 )=( u ' adj,1 v ' adj,1 ),
( a 11 a 12 a 21 a 22 )( u ' ref,2 v ' ref,2 )+( b 1 b 2 )=( u ' adj,2 v ' adj,2 ),
( a 11 a 12 a 21 a 22 )( u ' ref,3 v ' ref,3 )+( b 1 b 2 )=( u ' adj,3 v ' adj,3 ).
( u ' ref,1 v ' ref,1 0 0 1 0 0 0 u ' ref,1 v ' ref,1 0 1 u ' ref,2 v ' ref,2 0 0 1 0 0 0 u ' ref,2 v ' ref,2 0 1 u ' ref,3 v ' ref,3 0 0 1 0 0 0 u ' ref,3 v ' ref,3 0 1 )( a 11 a 12 a 21 a 22 b 1 b 2 )=( u ' adj,1 v ' adj,1 u ' adj,2 v ' adj,2 u ' adj,3 v ' adj,3 ),
( a 11 a 12 a 21 a 22 )( u ' ref v ' ref )+( b 1 b 2 )=( u ' adj v ' adj ).
F( u ' ref ,y ' ref )= A ref ( α ' ref ,β ' ref )exp [ j2π( α ' ref u ' ref +β ' ref v ' ref ) ]dα ' ref dβ ' ref ,
G( u ' adj ,v ' adj )= A adj ( α ' adj ,β ' adj )exp [ j2π( α ' adj u ' adj +β ' adj v ' adj ) ]dα ' adj dβ ' adj .
( u ' adj v ' adj )=( a 11 a 12 a 21 a 22 )( u ' ref v ' ref )+( b 1 b 2 ).
G( a 11 u ' ref + a 12 v ' ref + b 1 , a 21 u ' ref + a 22 v ' ref + b 2 ) = A adj ( α ' adj ,β ' adj ) ×exp{ j2π[ α ' adj ( a 11 u ' ref + a 12 v ' ref + b 1 )+β ' adj ( a 21 u ' ref + a 22 v ' ref + b 2 ) ] }dα ' adj dβ ' adj = { A adj ( α ' adj ,β ' adj )exp[ j2π( α ' adj b 1 +β ' adj b 2 ) ] } ×exp{ j2π[ ( a 11 α ' adj + a 21 β ' adj )u ' ref +( a 12 α ' adj + a 22 β ' adj )v ' ref ] }dα ' adj dβ ' adj .
( α ' ref β ' ref )=( a 11 α ' adj + a 21 β ' adj a 12 α ' adj + a 22 β ' adj ),
F( u ref , v ref ) = A ref ( α ' ref ,β ' ref )exp[ j2π( α ' ref u ' ref +β ' ref v ' ref ) ]dα ' ref dβ ' ref = A ref ( α ' ref ,β ' ref )exp[ j2π( α ' ref u ' ref +β ' ref v ' ref ) ]( a 11 a 22 a 12 a 21 )dα ' adj dβ ' adj =G( a 11 u ' ref + a 12 v ' ref + b 1 , a 21 u ' ref + a 22 v ' ref + b 2 ) = A adj ( α ' adj ,β ' adj )exp[ j2π( α ' adj b 1 +β ' adj b 2 ) ]exp[ j2π( α ' ref u ' ref +β ' ref v ' ref ) ]dα ' adj dβ ' adj .
A adj ( α ' adj ,β ' adj ) =exp[ j2π( α ' adj b 1 +β ' adj b 2 ) ]( a 11 a 22 a 12 a 21 ) A ref ( α ' ref ,β ' ref ) =exp[ j2π( α ' adj b 1 +β ' adj b 2 ) ]( a 11 a 22 a 12 a 21 ) A ref ( a 11 α ' adj + a 21 β ' adj , a 12 α ' adj + a 22 β ' adj ).
G( u ' adj ,v ' adj ) = exp[ j2π( α ' adj b 1 +β ' adj b 2 ) ]( a 11 a 22 a 12 a 21 ) A ref ( a 11 α ' adj + a 21 β ' adj , a 12 α ' adj + a 22 β ' adj ) ×exp[ j2π( α ' adj u ' adj +β ' adj v ' adj ) ]dα ' adj dβ ' adj = A adj ( α ' adj ,β ' adj )exp[ j2π( α ' adj u ' adj +β ' adj v ' adj ) ]dα ' adj dβ ' adj .
( u adj v adj w adj )=( cos θ adj 0 sin θ adj 0 1 0 sin θ adj 0 cos θ adj )×( cos ϕ adj sin ϕ adj 0 sin ϕ adj cos ϕ adj 0 0 0 1 )( u adj u adj,c v adj v adj,c w adj w adj,c ) =( cos θ adj cos ϕ adj cos θ adj sin ϕ adj sin θ adj sin ϕ adj cos ϕ adj 0 sin θ adj cos ϕ adj sin θ adj sin ϕ adj cos θ adj )( u adj u adj,c v adj v adj,c w adj w adj,c ).
W( u adj , v adj ,0 ) = η 0 exp{ j2π[ α 0 ( u adj + u adj,c )+ β 0 ( u adj + u adj,c )+ γ 0 ' w adj,c ] } × A adj@L ( α adj , β adj )exp[ j2π( α adj u adj + β adj v adj ) ]d α adj d β adj = η 0 exp[ j2π( α 0 u adj,c + β 0 u adj,c + γ 0 ' w adj,c ) ] × A adj@L ( α adj α 0 , β adj β 0 )exp[ j2π( α adj u adj + β adj v adj ) ]d α adj d β adj .
W( u adj , v adj , w adj ) = η 0 exp[ j2π( α 0 u adj,c + β 0 v adj,c + γ 0 w adj,c ) ] × A adj@L ( α adj α 0 , β adj β 0 )exp[ j2π( α adj u adj + β adj v adj + γ adj w adj ) ]d α adj d β adj .
α adj ( α adj , β adj )=cos θ adj cos ϕ adj α adj +cos θ adj sin ϕ adj β adj sin θ adj γ adj ,
β adj ( α adj , β adj )=sin ϕ adj α adj +cos ϕ adj β adj ,
γ adj ( α adj , β adj )=sin θ adj cos ϕ adj α adj +sin θ adj sin ϕ adj β adj +cos θ adj γ adj .
d α adj ( α adj , β adj )d β adj ( α adj , β adj ) =| J |d α adj d β adj =| cos θ adj + sin θ adj ( α adj cos ϕ adj + β adj sin ϕ adj )/ γ adj |d α adj d β adj .
W( u adj , v adj , w adj ) = η 0 exp[ j2π( α 0 u adj,c + β 0 v adj,c + γ 0 w adj,c ) ] × A adj@L ( α adj ( α adj , β adj ) α 0 ( α 0 , β 0 ), β adj ( α adj , β adj ) β 0 ( α 0 , β 0 ) ) H( γ adj ( α adj , β adj ) ) ×exp{ j2π[ α adj ( u adj u adj,c )+ β adj ( v adj v adj,c )+ γ adj ( w adj w adj,c ) ] } ×| cos θ adj + sin θ adj ( α adj cos ϕ adj + β adj sin ϕ adj )/ γ adj |d α adj d β adj .
A adj@G ( α adj , β adj ) = η 0 exp[ j2π( α 0 u adj,c + β 0 v adj,c + γ 0 w adj,c ) ] × A adj@L ( α ' adj ( α adj , β adj )α ' 0 ( α 0 , β 0 ),β ' adj ( α adj, β adj )β ' 0 ( α o , β 0 ) ) ×H( γ ' adj ( α adj , β adj ) )exp{ j2π[ α adj ( u adj,c )+ β adj ( v adj,c )+ γ adj ( w adj,c ) ] } ×| cos θ adj + sin θ adj ( α adj cos ϕ adj + β adj sin ϕ adj )/ γ adj |.
A adj@L ( α ' adj ( α adj , β adj )α ' 0 ( α 0 , β 0 ),β ' adj ( α adj, β adj )β ' 0 ( α o , β 0 ) ) = A adj@L ( α' ' adj ,β' ' adj ) =exp[ j2π( α' ' adj b 1 +β' ' adj b 2 ) ]( a 11 a 22 a 12 a 21 ) × A ref ( a 11 α' ' adj + a 21 β' ' adj , a 12 α' ' adj + a 22 β' ' adj ),
α' ' adj =α ' adj ( α adj , β adj )α ' 0 ( α 0 , β 0 ),
β' ' adj =β ' adj ( α adj, β adj )β ' 0 ( α o , β 0 ).
A adj@G ( α adj , β adj ) = η 0 exp[ j2π( α 0 u adj,c + β 0 v adj,c + γ 0 w adj,c ) ]H( γ ' adj ( α adj , β adj ) ) ×exp{ j2π[ α adj ( u adj,c )+ β adj ( v adj,c )+ γ adj ( w adj,c ) ] } ×| cos θ r + sin θ r ( α r cos ϕ r + β r sin ϕ r )/ γ r |exp[ j2π( α' ' adj b 1 +β' ' adj b 2 ) ] ×( a 11 a 22 a 12 a 21 ) A ref ( a 11 α' ' adj + a 21 β' ' adj , a 12 α' ' adj + a 22 β' ' adj ).
( x n y n z n )=( cos θ n cos ϕ n cos θ n sin ϕ n sin θ n sin ϕ n cos ϕ n 0 sin θ n cos ϕ n sin θ n sin ϕ n cos θ n )( x n x nc y n y nc z n z nc )=Gto L n ( x n x nc y n y nc z n z nc ),
( x n y n z n )=Gto L n [ ( x n x n0 y n y n0 z n z n0 )+( x n0 x nc y n0 y nc z n0 z nc ) ],
( x n y n z n )=Gto L n [ R( x x 0 y y 0 z z 0 )+( x n0 x nc y n0 y nc z n0 z nc ) ].
( x x c y y c z z c )=( cosθcosϕ sinϕ cosϕsinθ sinϕcosθ cosϕ sinϕsinθ sinθ 0 cosθ )( x' y' z' )=LtoG( x' y' z' ).
( x n y n z n )=Gto L n { R[ LtoG( x' y' z' )+( x c x 0 y c y 0 z c z 0 ) ]+( x n0 x nc y n0 y nc z n0 z nc ) } =Gto L n RLtoG( x' y' z' )+Gto L n [ R( x c x 0 y c y 0 z c z 0 )+( x n0 x nc y n0 y nc z n0 z nc ) ] =Gto L n RLtoG( x' y' z' ),

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