Abstract

A fast calculation method to obtain the full-analytical frequency spectrum of a spatial triangle based on the three-dimensional (3D) affine transformation is presented. Computer-generated holograms (CGHs) of an object can then be generated rapidly using the angular spectrum for propagation. The derivation process in the theory, which has more preciseness, indicates a difference from previous methods based on affine transformations ([Appl. Opt. 47, 1567 (2008) Appl. Opt. 52, A290 (2013)]). The proposed method to achieve 3D transformation from an arbitrary triangle to a primitive triangle includes two steps: 3D rotation and 2D affine transformation. The overall transform matrix is given by the product of a rotation matrix and a 2D affine matrix. A modified back-face culling is also introduced based on exterior normal for correct occlusion relation. Several complex 3D objects are implemented successfully using the proposed method in numerical simulations and optical experiments. The resulting computation time demonstrates that the efficiency of the proposed method is enhanced as compared to that of previous works.

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

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References

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2018 (1)

2017 (1)

2016 (1)

2015 (2)

2014 (2)

2013 (8)

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]

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]

Y. Pan, Y. Wang, J. Liu, X. Li, J. Jia, and Z. Zhang, “Analytical brightness compensation algorithm for traditional polygon-based method in computer-generated holography,” Appl. Opt. 52(18), 4391–4399 (2013).
[Crossref] [PubMed]

B. Lee, “Three-dimensional display, past and present,” Phys. Today 66(4), 36–41 (2013).
[Crossref]

T. Ichikawa and Y. Sakamoto, “A rendering method of background reflections on a specular surface for CGH,” J. Phys. Conf. Ser. 415, 012044 (2013).
[Crossref]

P. W. M. Tsang and T.-C. Poon, “Review on theory and applications of wavefront recording plane framework in generation and processing of digital holograms,” Chin. Opt. Lett. 11(1), 18–24 (2013).

Y.-P. Zhang, J. Q. Zhang, W. Chen, P. Wang, S. Wu, and J. C. Li, “Fast Computer Generated Hologram Algorithm of Triangle Mesh Models,” Chin. J. Lasers 40(7), 0709001 (2013).
[Crossref]

Y.-P. Zhang, J. Q. Zhang, W. Chen, J. L. Zhang, P. Wang, and W. Xu, “Research on three-dimensional computer-generated holographic algorithm based on conformal geometry theory,” Opt. Commun. 309(22), 196–200 (2013).
[Crossref]

2011 (1)

2010 (2)

2009 (2)

2008 (3)

2005 (1)

2003 (3)

2002 (1)

Y. Sakamoto and T. Nagao, “A fast computational method for computer-generated Fourier hologram using patch model,” Electron. Commun. Jpn. Part 2 Electron 85(11), 16–24 (2002).
[Crossref]

2000 (1)

1993 (3)

T. Tommasi and B. Bianco, “Computer-generated holograms of tilted planes by a spatial frequency approach,” J. Opt. Soc. Am. A 10(2), 299–305 (1993).
[Crossref]

R. N. Bracewell, K.-Y. Chang, A. K. Jha, and Y.-H. Wang, “Affine theorem for two-dimensional fourier transform,” Electron. Lett. 29(3), 304 (1993).
[Crossref]

M. E. Lucente, P. St-Hilaire, S. A. Benton, D. Arias, and J. A. Watlington, “New approaches to holographic video,” Proc. SPIE 1732, 377–386 (1993).
[Crossref]

1992 (3)

A. D. Stein, Z. Wang, and J. S. Leigh, “Computer-generated holograms: A simplified ray-tracing approach,” Comput. Phys. 6(4), 389–392 (1992).
[Crossref]

M. E. Lucente, “Optimization of hologram computation for real-time display,” Proc. SPIE 1667(5), 772–774 (1992).

T. Tommasi and B. Bianco, “Frequency analysis of light diffraction between rotated planes,” Opt. Lett. 17(8), 556–558 (1992).
[Crossref] [PubMed]

1988 (1)

1981 (1)

S. Ganci, “Fourier diffraction through a tilted slit,” Eur. J. Phys. 2(3), 158–160 (1981).
[Crossref]

Ahrenberg, L.

Arias, D.

M. E. Lucente, P. St-Hilaire, S. A. Benton, D. Arias, and J. A. Watlington, “New approaches to holographic video,” Proc. SPIE 1732, 377–386 (1993).
[Crossref]

Benton, S. A.

M. E. Lucente, P. St-Hilaire, S. A. Benton, D. Arias, and J. A. Watlington, “New approaches to holographic video,” Proc. SPIE 1732, 377–386 (1993).
[Crossref]

Benzie, P.

Bianco, B.

Bracewell, R. N.

R. N. Bracewell, K.-Y. Chang, A. K. Jha, and Y.-H. Wang, “Affine theorem for two-dimensional fourier transform,” Electron. Lett. 29(3), 304 (1993).
[Crossref]

Chang, K.-Y.

R. N. Bracewell, K.-Y. Chang, A. K. Jha, and Y.-H. Wang, “Affine theorem for two-dimensional fourier transform,” Electron. Lett. 29(3), 304 (1993).
[Crossref]

Chen, B. C.

Chen, N.

Chen, W.

Y.-P. Zhang, J. Q. Zhang, W. Chen, J. L. Zhang, P. Wang, and W. Xu, “Research on three-dimensional computer-generated holographic algorithm based on conformal geometry theory,” Opt. Commun. 309(22), 196–200 (2013).
[Crossref]

Y.-P. Zhang, J. Q. Zhang, W. Chen, P. Wang, S. Wu, and J. C. Li, “Fast Computer Generated Hologram Algorithm of Triangle Mesh Models,” Chin. J. Lasers 40(7), 0709001 (2013).
[Crossref]

Choi, H.-J.

Dong, J. W.

Ferraro, P.

Finizio, A.

Frère, C.

Ganci, S.

S. Ganci, “Fourier diffraction through a tilted slit,” Eur. J. Phys. 2(3), 158–160 (1981).
[Crossref]

Hahn, J.

Hanák, I.

He, H. X.

Hong, J.

Ichikawa, T.

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]

T. Ichikawa and Y. Sakamoto, “A rendering method of background reflections on a specular surface for CGH,” J. Phys. Conf. Ser. 415, 012044 (2013).
[Crossref]

Igarashi, S.

Im, D.

Ito, T.

Janda, M.

Javidi, B.

Jha, A. K.

R. N. Bracewell, K.-Y. Chang, A. K. Jha, and Y.-H. Wang, “Affine theorem for two-dimensional fourier transform,” Electron. Lett. 29(3), 304 (1993).
[Crossref]

Jia, J.

Kim, E.-S.

Kim, H.

Kim, S.-C.

Kim, Y.

Kishk, S.

Kondoh, A.

K. Matsushima and A. Kondoh, “Wave optical algorithm for creating digitally synthetic holograms of three-dimensional surface objects,” Proc. SPIE 5005, 190–197 (2003).
[Crossref]

Kwon, M.-W.

Lee, B.

Lee, W.

Leigh, J. S.

A. D. Stein, Z. Wang, and J. S. Leigh, “Computer-generated holograms: A simplified ray-tracing approach,” Comput. Phys. 6(4), 389–392 (1992).
[Crossref]

Leseberg, D.

Li, J. C.

Y.-P. Zhang, J. Q. Zhang, W. Chen, P. Wang, S. Wu, and J. C. Li, “Fast Computer Generated Hologram Algorithm of Triangle Mesh Models,” Chin. J. Lasers 40(7), 0709001 (2013).
[Crossref]

Li, X.

Lin, S.-F.

Liu, J.

Liu, Y. Z.

Lucente, M. E.

M. E. Lucente, P. St-Hilaire, S. A. Benton, D. Arias, and J. A. Watlington, “New approaches to holographic video,” Proc. SPIE 1732, 377–386 (1993).
[Crossref]

M. E. Lucente, “Optimization of hologram computation for real-time display,” Proc. SPIE 1667(5), 772–774 (1992).

Magnor, M.

Masuda, N.

Matsushima, K.

Memmolo, P.

Min, S.-W.

Nagao, T.

Y. Sakamoto and T. Nagao, “A fast computational method for computer-generated Fourier hologram using patch model,” Electron. Commun. Jpn. Part 2 Electron 85(11), 16–24 (2002).
[Crossref]

Nakamura, T.

Näsänen, R.

Naughton, T. J.

Ogihara, Y.

Onural, L.

Paek, J.

Pan, Y.

Park, J.-H.

Paturzo, M.

Poon, T.-C.

P. W. M. Tsang and T.-C. Poon, “Fast generation of digital holograms based on warping of the wavefront recording plane,” Opt. Express 23(6), 7667–7673 (2015).
[Crossref] [PubMed]

P. W. M. Tsang and T.-C. Poon, “Review on theory and applications of wavefront recording plane framework in generation and processing of digital holograms,” Chin. Opt. Lett. 11(1), 18–24 (2013).

Pu, Y. Y.

Sakamoto, Y.

Sakata, H.

Schimmel, H.

Shimobaba, T.

Shiraki, A.

Stein, A. D.

A. D. Stein, Z. Wang, and J. S. Leigh, “Computer-generated holograms: A simplified ray-tracing approach,” Comput. Phys. 6(4), 389–392 (1992).
[Crossref]

St-Hilaire, P.

M. E. Lucente, P. St-Hilaire, S. A. Benton, D. Arias, and J. A. Watlington, “New approaches to holographic video,” Proc. SPIE 1732, 377–386 (1993).
[Crossref]

Sugie, T.

Takai, M.

Tommasi, T.

Tsang, P. W. M.

P. W. M. Tsang and T.-C. Poon, “Fast generation of digital holograms based on warping of the wavefront recording plane,” Opt. Express 23(6), 7667–7673 (2015).
[Crossref] [PubMed]

P. W. M. Tsang and T.-C. Poon, “Review on theory and applications of wavefront recording plane framework in generation and processing of digital holograms,” Chin. Opt. Lett. 11(1), 18–24 (2013).

Wang, H. Z.

Wang, P.

Y.-P. Zhang, J. Q. Zhang, W. Chen, P. Wang, S. Wu, and J. C. Li, “Fast Computer Generated Hologram Algorithm of Triangle Mesh Models,” Chin. J. Lasers 40(7), 0709001 (2013).
[Crossref]

Y.-P. Zhang, J. Q. Zhang, W. Chen, J. L. Zhang, P. Wang, and W. Xu, “Research on three-dimensional computer-generated holographic algorithm based on conformal geometry theory,” Opt. Commun. 309(22), 196–200 (2013).
[Crossref]

Wang, Y.

Wang, Y.-H.

R. N. Bracewell, K.-Y. Chang, A. K. Jha, and Y.-H. Wang, “Affine theorem for two-dimensional fourier transform,” Electron. Lett. 29(3), 304 (1993).
[Crossref]

Wang, Z.

A. D. Stein, Z. Wang, and J. S. Leigh, “Computer-generated holograms: A simplified ray-tracing approach,” Comput. Phys. 6(4), 389–392 (1992).
[Crossref]

Watlington, J. A.

M. E. Lucente, P. St-Hilaire, S. A. Benton, D. Arias, and J. A. Watlington, “New approaches to holographic video,” Proc. SPIE 1732, 377–386 (1993).
[Crossref]

Watson, J.

Wu, S.

Y.-P. Zhang, J. Q. Zhang, W. Chen, P. Wang, S. Wu, and J. C. Li, “Fast Computer Generated Hologram Algorithm of Triangle Mesh Models,” Chin. J. Lasers 40(7), 0709001 (2013).
[Crossref]

Wyrowski, F.

Xu, W.

Y.-P. Zhang, J. Q. Zhang, W. Chen, J. L. Zhang, P. Wang, and W. Xu, “Research on three-dimensional computer-generated holographic algorithm based on conformal geometry theory,” Opt. Commun. 309(22), 196–200 (2013).
[Crossref]

Yamaguchi, K.

Yamaguchi, M.

Yoshimura, K.

Zhang, J. L.

Y.-P. Zhang, J. Q. Zhang, W. Chen, J. L. Zhang, P. Wang, and W. Xu, “Research on three-dimensional computer-generated holographic algorithm based on conformal geometry theory,” Opt. Commun. 309(22), 196–200 (2013).
[Crossref]

Zhang, J. Q.

Y.-P. Zhang, J. Q. Zhang, W. Chen, P. Wang, S. Wu, and J. C. Li, “Fast Computer Generated Hologram Algorithm of Triangle Mesh Models,” Chin. J. Lasers 40(7), 0709001 (2013).
[Crossref]

Y.-P. Zhang, J. Q. Zhang, W. Chen, J. L. Zhang, P. Wang, and W. Xu, “Research on three-dimensional computer-generated holographic algorithm based on conformal geometry theory,” Opt. Commun. 309(22), 196–200 (2013).
[Crossref]

Zhang, Y.-P.

Y.-P. Zhang, J. Q. Zhang, W. Chen, J. L. Zhang, P. Wang, and W. Xu, “Research on three-dimensional computer-generated holographic algorithm based on conformal geometry theory,” Opt. Commun. 309(22), 196–200 (2013).
[Crossref]

Y.-P. Zhang, J. Q. Zhang, W. Chen, P. Wang, S. Wu, and J. C. Li, “Fast Computer Generated Hologram Algorithm of Triangle Mesh Models,” Chin. J. Lasers 40(7), 0709001 (2013).
[Crossref]

Zhang, Z.

Appl. Opt. (12)

D. Leseberg and C. Frère, “Computer-generated holograms of 3-D objects composed of tilted planar segments,” Appl. Opt. 27(14), 3020–3024 (1988).
[Crossref] [PubMed]

K. Matsushima and M. Takai, “Recurrence formulas for fast creation of synthetic three-dimensional holograms,” Appl. Opt. 39(35), 6587–6594 (2000).
[Crossref] [PubMed]

L. Ahrenberg, P. Benzie, M. Magnor, and J. Watson, “Computer generated holograms from three dimensional meshes using an analytic light transport model,” Appl. Opt. 47(10), 1567–1574 (2008).

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]

H. Sakata and Y. Sakamoto, “Fast computation method for a Fresnel hologram using three-dimensional affine transformations in real space,” Appl. Opt. 48(34), H212–H221 (2009).
[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 [Invited],” Appl. Opt. 50(34), H87–H115 (2011).
[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]

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]

Y. Pan, Y. Wang, J. Liu, X. Li, J. Jia, and Z. Zhang, “Analytical brightness compensation algorithm for traditional polygon-based method in computer-generated holography,” Appl. Opt. 52(18), 4391–4399 (2013).
[Crossref] [PubMed]

Y. Pan, Y. Wang, J. Liu, X. Li, and J. Jia, “Improved full analytical polygon-based method using Fourier analysis of the three-dimensional affine transformation,” Appl. Opt. 53(7), 1354–1362 (2014).
[Crossref] [PubMed]

Y. Ogihara and Y. Sakamoto, “Fast calculation method of a CGH for a patch model using a point-based method,” Appl. Opt. 54(1), A76–A83 (2015).
[Crossref] [PubMed]

M.-W. Kwon, S.-C. Kim, and E.-S. Kim, “Three-directional motion-compensation mask-based novel look-up table on graphics processing units for video-rate generation of digital holographic videos of three-dimensional scenes,” Appl. Opt. 55(3), A22–A31 (2016).
[Crossref] [PubMed]

Chin. J. Lasers (1)

Y.-P. Zhang, J. Q. Zhang, W. Chen, P. Wang, S. Wu, and J. C. Li, “Fast Computer Generated Hologram Algorithm of Triangle Mesh Models,” Chin. J. Lasers 40(7), 0709001 (2013).
[Crossref]

Chin. Opt. Lett. (1)

P. W. M. Tsang and T.-C. Poon, “Review on theory and applications of wavefront recording plane framework in generation and processing of digital holograms,” Chin. Opt. Lett. 11(1), 18–24 (2013).

Comput. Phys. (1)

A. D. Stein, Z. Wang, and J. S. Leigh, “Computer-generated holograms: A simplified ray-tracing approach,” Comput. Phys. 6(4), 389–392 (1992).
[Crossref]

Electron. Commun. Jpn. Part 2 Electron (1)

Y. Sakamoto and T. Nagao, “A fast computational method for computer-generated Fourier hologram using patch model,” Electron. Commun. Jpn. Part 2 Electron 85(11), 16–24 (2002).
[Crossref]

Electron. Lett. (1)

R. N. Bracewell, K.-Y. Chang, A. K. Jha, and Y.-H. Wang, “Affine theorem for two-dimensional fourier transform,” Electron. Lett. 29(3), 304 (1993).
[Crossref]

Eur. J. Phys. (1)

S. Ganci, “Fourier diffraction through a tilted slit,” Eur. J. Phys. 2(3), 158–160 (1981).
[Crossref]

J. Opt. Soc. Am. A (3)

J. Phys. Conf. Ser. (1)

T. Ichikawa and Y. Sakamoto, “A rendering method of background reflections on a specular surface for CGH,” J. Phys. Conf. Ser. 415, 012044 (2013).
[Crossref]

Opt. Commun. (1)

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Figures (10)

Fig. 1
Fig. 1 Spatial arbitrary triangle in the global coordinate system (x, y, z) that is tilted from the hologram plane.
Fig. 2
Fig. 2 Rotating normal n T byαaround the z-axis and then rotating byθaround the y-axis.
Fig. 3
Fig. 3 Affine transformation principle on the 2D plane. The gray triangle is the rotated triangle, and the blue one is the primitive triangle. (a) The relative position of the local and global coordinate systems. (b) Affine transformation for the arbitrary triangle.
Fig. 4
Fig. 4 Frequency range (colored red stripe) that is recordable on hologram due to source rotation. (a) v w -perspective. (b) u w -perspective.
Fig. 5
Fig. 5 Simulation results for an arbitrary triangle using the proposed method. (a) Triangle position in 3D space. (b) Diffracted field of the triangle at the hologram plane z = 0, namely, the obtained computer-generated hologram. (c) Reconstruction of the computer-generated hologram at z = 134mm, where vertex C is located. The yellow squares point to enlarged local views. (d) Reconstruction of the computer-generated hologram at z = 86mm, where vertex A is located.
Fig. 6
Fig. 6 3D models applied to calculate CGH using the proposed method. (a) A ‘sphere’ with 320 triangles in which green triangles represent those hidden surfaces; in which red and blue meshes are enlarged and shown in (d). (b) A ‘cartoon’ character, Baymax, with 3060 triangles. (c) A ‘teapot’ with 6320 triangles. (d) The blue triangle, ΔABC, is used to judge whether it is hidden with the method of outer-normal, and the red ones are its adjacent triangles. n o , n i , and e z are outer-normal, inner-normal, and unit vector of the z-axis, respectively. The hologram plane is on the plane z = 0.
Fig. 7
Fig. 7 Reconstruction images of the sphere using different ways. (a) Numerical reconstruction of the complex hologram for all triangles on sphere that include hidden polygons. (b) Numerical reconstruction of the complex hologram for those triangles facing toward the hologram plane. (c) Optical reconstruction of the phase hologram (extracted from the complex hologram) with the removal of back faces. (d) Numerical reconstruction of the phase hologram with the removal of back faces.
Fig. 8
Fig. 8 Computer generated holograms based on the proposed method. (a) Sphere. (b) Baymax. (c) Teapot.
Fig. 9
Fig. 9 Numerical reconstruction results of the complex hologram. (a) Reconstruction of Baymax at the chest plane where z = 490mm. (b) Reconstruction of the teapot at the lid plane where z = 500mm. (c) Closeup of the marked area in (a). (d) Closeup of the marked area in (b).
Fig. 10
Fig. 10 Experimental and numerical reconstruction results of the phase hologram. (a) Reconstruction of Baymax optically. (b) Reconstruction of the teapot optically. (c) Reconstruction of Baymax numerically. (d) Reconstruction of the teapot numerically.

Equations (45)

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[ x y ]=[ a b d e ]·[ x y ]+[ c f ].
[ s t ]=[ a 11 a 12 a 21 a 22 ]·[ x y ]+[ a 13 a 23 ].
F Γ ( u s , v t )=J e j2π( a 13 u s + a 23 v t ) F Δ ( a 11 u s + a 21 v t , a 12 u s + a 22 v t ),
T=[ a 11 a 12 a 13 t 1 a 21 a 22 a 23 t 2 a 31 a 32 a 33 t 3 0 0 0 1 ],
{ x= a 11 x + a 12 y + a 13 z + t 1 y= a 21 x + a 22 y + a 23 z + t 2 z= a 31 x + a 32 y + a 33 z + t 3 .
{ x= a 11 x + a 12 y + t 1 y= a 21 x + a 22 y + t 2 z= a 31 x + a 32 y + t 3 .
P=[ a 11 a 12 0 t 1 a 21 a 22 0 t 2 a 31 a 32 0 t 3 0 0 0 1 ].
{ x T =x x 0 y T =y y 0 z T =z z 0 .
T R = R y · R z ,
R z =| cosα sinα 0 sinα cosα 0 0 0 1 |,
R y =| cosθ 0 sinθ 0 1 0 sinθ 0 cosθ |.
[ x y z ] T = T R · [ x T y T z T ] T ,
{ x R = x + x 0 y R = y + y 0 z R = z + z 0 .
[ x R y R z R 1 ]=[ a 11 a 12 a 13 x 0 a 21 a 22 a 23 y 0 a 31 a 32 a 33 z 0 0 0 0 1 ]·[ x x 0 y y 0 z z 0 1 ],
[ x R y R ]=[ b 11 b 12 b 21 b 22 ]·[ s t ]+[ b 13 b 23 ],
[ x 1R y 1R 1 x 2R y 2R 1 x 3R y 2R 1 ]=[ b 11 b 12 b 13 b 21 b 22 b 23 0 0 1 ]·[ s 1 t 1 1 s 2 t 2 1 s 3 t 3 1 ],
T a =[ x 1R y 1R 1 x 2R y 2R 1 x 3R y 2R 1 ]· [ s 1 t 1 1 s 2 t 2 1 s 3 t 3 1 ] 1 ,
T a =[ b 11 b 12 0 b 13 b 21 b 22 0 b 23 0 0 1 0 0 0 0 1 ],
[ x R y R z 0 1 ]=[ b 11 b 12 0 b 13 b 21 b 22 0 b 23 0 0 1 0 0 0 0 1 ]·[ s t z 0 1 ],
[ x y z 1 ]=[ c 11 c 12 c 13 c 14 + x 0 c 21 c 22 c 23 c 24 + y 0 c 31 c 32 c 33 c 34 + z 0 0 0 0 1 ]·[ s t z 0 1 ],
T= T R 1 · T a .
F sQt (u,v)= + + f(x,y,z= z 0 ) e j2π(ux+vy) dxdy .
g(s,t, z 0 )={ 1 0 (s,t) lies inside primitive triangle else ,
F sQt (u,v)= J st 0 1 0 s g(s,t, z 0 ) e j2π[u( c 11 s+ c 12 t+ c 13 z 0 + c 14 + x 0 )+v( c 21 s+ c 22 t+ c 23 z 0 + c 24 + y 0 ] dtds ,
F holo (u,v)= e j2πzw F sQt (u,v),
w= λ 2 u 2 v 2 .
F holo (u,v)={ J st 2 e j2π(p+q) ......................................................................................... u =0, v =0 J st j2π u e j2π(p+q) [ e j2π u + 1 j2π u ( e j2π u 1)]............................ u 0, v =0 J st j2π v e j2π(p+q) [ 1 j2π v ( e j2π v 1)1].................................. u =0, v 0 J st e j2π(p+q) [ 1 j2π v + 1 4 π 2 u v ( e j2π u 1)]............................... u 0, v 0, u + v =0 J st e j2π(p+q) [ 1 4 π 2 v ( u + v ) ( e j2π( u + v ) 1)+ 1 4 π 2 u v ( e j2π u 1)]...else ,
{ J st =| c 11 c 22 c 12 c 21 | u = c 11 u+ c 21 v+ c 31 w v = c 12 u+ c 22 v+ c 32 w q=( c 13 u+ c 23 v+ c 33 w) z 0 p=( c 14 + x 0 )u+( c 24 + y 0 )v+( c 34 + z 0 )w .
Δu= c 31 /λ ,Δv= c 32 /λ .
{ u = u Δu v = v Δv .
U object (x,y)= i tr i total F 1 ( F holo,i ) .
s= e z n o ,
{ s= a 11 x+ a 12 y+ a 13 t= a 21 x+ a 22 y+ a 23 ,
{ a 11 = s 2 s 1 a 12 = s 3 s 2 a 13 = s 1 a 21 = t 2 t 1 a 22 = t 3 t 2 a 23 = t 1 .
F Γ ( u s , v t )= + f Γ (s,t) e j2π( u s s+ v t t) dsdt .
F Γ ( u s , v t )=J 0 1 0 x f Δ (x,y) e j2π[ u s ( a 11 x+ a 12 y+ a 13 )+ v t ( a 21 x+ a 22 y+ a 23 )] dydx ,
J=| a 11 a 22 a 12 a 21 |.
f Δ (x,y)={ 1 0 (x,y) lies inside primitive triangle else ,
F Γ ( u s , v t )=J e j2π( a 13 u s + a 23 v t ) 0 1 0 x e j2π[( a 11 u s + a 21 v t )x+( a 12 u s + a 22 v t )y)] dydx .
0 1 0 x e j2π[( a 11 u s + a 21 v t )x+( a 12 u s + a 22 v t )y)] dydx = F Δ ( a 11 u s + a 21 v t , a 12 u s + a 22 v t ).
F Γ ( u s , v t )=J e j2π( a 13 u s + a 23 v t ) F Δ ( a 11 u s + a 21 v t , a 12 u s + a 22 v t ),
F holo (u,v)= J st 0 1 0 s g(s,t, z 0 ) e j2πzw e j2π[u( c 11 s+ c 12 t+ c 13 z 0 + c 14 + x 0 )+v( c 21 s+ c 22 t+ c 23 z 0 + c 24 + y 0 ] dtds,
F holo (u,v)= J st 0 1 0 s e j2π[u( c 11 s+ c 12 t+ c 13 z 0 + c 14 + x 0 )+v( c 21 s+ c 22 t+ c 23 z 0 + c 24 + y 0 )+w( c 31 s+ c 32 t+ c 33 z 0 + c 34 + z 0 )] dtds .
F holo (u,v)= J st 0 1 0 s e j2π[( c 11 u+ c 21 v+ c 31 w)s+( c 12 u+ c 22 v+ c 32 w)t+( c 13 u+ c 23 v+ c 33 w) z 0 e j2π[( c 14 + x 0 )u+( c 24 + y 0 )v+( c 34 + z 0 )w] dtds.
F holo (u,v)= J st e j2π(p+q) 0 1 0 s e j2π( u s+ v t) dtds ,

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