Abstract

Previous research [Appl. Opt. 52, A290 (2013) [CrossRef]  ] has revealed that Fourier analysis of three-dimensional affine transformation theory can be used to improve the computation speed of the traditional polygon-based method. In this paper, we continue our research and propose an improved full analytical polygon-based method developed upon this theory. Vertex vectors of primitive and arbitrary triangles and the pseudo-inverse matrix were used to obtain an affine transformation matrix representing the spatial relationship between the two triangles. With this relationship and the primitive spectrum, we analytically obtained the spectrum of the arbitrary triangle. This algorithm discards low-level angular dependent computations. In order to add diffusive reflection to each arbitrary surface, we also propose a whole matrix computation approach that takes advantage of the affine transformation matrix and uses matrix multiplication to calculate shifting parameters of similar sub-polygons. The proposed method improves hologram computation speed for the conventional full analytical approach. Optical experimental results are demonstrated which prove that the proposed method can effectively reconstruct three-dimensional scenes.

© 2014 Optical Society of America

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References

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    [CrossRef]
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2013 (5)

2012 (2)

2011 (4)

2010 (1)

P. A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W. Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature 468, 80–83 (2010).
[CrossRef]

2009 (3)

2008 (6)

2005 (1)

2003 (1)

1993 (2)

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

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

1992 (1)

1989 (1)

1988 (1)

1981 (1)

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

Ahrenberg, L.

Bablumian, A.

P. A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W. Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature 468, 80–83 (2010).
[CrossRef]

Barabas, J.

D. E. Smalley, Q. Y. J. Smithwick, V. M. Bove, J. Barabas, and S. Jolly, “Anisotropic leaky-mode modulator for holographic video displays,” Nature 498, 313–317 (2013).
[CrossRef]

Benzie, P.

Bianco, B.

Blanche, P. A.

P. A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W. Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature 468, 80–83 (2010).
[CrossRef]

Bove, V. M.

D. E. Smalley, Q. Y. J. Smithwick, V. M. Bove, J. Barabas, and S. Jolly, “Anisotropic leaky-mode modulator for holographic video displays,” Nature 498, 313–317 (2013).
[CrossRef]

Chen, B.-C.

Cheung, W.-K.

Chong, T.-C.

Christenson, C.

P. A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W. Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature 468, 80–83 (2010).
[CrossRef]

Dong, J.-W.

Finke, G.

Flores, D.

P. A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W. Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature 468, 80–83 (2010).
[CrossRef]

Frère, C.

Fuji, T.

Ganci, S.

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

Golub, G. H.

G. H. Golub and C. F. Loan, Matrix Computations, 3rd. ed. (Johns Hopkins University, 1996).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Gu, T.

P. A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W. Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature 468, 80–83 (2010).
[CrossRef]

Hahn, J.

Hartley, R.

R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision, 2nd ed. (Cambridge University, 2003).

He, H.-X.

Hennelly, B.

Hsieh, W. Y.

P. A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W. Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature 468, 80–83 (2010).
[CrossRef]

Ito, T.

Jia, J.

Jiang, S.

Jiang, W.

Jolly, S.

D. E. Smalley, Q. Y. J. Smithwick, V. M. Bove, J. Barabas, and S. Jolly, “Anisotropic leaky-mode modulator for holographic video displays,” Nature 498, 313–317 (2013).
[CrossRef]

Kang, H.

Kathaperumal, M.

P. A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W. Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature 468, 80–83 (2010).
[CrossRef]

Kim, E.-S.

Kim, H.

Kim, S.-C.

Kozacki, T.

Kujawinska, M.

Lee, B.

Leseberg, D.

Li, X.

Liang, X.

Lim, Y.

Lin, W.

P. A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W. Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature 468, 80–83 (2010).
[CrossRef]

Liu, J.

Liu, Y.-Z.

Loan, C. F.

G. H. Golub and C. F. Loan, Matrix Computations, 3rd. ed. (Johns Hopkins University, 1996).

Lucente, M.

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

Magnor, M.

Masuda, N.

Matsushima, K.

Nakahara, S.

Nakayama, H.

Nishi, H.

Norwood, R. A.

P. A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W. Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature 468, 80–83 (2010).
[CrossRef]

Oikawa, M.

Okada, N.

Onural, L.

Pan, Y.

Pang, X.-N.

Park, G.

Peyghambarian, N.

P. A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W. Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature 468, 80–83 (2010).
[CrossRef]

Poon, T.-C.

Pu, Y.-Y.

Rachwal, B.

P. A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W. Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature 468, 80–83 (2010).
[CrossRef]

Sakamoto, Y.

Sakata, H.

Schimmel, H.

Shimobaba, T.

Shiraki, A.

Siddiqui, O.

P. A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W. Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature 468, 80–83 (2010).
[CrossRef]

Smalley, D. E.

D. E. Smalley, Q. Y. J. Smithwick, V. M. Bove, J. Barabas, and S. Jolly, “Anisotropic leaky-mode modulator for holographic video displays,” Nature 498, 313–317 (2013).
[CrossRef]

Smithwick, Q. Y. J.

D. E. Smalley, Q. Y. J. Smithwick, V. M. Bove, J. Barabas, and S. Jolly, “Anisotropic leaky-mode modulator for holographic video displays,” Nature 498, 313–317 (2013).
[CrossRef]

Solanki, S.

Sun, Z.

Takada, N.

Tan, C.

Tanjung, R.

Thomas, J.

P. A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W. Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature 468, 80–83 (2010).
[CrossRef]

Tommasi, T.

Tsang, P.

Voorakaranam, R.

P. A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W. Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature 468, 80–83 (2010).
[CrossRef]

Wang, H.-Z.

Wang, P.

P. A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W. Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature 468, 80–83 (2010).
[CrossRef]

Wang, Y.

Watson, J.

Wyrowski, F.

Xin, L.

Xu, X.

Yamaguchi, T.

Yamamoto, M.

P. A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W. Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature 468, 80–83 (2010).
[CrossRef]

Yaras, F.

Yoshikawa, H.

Yu, Y.

Zaperty, W.

Zhang, B.

Zhang, Z.

Zhao, Q.

Zheng, H.

Zhou, C.

Zisserman, A.

R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision, 2nd ed. (Cambridge University, 2003).

Appl. Opt. (15)

T. Yamaguchi, T. Fuji, and H. Yoshikawa, “Fast calculation method for computer-generated cylindrical holograms,” Appl. Opt. 47, D63–D70 (2008).
[CrossRef]

S.-C. Kim and E.-S. Kim, “Effective generation of digital holograms of three-dimensional objects using a novel look-up table method,” Appl. Opt. 47, D55–D62 (2008).
[CrossRef]

J. Jia, Y. Wang, J. Liu, L. Xin, Y. Pan, Z. Sun, B. Zhang, Q. Zhao, and W. Jiang, “Reducing the memory usage for effective computer-generated hologram calculation using compressed look-up table in full-color holographic display,” Appl. Opt. 52, 1404–1412 (2013).
[CrossRef]

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, 7303–7307 (2012).
[CrossRef]

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

C. Frère and D. Leseberg, “Large objects reconstructed from computer-generated holograms,” Appl. Opt. 28, 2422–2425 (1989).
[CrossRef]

K. Matsushima, “Formulation of the rotational transformation of wave fields and their application to digital holography,” Appl. Opt. 47, D110–D116 (2008).
[CrossRef]

K. Matsushima, “Computer-generated holograms for three-dimensional surface objects with shade and texture,” Appl. Opt. 44, 4607–4614 (2005).
[CrossRef]

H. Nishi, K. Matsushima, and S. Nakahara, “Rendering of specular surfaces in polygon-based computer-generated holograms,” Appl. Opt. 50, H245–H252 (2011).
[CrossRef]

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

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

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

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, A290–A299 (2013).
[CrossRef]

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, 1567–1574 (2008).
[CrossRef]

H. Kim, J. Hahn, and B. Lee, “Mathematical modeling of triangle-mesh-modeled three-dimensional surface objects for digital holography,” Appl. Opt. 47, D117–D127 (2008).
[CrossRef]

Eur. J. Phys. (1)

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

J. Display Technol. (1)

J. Electron. Imaging (1)

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

J. Opt. Soc. Am. A (2)

Nature (2)

D. E. Smalley, Q. Y. J. Smithwick, V. M. Bove, J. Barabas, and S. Jolly, “Anisotropic leaky-mode modulator for holographic video displays,” Nature 498, 313–317 (2013).
[CrossRef]

P. A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W. Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature 468, 80–83 (2010).
[CrossRef]

Opt. Express (5)

Opt. Lett. (2)

Other (3)

R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision, 2nd ed. (Cambridge University, 2003).

G. H. Golub and C. F. Loan, Matrix Computations, 3rd. ed. (Johns Hopkins University, 1996).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

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Figures (8)

Fig. 1.
Fig. 1.

Primitive triangle under local coordinates and arbitrary triangle under global coordinates.

Fig. 2.
Fig. 2.

Specified primitive triangle.

Fig. 3.
Fig. 3.

(a) Primitive and arbitrary triangles were divided with the same strategy (for example M = 3 ). (b) Transformation relationships between Δ pri and Δ P , , ( 1 , 1 ) , Δ P , , ( 1 , 1 ) and Δ P , , ( m , n ) , and Δ pri and Δ arb , which are represented by T 1 , T 3 , and P .

Fig. 4.
Fig. 4.

Experimental setup: SF, spatial filter; CL, collimating lens; HWP, half-wave plate; SLM, spatial light modulator; FL, Fourier lens; HPF, high-pass filter; OP1, observation plane one; OP2, observation plane two. f = 543 mm , d 1 = 131 mm , d 2 = 262 mm .

Fig. 5.
Fig. 5.

Optical reconstruction results from the general performance test of the proposed method. (a) Large and small squares with side lengths equaling 10.9 and 1.09 mm, respectively, were observed at OP1. The large square rotated 45° about the x and y axes. For the large one, M = 40 ; for the small one, M = 10 . (b,c) Cubic and pyramid shapes in a 30 mm × 12 mm × 140 mm 3D scene were observed at OP1 and OP2, respectively. Both objects used M = 40 . (d) Complicated teapot in a 13.5 mm × 13.5 mm × 17.5 mm scene was reconstructed with M = 6 for all arbitrary triangles. It was observed at OP1.

Fig. 6.
Fig. 6.

Test results of diffusive reflection adding performance. Surfaces were divided by (a), (b)  M = 32 and (c), (d)  M = 64 . The same cubic and pyramid scene was reconstructed and observed at [(a), (c)] OP1 and [(b), (d)] OP2.

Fig. 7.
Fig. 7.

Comparison of optical reconstruction results among (a) the traditional method, (b) the improved traditional method, (c) the conventional full analytical method, and (d) the proposed improved full analytical methods. Same cubic and pyramid scene was used and a light source was added with directional vector equaling ( 2 , 1 , 1 ) . For the improved traditional method (b), a side length of 500 pixels was used in the primitive right angle triangle. For both full analytical methods [(c), (d)], M = 40 . All images were captured at OP1. The coordinates and illumination direction shown in (d) represent the identical environment of all four experiments.

Fig. 8.
Fig. 8.

Average computation time per polygon of the traditional method, the improved traditional method, the conventional full analytical method, and the proposed method over 50–1000 effective facets. Resolution of the holograms was 1920 × 1080 . M = 1 was used in the conventional full analytical method and the proposed method.

Equations (23)

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

[ a 1 , a 2 , a 3 , 1 ] T = T × [ a 1 , a 2 , a 3 , 1 ] 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 ] .
F Δ pri ( μ , ν ; z = 0 ) = f Δ pri ( x , y , z = 0 ) e i 2 π ( μ x + ν y ) d x d y ,
G Δ arb ( μ , ν ; z = 0 ) = | J | · E 2 ( μ , ν ) · F Δ pri ( μ , ν ; z = 0 ) ,
μ = a 11 μ + a 21 ν + a 31 o ( μ , ν ) ν = a 12 μ + a 22 ν + a 33 o ( μ , ν ) E 2 ( μ , ν ) = e i 2 π ( t 1 μ + t 2 ν + t 3 ϖ ( μ , ν ) ) J = a 11 a 22 a 12 a 21 .
ϖ ( μ , ν ) = ( λ 2 μ 2 ν 2 ) 1 / 2 .
μ a 11 μ + a 21 ν ν a 12 μ + a 22 ν .
P = V [ Δ arb ] × V [ Δ pri ] ,
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 ] .
V [ Δ arb ] = P × V [ Δ pri ] .
f Δ pri ( x , y ; z = 0 ) = { 1 if ( x , y ; z = 0 ) inside Δ pri 0 else .
F ( μ , ν ) = { 1 / 2 μ , ν = 0 e j π ν e j π ν 4 π 2 ν 2 j e j π ν 2 π ν μ = 0 , ν 0 ( 1 2 i 4 π μ ) ( e j π μ e j π μ ) μ 0 , ν = 0 1 2 π μ + 1 e j 2 π μ 4 π 2 μ 2 μ + ν = 0 , μ 0 e j π ( μ + ν ) e j π ( μ + ν ) 4 π 2 ν ( μ + ν ) + e j π ( μ ν ) e j π ( μ + ν ) 4 π 2 ν μ else .
W ( Δ A ) = m = 1 M n = 1 M i W ( Δ A , , ( m , n ) ) + m = 1 M n = 1 M i W ( Δ A , , ( m , n ) ) .
W ( Δ A , , ( m , n ) ) = W ( Δ A , , ( 1 , 1 ) ) · exp ( x offset μ + y offset ν + z offset ϖ ) .
V [ Δ P , , ( 1 , 1 ) ] = T 1 × V [ Δ pri ] V [ Δ P , , ( m , n ) ] = T 3 × V [ Δ P , , ( 1 , 1 ) ] V [ Δ A , , ( 1 , 1 ) ] = P × V [ Δ P , , ( 1 , 1 ) ] V [ Δ A , , ( m , n ) ] = P × V [ Δ P , , ( m , n ) ] .
V [ Δ A , , ( 1 , 1 ) ] = P × T 1 × V [ Δ pri ] .
V [ Δ A , , ( m , n ) ] = P × T 3 × T 1 × V [ Δ pri ] .
[ x offset , y offset , z offset , 1 ] T = P × T 3 × T 1 × V [ Δ pri ] P × T 1 × V [ Δ pri ] .
Computation time ratio = Computation time of the proposed method Computation time of reference method .
ϖ = 1 λ ( 1 λ 2 μ 2 λ 2 ν 2 ) 1 / 2 = 1 λ + [ λ u 2 + λ ν 2 2 ] + [ ( λ u 2 + λ ν 2 ) 2 8 ] + + 1 ( n + 1 ) ! f n + 1 ( 0 ) ( λ u 2 + λ ν 2 ) n + 1 ,
o ( μ , ν ) = λ u 2 + λ ν 2 2 .
T = [ K x cos θ cos ψ K x ( cos ϕ sin ψ + sin ϕ sin θ cos ψ ) K x ( sin ϕ sin ψ + cos ϕ sin θ cos ψ ) K y cos θ sin ψ K y ( cos ϕ cos ψ + sin ϕ sin θ sin ψ ) K y ( sin ϕ cos ψ + cos ϕ sin θ sin ψ ) K z sin θ K z sin ϕ cos θ K z cos ϕ cos θ ] ,
L = [ a 13 o ( μ , ν ) ] 2 + [ a 23 o ( μ , ν ) ] 2 K λ u 2 + λ ν 2 2 .

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