Abstract

Optical lenses with a freeform surface can be designed for diverse illumination profiles with uniformity. However, most of the previous studies formulate the problem for a single point source, and the lens topology has freeform top and spherical bottom surfaces. In this study, the formulation is extended for multiple point sources, and a flat surface is included in the lens bottom topology for ease of prototyping and manufacturing. The extended formulation for multiple point sources requires only a freeform surface to design. The formulation of overdetermined coupling equations is solved by applying the weighted least-square method. The weightings are correlated with the emitting intensities of sources in terms of an inverse gamma function. The weighting scheme gives a parameter space for designation of illumination profile fit and uniformity. The adequacy of the extended formulation is demonstrated by simulation. Examples of circular and rectangular illumination for single and multiple point sources are studied. The simulation results show that unbalanced luminance distribution can be induced by an offset source and collimated by a lens, which is designated by taking the offset into account. For multiple point sources, illumination profile fit and uniformity are designated in trade off based on the parameter design.

© 2012 Optical Society of America

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References

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  1. H. Ries, J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19, 590–595 (2002).
    [CrossRef]
  2. K. Wang, D. Wu, F. Chen, Z. Liu, X. Luo, S. Liu, “Freeform lens for white LEDs with high angular color uniformity,” in Proceedings of IEEE Electronic System-Integration Technology Conference (ESTC) (IEEE, 2010), pp. 1–5.
  3. Y. Ding, X. Liu, Z. R. Zheng, P. F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16, 12958–12966 (2008).
    [CrossRef]
  4. K. Wang, S. Liu, F. Chen, Z. Y. Liu, X. B. Luo, “Effect on manufacturing defects on optical performance of discontinuous freeform lens,” Opt. Express 17, 5457–5465 (2009).
    [CrossRef]
  5. V. Oliker, “Optical design of freeform two-mirror beam-shaping systems,” J. Opt. Soc. Am. A 24, 3741–3752 (2007).
    [CrossRef]
  6. Z. Zheng, X. Hao, X. Liu, “Freeform surface lens for LED uniform illumination,” Appl. Opt. 48, 6627–6634 (2009).
    [CrossRef]
  7. K. Wang, F. Chen, Z. Y. Liu, X. B. Luo, S. Liu, “Design of compact freeform lens for application specific lighting-emitting diode packaging,” Opt. Express 18, 413–425 (2010).
    [CrossRef]
  8. P. Benítez, J. C. Miňano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 431489–1502 (2004).
    [CrossRef]
  9. F. Chen, K. Wang, Z. Qin, D. Wu, X. Luo, S. Liu, “Design method of high-efficient LED headlamp lens,” Opt. Express 18, 20926–20938 (2010).
    [CrossRef]
  10. K. C. Lin, “Illumination variation under a freeform lens and misalignment of light source,” in Proceedings of 6th IEEE Conference on Industrial Electronics and Applications (ICIEA) (IEEE, 2011), pp. 1079–1084.
  11. W. H. Press, S. A. Teukoisky, W. T. Vettering, B. P. Flannery, Numerical Recipes, 3rd ed. (Cambridge University, 2007).

2010

K. Wang, F. Chen, Z. Y. Liu, X. B. Luo, S. Liu, “Design of compact freeform lens for application specific lighting-emitting diode packaging,” Opt. Express 18, 413–425 (2010).
[CrossRef]

F. Chen, K. Wang, Z. Qin, D. Wu, X. Luo, S. Liu, “Design method of high-efficient LED headlamp lens,” Opt. Express 18, 20926–20938 (2010).
[CrossRef]

2009

Z. Zheng, X. Hao, X. Liu, “Freeform surface lens for LED uniform illumination,” Appl. Opt. 48, 6627–6634 (2009).
[CrossRef]

K. Wang, S. Liu, F. Chen, Z. Y. Liu, X. B. Luo, “Effect on manufacturing defects on optical performance of discontinuous freeform lens,” Opt. Express 17, 5457–5465 (2009).
[CrossRef]

2008

Y. Ding, X. Liu, Z. R. Zheng, P. F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16, 12958–12966 (2008).
[CrossRef]

2007

V. Oliker, “Optical design of freeform two-mirror beam-shaping systems,” J. Opt. Soc. Am. A 24, 3741–3752 (2007).
[CrossRef]

2004

P. Benítez, J. C. Miňano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 431489–1502 (2004).
[CrossRef]

2002

H. Ries, J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19, 590–595 (2002).
[CrossRef]

Benítez, P.

P. Benítez, J. C. Miňano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 431489–1502 (2004).
[CrossRef]

Blen, J.

P. Benítez, J. C. Miňano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 431489–1502 (2004).
[CrossRef]

Chaves, J.

P. Benítez, J. C. Miňano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 431489–1502 (2004).
[CrossRef]

Chen, F.

K. Wang, F. Chen, Z. Y. Liu, X. B. Luo, S. Liu, “Design of compact freeform lens for application specific lighting-emitting diode packaging,” Opt. Express 18, 413–425 (2010).
[CrossRef]

F. Chen, K. Wang, Z. Qin, D. Wu, X. Luo, S. Liu, “Design method of high-efficient LED headlamp lens,” Opt. Express 18, 20926–20938 (2010).
[CrossRef]

K. Wang, S. Liu, F. Chen, Z. Y. Liu, X. B. Luo, “Effect on manufacturing defects on optical performance of discontinuous freeform lens,” Opt. Express 17, 5457–5465 (2009).
[CrossRef]

K. Wang, D. Wu, F. Chen, Z. Liu, X. Luo, S. Liu, “Freeform lens for white LEDs with high angular color uniformity,” in Proceedings of IEEE Electronic System-Integration Technology Conference (ESTC) (IEEE, 2010), pp. 1–5.

Ding, Y.

Y. Ding, X. Liu, Z. R. Zheng, P. F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16, 12958–12966 (2008).
[CrossRef]

Dross, O.

P. Benítez, J. C. Miňano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 431489–1502 (2004).
[CrossRef]

Falicoff, W.

P. Benítez, J. C. Miňano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 431489–1502 (2004).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukoisky, W. T. Vettering, B. P. Flannery, Numerical Recipes, 3rd ed. (Cambridge University, 2007).

Gu, P. F.

Y. Ding, X. Liu, Z. R. Zheng, P. F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16, 12958–12966 (2008).
[CrossRef]

Hao, X.

Z. Zheng, X. Hao, X. Liu, “Freeform surface lens for LED uniform illumination,” Appl. Opt. 48, 6627–6634 (2009).
[CrossRef]

Hernández, M.

P. Benítez, J. C. Miňano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 431489–1502 (2004).
[CrossRef]

Lin, K. C.

K. C. Lin, “Illumination variation under a freeform lens and misalignment of light source,” in Proceedings of 6th IEEE Conference on Industrial Electronics and Applications (ICIEA) (IEEE, 2011), pp. 1079–1084.

Liu, S.

K. Wang, F. Chen, Z. Y. Liu, X. B. Luo, S. Liu, “Design of compact freeform lens for application specific lighting-emitting diode packaging,” Opt. Express 18, 413–425 (2010).
[CrossRef]

F. Chen, K. Wang, Z. Qin, D. Wu, X. Luo, S. Liu, “Design method of high-efficient LED headlamp lens,” Opt. Express 18, 20926–20938 (2010).
[CrossRef]

K. Wang, S. Liu, F. Chen, Z. Y. Liu, X. B. Luo, “Effect on manufacturing defects on optical performance of discontinuous freeform lens,” Opt. Express 17, 5457–5465 (2009).
[CrossRef]

K. Wang, D. Wu, F. Chen, Z. Liu, X. Luo, S. Liu, “Freeform lens for white LEDs with high angular color uniformity,” in Proceedings of IEEE Electronic System-Integration Technology Conference (ESTC) (IEEE, 2010), pp. 1–5.

Liu, X.

Z. Zheng, X. Hao, X. Liu, “Freeform surface lens for LED uniform illumination,” Appl. Opt. 48, 6627–6634 (2009).
[CrossRef]

Y. Ding, X. Liu, Z. R. Zheng, P. F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16, 12958–12966 (2008).
[CrossRef]

Liu, Z.

K. Wang, D. Wu, F. Chen, Z. Liu, X. Luo, S. Liu, “Freeform lens for white LEDs with high angular color uniformity,” in Proceedings of IEEE Electronic System-Integration Technology Conference (ESTC) (IEEE, 2010), pp. 1–5.

Liu, Z. Y.

K. Wang, F. Chen, Z. Y. Liu, X. B. Luo, S. Liu, “Design of compact freeform lens for application specific lighting-emitting diode packaging,” Opt. Express 18, 413–425 (2010).
[CrossRef]

K. Wang, S. Liu, F. Chen, Z. Y. Liu, X. B. Luo, “Effect on manufacturing defects on optical performance of discontinuous freeform lens,” Opt. Express 17, 5457–5465 (2009).
[CrossRef]

Luo, X.

F. Chen, K. Wang, Z. Qin, D. Wu, X. Luo, S. Liu, “Design method of high-efficient LED headlamp lens,” Opt. Express 18, 20926–20938 (2010).
[CrossRef]

K. Wang, D. Wu, F. Chen, Z. Liu, X. Luo, S. Liu, “Freeform lens for white LEDs with high angular color uniformity,” in Proceedings of IEEE Electronic System-Integration Technology Conference (ESTC) (IEEE, 2010), pp. 1–5.

Luo, X. B.

K. Wang, F. Chen, Z. Y. Liu, X. B. Luo, S. Liu, “Design of compact freeform lens for application specific lighting-emitting diode packaging,” Opt. Express 18, 413–425 (2010).
[CrossRef]

K. Wang, S. Liu, F. Chen, Z. Y. Liu, X. B. Luo, “Effect on manufacturing defects on optical performance of discontinuous freeform lens,” Opt. Express 17, 5457–5465 (2009).
[CrossRef]

Minano, J. C.

P. Benítez, J. C. Miňano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 431489–1502 (2004).
[CrossRef]

Mohedano, R.

P. Benítez, J. C. Miňano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 431489–1502 (2004).
[CrossRef]

Muschaweck, J.

H. Ries, J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19, 590–595 (2002).
[CrossRef]

Oliker, V.

V. Oliker, “Optical design of freeform two-mirror beam-shaping systems,” J. Opt. Soc. Am. A 24, 3741–3752 (2007).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukoisky, W. T. Vettering, B. P. Flannery, Numerical Recipes, 3rd ed. (Cambridge University, 2007).

Qin, Z.

F. Chen, K. Wang, Z. Qin, D. Wu, X. Luo, S. Liu, “Design method of high-efficient LED headlamp lens,” Opt. Express 18, 20926–20938 (2010).
[CrossRef]

Ries, H.

H. Ries, J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19, 590–595 (2002).
[CrossRef]

Teukoisky, S. A.

W. H. Press, S. A. Teukoisky, W. T. Vettering, B. P. Flannery, Numerical Recipes, 3rd ed. (Cambridge University, 2007).

Vettering, W. T.

W. H. Press, S. A. Teukoisky, W. T. Vettering, B. P. Flannery, Numerical Recipes, 3rd ed. (Cambridge University, 2007).

Wang, K.

F. Chen, K. Wang, Z. Qin, D. Wu, X. Luo, S. Liu, “Design method of high-efficient LED headlamp lens,” Opt. Express 18, 20926–20938 (2010).
[CrossRef]

K. Wang, F. Chen, Z. Y. Liu, X. B. Luo, S. Liu, “Design of compact freeform lens for application specific lighting-emitting diode packaging,” Opt. Express 18, 413–425 (2010).
[CrossRef]

K. Wang, S. Liu, F. Chen, Z. Y. Liu, X. B. Luo, “Effect on manufacturing defects on optical performance of discontinuous freeform lens,” Opt. Express 17, 5457–5465 (2009).
[CrossRef]

K. Wang, D. Wu, F. Chen, Z. Liu, X. Luo, S. Liu, “Freeform lens for white LEDs with high angular color uniformity,” in Proceedings of IEEE Electronic System-Integration Technology Conference (ESTC) (IEEE, 2010), pp. 1–5.

Wu, D.

F. Chen, K. Wang, Z. Qin, D. Wu, X. Luo, S. Liu, “Design method of high-efficient LED headlamp lens,” Opt. Express 18, 20926–20938 (2010).
[CrossRef]

K. Wang, D. Wu, F. Chen, Z. Liu, X. Luo, S. Liu, “Freeform lens for white LEDs with high angular color uniformity,” in Proceedings of IEEE Electronic System-Integration Technology Conference (ESTC) (IEEE, 2010), pp. 1–5.

Zheng, Z.

Z. Zheng, X. Hao, X. Liu, “Freeform surface lens for LED uniform illumination,” Appl. Opt. 48, 6627–6634 (2009).
[CrossRef]

Zheng, Z. R.

Y. Ding, X. Liu, Z. R. Zheng, P. F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16, 12958–12966 (2008).
[CrossRef]

Appl. Opt.

Z. Zheng, X. Hao, X. Liu, “Freeform surface lens for LED uniform illumination,” Appl. Opt. 48, 6627–6634 (2009).
[CrossRef]

J. Opt. Soc. Am. A

V. Oliker, “Optical design of freeform two-mirror beam-shaping systems,” J. Opt. Soc. Am. A 24, 3741–3752 (2007).
[CrossRef]

H. Ries, J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19, 590–595 (2002).
[CrossRef]

Opt. Eng.

P. Benítez, J. C. Miňano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 431489–1502 (2004).
[CrossRef]

Opt. Express

F. Chen, K. Wang, Z. Qin, D. Wu, X. Luo, S. Liu, “Design method of high-efficient LED headlamp lens,” Opt. Express 18, 20926–20938 (2010).
[CrossRef]

Y. Ding, X. Liu, Z. R. Zheng, P. F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16, 12958–12966 (2008).
[CrossRef]

K. Wang, S. Liu, F. Chen, Z. Y. Liu, X. B. Luo, “Effect on manufacturing defects on optical performance of discontinuous freeform lens,” Opt. Express 17, 5457–5465 (2009).
[CrossRef]

K. Wang, F. Chen, Z. Y. Liu, X. B. Luo, S. Liu, “Design of compact freeform lens for application specific lighting-emitting diode packaging,” Opt. Express 18, 413–425 (2010).
[CrossRef]

Other

K. Wang, D. Wu, F. Chen, Z. Liu, X. Luo, S. Liu, “Freeform lens for white LEDs with high angular color uniformity,” in Proceedings of IEEE Electronic System-Integration Technology Conference (ESTC) (IEEE, 2010), pp. 1–5.

K. C. Lin, “Illumination variation under a freeform lens and misalignment of light source,” in Proceedings of 6th IEEE Conference on Industrial Electronics and Applications (ICIEA) (IEEE, 2011), pp. 1079–1084.

W. H. Press, S. A. Teukoisky, W. T. Vettering, B. P. Flannery, Numerical Recipes, 3rd ed. (Cambridge University, 2007).

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Figures (17)

Fig. 1.
Fig. 1.

Schematic of an optical system of a freeform lens and multiple point sources.

Fig. 2.
Fig. 2.

Schematics of trigonometry at the bottom.

Fig. 3.
Fig. 3.

Accumulative intensity of source used in simulation.

Fig. 4.
Fig. 4.

Designated freeform lenses and dimensions (a)  L C 1 ( 42.4 × 42.4 × 13.0 mm 2 ) , (b)  L C 1 x ( 42.3 × 42.2 × 13.0 mm 2 ) , (c)  L C 00 ( 40.7 × 41.0 × 13.0 mm 2 ) , (d)  L R 1 ( 43.0 × 32.8 × 13.0 mm 2 ) , (e)  L R 1 x ( 42.9 × 34.2 × 13.0 mm 2 ) , and (f)  L R 100 ( 40.8 × 31.4 × 13.0 mm 2 ) .

Fig. 5.
Fig. 5.

Illumination under L C 1 for an unoffset source. (a) luminance distribution, (b) 3D luminance distribution.

Fig. 6.
Fig. 6.

Illumination under L C 1 for an offset source. (a) luminance distribution, Δ x = 3 mm , (b) 3D luminance distribution, Δ x = 3 mm , (c) luminance distribution, Δ x = 5 mm , and (d) 3D luminance distribution, Δ x = 5 mm .

Fig. 7.
Fig. 7.

Illumination under L C 1 x for an offset source. (a) luminance distribution, (b) 3D luminance distribution.

Fig. 8.
Fig. 8.

Illumination under L R 1 for an unoffset source. (a) luminance distribution, (b) 3D luminance distribution.

Fig. 9.
Fig. 9.

Illumination under L R 1 for an offset source. (a) luminance distribution, Δ x = 3 mm , (b) 3D luminance distribution, Δ x = 3 mm , (c) luminance distribution, Δ x = 5 mm , and (d) 3D luminance distribution, Δ x = 5 mm .

Fig. 10.
Fig. 10.

Illumination under L R 1 x for an offset source. (a) luminance distribution, (b) 3D luminance distribution.

Fig. 11.
Fig. 11.

Illumination under L C 1 for two offset sources. (a) luminance distribution, (b) 3D luminance distribution.

Fig. 12.
Fig. 12.

Profiles of the lenses L C γ 0 γ 1 .

Fig. 13.
Fig. 13.

Luminance distribution under L C γ 0 γ 1 for two offset sources.

Fig. 14.
Fig. 14.

Illumination under L R 1 for two offset sources. (a) luminance distribution, (b) 3D luminance distribution.

Fig. 15.
Fig. 15.

Luminance distribution under L R γ 0 γ 1 for two offset sources.

Fig. 16.
Fig. 16.

Luminance distribution under L C 1 for nine sources.

Fig. 17.
Fig. 17.

Luminance distribution under L 9 C γ 0 γ 1 for nine sources.

Equations (28)

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

x d k x t = N x N z [ ( z d z t ) n I z k p t p d k ¯ ] + n I x k p t p d k ¯ ,
y d k y t = N y N z [ ( z d z t ) n I z k p t p d k ¯ ] + n I y k p t p d k ¯ , k = 1 , 2 , 3 , , N s ,
p s k p t ¯ sin ( α i k α o k β 2 k ) sin ( α i k α o k ) = R b sin ( α i k + β 1 k ) sin β 1 k ,
sin α i k = Δ x k 2 + Δ y k 2 + ( R b p o p b ¯ + Δ z k ) 2 R b sin β 1 k , k = 1 , 2 , 3 , , N s ,
p s k p t ¯ sin ( φ φ I k ) cos φ s k = p o p b ¯ sin ( φ s k φ I k ) , k = 1 , 2 , 3 , , N s ,
β 1 k + β 2 k = cos 1 ( p s k p t ¯ 2 + p s k p c ¯ 2 p c p t ¯ 2 2 p s k p t ¯ p s k p c ¯ ) ,
[ y t ( R b p o p b ¯ + Δ z k ) + z t Δ y k ] x b k + [ x t ( R b p o p b ¯ + Δ z k ) z t Δ x k ] y b k + [ y t Δ x k x t Δ y k ] z b k = 0 ,
x b k 2 + y b k 2 + z b k 2 = [ R b sin ( β 1 k + α i k ) sin β 1 k ] 2 ,
( x b k + Δ x k ) 2 + ( y b k + Δ y k ) 2 + [ z b k + ( R b p o p b ¯ + Δ z k ) ] 2 = R b 2 ,
φ I k = tan 1 [ z t z b k ( x t x b k ) 2 + ( y t y b k ) 2 ] , k = 1 , 2 , 3 , , N s ,
ρ φ ρ i + 1 , j ρ i 1 , j 2 Δ φ , i = 0 , 1 , 2 , , N φ ,
ρ θ ρ i , j + 1 ρ i , j 1 2 Δ θ , j = 0 , 1 , 2 , , N θ ,
A [ ρ i + 1 , j ρ i , j + 1 ] = b ,
A = [ a 11 , 1 a 12 , 1 a 21 , 1 a 22 , 1 a 11 , N s a 12 , N s a 21 , N s a 22 , N s ] , b = [ b 1 , 1 b 2 , 1 b 1 , N s b 2 , N s ] ,
a 11 , k = 2 Δ θ sin φ [ ( z d cos φ ρ i , j ) cos θ n ( cos φ cos φ I k cos θ + sin φ sin φ I k cos θ I k ) p t p d k ¯ + x d k , i j sin φ ] ,
a 12 , k = ( Δ φ i + 1 + Δ φ i ) ( z d ρ i , j cos φ n cos φ I k p t p d k ¯ ) sin θ ,
a 21 , k = 2 Δ θ sin φ [ ( z d cos φ ρ i , j ) sin θ n ( cos φ cos φ I k sin θ + sin φ I k sin φ sin θ I k ) p t p d k ¯ + y d k , i j sin φ ] ,
a 22 , k = ( Δ φ i + 1 + Δ φ i ) ( z d ρ cos φ n cos φ I k p t p d k ¯ ) cos θ ,
b 1 , k = a 11 , k ρ i 1 , j + a 12 , k ρ i , j 1 + 2 Δ θ ( Δ φ i + 1 + Δ φ i ) ρ i , j sin φ [ z d cos θ sin φ x d k , i j cos φ n p t p d k ¯ ( cos φ I k sin φ cos θ cos φ sin φ I k cos θ I k ) ] ,
b 2 , k = a 21 , k ρ i 1 , j + a 22 , k ρ i , j 1 + 2 Δ θ ( Δ φ i + 1 + Δ φ i ) ρ i , j sin φ [ z d sin φ sin θ y d k , i j cos φ + n p t p d k ¯ ( sin φ I k cos φ sin θ I k cos φ I k sin φ sin θ ) ] .
[ ρ i + 1 , j ρ i , j + 1 ] = ( A T W A ) 1 A T W b , i = 0 , 1 , 2 , , N φ , j = 0 , 1 , 2 , , N θ ,
[ ρ 1 , j ρ 0 , j + 1 ] = [ k = 1 N s w k , 0 j p o p k ¯ ρ 0 , 0 ] , j = 0 , 1 , 2 , , N θ ,
x d k , i j = R d φ s k , i j * φ s k , max cos θ I k , i j ,
y d k , i j = R d φ s k , i j * φ s k , max sin θ I k , i j ,
φ s k , i j * = F k 1 ( i N φ ) , i = 0 , 1 , 2 , , N φ ,
F k ( φ s k , i j ) = h = 1 i f k ( φ s k , h j ) / h = 1 N φ f k ( φ s k , h j ) ,
w k , i j = f k γ ( φ s k , i j ) , k = 1 , 2 , 3 , , N s ,
γ = γ 0 + i N φ ( γ 1 γ 0 ) , i = 0 , 1 , , N φ ,

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