Abstract

A retina-like image sensor based on a space-variant lens array is proposed. First, the mathematical models of the proposed image sensor and its space-variant lens array are developed and verified. Second, the relationships among the parameters of the space-variant lens have been simulated and discussed. Finally, some conclusions are deduced, which will help to result in a retina-like image sensor with the characteristics of high speed, large resolution, high sensitivity, and big planar array, etc.

© 2013 Optical Society of America

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  1. E. L. Schwartz, “A quantitative model of the functional architecture of human striate cortex with application to visual illusion and cortical texture analysis,” Biol. Cybern. 37, 63–76 (1980).
    [CrossRef]
  2. N. Onkarappa and A. D. Sappa, “Space variant representations for mobile platform vision applications,” in Computer Analysis of Images and Patterns: 14th International Conference, CAIP 2011, A. Berciano, D. Diaz-Pernil, W. Kropatsch, and P. Molina-Abril Real, eds., (Springer, 2011), Vol. 2, pp. 146–154.
  3. V. J. Traver and A. Bernardino, “A review of log-polar imaging for visual perception in robotics,” Robot. Auton. Syst. 58, 378–398 (2010).
    [CrossRef]
  4. C. Posch, “Bio-inspired vision,” J. Instrum. 7, C01054 (2012).
    [CrossRef]
  5. R. Maldonado-Lopez, F. Vidal-Verdu, G. Linan, E. Roca, and A. Rodriguez-Vazquez, “Early slip detection with a tactile sensor based on retina,” Analog Integr. Circuits Signal Process. 53, 97–108 (2007).
    [CrossRef]
  6. R. Wodnicki, G. W. Roberts, and M. D. Levine, “A foveated image sensor in standard CMOS technology,” in Custom Integrated Circuits Conference (IEEE, 1995), pp. 357–360.
  7. J. Perez, F. Pardo, J. Boluda, S. Felici, and B. Diericki, “Design of a foveated log-polar image sensor in standard CMOS technology,” in Design of Integrated Circuits and Systems (DCIS’96) (IEEE, 2011).
  8. F. Pardo, J. A. Boluda, J. J. Perez, S. Felici, B. Diericki, and D. Scheffer, “Response properties of a foveated space-variant CMOS image sensor,” in Circuits and Systems (IEEE, 1996), pp. 373–376.
  9. D. S. Park, J. H. Kim, H. S. Kim, J. H. Park, J. K. Shin, and M. Lee, “A foveated-structure CMOS retina chip for edge detection with local light adaptation,” Sens. Actuators A Phys. 108, 75–80 (2003).
    [CrossRef]
  10. M. Bigas, E. Cabruja, J. Forest, and J. Salvi, “Review of CMOS image sensors,” Microelectron. J. 37, 433–451 (2006).
    [CrossRef]
  11. R. Stevens and T. Miyashita, “Review of standards for micro lenses and micro lens arrays,” J. Imaging Sci. 58, 202–212 (2010).
  12. X. Jin, Y. Liu, and J. Yang, “Design, characterization and evaluation of high performance 2.8 μm pitch zero space microlens,” Opt. Commun. 284, 2357–2362 (2011).
    [CrossRef]
  13. L. Nino-de-Rivera, R. Calzada Salas, G. Duchen Sanchez, and J. A. Loaiza Brito, “Visual simulation of retinal images through microstructures,” Microelectron. Eng. 90, 159–162 (2012).
    [CrossRef]
  14. S. Donati, G. Martini, and E. Randone, “Improving photodetector performance by means of microoptics concentrators,” J. Lightwave Technol. 29, 661–665 (2011).
    [CrossRef]
  15. Y. Huo, C. C. Fesenmaier, and P. B. Catrysse, “Microlens performance limits in sub-2 μm pixel CMOS image sensors,” Opt. Express 18, 5861–5872 (2010).
    [CrossRef]
  16. V. J. Traver and F. Pla, “Log-polar mapping template design: from task-level requirements to geometry parameters,” Image Vis. Comput. 26, 1354–1370 (2008).
    [CrossRef]
  17. E. Hecht, Optics (Addison-Wesley, 2002).

2012 (2)

C. Posch, “Bio-inspired vision,” J. Instrum. 7, C01054 (2012).
[CrossRef]

L. Nino-de-Rivera, R. Calzada Salas, G. Duchen Sanchez, and J. A. Loaiza Brito, “Visual simulation of retinal images through microstructures,” Microelectron. Eng. 90, 159–162 (2012).
[CrossRef]

2011 (2)

X. Jin, Y. Liu, and J. Yang, “Design, characterization and evaluation of high performance 2.8 μm pitch zero space microlens,” Opt. Commun. 284, 2357–2362 (2011).
[CrossRef]

S. Donati, G. Martini, and E. Randone, “Improving photodetector performance by means of microoptics concentrators,” J. Lightwave Technol. 29, 661–665 (2011).
[CrossRef]

2010 (3)

Y. Huo, C. C. Fesenmaier, and P. B. Catrysse, “Microlens performance limits in sub-2 μm pixel CMOS image sensors,” Opt. Express 18, 5861–5872 (2010).
[CrossRef]

R. Stevens and T. Miyashita, “Review of standards for micro lenses and micro lens arrays,” J. Imaging Sci. 58, 202–212 (2010).

V. J. Traver and A. Bernardino, “A review of log-polar imaging for visual perception in robotics,” Robot. Auton. Syst. 58, 378–398 (2010).
[CrossRef]

2008 (1)

V. J. Traver and F. Pla, “Log-polar mapping template design: from task-level requirements to geometry parameters,” Image Vis. Comput. 26, 1354–1370 (2008).
[CrossRef]

2007 (1)

R. Maldonado-Lopez, F. Vidal-Verdu, G. Linan, E. Roca, and A. Rodriguez-Vazquez, “Early slip detection with a tactile sensor based on retina,” Analog Integr. Circuits Signal Process. 53, 97–108 (2007).
[CrossRef]

2006 (1)

M. Bigas, E. Cabruja, J. Forest, and J. Salvi, “Review of CMOS image sensors,” Microelectron. J. 37, 433–451 (2006).
[CrossRef]

2003 (1)

D. S. Park, J. H. Kim, H. S. Kim, J. H. Park, J. K. Shin, and M. Lee, “A foveated-structure CMOS retina chip for edge detection with local light adaptation,” Sens. Actuators A Phys. 108, 75–80 (2003).
[CrossRef]

1980 (1)

E. L. Schwartz, “A quantitative model of the functional architecture of human striate cortex with application to visual illusion and cortical texture analysis,” Biol. Cybern. 37, 63–76 (1980).
[CrossRef]

Bernardino, A.

V. J. Traver and A. Bernardino, “A review of log-polar imaging for visual perception in robotics,” Robot. Auton. Syst. 58, 378–398 (2010).
[CrossRef]

Bigas, M.

M. Bigas, E. Cabruja, J. Forest, and J. Salvi, “Review of CMOS image sensors,” Microelectron. J. 37, 433–451 (2006).
[CrossRef]

Boluda, J.

J. Perez, F. Pardo, J. Boluda, S. Felici, and B. Diericki, “Design of a foveated log-polar image sensor in standard CMOS technology,” in Design of Integrated Circuits and Systems (DCIS’96) (IEEE, 2011).

Boluda, J. A.

F. Pardo, J. A. Boluda, J. J. Perez, S. Felici, B. Diericki, and D. Scheffer, “Response properties of a foveated space-variant CMOS image sensor,” in Circuits and Systems (IEEE, 1996), pp. 373–376.

Cabruja, E.

M. Bigas, E. Cabruja, J. Forest, and J. Salvi, “Review of CMOS image sensors,” Microelectron. J. 37, 433–451 (2006).
[CrossRef]

Calzada Salas, R.

L. Nino-de-Rivera, R. Calzada Salas, G. Duchen Sanchez, and J. A. Loaiza Brito, “Visual simulation of retinal images through microstructures,” Microelectron. Eng. 90, 159–162 (2012).
[CrossRef]

Catrysse, P. B.

Diericki, B.

F. Pardo, J. A. Boluda, J. J. Perez, S. Felici, B. Diericki, and D. Scheffer, “Response properties of a foveated space-variant CMOS image sensor,” in Circuits and Systems (IEEE, 1996), pp. 373–376.

J. Perez, F. Pardo, J. Boluda, S. Felici, and B. Diericki, “Design of a foveated log-polar image sensor in standard CMOS technology,” in Design of Integrated Circuits and Systems (DCIS’96) (IEEE, 2011).

Donati, S.

Duchen Sanchez, G.

L. Nino-de-Rivera, R. Calzada Salas, G. Duchen Sanchez, and J. A. Loaiza Brito, “Visual simulation of retinal images through microstructures,” Microelectron. Eng. 90, 159–162 (2012).
[CrossRef]

Felici, S.

J. Perez, F. Pardo, J. Boluda, S. Felici, and B. Diericki, “Design of a foveated log-polar image sensor in standard CMOS technology,” in Design of Integrated Circuits and Systems (DCIS’96) (IEEE, 2011).

F. Pardo, J. A. Boluda, J. J. Perez, S. Felici, B. Diericki, and D. Scheffer, “Response properties of a foveated space-variant CMOS image sensor,” in Circuits and Systems (IEEE, 1996), pp. 373–376.

Fesenmaier, C. C.

Forest, J.

M. Bigas, E. Cabruja, J. Forest, and J. Salvi, “Review of CMOS image sensors,” Microelectron. J. 37, 433–451 (2006).
[CrossRef]

Hecht, E.

E. Hecht, Optics (Addison-Wesley, 2002).

Huo, Y.

Jin, X.

X. Jin, Y. Liu, and J. Yang, “Design, characterization and evaluation of high performance 2.8 μm pitch zero space microlens,” Opt. Commun. 284, 2357–2362 (2011).
[CrossRef]

Kim, H. S.

D. S. Park, J. H. Kim, H. S. Kim, J. H. Park, J. K. Shin, and M. Lee, “A foveated-structure CMOS retina chip for edge detection with local light adaptation,” Sens. Actuators A Phys. 108, 75–80 (2003).
[CrossRef]

Kim, J. H.

D. S. Park, J. H. Kim, H. S. Kim, J. H. Park, J. K. Shin, and M. Lee, “A foveated-structure CMOS retina chip for edge detection with local light adaptation,” Sens. Actuators A Phys. 108, 75–80 (2003).
[CrossRef]

Lee, M.

D. S. Park, J. H. Kim, H. S. Kim, J. H. Park, J. K. Shin, and M. Lee, “A foveated-structure CMOS retina chip for edge detection with local light adaptation,” Sens. Actuators A Phys. 108, 75–80 (2003).
[CrossRef]

Levine, M. D.

R. Wodnicki, G. W. Roberts, and M. D. Levine, “A foveated image sensor in standard CMOS technology,” in Custom Integrated Circuits Conference (IEEE, 1995), pp. 357–360.

Linan, G.

R. Maldonado-Lopez, F. Vidal-Verdu, G. Linan, E. Roca, and A. Rodriguez-Vazquez, “Early slip detection with a tactile sensor based on retina,” Analog Integr. Circuits Signal Process. 53, 97–108 (2007).
[CrossRef]

Liu, Y.

X. Jin, Y. Liu, and J. Yang, “Design, characterization and evaluation of high performance 2.8 μm pitch zero space microlens,” Opt. Commun. 284, 2357–2362 (2011).
[CrossRef]

Loaiza Brito, J. A.

L. Nino-de-Rivera, R. Calzada Salas, G. Duchen Sanchez, and J. A. Loaiza Brito, “Visual simulation of retinal images through microstructures,” Microelectron. Eng. 90, 159–162 (2012).
[CrossRef]

Maldonado-Lopez, R.

R. Maldonado-Lopez, F. Vidal-Verdu, G. Linan, E. Roca, and A. Rodriguez-Vazquez, “Early slip detection with a tactile sensor based on retina,” Analog Integr. Circuits Signal Process. 53, 97–108 (2007).
[CrossRef]

Martini, G.

Miyashita, T.

R. Stevens and T. Miyashita, “Review of standards for micro lenses and micro lens arrays,” J. Imaging Sci. 58, 202–212 (2010).

Nino-de-Rivera, L.

L. Nino-de-Rivera, R. Calzada Salas, G. Duchen Sanchez, and J. A. Loaiza Brito, “Visual simulation of retinal images through microstructures,” Microelectron. Eng. 90, 159–162 (2012).
[CrossRef]

Onkarappa, N.

N. Onkarappa and A. D. Sappa, “Space variant representations for mobile platform vision applications,” in Computer Analysis of Images and Patterns: 14th International Conference, CAIP 2011, A. Berciano, D. Diaz-Pernil, W. Kropatsch, and P. Molina-Abril Real, eds., (Springer, 2011), Vol. 2, pp. 146–154.

Pardo, F.

J. Perez, F. Pardo, J. Boluda, S. Felici, and B. Diericki, “Design of a foveated log-polar image sensor in standard CMOS technology,” in Design of Integrated Circuits and Systems (DCIS’96) (IEEE, 2011).

F. Pardo, J. A. Boluda, J. J. Perez, S. Felici, B. Diericki, and D. Scheffer, “Response properties of a foveated space-variant CMOS image sensor,” in Circuits and Systems (IEEE, 1996), pp. 373–376.

Park, D. S.

D. S. Park, J. H. Kim, H. S. Kim, J. H. Park, J. K. Shin, and M. Lee, “A foveated-structure CMOS retina chip for edge detection with local light adaptation,” Sens. Actuators A Phys. 108, 75–80 (2003).
[CrossRef]

Park, J. H.

D. S. Park, J. H. Kim, H. S. Kim, J. H. Park, J. K. Shin, and M. Lee, “A foveated-structure CMOS retina chip for edge detection with local light adaptation,” Sens. Actuators A Phys. 108, 75–80 (2003).
[CrossRef]

Perez, J.

J. Perez, F. Pardo, J. Boluda, S. Felici, and B. Diericki, “Design of a foveated log-polar image sensor in standard CMOS technology,” in Design of Integrated Circuits and Systems (DCIS’96) (IEEE, 2011).

Perez, J. J.

F. Pardo, J. A. Boluda, J. J. Perez, S. Felici, B. Diericki, and D. Scheffer, “Response properties of a foveated space-variant CMOS image sensor,” in Circuits and Systems (IEEE, 1996), pp. 373–376.

Pla, F.

V. J. Traver and F. Pla, “Log-polar mapping template design: from task-level requirements to geometry parameters,” Image Vis. Comput. 26, 1354–1370 (2008).
[CrossRef]

Posch, C.

C. Posch, “Bio-inspired vision,” J. Instrum. 7, C01054 (2012).
[CrossRef]

Randone, E.

Roberts, G. W.

R. Wodnicki, G. W. Roberts, and M. D. Levine, “A foveated image sensor in standard CMOS technology,” in Custom Integrated Circuits Conference (IEEE, 1995), pp. 357–360.

Roca, E.

R. Maldonado-Lopez, F. Vidal-Verdu, G. Linan, E. Roca, and A. Rodriguez-Vazquez, “Early slip detection with a tactile sensor based on retina,” Analog Integr. Circuits Signal Process. 53, 97–108 (2007).
[CrossRef]

Rodriguez-Vazquez, A.

R. Maldonado-Lopez, F. Vidal-Verdu, G. Linan, E. Roca, and A. Rodriguez-Vazquez, “Early slip detection with a tactile sensor based on retina,” Analog Integr. Circuits Signal Process. 53, 97–108 (2007).
[CrossRef]

Salvi, J.

M. Bigas, E. Cabruja, J. Forest, and J. Salvi, “Review of CMOS image sensors,” Microelectron. J. 37, 433–451 (2006).
[CrossRef]

Sappa, A. D.

N. Onkarappa and A. D. Sappa, “Space variant representations for mobile platform vision applications,” in Computer Analysis of Images and Patterns: 14th International Conference, CAIP 2011, A. Berciano, D. Diaz-Pernil, W. Kropatsch, and P. Molina-Abril Real, eds., (Springer, 2011), Vol. 2, pp. 146–154.

Scheffer, D.

F. Pardo, J. A. Boluda, J. J. Perez, S. Felici, B. Diericki, and D. Scheffer, “Response properties of a foveated space-variant CMOS image sensor,” in Circuits and Systems (IEEE, 1996), pp. 373–376.

Schwartz, E. L.

E. L. Schwartz, “A quantitative model of the functional architecture of human striate cortex with application to visual illusion and cortical texture analysis,” Biol. Cybern. 37, 63–76 (1980).
[CrossRef]

Shin, J. K.

D. S. Park, J. H. Kim, H. S. Kim, J. H. Park, J. K. Shin, and M. Lee, “A foveated-structure CMOS retina chip for edge detection with local light adaptation,” Sens. Actuators A Phys. 108, 75–80 (2003).
[CrossRef]

Stevens, R.

R. Stevens and T. Miyashita, “Review of standards for micro lenses and micro lens arrays,” J. Imaging Sci. 58, 202–212 (2010).

Traver, V. J.

V. J. Traver and A. Bernardino, “A review of log-polar imaging for visual perception in robotics,” Robot. Auton. Syst. 58, 378–398 (2010).
[CrossRef]

V. J. Traver and F. Pla, “Log-polar mapping template design: from task-level requirements to geometry parameters,” Image Vis. Comput. 26, 1354–1370 (2008).
[CrossRef]

Vidal-Verdu, F.

R. Maldonado-Lopez, F. Vidal-Verdu, G. Linan, E. Roca, and A. Rodriguez-Vazquez, “Early slip detection with a tactile sensor based on retina,” Analog Integr. Circuits Signal Process. 53, 97–108 (2007).
[CrossRef]

Wodnicki, R.

R. Wodnicki, G. W. Roberts, and M. D. Levine, “A foveated image sensor in standard CMOS technology,” in Custom Integrated Circuits Conference (IEEE, 1995), pp. 357–360.

Yang, J.

X. Jin, Y. Liu, and J. Yang, “Design, characterization and evaluation of high performance 2.8 μm pitch zero space microlens,” Opt. Commun. 284, 2357–2362 (2011).
[CrossRef]

Analog Integr. Circuits Signal Process. (1)

R. Maldonado-Lopez, F. Vidal-Verdu, G. Linan, E. Roca, and A. Rodriguez-Vazquez, “Early slip detection with a tactile sensor based on retina,” Analog Integr. Circuits Signal Process. 53, 97–108 (2007).
[CrossRef]

Biol. Cybern. (1)

E. L. Schwartz, “A quantitative model of the functional architecture of human striate cortex with application to visual illusion and cortical texture analysis,” Biol. Cybern. 37, 63–76 (1980).
[CrossRef]

Image Vis. Comput. (1)

V. J. Traver and F. Pla, “Log-polar mapping template design: from task-level requirements to geometry parameters,” Image Vis. Comput. 26, 1354–1370 (2008).
[CrossRef]

J. Imaging Sci. (1)

R. Stevens and T. Miyashita, “Review of standards for micro lenses and micro lens arrays,” J. Imaging Sci. 58, 202–212 (2010).

J. Instrum. (1)

C. Posch, “Bio-inspired vision,” J. Instrum. 7, C01054 (2012).
[CrossRef]

J. Lightwave Technol. (1)

Microelectron. Eng. (1)

L. Nino-de-Rivera, R. Calzada Salas, G. Duchen Sanchez, and J. A. Loaiza Brito, “Visual simulation of retinal images through microstructures,” Microelectron. Eng. 90, 159–162 (2012).
[CrossRef]

Microelectron. J. (1)

M. Bigas, E. Cabruja, J. Forest, and J. Salvi, “Review of CMOS image sensors,” Microelectron. J. 37, 433–451 (2006).
[CrossRef]

Opt. Commun. (1)

X. Jin, Y. Liu, and J. Yang, “Design, characterization and evaluation of high performance 2.8 μm pitch zero space microlens,” Opt. Commun. 284, 2357–2362 (2011).
[CrossRef]

Opt. Express (1)

Robot. Auton. Syst. (1)

V. J. Traver and A. Bernardino, “A review of log-polar imaging for visual perception in robotics,” Robot. Auton. Syst. 58, 378–398 (2010).
[CrossRef]

Sens. Actuators A Phys. (1)

D. S. Park, J. H. Kim, H. S. Kim, J. H. Park, J. K. Shin, and M. Lee, “A foveated-structure CMOS retina chip for edge detection with local light adaptation,” Sens. Actuators A Phys. 108, 75–80 (2003).
[CrossRef]

Other (5)

E. Hecht, Optics (Addison-Wesley, 2002).

N. Onkarappa and A. D. Sappa, “Space variant representations for mobile platform vision applications,” in Computer Analysis of Images and Patterns: 14th International Conference, CAIP 2011, A. Berciano, D. Diaz-Pernil, W. Kropatsch, and P. Molina-Abril Real, eds., (Springer, 2011), Vol. 2, pp. 146–154.

R. Wodnicki, G. W. Roberts, and M. D. Levine, “A foveated image sensor in standard CMOS technology,” in Custom Integrated Circuits Conference (IEEE, 1995), pp. 357–360.

J. Perez, F. Pardo, J. Boluda, S. Felici, and B. Diericki, “Design of a foveated log-polar image sensor in standard CMOS technology,” in Design of Integrated Circuits and Systems (DCIS’96) (IEEE, 2011).

F. Pardo, J. A. Boluda, J. J. Perez, S. Felici, B. Diericki, and D. Scheffer, “Response properties of a foveated space-variant CMOS image sensor,” in Circuits and Systems (IEEE, 1996), pp. 373–376.

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Figures (16)

Fig. 1.
Fig. 1.

Space-variant distribution of photodetectors in retina-like image sensor.

Fig. 2.
Fig. 2.

Rotation and scaling invariance property of retina-like image sensor. (b) is the original Cartesian image with a target “ D ,” while (e) is the corresponding log-polar image. (a) is the Cartesian image with a rotated target compared with (b), while (d) is the corresponding log-polar image. (c) is the Cartesian image with a scaled target compared with (b), while (f) is the corresponding log-polar image of (c).

Fig. 3.
Fig. 3.

Proposed retina-like sensor based on space-variant lens array.

Fig. 4.
Fig. 4.

System structure of the retina-like image sensor based on space-variant lens array and its application.

Fig. 5.
Fig. 5.

Working process of the retina-like image sensor based on space-variant lens array, in which (a) is the cross section of emergent light of telescope optical system, (b) is the space-variant lens array, (c) is the photodetector array, and (d) is the corresponding log-polar mapping result.

Fig. 6.
Fig. 6.

Parameters of the space-variant lens array.

Fig. 7.
Fig. 7.

Parameters of the plano-convex lens with respect to the i th ring of the space-variant lens array.

Fig. 8.
Fig. 8.

Fill factor of the space-variant lens array. (a) Gap among the adjacent lenses. (b) Definition of the fill factor, which is the ratio of the total area of lenses ( s f ) and the area ( s ) of the ring with the minimum radius r 0 and the maximum radius r max .

Fig. 9.
Fig. 9.

Simulation results of the space-variant lens array with the input parameters of { M , N , r max , n , d , f } = { 20 , 50 , 2.3 , 1.3 , 0.004 , 0.5 } . (a) 2D view. (b) 3D view.

Fig. 10.
Fig. 10.

Simulation results of the space-variant lens array with the input parameters of { M , N , r max , n , d , f } = { 12 , 25 , 3.3 , 1.3 , 0.004 , 1.6 } . (a) 2D view. (b) 3D view.

Fig. 11.
Fig. 11.

Simulation results of the space-variant lens array with the input parameters of { M , N , r 0 , n , d , f } = { 10 , 30 , 3 , 1.5 , 0.1 , 10 } . (a) 2D view. (b) 3D view.

Fig. 12.
Fig. 12.

Relationship among r 0 , M , and N under the condition that r max = 3.3 mm . (a) 2D view. (b) 3D view.

Fig. 13.
Fig. 13.

Influence of the radii of the medial rings ( r i ). (a) Relationship between q and N in the case that N increases from 5 to 50. (b) Relationships among r i , M and q .

Fig. 14.
Fig. 14.

Relationship between the lens diameter ( D i ) and M in cases that N = 10 , 20, 30, 40, and 50.

Fig. 15.
Fig. 15.

Relationship among h 1 , N , and f under the conditions that r 0 = 6 , n = 1.5 nm , N = 10 100 and f = 6 , 8, 10, 12, and 14 mm.

Fig. 16.
Fig. 16.

Relationship between the fill factor ( η f ) and N .

Equations (16)

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

{ r i = x 2 + y 2 α j = arctan ( y x ) .
z = x + i y = r i cos α j + r i sin α j = r i q i α j ,
w = log q ( z ) = log q ( r i q i α j ) = log q ( r i ) + i α j = ρ i + i θ j .
ρ i = log q ( r i ) = log q ( k r i ) = log q k + ρ i .
r 1 = r 0 1 sin β .
D 1 = 2 r 1 sin β = 2 r 0 sin β 1 sin β .
{ r i = r 1 · q i 1 = r 0 ( 1 + sin β ) i 1 ( 1 sin β ) i D i = D 1 · q i 1 = 2 r 0 sin β ( 1 + sin β ) i 1 ( 1 sin β ) i i = 1 , 2 , 3 M .
r max = r 0 + i = 1 M D i = r 0 · q M = r 0 · ( 1 + sin β 1 sin β ) M .
{ R = f · ( n 1 ) h i = R 1 2 4 R 2 D i 2 .
f D M 2 ( n 1 ) = r 0 · q M 1 · sin ( π N ) ( n 1 ) · [ 1 sin ( π N ) ] .
u = arctan ( d 2 f ) .
s = π ( r max 2 r 0 2 ) = π r 0 2 ( q 2 M 1 ) .
s f = N · π i = 1 M ( D i 2 ) 2 = π 4 N r 0 2 ( q 2 M 1 ) sin ( π N ) .
η f = s f s = π 4 N r 0 2 ( q 2 M 1 ) sin ( π N ) π r 0 2 ( q 2 M 1 ) = N 4 sin ( π N ) .
η f = lim n N 4 sin ( π N ) = N 4 · ( π N ) = π 4 .
{ β = π N r i = r 0 ( 1 + sin β ) i 1 ( 1 sin β ) i r max = r 0 · ( 1 + sin β 1 sin β ) M D i = 2 r 0 sin β ( 1 + sin β ) i 1 ( 1 sin β ) i f r 0 · ( 1 + sin β 1 sin β ) M 1 · sin β ( n 1 ) ( 1 sin β ) R = f · ( n 1 ) h i = R 1 2 4 R 2 D i 2 u = arctan ( d 2 f ) η f = N 4 sin β i = 1 , 2 , 3 M .

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