Abstract

In the field of industrial vision, the three-dimensional inspection of highly reflective metallic objects is still a delicate task. We deal with a new automated three-dimensional inspection system based on polarization analysis. We first present an extension of the shape-from-polarization method for dielectric surfaces to metallic surfaces. Then, we describe what we believe to be a new way of solving the ambiguity concerning the normal orientation with an active lighting system. Finally, applications to shape-defect detection are discussed, and the efficiency of the system to discriminate defects on specular metallic objects made by stamping and polishing is presented.

© 2006 Optical Society of America

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  1. I. Yun, E. Jung, and S. Lee, "On the fast shape recovery technique using multiple ring lights," Pattern Recogn. 30, 883-893 (1997).
    [Crossref]
  2. S. Nayar, A. Sanderson, L. Weiss, and D. Simon, "Specular surface inspection using structured highlight and Gaussian images," IEEE Trans. Rob. Autom. 6, 208-218 (1990).
    [Crossref]
  3. S. Savarese and P. Perona, "Local analysis for 3D reconstruction of specular surfaces," in IEEE Computer Vision and Pattern Recognition (IEEE, 2001), Vol. 2, pp. 738-745.
  4. S. Savarese and P. Perona, "Local analysis for 3D reconstruction of specular surfaces: Part II," in Proceeding of the European Conference on Computer Vision (Springer-Verlag, 2002), pp. 759-774.
  5. J. Zheng and A. Murata, "Acquiring a complete 3D model from specular motion under the illumination of circular-shaped light sources," IEEE Trans. Pattern Anal. Mach. Intell. 22, 913-920 (2000).
    [Crossref]
  6. K. Koshikawa and Y. Shirai, "A model-based recognition of glossy objects using their polarimetrical properties," Adv. Rob. 2, 137-147 (1987).
    [Crossref]
  7. L. B. Wolff and T. E. Boult, "Constraining object features using a polarization reflectance model," IEEE Trans. Pattern Anal. Mach. Intell. 13, 635-657 (1991).
    [Crossref]
  8. D. Miyazaki, M. Kagesawa, and K. Ikeuchi, "Determining shapes of transparent objects from two polarization images," in Proceedings of the International Association for Pattern Recognition Workshop on Machine Vision Applications (Springer-Verlag, 2002), pp. 26-31.
  9. D. Miyazaki, M. Kagesawa, and K. Ikeuchi, "Transparent surface modeling from a pair of polarization images," IEEE Trans. Pattern Anal. Mach. Intell. 26, 73-82 (2004).
    [Crossref] [PubMed]
  10. S. Rahmann, "Inferring 3D scene structure from a single polarization image," in Proc. SPIE 3826, 22-33 (1999).
    [Crossref]
  11. S. Rahmann and N. Canterakis, "Reconstruction of specular surfaces using polarization imaging," in IEEE Computer Vision and Pattern Recognition (IEEE, 2001), Vol. 1, pp. 149-155.
  12. O. Morel, C. Stolz, and P. Gorria, "Application of polarimetric imaging to 3D inspection of highly reflective metallic surface," in Proc. SPIE 5606, 82-89 (2004).
    [Crossref]
  13. O. Morel, C. Stolz, F. Meriaudeau, and P. Gorria, "Polarization imaging applied to 3D inspection of specular metallic surfaces," in Proc. SPIE 5679,178-186 (2005).
  14. O. Morel, C. Stolz, F. Meriaudeau, and P. Gorria, "Three-dimensional inspection of highly-reflective metallic objects by polarization imaging," Electron. Imag. Newslett. 15(2), 4 (2005).
  15. L. B. Wolff, "Polarization vision: a new sensory approach to image understanding," Image Vision Comput. 15, 81-93 (1997).
    [Crossref]
  16. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).
  17. R. Frankot and R. Chellappa, "A method for enforcing integrability in shape from shading algorithms," IEEE Trans. Pattern Anal. Mach. Intell. 10, 439-451 (1988).
    [Crossref]
  18. K. Ikeuchi, "Determining surface orientations of specular surfaces by using the photometric stereo method," IEEE Trans. Pattern Anal. Mach. Intell. 3, 661-669 (1981).
    [Crossref] [PubMed]
  19. R. Seulin, F. Merienne, and P. Gorria, "Simulation of specular surface imaging based on computer graphics: application on a vision inspection system," EURASIP J. Appl. Signal Process. 2002, 649-658 (2002).
    [Crossref]
  20. P. Besl and N. McKay, "A method for registration of 3-d shapes," IEEE Trans. Pattern Anal. Mach. Intell. 14, 239-256 (1992).
    [Crossref]
  21. C. Matabosch, J. Salvi, D. Fofi, and F. Meriaudeau, "Range image registration for industrial inspection," in Proc. SPIE 5679, 216-227 (2005).

2005 (1)

O. Morel, C. Stolz, F. Meriaudeau, and P. Gorria, "Three-dimensional inspection of highly-reflective metallic objects by polarization imaging," Electron. Imag. Newslett. 15(2), 4 (2005).

2004 (2)

D. Miyazaki, M. Kagesawa, and K. Ikeuchi, "Transparent surface modeling from a pair of polarization images," IEEE Trans. Pattern Anal. Mach. Intell. 26, 73-82 (2004).
[Crossref] [PubMed]

O. Morel, C. Stolz, and P. Gorria, "Application of polarimetric imaging to 3D inspection of highly reflective metallic surface," in Proc. SPIE 5606, 82-89 (2004).
[Crossref]

2002 (1)

R. Seulin, F. Merienne, and P. Gorria, "Simulation of specular surface imaging based on computer graphics: application on a vision inspection system," EURASIP J. Appl. Signal Process. 2002, 649-658 (2002).
[Crossref]

2000 (1)

J. Zheng and A. Murata, "Acquiring a complete 3D model from specular motion under the illumination of circular-shaped light sources," IEEE Trans. Pattern Anal. Mach. Intell. 22, 913-920 (2000).
[Crossref]

1999 (1)

S. Rahmann, "Inferring 3D scene structure from a single polarization image," in Proc. SPIE 3826, 22-33 (1999).
[Crossref]

1997 (2)

I. Yun, E. Jung, and S. Lee, "On the fast shape recovery technique using multiple ring lights," Pattern Recogn. 30, 883-893 (1997).
[Crossref]

L. B. Wolff, "Polarization vision: a new sensory approach to image understanding," Image Vision Comput. 15, 81-93 (1997).
[Crossref]

1992 (1)

P. Besl and N. McKay, "A method for registration of 3-d shapes," IEEE Trans. Pattern Anal. Mach. Intell. 14, 239-256 (1992).
[Crossref]

1991 (1)

L. B. Wolff and T. E. Boult, "Constraining object features using a polarization reflectance model," IEEE Trans. Pattern Anal. Mach. Intell. 13, 635-657 (1991).
[Crossref]

1990 (1)

S. Nayar, A. Sanderson, L. Weiss, and D. Simon, "Specular surface inspection using structured highlight and Gaussian images," IEEE Trans. Rob. Autom. 6, 208-218 (1990).
[Crossref]

1988 (1)

R. Frankot and R. Chellappa, "A method for enforcing integrability in shape from shading algorithms," IEEE Trans. Pattern Anal. Mach. Intell. 10, 439-451 (1988).
[Crossref]

1987 (1)

K. Koshikawa and Y. Shirai, "A model-based recognition of glossy objects using their polarimetrical properties," Adv. Rob. 2, 137-147 (1987).
[Crossref]

1981 (1)

K. Ikeuchi, "Determining surface orientations of specular surfaces by using the photometric stereo method," IEEE Trans. Pattern Anal. Mach. Intell. 3, 661-669 (1981).
[Crossref] [PubMed]

Besl, P.

P. Besl and N. McKay, "A method for registration of 3-d shapes," IEEE Trans. Pattern Anal. Mach. Intell. 14, 239-256 (1992).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Boult, T. E.

L. B. Wolff and T. E. Boult, "Constraining object features using a polarization reflectance model," IEEE Trans. Pattern Anal. Mach. Intell. 13, 635-657 (1991).
[Crossref]

Canterakis, N.

S. Rahmann and N. Canterakis, "Reconstruction of specular surfaces using polarization imaging," in IEEE Computer Vision and Pattern Recognition (IEEE, 2001), Vol. 1, pp. 149-155.

Chellappa, R.

R. Frankot and R. Chellappa, "A method for enforcing integrability in shape from shading algorithms," IEEE Trans. Pattern Anal. Mach. Intell. 10, 439-451 (1988).
[Crossref]

Fofi, D.

C. Matabosch, J. Salvi, D. Fofi, and F. Meriaudeau, "Range image registration for industrial inspection," in Proc. SPIE 5679, 216-227 (2005).

Frankot, R.

R. Frankot and R. Chellappa, "A method for enforcing integrability in shape from shading algorithms," IEEE Trans. Pattern Anal. Mach. Intell. 10, 439-451 (1988).
[Crossref]

Gorria, P.

O. Morel, C. Stolz, F. Meriaudeau, and P. Gorria, "Three-dimensional inspection of highly-reflective metallic objects by polarization imaging," Electron. Imag. Newslett. 15(2), 4 (2005).

O. Morel, C. Stolz, and P. Gorria, "Application of polarimetric imaging to 3D inspection of highly reflective metallic surface," in Proc. SPIE 5606, 82-89 (2004).
[Crossref]

R. Seulin, F. Merienne, and P. Gorria, "Simulation of specular surface imaging based on computer graphics: application on a vision inspection system," EURASIP J. Appl. Signal Process. 2002, 649-658 (2002).
[Crossref]

O. Morel, C. Stolz, F. Meriaudeau, and P. Gorria, "Polarization imaging applied to 3D inspection of specular metallic surfaces," in Proc. SPIE 5679,178-186 (2005).

Ikeuchi, K.

D. Miyazaki, M. Kagesawa, and K. Ikeuchi, "Transparent surface modeling from a pair of polarization images," IEEE Trans. Pattern Anal. Mach. Intell. 26, 73-82 (2004).
[Crossref] [PubMed]

K. Ikeuchi, "Determining surface orientations of specular surfaces by using the photometric stereo method," IEEE Trans. Pattern Anal. Mach. Intell. 3, 661-669 (1981).
[Crossref] [PubMed]

D. Miyazaki, M. Kagesawa, and K. Ikeuchi, "Determining shapes of transparent objects from two polarization images," in Proceedings of the International Association for Pattern Recognition Workshop on Machine Vision Applications (Springer-Verlag, 2002), pp. 26-31.

Jung, E.

I. Yun, E. Jung, and S. Lee, "On the fast shape recovery technique using multiple ring lights," Pattern Recogn. 30, 883-893 (1997).
[Crossref]

Kagesawa, M.

D. Miyazaki, M. Kagesawa, and K. Ikeuchi, "Transparent surface modeling from a pair of polarization images," IEEE Trans. Pattern Anal. Mach. Intell. 26, 73-82 (2004).
[Crossref] [PubMed]

D. Miyazaki, M. Kagesawa, and K. Ikeuchi, "Determining shapes of transparent objects from two polarization images," in Proceedings of the International Association for Pattern Recognition Workshop on Machine Vision Applications (Springer-Verlag, 2002), pp. 26-31.

Koshikawa, K.

K. Koshikawa and Y. Shirai, "A model-based recognition of glossy objects using their polarimetrical properties," Adv. Rob. 2, 137-147 (1987).
[Crossref]

Lee, S.

I. Yun, E. Jung, and S. Lee, "On the fast shape recovery technique using multiple ring lights," Pattern Recogn. 30, 883-893 (1997).
[Crossref]

Matabosch, C.

C. Matabosch, J. Salvi, D. Fofi, and F. Meriaudeau, "Range image registration for industrial inspection," in Proc. SPIE 5679, 216-227 (2005).

McKay, N.

P. Besl and N. McKay, "A method for registration of 3-d shapes," IEEE Trans. Pattern Anal. Mach. Intell. 14, 239-256 (1992).
[Crossref]

Meriaudeau, F.

O. Morel, C. Stolz, F. Meriaudeau, and P. Gorria, "Three-dimensional inspection of highly-reflective metallic objects by polarization imaging," Electron. Imag. Newslett. 15(2), 4 (2005).

O. Morel, C. Stolz, F. Meriaudeau, and P. Gorria, "Polarization imaging applied to 3D inspection of specular metallic surfaces," in Proc. SPIE 5679,178-186 (2005).

C. Matabosch, J. Salvi, D. Fofi, and F. Meriaudeau, "Range image registration for industrial inspection," in Proc. SPIE 5679, 216-227 (2005).

Merienne, F.

R. Seulin, F. Merienne, and P. Gorria, "Simulation of specular surface imaging based on computer graphics: application on a vision inspection system," EURASIP J. Appl. Signal Process. 2002, 649-658 (2002).
[Crossref]

Miyazaki, D.

D. Miyazaki, M. Kagesawa, and K. Ikeuchi, "Transparent surface modeling from a pair of polarization images," IEEE Trans. Pattern Anal. Mach. Intell. 26, 73-82 (2004).
[Crossref] [PubMed]

D. Miyazaki, M. Kagesawa, and K. Ikeuchi, "Determining shapes of transparent objects from two polarization images," in Proceedings of the International Association for Pattern Recognition Workshop on Machine Vision Applications (Springer-Verlag, 2002), pp. 26-31.

Morel, O.

O. Morel, C. Stolz, F. Meriaudeau, and P. Gorria, "Three-dimensional inspection of highly-reflective metallic objects by polarization imaging," Electron. Imag. Newslett. 15(2), 4 (2005).

O. Morel, C. Stolz, and P. Gorria, "Application of polarimetric imaging to 3D inspection of highly reflective metallic surface," in Proc. SPIE 5606, 82-89 (2004).
[Crossref]

O. Morel, C. Stolz, F. Meriaudeau, and P. Gorria, "Polarization imaging applied to 3D inspection of specular metallic surfaces," in Proc. SPIE 5679,178-186 (2005).

Murata, A.

J. Zheng and A. Murata, "Acquiring a complete 3D model from specular motion under the illumination of circular-shaped light sources," IEEE Trans. Pattern Anal. Mach. Intell. 22, 913-920 (2000).
[Crossref]

Nayar, S.

S. Nayar, A. Sanderson, L. Weiss, and D. Simon, "Specular surface inspection using structured highlight and Gaussian images," IEEE Trans. Rob. Autom. 6, 208-218 (1990).
[Crossref]

Perona, P.

S. Savarese and P. Perona, "Local analysis for 3D reconstruction of specular surfaces," in IEEE Computer Vision and Pattern Recognition (IEEE, 2001), Vol. 2, pp. 738-745.

S. Savarese and P. Perona, "Local analysis for 3D reconstruction of specular surfaces: Part II," in Proceeding of the European Conference on Computer Vision (Springer-Verlag, 2002), pp. 759-774.

Rahmann, S.

S. Rahmann, "Inferring 3D scene structure from a single polarization image," in Proc. SPIE 3826, 22-33 (1999).
[Crossref]

S. Rahmann and N. Canterakis, "Reconstruction of specular surfaces using polarization imaging," in IEEE Computer Vision and Pattern Recognition (IEEE, 2001), Vol. 1, pp. 149-155.

Salvi, J.

C. Matabosch, J. Salvi, D. Fofi, and F. Meriaudeau, "Range image registration for industrial inspection," in Proc. SPIE 5679, 216-227 (2005).

Sanderson, A.

S. Nayar, A. Sanderson, L. Weiss, and D. Simon, "Specular surface inspection using structured highlight and Gaussian images," IEEE Trans. Rob. Autom. 6, 208-218 (1990).
[Crossref]

Savarese, S.

S. Savarese and P. Perona, "Local analysis for 3D reconstruction of specular surfaces: Part II," in Proceeding of the European Conference on Computer Vision (Springer-Verlag, 2002), pp. 759-774.

S. Savarese and P. Perona, "Local analysis for 3D reconstruction of specular surfaces," in IEEE Computer Vision and Pattern Recognition (IEEE, 2001), Vol. 2, pp. 738-745.

Seulin, R.

R. Seulin, F. Merienne, and P. Gorria, "Simulation of specular surface imaging based on computer graphics: application on a vision inspection system," EURASIP J. Appl. Signal Process. 2002, 649-658 (2002).
[Crossref]

Shirai, Y.

K. Koshikawa and Y. Shirai, "A model-based recognition of glossy objects using their polarimetrical properties," Adv. Rob. 2, 137-147 (1987).
[Crossref]

Simon, D.

S. Nayar, A. Sanderson, L. Weiss, and D. Simon, "Specular surface inspection using structured highlight and Gaussian images," IEEE Trans. Rob. Autom. 6, 208-218 (1990).
[Crossref]

Stolz, C.

O. Morel, C. Stolz, F. Meriaudeau, and P. Gorria, "Three-dimensional inspection of highly-reflective metallic objects by polarization imaging," Electron. Imag. Newslett. 15(2), 4 (2005).

O. Morel, C. Stolz, and P. Gorria, "Application of polarimetric imaging to 3D inspection of highly reflective metallic surface," in Proc. SPIE 5606, 82-89 (2004).
[Crossref]

O. Morel, C. Stolz, F. Meriaudeau, and P. Gorria, "Polarization imaging applied to 3D inspection of specular metallic surfaces," in Proc. SPIE 5679,178-186 (2005).

Weiss, L.

S. Nayar, A. Sanderson, L. Weiss, and D. Simon, "Specular surface inspection using structured highlight and Gaussian images," IEEE Trans. Rob. Autom. 6, 208-218 (1990).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Wolff, L. B.

L. B. Wolff, "Polarization vision: a new sensory approach to image understanding," Image Vision Comput. 15, 81-93 (1997).
[Crossref]

L. B. Wolff and T. E. Boult, "Constraining object features using a polarization reflectance model," IEEE Trans. Pattern Anal. Mach. Intell. 13, 635-657 (1991).
[Crossref]

Yun, I.

I. Yun, E. Jung, and S. Lee, "On the fast shape recovery technique using multiple ring lights," Pattern Recogn. 30, 883-893 (1997).
[Crossref]

Zheng, J.

J. Zheng and A. Murata, "Acquiring a complete 3D model from specular motion under the illumination of circular-shaped light sources," IEEE Trans. Pattern Anal. Mach. Intell. 22, 913-920 (2000).
[Crossref]

Adv. Rob. (1)

K. Koshikawa and Y. Shirai, "A model-based recognition of glossy objects using their polarimetrical properties," Adv. Rob. 2, 137-147 (1987).
[Crossref]

Electron. Imag. Newslett. (1)

O. Morel, C. Stolz, F. Meriaudeau, and P. Gorria, "Three-dimensional inspection of highly-reflective metallic objects by polarization imaging," Electron. Imag. Newslett. 15(2), 4 (2005).

EURASIP J. Appl. Signal Process. (1)

R. Seulin, F. Merienne, and P. Gorria, "Simulation of specular surface imaging based on computer graphics: application on a vision inspection system," EURASIP J. Appl. Signal Process. 2002, 649-658 (2002).
[Crossref]

IEEE Trans. Pattern Anal. Mach. Intell. (6)

P. Besl and N. McKay, "A method for registration of 3-d shapes," IEEE Trans. Pattern Anal. Mach. Intell. 14, 239-256 (1992).
[Crossref]

R. Frankot and R. Chellappa, "A method for enforcing integrability in shape from shading algorithms," IEEE Trans. Pattern Anal. Mach. Intell. 10, 439-451 (1988).
[Crossref]

K. Ikeuchi, "Determining surface orientations of specular surfaces by using the photometric stereo method," IEEE Trans. Pattern Anal. Mach. Intell. 3, 661-669 (1981).
[Crossref] [PubMed]

L. B. Wolff and T. E. Boult, "Constraining object features using a polarization reflectance model," IEEE Trans. Pattern Anal. Mach. Intell. 13, 635-657 (1991).
[Crossref]

D. Miyazaki, M. Kagesawa, and K. Ikeuchi, "Transparent surface modeling from a pair of polarization images," IEEE Trans. Pattern Anal. Mach. Intell. 26, 73-82 (2004).
[Crossref] [PubMed]

J. Zheng and A. Murata, "Acquiring a complete 3D model from specular motion under the illumination of circular-shaped light sources," IEEE Trans. Pattern Anal. Mach. Intell. 22, 913-920 (2000).
[Crossref]

IEEE Trans. Rob. Autom. (1)

S. Nayar, A. Sanderson, L. Weiss, and D. Simon, "Specular surface inspection using structured highlight and Gaussian images," IEEE Trans. Rob. Autom. 6, 208-218 (1990).
[Crossref]

Image Vision Comput. (1)

L. B. Wolff, "Polarization vision: a new sensory approach to image understanding," Image Vision Comput. 15, 81-93 (1997).
[Crossref]

Pattern Recogn. (1)

I. Yun, E. Jung, and S. Lee, "On the fast shape recovery technique using multiple ring lights," Pattern Recogn. 30, 883-893 (1997).
[Crossref]

Proc. SPIE (2)

S. Rahmann, "Inferring 3D scene structure from a single polarization image," in Proc. SPIE 3826, 22-33 (1999).
[Crossref]

O. Morel, C. Stolz, and P. Gorria, "Application of polarimetric imaging to 3D inspection of highly reflective metallic surface," in Proc. SPIE 5606, 82-89 (2004).
[Crossref]

Other (7)

O. Morel, C. Stolz, F. Meriaudeau, and P. Gorria, "Polarization imaging applied to 3D inspection of specular metallic surfaces," in Proc. SPIE 5679,178-186 (2005).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

C. Matabosch, J. Salvi, D. Fofi, and F. Meriaudeau, "Range image registration for industrial inspection," in Proc. SPIE 5679, 216-227 (2005).

S. Rahmann and N. Canterakis, "Reconstruction of specular surfaces using polarization imaging," in IEEE Computer Vision and Pattern Recognition (IEEE, 2001), Vol. 1, pp. 149-155.

D. Miyazaki, M. Kagesawa, and K. Ikeuchi, "Determining shapes of transparent objects from two polarization images," in Proceedings of the International Association for Pattern Recognition Workshop on Machine Vision Applications (Springer-Verlag, 2002), pp. 26-31.

S. Savarese and P. Perona, "Local analysis for 3D reconstruction of specular surfaces," in IEEE Computer Vision and Pattern Recognition (IEEE, 2001), Vol. 2, pp. 738-745.

S. Savarese and P. Perona, "Local analysis for 3D reconstruction of specular surfaces: Part II," in Proceeding of the European Conference on Computer Vision (Springer-Verlag, 2002), pp. 759-774.

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Figures (14)

Fig. 1
Fig. 1

Light-wave reflection on a specular surface.

Fig. 2
Fig. 2

Variation of the light intensity according to the angle of the polarizer α.

Fig. 3
Fig. 3

Comparison between the approximated degree of polarization and the real degree of polarization for a metallic surface ( n ^ = 1.94 + 5.28 i ) .

Fig. 4
Fig. 4

Expanded view of the active lighting system.

Fig. 5
Fig. 5

Disambiguation of the azimuth angle: (a) without disambiguation (ϕ is defined modulo π), (b) theoretical values (ϕ is defined modulo 2π), (c) result of the disambiguation (ϕ is defined modulo 2π).

Fig. 6
Fig. 6

Acquisition principle of the segmented image I quad.

Fig. 7
Fig. 7

Experimental setup.

Fig. 8
Fig. 8

Error images of the normals e : (a) polarimetric system, (b) Replica scanner, (c) Minolta VI-910 scanner.

Fig. 9
Fig. 9

Calibration curve.

Fig. 10
Fig. 10

3D reconstruction of a specular metallic object:    (a) surface from the 3D scanner, (b) surface from our method.

Fig. 11
Fig. 11

Comparison of the cross sections (scales given in millimeters).

Fig. 12
Fig. 12

(a) Object reference photograph and (b) its 3D reconstructed surface.

Fig. 13
Fig. 13

(a) Photograph of an object with a shape defect and (b) its 3D reconstructed surface.

Fig. 14
Fig. 14

Mean deviation between the two reconstructed surfaces.

Tables (1)

Tables Icon

Table 1 Mean Error on Normals Acquisition

Equations (15)

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

n = ( f ( x , y ) x f ( x , y ) y 1 ) = ( p = tan θ r cos ϕ q = tan θ r sin ϕ 1 ) .
I p ( α ) = I max I min 2 cos ( 2 α 2 φ ) + I max + I min 2 ,
I p ( α ) = I 2 [ ρ cos ( 2 α 2 φ ) + 1 ] ,
I = I max + I min , ρ = I max I min I max + I min .
F = | sin ( θ i θ t ) sin ( θ i + θ t ) | 2 F = | tan ( θ i θ t ) tan ( θ i + θ t ) | 2 } ,
ϕ = φ ± π 2 .
ρ = F F F + F .
sin θ i = n ^ sin θ t ,
| n ^ | 2 = n 2 ( 1 + κ 2 ) 1.
ρ ( θ r ) = 2 n tan θ r sin θ r tan 2 θ r sin 2 θ r + | n ^ | 2 .
( u , v ) ( 0 , 0 ) , f ˜ ( u , v ) = j u p ˜ j v q ˜ u 2 + v 2 .
f ( x , y ) = f 0 ( x , y ) + g ( x , y ) ,
g ( x , y ) = x p ˜ ( 0 , 0 ) + y q ˜ ( 0 , 0 ) + cst ,
ϕ = φ π / 2 + { 0 π .
e = n th n m ,

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