Abstract

The image of an object can vary dramatically, depending on lighting, specularities, reflections, and shadows. It is often advantageous to separate these incidental variations from the intrinsic aspects of an image. We describe a method for photographing objects behind glass and digitally removing the reflections from the surface of the glass, leaving the image of the objects behind the glass intact. We describe the details of this method, which employs simple optical techniques and independent component analysis and show its efficacy with several examples.

© 1999 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1965).
  2. S. Ullman, “On visual detection of light sources,” Bio. Cybern. 21, 205–212 (1976).
    [CrossRef]
  3. G. Brelstaff, A. Blake, “Detecting specular reflections using Lambertian constraints,” in International Conference on Computer Vision (IEEE, New York, 1988), pp. 297–302.
  4. S. Shafer, “Using color to separate reflection components,” Color Res. Appl. 10, 210–218 (1985).
    [CrossRef]
  5. R. Bajcsy, S. W. Lee, A. Leonardis, “Color image segmentation with detection of highlights and local illumination induced by interreflections,” in Proceedings of International Conference on Pattern Recognition (IEEE, Piscataway, N.J., 1990), pp. 785–790.
  6. G. J. Klinker, S. A. Shafer, T. Kanade, “The measurement of highlights in color images,” Int. J. Comput. Vis. 2(1), 7–32 (1990).
    [CrossRef]
  7. L. B. Wolff, “Polarization-based material classification from specular reflection,” IEEE Trans. Pattern. Anal. Mach. Intell. 12, 1059–1071 (1990).
    [CrossRef]
  8. L. B. Wolff, T. E. Boult, “Constraining object features using a polarization reflectance model,” IEEE Trans. Pattern. Anal. Mach. Intell. 13, 635–657 (1991).
    [CrossRef]
  9. Y. Y. Schechner, N. Kiryati, J. Shamir, “Separation of transparent layers by polarization analysis,” presented at the IAPR 11th Scandinavian Conference on Image Analysis, June 7–11, 1999, Kangerlussuaq, Greeland.
  10. S. K. Nayar, X. Fang, B. Terrance, “Separation of reflection components using color and polarization,” Int. J. Comput. Vis. 21(3), 163–186 (1997).
    [CrossRef]
  11. J. Cardoso, “Source separation using higher order moments,” in International Conference on Acoustics, Speech and Signal Processing (IEEE, Piscataway, N.J., 1989), pp. 2109–2112.
  12. P. Comon, “Separation of stochastic processes,” in Workshop on Higher-Order Spectral Analysis (IEEE, New York, 1989), pp. 174–179.
  13. Y. Bar-Ness, “Bootstrapping adaptive interference cancelers: some practical limitations,” in The Globecom Conference (IEEE, New York, 1982), pp. 1251–1255.
  14. Y. Inouye, T. Matsui, “Cumulant based parameter estimation of linear systems,” in Workshop on Higher-Order Spectral Analysis (IEEE, New York, 1989), pp. 180–185.
  15. M. Gaeta, J. L. Lacoume, “Source separation without a priori knowledge: the maximum likelihood solution,” in Proceedings of the EUSIPCO Conference, Masgrau, Torres, Lagunas, eds. (Elsevier, Amsterdam, 1990), pp. 621–624.
  16. V. C. Soon, L. Tong, Y. F. Huang, R. Liu, “An extended fourth order blind identification algorithm in spatially correlated noise,” in International Conference on Acoustics, Speech and Signal Processing (IEEE, Piscataway, N.J., 1990), pp. 1365–1368.
  17. C. Jutten, J. Herault, “Blind separation of sources. I. An adaptive algorithm based on neuromimetic architecture,” IEEE Trans. Signal Process. 24, 1–10 (1991).
    [CrossRef]
  18. J. L. Lacoume, P. Ruiz, “Separation of independent sources from correlated inputs,” IEEE Trans. Signal Process. 40, 3074–3078 (1992).
    [CrossRef]
  19. A. J. Bell, T. J. Sejnowski, “An information maximisation approach to blind separation and blind deconvolution,” Neural Comput. 7, 1129–1159 (1995).
    [CrossRef] [PubMed]
  20. R. A. Redner, H. F. Walker, “Mixture densities, maximum likelihood and the EM algorithm,” SIAM (Soc. Ind. Appl. Math.) Rev. 26, 195–239 (1994).
  21. J. Cardoso, “Super-symmetric decomposition of the fourth-order cumulant tensor. Blind identification of more sources than sensors,” in International Conference on Acoustics, Speech and Signal Processing (IEEE, Piscataway, N.J., 1991), pp. 3109–3112.
  22. L. B. Wolff, T. A. Mancini, P. Pouliquen, A. G. Andreou, “Liquid crystal polarization camera,” IEEE Trans. Rob. Autom. 13, 195–203 (1997).
    [CrossRef]

1997 (2)

S. K. Nayar, X. Fang, B. Terrance, “Separation of reflection components using color and polarization,” Int. J. Comput. Vis. 21(3), 163–186 (1997).
[CrossRef]

L. B. Wolff, T. A. Mancini, P. Pouliquen, A. G. Andreou, “Liquid crystal polarization camera,” IEEE Trans. Rob. Autom. 13, 195–203 (1997).
[CrossRef]

1995 (1)

A. J. Bell, T. J. Sejnowski, “An information maximisation approach to blind separation and blind deconvolution,” Neural Comput. 7, 1129–1159 (1995).
[CrossRef] [PubMed]

1994 (1)

R. A. Redner, H. F. Walker, “Mixture densities, maximum likelihood and the EM algorithm,” SIAM (Soc. Ind. Appl. Math.) Rev. 26, 195–239 (1994).

1992 (1)

J. L. Lacoume, P. Ruiz, “Separation of independent sources from correlated inputs,” IEEE Trans. Signal Process. 40, 3074–3078 (1992).
[CrossRef]

1991 (2)

L. B. Wolff, T. E. Boult, “Constraining object features using a polarization reflectance model,” IEEE Trans. Pattern. Anal. Mach. Intell. 13, 635–657 (1991).
[CrossRef]

C. Jutten, J. Herault, “Blind separation of sources. I. An adaptive algorithm based on neuromimetic architecture,” IEEE Trans. Signal Process. 24, 1–10 (1991).
[CrossRef]

1990 (2)

G. J. Klinker, S. A. Shafer, T. Kanade, “The measurement of highlights in color images,” Int. J. Comput. Vis. 2(1), 7–32 (1990).
[CrossRef]

L. B. Wolff, “Polarization-based material classification from specular reflection,” IEEE Trans. Pattern. Anal. Mach. Intell. 12, 1059–1071 (1990).
[CrossRef]

1985 (1)

S. Shafer, “Using color to separate reflection components,” Color Res. Appl. 10, 210–218 (1985).
[CrossRef]

1976 (1)

S. Ullman, “On visual detection of light sources,” Bio. Cybern. 21, 205–212 (1976).
[CrossRef]

Andreou, A. G.

L. B. Wolff, T. A. Mancini, P. Pouliquen, A. G. Andreou, “Liquid crystal polarization camera,” IEEE Trans. Rob. Autom. 13, 195–203 (1997).
[CrossRef]

Bajcsy, R.

R. Bajcsy, S. W. Lee, A. Leonardis, “Color image segmentation with detection of highlights and local illumination induced by interreflections,” in Proceedings of International Conference on Pattern Recognition (IEEE, Piscataway, N.J., 1990), pp. 785–790.

Bar-Ness, Y.

Y. Bar-Ness, “Bootstrapping adaptive interference cancelers: some practical limitations,” in The Globecom Conference (IEEE, New York, 1982), pp. 1251–1255.

Bell, A. J.

A. J. Bell, T. J. Sejnowski, “An information maximisation approach to blind separation and blind deconvolution,” Neural Comput. 7, 1129–1159 (1995).
[CrossRef] [PubMed]

Blake, A.

G. Brelstaff, A. Blake, “Detecting specular reflections using Lambertian constraints,” in International Conference on Computer Vision (IEEE, New York, 1988), pp. 297–302.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1965).

Boult, T. E.

L. B. Wolff, T. E. Boult, “Constraining object features using a polarization reflectance model,” IEEE Trans. Pattern. Anal. Mach. Intell. 13, 635–657 (1991).
[CrossRef]

Brelstaff, G.

G. Brelstaff, A. Blake, “Detecting specular reflections using Lambertian constraints,” in International Conference on Computer Vision (IEEE, New York, 1988), pp. 297–302.

Cardoso, J.

J. Cardoso, “Super-symmetric decomposition of the fourth-order cumulant tensor. Blind identification of more sources than sensors,” in International Conference on Acoustics, Speech and Signal Processing (IEEE, Piscataway, N.J., 1991), pp. 3109–3112.

J. Cardoso, “Source separation using higher order moments,” in International Conference on Acoustics, Speech and Signal Processing (IEEE, Piscataway, N.J., 1989), pp. 2109–2112.

Comon, P.

P. Comon, “Separation of stochastic processes,” in Workshop on Higher-Order Spectral Analysis (IEEE, New York, 1989), pp. 174–179.

Fang, X.

S. K. Nayar, X. Fang, B. Terrance, “Separation of reflection components using color and polarization,” Int. J. Comput. Vis. 21(3), 163–186 (1997).
[CrossRef]

Gaeta, M.

M. Gaeta, J. L. Lacoume, “Source separation without a priori knowledge: the maximum likelihood solution,” in Proceedings of the EUSIPCO Conference, Masgrau, Torres, Lagunas, eds. (Elsevier, Amsterdam, 1990), pp. 621–624.

Herault, J.

C. Jutten, J. Herault, “Blind separation of sources. I. An adaptive algorithm based on neuromimetic architecture,” IEEE Trans. Signal Process. 24, 1–10 (1991).
[CrossRef]

Huang, Y. F.

V. C. Soon, L. Tong, Y. F. Huang, R. Liu, “An extended fourth order blind identification algorithm in spatially correlated noise,” in International Conference on Acoustics, Speech and Signal Processing (IEEE, Piscataway, N.J., 1990), pp. 1365–1368.

Inouye, Y.

Y. Inouye, T. Matsui, “Cumulant based parameter estimation of linear systems,” in Workshop on Higher-Order Spectral Analysis (IEEE, New York, 1989), pp. 180–185.

Jutten, C.

C. Jutten, J. Herault, “Blind separation of sources. I. An adaptive algorithm based on neuromimetic architecture,” IEEE Trans. Signal Process. 24, 1–10 (1991).
[CrossRef]

Kanade, T.

G. J. Klinker, S. A. Shafer, T. Kanade, “The measurement of highlights in color images,” Int. J. Comput. Vis. 2(1), 7–32 (1990).
[CrossRef]

Kiryati, N.

Y. Y. Schechner, N. Kiryati, J. Shamir, “Separation of transparent layers by polarization analysis,” presented at the IAPR 11th Scandinavian Conference on Image Analysis, June 7–11, 1999, Kangerlussuaq, Greeland.

Klinker, G. J.

G. J. Klinker, S. A. Shafer, T. Kanade, “The measurement of highlights in color images,” Int. J. Comput. Vis. 2(1), 7–32 (1990).
[CrossRef]

Lacoume, J. L.

J. L. Lacoume, P. Ruiz, “Separation of independent sources from correlated inputs,” IEEE Trans. Signal Process. 40, 3074–3078 (1992).
[CrossRef]

M. Gaeta, J. L. Lacoume, “Source separation without a priori knowledge: the maximum likelihood solution,” in Proceedings of the EUSIPCO Conference, Masgrau, Torres, Lagunas, eds. (Elsevier, Amsterdam, 1990), pp. 621–624.

Lee, S. W.

R. Bajcsy, S. W. Lee, A. Leonardis, “Color image segmentation with detection of highlights and local illumination induced by interreflections,” in Proceedings of International Conference on Pattern Recognition (IEEE, Piscataway, N.J., 1990), pp. 785–790.

Leonardis, A.

R. Bajcsy, S. W. Lee, A. Leonardis, “Color image segmentation with detection of highlights and local illumination induced by interreflections,” in Proceedings of International Conference on Pattern Recognition (IEEE, Piscataway, N.J., 1990), pp. 785–790.

Liu, R.

V. C. Soon, L. Tong, Y. F. Huang, R. Liu, “An extended fourth order blind identification algorithm in spatially correlated noise,” in International Conference on Acoustics, Speech and Signal Processing (IEEE, Piscataway, N.J., 1990), pp. 1365–1368.

Mancini, T. A.

L. B. Wolff, T. A. Mancini, P. Pouliquen, A. G. Andreou, “Liquid crystal polarization camera,” IEEE Trans. Rob. Autom. 13, 195–203 (1997).
[CrossRef]

Matsui, T.

Y. Inouye, T. Matsui, “Cumulant based parameter estimation of linear systems,” in Workshop on Higher-Order Spectral Analysis (IEEE, New York, 1989), pp. 180–185.

Nayar, S. K.

S. K. Nayar, X. Fang, B. Terrance, “Separation of reflection components using color and polarization,” Int. J. Comput. Vis. 21(3), 163–186 (1997).
[CrossRef]

Pouliquen, P.

L. B. Wolff, T. A. Mancini, P. Pouliquen, A. G. Andreou, “Liquid crystal polarization camera,” IEEE Trans. Rob. Autom. 13, 195–203 (1997).
[CrossRef]

Redner, R. A.

R. A. Redner, H. F. Walker, “Mixture densities, maximum likelihood and the EM algorithm,” SIAM (Soc. Ind. Appl. Math.) Rev. 26, 195–239 (1994).

Ruiz, P.

J. L. Lacoume, P. Ruiz, “Separation of independent sources from correlated inputs,” IEEE Trans. Signal Process. 40, 3074–3078 (1992).
[CrossRef]

Schechner, Y. Y.

Y. Y. Schechner, N. Kiryati, J. Shamir, “Separation of transparent layers by polarization analysis,” presented at the IAPR 11th Scandinavian Conference on Image Analysis, June 7–11, 1999, Kangerlussuaq, Greeland.

Sejnowski, T. J.

A. J. Bell, T. J. Sejnowski, “An information maximisation approach to blind separation and blind deconvolution,” Neural Comput. 7, 1129–1159 (1995).
[CrossRef] [PubMed]

Shafer, S.

S. Shafer, “Using color to separate reflection components,” Color Res. Appl. 10, 210–218 (1985).
[CrossRef]

Shafer, S. A.

G. J. Klinker, S. A. Shafer, T. Kanade, “The measurement of highlights in color images,” Int. J. Comput. Vis. 2(1), 7–32 (1990).
[CrossRef]

Shamir, J.

Y. Y. Schechner, N. Kiryati, J. Shamir, “Separation of transparent layers by polarization analysis,” presented at the IAPR 11th Scandinavian Conference on Image Analysis, June 7–11, 1999, Kangerlussuaq, Greeland.

Soon, V. C.

V. C. Soon, L. Tong, Y. F. Huang, R. Liu, “An extended fourth order blind identification algorithm in spatially correlated noise,” in International Conference on Acoustics, Speech and Signal Processing (IEEE, Piscataway, N.J., 1990), pp. 1365–1368.

Terrance, B.

S. K. Nayar, X. Fang, B. Terrance, “Separation of reflection components using color and polarization,” Int. J. Comput. Vis. 21(3), 163–186 (1997).
[CrossRef]

Tong, L.

V. C. Soon, L. Tong, Y. F. Huang, R. Liu, “An extended fourth order blind identification algorithm in spatially correlated noise,” in International Conference on Acoustics, Speech and Signal Processing (IEEE, Piscataway, N.J., 1990), pp. 1365–1368.

Ullman, S.

S. Ullman, “On visual detection of light sources,” Bio. Cybern. 21, 205–212 (1976).
[CrossRef]

Walker, H. F.

R. A. Redner, H. F. Walker, “Mixture densities, maximum likelihood and the EM algorithm,” SIAM (Soc. Ind. Appl. Math.) Rev. 26, 195–239 (1994).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1965).

Wolff, L. B.

L. B. Wolff, T. A. Mancini, P. Pouliquen, A. G. Andreou, “Liquid crystal polarization camera,” IEEE Trans. Rob. Autom. 13, 195–203 (1997).
[CrossRef]

L. B. Wolff, T. E. Boult, “Constraining object features using a polarization reflectance model,” IEEE Trans. Pattern. Anal. Mach. Intell. 13, 635–657 (1991).
[CrossRef]

L. B. Wolff, “Polarization-based material classification from specular reflection,” IEEE Trans. Pattern. Anal. Mach. Intell. 12, 1059–1071 (1990).
[CrossRef]

Bio. Cybern. (1)

S. Ullman, “On visual detection of light sources,” Bio. Cybern. 21, 205–212 (1976).
[CrossRef]

Color Res. Appl. (1)

S. Shafer, “Using color to separate reflection components,” Color Res. Appl. 10, 210–218 (1985).
[CrossRef]

IEEE Trans. Pattern. Anal. Mach. Intell. (2)

L. B. Wolff, “Polarization-based material classification from specular reflection,” IEEE Trans. Pattern. Anal. Mach. Intell. 12, 1059–1071 (1990).
[CrossRef]

L. B. Wolff, T. E. Boult, “Constraining object features using a polarization reflectance model,” IEEE Trans. Pattern. Anal. Mach. Intell. 13, 635–657 (1991).
[CrossRef]

IEEE Trans. Rob. Autom. (1)

L. B. Wolff, T. A. Mancini, P. Pouliquen, A. G. Andreou, “Liquid crystal polarization camera,” IEEE Trans. Rob. Autom. 13, 195–203 (1997).
[CrossRef]

IEEE Trans. Signal Process. (2)

C. Jutten, J. Herault, “Blind separation of sources. I. An adaptive algorithm based on neuromimetic architecture,” IEEE Trans. Signal Process. 24, 1–10 (1991).
[CrossRef]

J. L. Lacoume, P. Ruiz, “Separation of independent sources from correlated inputs,” IEEE Trans. Signal Process. 40, 3074–3078 (1992).
[CrossRef]

Int. J. Comput. Vis. (2)

S. K. Nayar, X. Fang, B. Terrance, “Separation of reflection components using color and polarization,” Int. J. Comput. Vis. 21(3), 163–186 (1997).
[CrossRef]

G. J. Klinker, S. A. Shafer, T. Kanade, “The measurement of highlights in color images,” Int. J. Comput. Vis. 2(1), 7–32 (1990).
[CrossRef]

Neural Comput. (1)

A. J. Bell, T. J. Sejnowski, “An information maximisation approach to blind separation and blind deconvolution,” Neural Comput. 7, 1129–1159 (1995).
[CrossRef] [PubMed]

SIAM (Soc. Ind. Appl. Math.) Rev. (1)

R. A. Redner, H. F. Walker, “Mixture densities, maximum likelihood and the EM algorithm,” SIAM (Soc. Ind. Appl. Math.) Rev. 26, 195–239 (1994).

Other (11)

J. Cardoso, “Super-symmetric decomposition of the fourth-order cumulant tensor. Blind identification of more sources than sensors,” in International Conference on Acoustics, Speech and Signal Processing (IEEE, Piscataway, N.J., 1991), pp. 3109–3112.

J. Cardoso, “Source separation using higher order moments,” in International Conference on Acoustics, Speech and Signal Processing (IEEE, Piscataway, N.J., 1989), pp. 2109–2112.

P. Comon, “Separation of stochastic processes,” in Workshop on Higher-Order Spectral Analysis (IEEE, New York, 1989), pp. 174–179.

Y. Bar-Ness, “Bootstrapping adaptive interference cancelers: some practical limitations,” in The Globecom Conference (IEEE, New York, 1982), pp. 1251–1255.

Y. Inouye, T. Matsui, “Cumulant based parameter estimation of linear systems,” in Workshop on Higher-Order Spectral Analysis (IEEE, New York, 1989), pp. 180–185.

M. Gaeta, J. L. Lacoume, “Source separation without a priori knowledge: the maximum likelihood solution,” in Proceedings of the EUSIPCO Conference, Masgrau, Torres, Lagunas, eds. (Elsevier, Amsterdam, 1990), pp. 621–624.

V. C. Soon, L. Tong, Y. F. Huang, R. Liu, “An extended fourth order blind identification algorithm in spatially correlated noise,” in International Conference on Acoustics, Speech and Signal Processing (IEEE, Piscataway, N.J., 1990), pp. 1365–1368.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1965).

Y. Y. Schechner, N. Kiryati, J. Shamir, “Separation of transparent layers by polarization analysis,” presented at the IAPR 11th Scandinavian Conference on Image Analysis, June 7–11, 1999, Kangerlussuaq, Greeland.

R. Bajcsy, S. W. Lee, A. Leonardis, “Color image segmentation with detection of highlights and local illumination induced by interreflections,” in Proceedings of International Conference on Pattern Recognition (IEEE, Piscataway, N.J., 1990), pp. 785–790.

G. Brelstaff, A. Blake, “Detecting specular reflections using Lambertian constraints,” in International Conference on Computer Vision (IEEE, New York, 1988), pp. 297–302.

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Figures (11)

Fig. 1
Fig. 1

Renoir’s On the Terrace with a reflection of Sheila photographed through a linear polarizer at orthogonal orientations, maximizing and minimizing the reflection.

Fig. 2
Fig. 2

A photograph of a painting behind glass contains a superposition of the light that is reflected by the painting and the light that is reflected directly off the glass.

Fig. 3
Fig. 3

Bottom left, an idealized joint probability distribution for a pair of independent images. The linear mixing of these images, Eq. (3), transforms this distribution, by means of a rotation, scaling, and a rotation, from a square into a parallelogram (left). The goal of the ICA is to transform this parallelogram back into a square, thus yielding the original independent images (right).

Fig. 4
Fig. 4

Variation in the second moment (variance) as the joint probability distribution is rotated and projected onto the horizontal axis [Eq. (6)]. The variance is minimal along the short axis of the parallelogram and maximal at the orthogonal orientation. Note that this error function is a two-cycle sinusoid.

Fig. 5
Fig. 5

Variation in the fourth moment as the joint probability distribution is rotated and projected onto the horizontal axis [Eq. (13)]. The fourth moment is minimal at the orientation required for transforming the diamond into a square, which in turn yields the independent images. Note that this error function is a four-cycle sinusoid.

Fig. 6
Fig. 6

Top row, original images and their normalized joint histogram. Second row, the mixed images, lower rows, the intermediate steps that lead to the separation of the mixed images. See also Table 1.

Fig. 7
Fig. 7

Separating in the presence of uniform zero-mean white noise. Shown is the root-mean-square error between the separated and the original images plotted as a function of the signal-to-noise ratio.

Fig. 8
Fig. 8

Renoir’s On the Terrace, Sheila, and Sheila’s reflection.  

Fig. 9
Fig. 9

Top row, a pair of images of Renoir’s On the Terrace with a reflection of Sheila photographed through a linear polarizer at orthogonal orientations. Bottom row, the independent components. Also shown are the intermediate steps that lead to the separation of the independent components. The rightmost column shows the normalized joint histogram of the pair of images to its left.

Fig. 10
Fig. 10

Top row, a pair of images from Renoir’s Luncheon of the Boating Party with a reflection of Sheila photographed through a linear polarizer at orthogonal orientations. Bottom row, the independent components. Also shown are the intermediate steps that lead to the separation of the independent components. The rightmost column shows the normalized joint histogram of the pair of images to its left.

Fig. 11
Fig. 11

Top, a pair of images photographed through a linear polarizer at orthogonal orientations; bottom, the separated components.  

Tables (1)

Tables Icon

Table 1 Results from the Separation of the Cats (Fig. 6)

Equations (37)

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

I1(x, y)=aP(x, y)+bR(x, y),
I2(x, y)=cP(x, y)+dR(x, y).
y1y2=abcd x1x2,
Y=MX,
X˜=M-1Y.
M=R1SR2,
E(θ2)=i=1N[y1(i)y2(i)]cos(θ2)sin(θ2)2.
θ2=12 tan-1i=1Nr2(i)sin[2ϕ(i)]i=1Nr2(i)cos[2ϕ(i)],
R˜1=cos(θ2)sin(θ2)-sin(θ2)cos(θ2).
s1=i=1N[y1(i)y2(i)]cos(θ2)sin(θ2)2,
s2=i=1N[y1(i)y2(i)]cos(θ2-π/2)sin(θ2-π/2)2,
S˜=1s1001/s2.
E(θ4)=i=1N[y1(i)y2(i)]cos(θ4)sin(θ4)4,
E(θ4)=i=1N1y1(i)2+y2(i)2[y1(i)y2(i)]cos(θ4)sin(θ4)4.
θ4=14 tan-1i=1Nr2(i)sin[4ϕ(i)]i=1Nr2(i)cos[4ϕ(i)],
R˜2=cos(θ4)sin(θ4)-sin(θ4)cos(θ4).
X˜=(R˜2S˜R˜1)Y.
abcd x1x2=badc x2x1.
abcd x1x2=a/γb/δc/γd/δ γx1δx1.
E(θ2)=i=1N[y1(i)y2(i)]cos(θ2)sin(θ2)2=i=1N[y1(i)cos(θ2)+y2(i)sin(θ2)]2=i=1Ny12(i)cos2(θ2)+2y1(i)y2(i)cos(θ2)sin(θ2)+y22(i)sin2(θ2).
dE(θ2)dθ2=i=1N-2y12(i)sin(θ2)cos(θ2)+2y1(i)y2(i)[cos2(θ2)-sin2(θ2)]+2y22(i) sin(θ2)cos(θ2)=2i=1N[y22(i)-y12(i)]sin(θ2)cos(θ2)+y1(i)y2(i)[cos2(θ2)-sin2(θ2)]=2i=1N[y22(i)-y12(i)]12 sin(2θ2)+y1(i)y2(i)×12 [1+cos(2θ2)]-12 [1-cos(2θ2)]=i=1N[y22(i)-y12(i)]sin(2θ2)+2y1(i)y2(i)cos(2θ2),
dE(θ2)dθ2=0,
sin(2θ2)cos(2θ2)=-2i=1Ny1(i)y2(i)i=1Ny22(i)-y12(i),
θ2=12 tan-1-2i=1Ny1(i)y2(i)i=1Ny22(i)-y12(i).
θ2=tan-1-2i=1Nr(i)cos[ϕ(i)]r(i)sin[ϕ(i)]i=1Nr2(i)sin2[ϕ(i)]-r2(i)cos2[ϕ(i)]
=tan-1-2i=1N1/2r2(i)sin[2ϕ(i)]i=1Nr2(i)12 {1-cos[2ϕ(i)]}-12 {1+cos[2ϕ(i)]}
=12 tan-1i=1Nr2(i)sin[2ϕ(i)]i=1Nr2(i)cos[2ϕ(i)].
E(θ4)=i=1N[y1(i)y2(i)]cos(θ4)sin(θ4)4=i=1N[y1(i)cos(θ4)+y2(i)sin(θ4)]4=i=1Ny14(i)cos4(θ4)+4y13(i)y2(i)cos3(θ4)sin(θ4)+6y12(i)y22(i)cos2(θ4)sin2(θ4)+4y1(i)y23(i)cos(θ4)sin3(θ4)+y24(i)sin4(θ4)=i=1N18y14(i)[3+4 cos(2θ4)+cos(4θ4)]+y13(i)y2(i)[sin(2θ4)+12 sin(4θ4)]+64y12(i)y22(i)[ 12-12 cos(4θ4)]+y1(i)y23(i)[sin(2θ4)-12 sin(4θ4)]+18y12(i)[3-4 cos(2θ4)+cos(4θ4)].
dE(θ4)dθ4=i=1N18y14(i)[-8 sin(2θ4)-4 sin(4θ4)]+y13(i)y2(i)[2 cos(2θ4)+2 cos(4θ4)]+32y12(i)y22(i)[2 sin(4θ4)]+y1(i)y23(i)[2 cos(2θ4)-2 cos(4θ4)]+18y12(i)[8 sin(2θ4)-4 sin(4θ4)].
E(θ4)=i=1N1y12(i)+y22(i)[y1(i)y2(i)]cos(θ4)sin(θ4)4,
dE(θ4)dθ4=i=1N 1y12(i)+y22(i) {[y24(i)-y14(i)]sin(2θ4)+[2y1(i)y23(i)+2y13(i)y2(i)]cos(2θ4)}+1y12(i)+y22(i) {[-12y14(i)-12y24(i)+3y12(i)y22(i)]sin(4θ4)+[2y13(i)y2(i)-2y1(i)y23(i)]cos(4θ4)}=i=1N 1y12(i)+y22(i) {[-12y14(i)-12y24(i)+3y12(i)y22(i)]sin(4θ4)+[2y13(i)y2(i)-2y1(i)y23(i)]cos(4θ4)},
dE(θ4)dθ4=0,
sin(4θ4)cos(4θ4)
=-i=1N N[2y13(i)y2(i)-2y1(i)y23(i)]i=1N N[-12y14(i)-12y24(i)+3y12(i)y22(i)]
θ4=14 tan-1-i=1N N [2y13(i)y2(i)-2y1(i)y23(i)]i=1NN [-12y14(i)-12y24(i)+3y12(i)y22(i)],
θ4=14 tan-1i=1Nr2(i)sin[4ϕ(i)]i=1Nr2(i)cos[4ϕ(i)].

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