Abstract

We evaluate a method based on the two-dimensional directional wavelet transform and the introduction of a spatial carrier to retrieve optical phase distributions in singular scalar light fields. The performance of the proposed phase-retrieval method is compared with an approach based on Fourier transform. The advantages and limitations of the proposed method are discussed.

© 2008 Optical Society of America

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References

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  1. M. S. Soskin and M. V. Vasnetsov, “Singular Optics,” in Progress in Optics, E. Wolf, ed. (North-Holland, 2001), Vol. 42.
  2. M.V.Vasnetsov and K.Staliunas, eds., Optical Vortices, Vol. 228 of Horizons in World Physics (Nova Science, 1999).
  3. I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422-428 (1993).
    [CrossRef]
  4. A. V. Martin and L. J. Allen, “Phase imaging from a diffraction pattern in the presence of vortices,” Opt. Commun. 277, 288-294 (2007).
    [CrossRef]
  5. V. G. Denisenko, A. Minovich, A. S. Desyatnikov, W. Krolikowski, M. S. Soskin, and Y. Kivshar, “Mapping phases of singular scalar light fields,” Opt. Lett. 33, 89-91(2008).
    [CrossRef]
  6. F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749-3752 (1991).
    [CrossRef]
  7. W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental investigation of local properties and statistics of optical vortices in random wave fields,” Phys. Rev. Lett. 94, 103902-103904 (2005).
    [CrossRef]
  8. J. Lin, X.-C. Yuan, J. Bu, B. P. S. Ahluwalia, Y. Y. Sun, and R. E. Burge, “Selective generation of high-order optical vortices from a single phase wedge,” Opt. Lett. 32, 2927-2929 (2007).
    [CrossRef]
  9. Q. Xie and D. Zhao, “Optical vortices generated by multi-level achromatic spiral phase plates for broadband beams,” Opt. Commun. 281, 7-11 (2008).
    [CrossRef]
  10. J. W. Goodman, Speckle Phenomena in Optics. Theory and Applications (Roberts, 2006).
  11. L. R. Watkins, “Phase recovery from fringe patterns using the continuous wavelet transform,” Opt. Lasers Eng. 45, 298-303 (2007).
    [CrossRef]
  12. M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Spatial carrier fringe pattern demodulation by use of a two-dimensional continuous wavelet transform,” Appl. Opt. 45, 8722-8732 (2006).
    [CrossRef]
  13. A. Z. Abid, M. A. Gdeisat, D. R. Burton, M. J. Lalor, and F. Lilley, “Spatial fringe pattern analysis using the two-dimensional continuous wavelet transform employing a cost function,” Appl. Opt. 46, 6120-6126 (2007).
    [CrossRef]
  14. N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for non-linear and non-stationary time series analysis,” Proc. R. Soc. A 454, 903-995 (1998).
    [CrossRef]
  15. J.-P. Antoine and R. Murenzi, “Two-dimensional directional wavelets and the scale-angle representation,” Signal Process. 52, 259-281 (1996).
    [CrossRef]
  16. A. Federico and G. H. Kaufmann, “Comparative study of wavelet thresholding methods for denoising electronic speckle pattern interferometry fringes,” Opt. Eng. 40, 2598-2604 (2001).
    [CrossRef]
  17. M. S. Soskin, V. N. Gorshkov, and M. V. Vasnetsov, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064-4075 (1997).
    [CrossRef]
  18. Z. Wang and A. C. Bovik, “A universal quality index,” IEEE Signal Process. Lett. 9, 81-84 (2002).
    [CrossRef]
  19. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156-160 (1982).
  20. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, 1989).
  21. A. Federico and G. H. Kaufmann, “Phase recovery in temporal speckle pattern interferometry using the generalized S-transform,” Opt. Lett. 33, 866-868 (2008).
    [CrossRef]

2008

2007

2006

2005

W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental investigation of local properties and statistics of optical vortices in random wave fields,” Phys. Rev. Lett. 94, 103902-103904 (2005).
[CrossRef]

2002

Z. Wang and A. C. Bovik, “A universal quality index,” IEEE Signal Process. Lett. 9, 81-84 (2002).
[CrossRef]

2001

A. Federico and G. H. Kaufmann, “Comparative study of wavelet thresholding methods for denoising electronic speckle pattern interferometry fringes,” Opt. Eng. 40, 2598-2604 (2001).
[CrossRef]

1998

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for non-linear and non-stationary time series analysis,” Proc. R. Soc. A 454, 903-995 (1998).
[CrossRef]

1997

M. S. Soskin, V. N. Gorshkov, and M. V. Vasnetsov, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064-4075 (1997).
[CrossRef]

1996

J.-P. Antoine and R. Murenzi, “Two-dimensional directional wavelets and the scale-angle representation,” Signal Process. 52, 259-281 (1996).
[CrossRef]

1993

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422-428 (1993).
[CrossRef]

1991

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749-3752 (1991).
[CrossRef]

1982

Abid, A. Z.

Ahluwalia, B. P. S.

Allen, L. J.

A. V. Martin and L. J. Allen, “Phase imaging from a diffraction pattern in the presence of vortices,” Opt. Commun. 277, 288-294 (2007).
[CrossRef]

Antoine, J.-P.

J.-P. Antoine and R. Murenzi, “Two-dimensional directional wavelets and the scale-angle representation,” Signal Process. 52, 259-281 (1996).
[CrossRef]

Arecchi, F. T.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749-3752 (1991).
[CrossRef]

Basistiy, I. V.

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422-428 (1993).
[CrossRef]

Bazhenov, V. Y.

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422-428 (1993).
[CrossRef]

Bovik, A. C.

Z. Wang and A. C. Bovik, “A universal quality index,” IEEE Signal Process. Lett. 9, 81-84 (2002).
[CrossRef]

Bu, J.

Burge, R. E.

Burton, D. R.

Denisenko, V. G.

Desyatnikov, A. S.

Federico, A.

A. Federico and G. H. Kaufmann, “Phase recovery in temporal speckle pattern interferometry using the generalized S-transform,” Opt. Lett. 33, 866-868 (2008).
[CrossRef]

A. Federico and G. H. Kaufmann, “Comparative study of wavelet thresholding methods for denoising electronic speckle pattern interferometry fringes,” Opt. Eng. 40, 2598-2604 (2001).
[CrossRef]

Gdeisat, M. A.

Ghiglia, D. C.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, 1989).

Giacomelli, G.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749-3752 (1991).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Speckle Phenomena in Optics. Theory and Applications (Roberts, 2006).

Gorshkov, V. N.

M. S. Soskin, V. N. Gorshkov, and M. V. Vasnetsov, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064-4075 (1997).
[CrossRef]

Hanson, S. G.

W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental investigation of local properties and statistics of optical vortices in random wave fields,” Phys. Rev. Lett. 94, 103902-103904 (2005).
[CrossRef]

Huang, N. E.

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for non-linear and non-stationary time series analysis,” Proc. R. Soc. A 454, 903-995 (1998).
[CrossRef]

Ina, H.

Kaufmann, G. H.

A. Federico and G. H. Kaufmann, “Phase recovery in temporal speckle pattern interferometry using the generalized S-transform,” Opt. Lett. 33, 866-868 (2008).
[CrossRef]

A. Federico and G. H. Kaufmann, “Comparative study of wavelet thresholding methods for denoising electronic speckle pattern interferometry fringes,” Opt. Eng. 40, 2598-2604 (2001).
[CrossRef]

Kivshar, Y.

Kobayashi, S.

Krolikowski, W.

Lalor, M. J.

Lilley, F.

Lin, J.

Liu, H. H.

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for non-linear and non-stationary time series analysis,” Proc. R. Soc. A 454, 903-995 (1998).
[CrossRef]

Long, S. R.

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for non-linear and non-stationary time series analysis,” Proc. R. Soc. A 454, 903-995 (1998).
[CrossRef]

Martin, A. V.

A. V. Martin and L. J. Allen, “Phase imaging from a diffraction pattern in the presence of vortices,” Opt. Commun. 277, 288-294 (2007).
[CrossRef]

Minovich, A.

Miyamoto, Y.

W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental investigation of local properties and statistics of optical vortices in random wave fields,” Phys. Rev. Lett. 94, 103902-103904 (2005).
[CrossRef]

Murenzi, R.

J.-P. Antoine and R. Murenzi, “Two-dimensional directional wavelets and the scale-angle representation,” Signal Process. 52, 259-281 (1996).
[CrossRef]

Pritt, M. D.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, 1989).

Ramazza, P. L.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749-3752 (1991).
[CrossRef]

Residori, S.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749-3752 (1991).
[CrossRef]

Sheng, Z.

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for non-linear and non-stationary time series analysis,” Proc. R. Soc. A 454, 903-995 (1998).
[CrossRef]

Shih, H. H.

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for non-linear and non-stationary time series analysis,” Proc. R. Soc. A 454, 903-995 (1998).
[CrossRef]

Soskin, M. S.

V. G. Denisenko, A. Minovich, A. S. Desyatnikov, W. Krolikowski, M. S. Soskin, and Y. Kivshar, “Mapping phases of singular scalar light fields,” Opt. Lett. 33, 89-91(2008).
[CrossRef]

M. S. Soskin, V. N. Gorshkov, and M. V. Vasnetsov, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064-4075 (1997).
[CrossRef]

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422-428 (1993).
[CrossRef]

M. S. Soskin and M. V. Vasnetsov, “Singular Optics,” in Progress in Optics, E. Wolf, ed. (North-Holland, 2001), Vol. 42.

Sun, Y. Y.

Takeda, M.

W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental investigation of local properties and statistics of optical vortices in random wave fields,” Phys. Rev. Lett. 94, 103902-103904 (2005).
[CrossRef]

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156-160 (1982).

Tung, C. C.

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for non-linear and non-stationary time series analysis,” Proc. R. Soc. A 454, 903-995 (1998).
[CrossRef]

Vasnetsov, M. V.

M. S. Soskin, V. N. Gorshkov, and M. V. Vasnetsov, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064-4075 (1997).
[CrossRef]

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422-428 (1993).
[CrossRef]

M. S. Soskin and M. V. Vasnetsov, “Singular Optics,” in Progress in Optics, E. Wolf, ed. (North-Holland, 2001), Vol. 42.

Wang, W.

W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental investigation of local properties and statistics of optical vortices in random wave fields,” Phys. Rev. Lett. 94, 103902-103904 (2005).
[CrossRef]

Wang, Z.

Z. Wang and A. C. Bovik, “A universal quality index,” IEEE Signal Process. Lett. 9, 81-84 (2002).
[CrossRef]

Watkins, L. R.

L. R. Watkins, “Phase recovery from fringe patterns using the continuous wavelet transform,” Opt. Lasers Eng. 45, 298-303 (2007).
[CrossRef]

Wu, M. C.

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for non-linear and non-stationary time series analysis,” Proc. R. Soc. A 454, 903-995 (1998).
[CrossRef]

Xie, Q.

Q. Xie and D. Zhao, “Optical vortices generated by multi-level achromatic spiral phase plates for broadband beams,” Opt. Commun. 281, 7-11 (2008).
[CrossRef]

Yen, N. C.

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for non-linear and non-stationary time series analysis,” Proc. R. Soc. A 454, 903-995 (1998).
[CrossRef]

Yuan, X.-C.

Zhao, D.

Q. Xie and D. Zhao, “Optical vortices generated by multi-level achromatic spiral phase plates for broadband beams,” Opt. Commun. 281, 7-11 (2008).
[CrossRef]

Zheng, Q.

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for non-linear and non-stationary time series analysis,” Proc. R. Soc. A 454, 903-995 (1998).
[CrossRef]

Appl. Opt.

IEEE Signal Process. Lett.

Z. Wang and A. C. Bovik, “A universal quality index,” IEEE Signal Process. Lett. 9, 81-84 (2002).
[CrossRef]

J. Opt. Soc. Am.

Opt. Commun.

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422-428 (1993).
[CrossRef]

A. V. Martin and L. J. Allen, “Phase imaging from a diffraction pattern in the presence of vortices,” Opt. Commun. 277, 288-294 (2007).
[CrossRef]

Q. Xie and D. Zhao, “Optical vortices generated by multi-level achromatic spiral phase plates for broadband beams,” Opt. Commun. 281, 7-11 (2008).
[CrossRef]

Opt. Eng.

A. Federico and G. H. Kaufmann, “Comparative study of wavelet thresholding methods for denoising electronic speckle pattern interferometry fringes,” Opt. Eng. 40, 2598-2604 (2001).
[CrossRef]

Opt. Lasers Eng.

L. R. Watkins, “Phase recovery from fringe patterns using the continuous wavelet transform,” Opt. Lasers Eng. 45, 298-303 (2007).
[CrossRef]

Opt. Lett.

Phys. Rev. A

M. S. Soskin, V. N. Gorshkov, and M. V. Vasnetsov, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064-4075 (1997).
[CrossRef]

Phys. Rev. Lett.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749-3752 (1991).
[CrossRef]

W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental investigation of local properties and statistics of optical vortices in random wave fields,” Phys. Rev. Lett. 94, 103902-103904 (2005).
[CrossRef]

Proc. R. Soc. A

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for non-linear and non-stationary time series analysis,” Proc. R. Soc. A 454, 903-995 (1998).
[CrossRef]

Signal Process.

J.-P. Antoine and R. Murenzi, “Two-dimensional directional wavelets and the scale-angle representation,” Signal Process. 52, 259-281 (1996).
[CrossRef]

Other

J. W. Goodman, Speckle Phenomena in Optics. Theory and Applications (Roberts, 2006).

M. S. Soskin and M. V. Vasnetsov, “Singular Optics,” in Progress in Optics, E. Wolf, ed. (North-Holland, 2001), Vol. 42.

M.V.Vasnetsov and K.Staliunas, eds., Optical Vortices, Vol. 228 of Horizons in World Physics (Nova Science, 1999).

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, 1989).

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Figures (5)

Fig. 1
Fig. 1

Phase retrieval in a computer-simulated noisy speckle pattern with an average speckle size of 128 pixels: (a) original pattern, (b) fringe pattern obtained after the introduction of a horizontal spatial carrier to (a), (c) wrapped phase map obtained from (b) using the proposed 2D CWT method, (d) and (e) rewrapped phase maps obtained after phase unwrapping and carrier subtraction by using the Fourier and 2D CWT methods, respectively, (f) ridge of the continuous wavelet transform of (b).

Fig. 2
Fig. 2

Typical result from the normalization preprocess: (a) intensity along a column located in the part of Fig. 1b that contains a vortex, (b) normalized intensity from (a).

Fig. 3
Fig. 3

Phase retrieval in a noisy speckled speckle pattern generated by two computer-simulated speckle distributions having an average speckle size of 128 and 8 pixels: (a) original pattern, (b) fringe pattern obtained after the introduction of a horizontal spatial carrier to (a), (c) wrapped phase map obtained from (b) using the proposed 2D CWT method, (d) and (e) rewrapped phase maps obtained after phase unwrapping and carrier subtraction using the Fourier and the 2D CWT methods, respectively, (f) ridge of the 2D CWT of (b).

Fig. 4
Fig. 4

Phase retrieval from two optical vortices with charges m 1 = + 3 and m 2 = + 1 when the simulated phase mask was modified by uniformly distributed random noise: (a) intensity distribution, (b) wrapped phase map corresponding to (a), (c) fringe pattern obtained after the introduction of a horizontal spatial carrier to (a), (d) rewrapped phase map obtained from (c) after phase unwrapping and carrier subtraction using the Fourier transform method, (e) rewrapped phase map obtained from (c) after phase unwrapping and carrier subtraction using the proposed 2D CWT method, (f) ridge of the 2D CWT of (c).

Fig. 5
Fig. 5

Phase recovery from two optical vortices in combined beams with charges m 1 = + 3 and m 2 = 2 when the simulated phase mask was modified by uniformly distributed random noise: (a) intensity distribution, (b) wrapped phase map corresponding to (a), (c) fringe pattern obtained after the introduction of a horizontal spatial carrier to (a), (d) rewrapped phase map obtained from (c) after phase unwrapping and carrier subtraction using the Fourier transform method, (e) rewrapped phase map obtained from (c) after phase unwrapping and carrier subtraction using the proposed 2D CWT method, (f) ridge of the 2D CWT of (c).

Tables (1)

Tables Icon

Table 1 Quality Index Q Given by Fourier Transform and the Proposed 2D CWT Methods Using a Sliding Window with Size S = 5 Pixels

Equations (9)

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

S ( a , θ , b ) = 1 a 2 d x d y ψ * [ a 1 R θ ( r b ) ] I ( r ) ,
ψ ( r ) = exp ( i k 0 · r ) exp ( 1 2 | A r | 2 ) + corr . term ,
ψ ( r ) = exp ( i k 0 y ) exp [ 1 2 ( x 2 ϵ + y 2 ) ] ,
ψ ^ ( k ) = F [ ψ ( r ) ] = ϵ exp { 1 2 [ ϵ k x 2 + ( k y k 0 ) 2 ] } ,
Q 4 σ E O E ¯ O ¯ ( σ E 2 + σ O 2 ) [ E ¯ 2 + O ¯ 2 ] = σ E O σ E σ O · 2 E ¯ O ¯ E ¯ 2 + O ¯ 2 · 2 σ E σ O σ E 2 + σ O 2 ,
E ¯ = 1 N 2 x , y = 1 N E ( x , y ) , O ¯ = 1 N 2 x , y = 1 N O ( x , y ) ,
σ E 2 = 1 N 2 1 x , y = 1 N [ E ( x , y ) E ¯ ] 2 ,
σ O 2 = 1 N 2 1 x , y = 1 N [ O ( x , y ) O ¯ ] 2 ,
σ E O = 1 N 2 1 x , y = 1 N [ E ( x , y ) E ¯ ] [ O ( x , y ) O ¯ ] ,

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