Abstract

Presented method for fringe pattern enhancement has been designed for processing and analyzing low quality fringe patterns. It uses a modified fast and adaptive bidimensional empirical mode decomposition (FABEMD) for the extraction of bidimensional intrinsic mode functions (BIMFs) from an interferogram. Fringe pattern is then selectively reconstructed (SR) taking the regions of selected BIMFs with high modulation values only. Amplitude demodulation and normalization of the reconstructed image is conducted using the spiral phase Hilbert transform (HS). It has been tested using computer generated interferograms and real data. The performance of the presented SR-FABEMD-HS method is compared with other normalization techniques. Its superiority, potential and robustness to high fringe density variations and the presence of noise, modulation and background illumination defects in analyzed fringe patterns has been corroborated.

© 2012 OSA

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2011

2010

2009

2008

S. M. A. Bhuiyan, R. R. Adhami, and J. F. Khan, “Fast and adaptive bidimensional empirical mode decomposition using order-statistics filter based envelope estimation,” EURASIP J. Adv. Signal Process.2008(164), 725356 (2008).
[CrossRef]

M. B. Bernini, A. Federico, and G. H. Kaufmann, “Noise reduction in digital speckle pattern interferometry using bidimensional empirical mode decomposition,” Appl. Opt.47(14), 2592–2598 (2008).
[CrossRef] [PubMed]

2007

N. A. Ochoa and A. A. Silva-Moreno, “Normalization and noise reduction algorithm for fringe patterns,” Opt. Commun.270(2), 161–168 (2007).
[CrossRef]

2006

Z. Wang and H. Ma, “Advanced continuous wavelet transform algorithm for digital interferogram analysis and processing,” Opt. Eng.45(4), 045601 (2006).
[CrossRef]

2005

C. Damerval, S. Meignen, and V. Perrier, “A fast algorithm for bidimensional EMD,” IEEE Signal Process. Lett.12(10), 701–704 (2005).
[CrossRef]

J. A. Guerrero, J. L. Marroquin, M. Rivera, and J. A. Quiroga, “Adaptive monogenic filtering and normalization of ESPI fringe patterns,” Opt. Lett.30(22), 3018–3020 (2005).
[CrossRef] [PubMed]

2004

2003

J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and Ph. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vis. Comput.21(12), 1019–1026 (2003).
[CrossRef]

J. A. Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun.224(4–6), 221–227 (2003).
[CrossRef]

2002

Z. Wang and A. C. Bovik, “A universal image quality index,” IEEE Signal Process. Lett.9(3), 81–84 (2002).
[CrossRef]

2001

1998

N. E. Huang, Z. Shen, 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. Lond. A454(1971), 903–995 (1998).
[CrossRef]

1997

1996

Q. Yu, K. Andresen, W. Osten, and W. Jueptner, “Noise-free normalized fringe patterns and local pixel transforms for strain extraction,” Appl. Opt.35(20), 3783–3790 (1996).
[CrossRef] [PubMed]

C. B. Barber, D. P. Dobkin, and H. Huhdanpaa, “The quickhull algorithm for convex hulls,” ACM Trans. Math. Softw.22(4), 469–483 (1996).
[CrossRef]

Adhami, R. R.

S. M. A. Bhuiyan, N. O. Attoh-Okine, K. E. Barner, A. Y. Ayenu-Prah, and R. R. Adhami, “Bidimensional empirical mode decomposition using various interpolation techniques,” Adv. Adapt. Data Anal.01(02), 309–338 (2009).
[CrossRef]

S. M. A. Bhuiyan, R. R. Adhami, and J. F. Khan, “Fast and adaptive bidimensional empirical mode decomposition using order-statistics filter based envelope estimation,” EURASIP J. Adv. Signal Process.2008(164), 725356 (2008).
[CrossRef]

Andresen, K.

Antonio Gómez-Pedrero, J.

J. A. Quiroga, J. Antonio Gómez-Pedrero, and Á. García-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun.197(1–3), 43–51 (2001).
[CrossRef]

Attoh-Okine, N. O.

S. M. A. Bhuiyan, N. O. Attoh-Okine, K. E. Barner, A. Y. Ayenu-Prah, and R. R. Adhami, “Bidimensional empirical mode decomposition using various interpolation techniques,” Adv. Adapt. Data Anal.01(02), 309–338 (2009).
[CrossRef]

Ayenu-Prah, A. Y.

S. M. A. Bhuiyan, N. O. Attoh-Okine, K. E. Barner, A. Y. Ayenu-Prah, and R. R. Adhami, “Bidimensional empirical mode decomposition using various interpolation techniques,” Adv. Adapt. Data Anal.01(02), 309–338 (2009).
[CrossRef]

Barber, C. B.

C. B. Barber, D. P. Dobkin, and H. Huhdanpaa, “The quickhull algorithm for convex hulls,” ACM Trans. Math. Softw.22(4), 469–483 (1996).
[CrossRef]

Barner, K. E.

S. M. A. Bhuiyan, N. O. Attoh-Okine, K. E. Barner, A. Y. Ayenu-Prah, and R. R. Adhami, “Bidimensional empirical mode decomposition using various interpolation techniques,” Adv. Adapt. Data Anal.01(02), 309–338 (2009).
[CrossRef]

Bernini, M. B.

Bhuiyan, S. M. A.

S. M. A. Bhuiyan, N. O. Attoh-Okine, K. E. Barner, A. Y. Ayenu-Prah, and R. R. Adhami, “Bidimensional empirical mode decomposition using various interpolation techniques,” Adv. Adapt. Data Anal.01(02), 309–338 (2009).
[CrossRef]

S. M. A. Bhuiyan, R. R. Adhami, and J. F. Khan, “Fast and adaptive bidimensional empirical mode decomposition using order-statistics filter based envelope estimation,” EURASIP J. Adv. Signal Process.2008(164), 725356 (2008).
[CrossRef]

Bone, D. J.

Bouaoune, Y.

J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and Ph. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vis. Comput.21(12), 1019–1026 (2003).
[CrossRef]

Bovik, A. C.

Z. Wang and A. C. Bovik, “A universal image quality index,” IEEE Signal Process. Lett.9(3), 81–84 (2002).
[CrossRef]

Bunel, Ph.

J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and Ph. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vis. Comput.21(12), 1019–1026 (2003).
[CrossRef]

Cuevas, F. J.

Damerval, C.

C. Damerval, S. Meignen, and V. Perrier, “A fast algorithm for bidimensional EMD,” IEEE Signal Process. Lett.12(10), 701–704 (2005).
[CrossRef]

Delechelle, E.

J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and Ph. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vis. Comput.21(12), 1019–1026 (2003).
[CrossRef]

Dobkin, D. P.

C. B. Barber, D. P. Dobkin, and H. Huhdanpaa, “The quickhull algorithm for convex hulls,” ACM Trans. Math. Softw.22(4), 469–483 (1996).
[CrossRef]

Federico, A.

García-Botella, Á.

J. A. Quiroga, J. Antonio Gómez-Pedrero, and Á. García-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun.197(1–3), 43–51 (2001).
[CrossRef]

Guerrero, J. A.

Hoang, T.

Huang, N. E.

N. E. Huang, Z. Shen, 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. Lond. A454(1971), 903–995 (1998).
[CrossRef]

Huhdanpaa, H.

C. B. Barber, D. P. Dobkin, and H. Huhdanpaa, “The quickhull algorithm for convex hulls,” ACM Trans. Math. Softw.22(4), 469–483 (1996).
[CrossRef]

Jueptner, W.

Kai, L.

Kaufmann, G. H.

Kemao, Q.

Khan, J. F.

S. M. A. Bhuiyan, R. R. Adhami, and J. F. Khan, “Fast and adaptive bidimensional empirical mode decomposition using order-statistics filter based envelope estimation,” EURASIP J. Adv. Signal Process.2008(164), 725356 (2008).
[CrossRef]

Larkin, K. G.

Li, H.

Li, K.

Liu, D.

Liu, H. H.

N. E. Huang, Z. Shen, 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. Lond. A454(1971), 903–995 (1998).
[CrossRef]

Long, S. R.

N. E. Huang, Z. Shen, 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. Lond. A454(1971), 903–995 (1998).
[CrossRef]

Luo, Y.

Luu, L.

Ma, H.

Z. Wang and H. Ma, “Advanced continuous wavelet transform algorithm for digital interferogram analysis and processing,” Opt. Eng.45(4), 045601 (2006).
[CrossRef]

Ma, J.

Marroquin, J. L.

Meignen, S.

C. Damerval, S. Meignen, and V. Perrier, “A fast algorithm for bidimensional EMD,” IEEE Signal Process. Lett.12(10), 701–704 (2005).
[CrossRef]

Niang, O.

J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and Ph. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vis. Comput.21(12), 1019–1026 (2003).
[CrossRef]

Nunes, J. C.

J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and Ph. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vis. Comput.21(12), 1019–1026 (2003).
[CrossRef]

Ochoa, N. A.

N. A. Ochoa and A. A. Silva-Moreno, “Normalization and noise reduction algorithm for fringe patterns,” Opt. Commun.270(2), 161–168 (2007).
[CrossRef]

Oldfield, M. A.

Olszak, A.

K. Patorski and A. Olszak, “Digital in-plane electronic speckle pattern shearing interferometry,” Opt. Eng.36(7), 2010–2015 (1997).
[CrossRef]

Osten, W.

Pan, B.

Patorski, K.

Perrier, V.

C. Damerval, S. Meignen, and V. Perrier, “A fast algorithm for bidimensional EMD,” IEEE Signal Process. Lett.12(10), 701–704 (2005).
[CrossRef]

Pokorski, K.

Quiroga, J. A.

J. A. Guerrero, J. L. Marroquin, M. Rivera, and J. A. Quiroga, “Adaptive monogenic filtering and normalization of ESPI fringe patterns,” Opt. Lett.30(22), 3018–3020 (2005).
[CrossRef] [PubMed]

M. Servin, J. L. Marroquin, and J. A. Quiroga, “Regularized quadrature and phase tracking from a single closed-fringe interferogram,” J. Opt. Soc. Am. A21(3), 411–419 (2004).
[CrossRef] [PubMed]

J. A. Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun.224(4–6), 221–227 (2003).
[CrossRef]

J. A. Quiroga, J. Antonio Gómez-Pedrero, and Á. García-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun.197(1–3), 43–51 (2001).
[CrossRef]

Rivera, M.

Servin, M.

Shen, Z.

N. E. Huang, Z. Shen, 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. Lond. A454(1971), 903–995 (1998).
[CrossRef]

Shih, H. H.

N. E. Huang, Z. Shen, 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. Lond. A454(1971), 903–995 (1998).
[CrossRef]

Silva-Moreno, A. A.

N. A. Ochoa and A. A. Silva-Moreno, “Normalization and noise reduction algorithm for fringe patterns,” Opt. Commun.270(2), 161–168 (2007).
[CrossRef]

Tian, C.

Trusiak, M.

Tung, C. C.

N. E. Huang, Z. Shen, 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. Lond. A454(1971), 903–995 (1998).
[CrossRef]

Vo, M.

Wang, H.

Wang, Z.

J. Ma, Z. Wang, B. Pan, T. Hoang, M. Vo, and L. Luu, “Two-dimensional continuous wavelet transform for phase determination of complex interferograms,” Appl. Opt.50(16), 2425–2430 (2011).
[CrossRef] [PubMed]

Z. Wang and H. Ma, “Advanced continuous wavelet transform algorithm for digital interferogram analysis and processing,” Opt. Eng.45(4), 045601 (2006).
[CrossRef]

Z. Wang and A. C. Bovik, “A universal image quality index,” IEEE Signal Process. Lett.9(3), 81–84 (2002).
[CrossRef]

Wielgus, M.

Wu, M. C.

N. E. Huang, Z. Shen, 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. Lond. A454(1971), 903–995 (1998).
[CrossRef]

Yang, Y.

Yen, N.-C.

N. E. Huang, Z. Shen, 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. Lond. A454(1971), 903–995 (1998).
[CrossRef]

Yu, Q.

Zheng, Q.

N. E. Huang, Z. Shen, 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. Lond. A454(1971), 903–995 (1998).
[CrossRef]

Zhou, Y.

Zhuo, Y.

ACM Trans. Math. Softw.

C. B. Barber, D. P. Dobkin, and H. Huhdanpaa, “The quickhull algorithm for convex hulls,” ACM Trans. Math. Softw.22(4), 469–483 (1996).
[CrossRef]

Adv. Adapt. Data Anal.

S. M. A. Bhuiyan, N. O. Attoh-Okine, K. E. Barner, A. Y. Ayenu-Prah, and R. R. Adhami, “Bidimensional empirical mode decomposition using various interpolation techniques,” Adv. Adapt. Data Anal.01(02), 309–338 (2009).
[CrossRef]

Appl. Opt.

M. B. Bernini, A. Federico, and G. H. Kaufmann, “Noise reduction in digital speckle pattern interferometry using bidimensional empirical mode decomposition,” Appl. Opt.47(14), 2592–2598 (2008).
[CrossRef] [PubMed]

M. B. Bernini, A. Federico, and G. H. Kaufmann, “Normalization of fringe patterns using the bidimensional empirical mode decomposition and the Hilbert transform,” Appl. Opt.48(36), 6862–6869 (2009).
[CrossRef] [PubMed]

K. Li and B. Pan, “Frequency-guided windowed Fourier ridges technique for automatic demodulation of a single closed fringe pattern,” Appl. Opt.49(1), 56–60 (2010).
[CrossRef] [PubMed]

C. Tian, Y. Yang, D. Liu, Y. Luo, and Y. Zhuo, “Demodulation of a single complex fringe interferogram with a path-independent regularized phase-tracking technique,” Appl. Opt.49(2), 170–179 (2010).
[CrossRef] [PubMed]

M. B. Bernini, A. Federico, and G. H. Kaufmann, “Phase measurement in temporal speckle pattern interferometry signals presenting low-modulated regions by means of the bidimensional empirical mode decomposition,” Appl. Opt.50(5), 641–647 (2011).
[CrossRef] [PubMed]

K. Patorski and K. Pokorski, “Examination of singular scalar light fields using wavelet processing of fork fringes,” Appl. Opt.50(5), 773–781 (2011).

J. Ma, Z. Wang, B. Pan, T. Hoang, M. Vo, and L. Luu, “Two-dimensional continuous wavelet transform for phase determination of complex interferograms,” Appl. Opt.50(16), 2425–2430 (2011).
[CrossRef] [PubMed]

M. Wielgus and K. Patorski, “Evaluation of amplitude encoded fringe patterns using the bidimensional empirical mode decomposition and the 2D Hilbert transform generalizations,” Appl. Opt.50(28), 5513–5523 (2011).
[CrossRef] [PubMed]

M. Servin, J. L. Marroquin, and F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt.36(19), 4540–4548 (1997).
[CrossRef] [PubMed]

Q. Yu, K. Andresen, W. Osten, and W. Jueptner, “Noise-free normalized fringe patterns and local pixel transforms for strain extraction,” Appl. Opt.35(20), 3783–3790 (1996).
[CrossRef] [PubMed]

EURASIP J. Adv. Signal Process.

S. M. A. Bhuiyan, R. R. Adhami, and J. F. Khan, “Fast and adaptive bidimensional empirical mode decomposition using order-statistics filter based envelope estimation,” EURASIP J. Adv. Signal Process.2008(164), 725356 (2008).
[CrossRef]

IEEE Signal Process. Lett.

Z. Wang and A. C. Bovik, “A universal image quality index,” IEEE Signal Process. Lett.9(3), 81–84 (2002).
[CrossRef]

C. Damerval, S. Meignen, and V. Perrier, “A fast algorithm for bidimensional EMD,” IEEE Signal Process. Lett.12(10), 701–704 (2005).
[CrossRef]

Image Vis. Comput.

J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and Ph. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vis. Comput.21(12), 1019–1026 (2003).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

J. A. Quiroga, J. Antonio Gómez-Pedrero, and Á. García-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun.197(1–3), 43–51 (2001).
[CrossRef]

J. A. Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun.224(4–6), 221–227 (2003).
[CrossRef]

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Figures (17)

Fig. 1
Fig. 1

Simulated simple cosine term of constant period circular fringes (a) and modified patterns SI1 (b) and SI2 (c).

Fig. 2
Fig. 2

SI1 and SI2 synthetic patterns normalized using the SR-FABEMD-HS method (QSI1 = 0.9348 and QSI2 = 0.9375).

Fig. 3
Fig. 3

SI1 and SI2 synthetic patterns normalized using the BEMD-PHT method (QSI1 = 0.7930 and QSI2 = 0.6801).

Fig. 4
Fig. 4

Simulated complex cosine term (a) and modified fringe patterns SI3 (b) and SI4 (c).

Fig. 5
Fig. 5

SI3 (a) and SI4 (b) synthetic fringe patterns normalized using the BEMD-PHT method (QSI3 = 0.8523 and QSI2 = 0.7386).

Fig. 6
Fig. 6

(a) SI3 synthetic pattern without the bias (the residual part of SI3 calculated with the FABEMD OSFW type 1 decomposition) and (b) the bias-free SI3 normalized using the HS method (QSI3 = 0.9228).

Fig. 7
Fig. 7

First eleven BIMFs and the residue of SI4 obtained using the FABEMD OSFW type 1 algorithm.

Fig. 8
Fig. 8

(a) SI4 reconstructed using the sum from BIMF2 to BIMF8 and (b) normalized fringe pattern (Q = 0.8228).

Fig. 9
Fig. 9

Top: (a) modulation distribution of BIMF4 (Fig. 7d), (b) smoothed modulation distribution with the FABEMD OSFW type 1 method (BIMF10) and (c) isolated regions of BIMF4 with high modulation values (threshold set to −0.004). Bottom: (d) modulation distribution of BIMF7 (Fig. 7g), (e) smoothed modulation distribution with the FABEMD OSFW type 1 method (BIMF8) and (f) isolated regions of BIMF7 with high modulation values (threshold set to 0.0026).

Fig. 10
Fig. 10

(a) SI4 pattern selectively reconstructed from isolated regions of BIMF2-BIMF8 with high modulation values and (b) normalized fringe pattern (Q = 0.8631).

Fig. 11
Fig. 11

Simulated cosine term (a) ; corrupted fringe pattern, SI5 (b); fringe pattern normalized using the TOBF (c), DD (d), BEMD-PHT (e) FABEMD-HS without (f) and with selective reconstruction (g).

Fig. 12
Fig. 12

Ideal phase distribution modulo 2π calculated using analytic signal paradigm from π/2 phase-shifted defect free interferograms (a) and phase map obtained from corrupted interferograms normalized using the TOBF (b), DD (c), BEMD-PHT (d) FABEMD-HS methods without (e) and with selective reconstruction (f).

Fig. 13
Fig. 13

Experimental DSPI fringes.

Fig. 14
Fig. 14

DSPI fringes normalized using (a) the DD method (spacing 35) [14] and (b) the BEMD-PHT method (with 3x3 sliding window averaging) [15].

Fig. 15
Fig. 15

(a) DSPI fringe pattern selectively reconstructed from isolated regions of its empirical modes BIMF5 to BIMF9 with high modulation values and (b) the normalized fringe pattern; (c) DSPI fringe pattern reconstructed from its empirical modes BIMF5 to BIMF9 without SR and (d) normalized correlation fringes.

Fig. 16
Fig. 16

Image (996x793 pixels) of the statue with structured light illumination (a) and the normalized fringe pattern (b).

Fig. 17
Fig. 17

Intensity distribution cross-sections (900th row) of the statue image upper plinth fragment: original image (black), normalized image (blue) and simulated ideal sinusoid with period of 12 pixels (red).

Tables (4)

Tables Icon

Table 1 Normalization performance comparison of various methods using quality index Q

Tables Icon

Table 2 Threshold values and numbers of smoothed modulation distribution BIMFs used for selective reconstruction of SI4.

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Table 3 Threshold values and numbers of smoothed modulation distribution BIMFs used for selective reconstruction of SI5 and SI6.

Tables Icon

Table 4 Normalization and phase demodulation performance comparison of various methods using quality index Q

Equations (5)

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s A (x,y)=s(x,y)+i s H (x,y).
P( ζ 1 , ζ 2 )= ζ 1 +i ζ 2 ζ 1 2 + ζ 2 2
s H =iexp(iβ) F 1 {P( ζ 1 , ζ 2 )F[s(x,y)]},
| A(x,y) |= s 2 (x,y)+ | F 1 {P( ζ 1 , ζ 2 )F[s(x,y)]} | 2 .
Q= 4 σ EO E ¯ O ¯ ( σ E 2 + σ O 2 )( E ¯ 2 + O ¯ 2 ) ,

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