Abstract

This article is concerned with frequency filtering for electronic speckle pattern interferometry wrapped phase patterns. We propose a robust localized Fourier transform filter which is an extension of the root filtering method (RFM). We improve the RFM from a simple technical process and a filter function in the frequency domain. In our method, the proposed filter function is taken as the power spectrum of the convolution of an image and a Gaussian function to the power α. We test the proposed method on two computer-simulated wrapped phase fringe patterns and one experimentally obtained wrapped phase pattern, and compare our models with the widely used, well-known RFM and windowed Fourier filtering (WFF). The experimental results have demonstrated that our localized Fourier transform filter out performs the RFM and is comparable to WFF. Our method depends on fewer parameters, as compared with WFF, and can achieve a better balance between the computational complexity and the filtered results.

© 2011 Optical Society of America

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References

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  1. S. Nakadate and H. Saito, “Fringe scanning speckle-pattern interferometry,” Appl. Opt. 24, 2172–2180 (1985).
    [CrossRef] [PubMed]
  2. K. Creath, “Phase-shifting speckle interferometry,” Appl. Opt. 24, 3053–3058 (1985).
    [CrossRef] [PubMed]
  3. D. W. Robinson and D. C. Williams, “Digital phase stepping speckle interferometry,” Opt. Commun. 57, 26–30(1986).
    [CrossRef]
  4. C. C. Kao, G. B. Yeh, S. S. Lee, C. K. Lee, C. S. Yang, and K. C. Wu, “Phase shifting algorithms for electronic speckle pattern interferometry,” Appl. Opt. 41, 46–54 (2002).
    [CrossRef] [PubMed]
  5. D. Kerr, F. Mendoza Santoyo, and J. R. Tyrer, “Extraction of phase data from electronic speckle pattern interferometric fringes using a single-phase-step method: a novel approach,” J. Opt. Soc. Am. A 7, 820–826 (1990).
    [CrossRef]
  6. T. I. Voronyak, A. B. Kmet’, and O. V. Lychak, “Single-step phase-shifting speckle interferometry,” Mater Sci. 43, 554–567 (2007).
    [CrossRef]
  7. M. Servin, F. J. Cuevas, D. Malacara, J. L. Marroguin, and R. Rodriguez-Vera, “Phase unwrapping through demodulation by use of the regularized phase-tracking technique,” Appl. Opt. 38, 1934–1941 (1999).
    [CrossRef]
  8. C. K. Hong, H. S. Ryu, and H. C. Lim, “Least-squares fitting of the phase map obtained in phase-shifting electronic speckle pattern interferometry,” Opt. Lett. 20, 931–933 (1995).
    [CrossRef] [PubMed]
  9. C. Tang, W. Wang, H. Yan, and X. Gu, “Tangent least-squares fitting filtering method for electrical speckle pattern interferometry phase fringe patterns,” Appl. Opt. 46, 2907–2913 (2007).
    [CrossRef] [PubMed]
  10. H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999).
    [CrossRef]
  11. G. H. Kaufmann, A. Davila, and D. Kerr, “Speckle noise reduction in TV holography,” Proc. SPIE. 2730, 96–100 (1996).
    [CrossRef]
  12. Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43, 2695–2702 (2004).
    [CrossRef] [PubMed]
  13. J. Villa, R. R. Vera, J. A. Quiroga, I. de la Rosa, and E. González, “Anisotropic phase-map denoising using a regularized cost-function with complex-valued Markov-random-fields,” Opt. Lasers Eng. 48, 650–656 (2010).
    [CrossRef]
  14. C. Tang, L. Han, H. Ren, T. Gao, Z. Wang, and K. Tang, “The oriented-couple partial differential equations for filtering in wrapped phase patterns,” Opt. Express. 17, 5606–5617(2009).
    [CrossRef] [PubMed]
  15. C. Tang, T. Gao, S. Yan, L. Wang, and J. Wu, “The oriented spatial filter masks for electronic speckle pattern interferometry phase patterns,” Opt. Express. 18, 8942–8947 (2010).
    [CrossRef] [PubMed]
  16. R. Fisher, S. Perkins, A. Walker, and E. Wolfart, “Frequency filter,” http://homepages.inf.ed.ac.uk/rbf/HIPR2/freqfilt.htm.
  17. S. Chikkerur, A. N. Cartwright, V. Govindaraju, “Fingerprint enhancement using STFT analysis,” Pattern Recogn. 40, 198–211 (2007).
    [CrossRef]
  18. Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304–317 (2007).
    [CrossRef]
  19. Q. Kemao, L. T. H. Nam, L. Feng, and S. H. Soon, “Comparative analysis on some filters for wrapped phase maps,” Appl. Opt. 46, 7412–7418 (2007).
    [CrossRef] [PubMed]
  20. D. C. Ghiglia and L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A. 11, 107–117 (1994).
    [CrossRef]

2010 (2)

C. Tang, T. Gao, S. Yan, L. Wang, and J. Wu, “The oriented spatial filter masks for electronic speckle pattern interferometry phase patterns,” Opt. Express. 18, 8942–8947 (2010).
[CrossRef] [PubMed]

J. Villa, R. R. Vera, J. A. Quiroga, I. de la Rosa, and E. González, “Anisotropic phase-map denoising using a regularized cost-function with complex-valued Markov-random-fields,” Opt. Lasers Eng. 48, 650–656 (2010).
[CrossRef]

2009 (1)

C. Tang, L. Han, H. Ren, T. Gao, Z. Wang, and K. Tang, “The oriented-couple partial differential equations for filtering in wrapped phase patterns,” Opt. Express. 17, 5606–5617(2009).
[CrossRef] [PubMed]

2007 (5)

T. I. Voronyak, A. B. Kmet’, and O. V. Lychak, “Single-step phase-shifting speckle interferometry,” Mater Sci. 43, 554–567 (2007).
[CrossRef]

S. Chikkerur, A. N. Cartwright, V. Govindaraju, “Fingerprint enhancement using STFT analysis,” Pattern Recogn. 40, 198–211 (2007).
[CrossRef]

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304–317 (2007).
[CrossRef]

C. Tang, W. Wang, H. Yan, and X. Gu, “Tangent least-squares fitting filtering method for electrical speckle pattern interferometry phase fringe patterns,” Appl. Opt. 46, 2907–2913 (2007).
[CrossRef] [PubMed]

Q. Kemao, L. T. H. Nam, L. Feng, and S. H. Soon, “Comparative analysis on some filters for wrapped phase maps,” Appl. Opt. 46, 7412–7418 (2007).
[CrossRef] [PubMed]

2004 (1)

2002 (1)

1999 (2)

M. Servin, F. J. Cuevas, D. Malacara, J. L. Marroguin, and R. Rodriguez-Vera, “Phase unwrapping through demodulation by use of the regularized phase-tracking technique,” Appl. Opt. 38, 1934–1941 (1999).
[CrossRef]

H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999).
[CrossRef]

1996 (1)

G. H. Kaufmann, A. Davila, and D. Kerr, “Speckle noise reduction in TV holography,” Proc. SPIE. 2730, 96–100 (1996).
[CrossRef]

1995 (1)

1994 (1)

D. C. Ghiglia and L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A. 11, 107–117 (1994).
[CrossRef]

1990 (1)

1986 (1)

D. W. Robinson and D. C. Williams, “Digital phase stepping speckle interferometry,” Opt. Commun. 57, 26–30(1986).
[CrossRef]

1985 (2)

Aebischer, H. A.

H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999).
[CrossRef]

Cartwright, A. N.

S. Chikkerur, A. N. Cartwright, V. Govindaraju, “Fingerprint enhancement using STFT analysis,” Pattern Recogn. 40, 198–211 (2007).
[CrossRef]

Chikkerur, S.

S. Chikkerur, A. N. Cartwright, V. Govindaraju, “Fingerprint enhancement using STFT analysis,” Pattern Recogn. 40, 198–211 (2007).
[CrossRef]

Creath, K.

Cuevas, F. J.

Davila, A.

G. H. Kaufmann, A. Davila, and D. Kerr, “Speckle noise reduction in TV holography,” Proc. SPIE. 2730, 96–100 (1996).
[CrossRef]

de la Rosa, I.

J. Villa, R. R. Vera, J. A. Quiroga, I. de la Rosa, and E. González, “Anisotropic phase-map denoising using a regularized cost-function with complex-valued Markov-random-fields,” Opt. Lasers Eng. 48, 650–656 (2010).
[CrossRef]

Feng, L.

Fisher, R.

R. Fisher, S. Perkins, A. Walker, and E. Wolfart, “Frequency filter,” http://homepages.inf.ed.ac.uk/rbf/HIPR2/freqfilt.htm.

Gao, T.

C. Tang, T. Gao, S. Yan, L. Wang, and J. Wu, “The oriented spatial filter masks for electronic speckle pattern interferometry phase patterns,” Opt. Express. 18, 8942–8947 (2010).
[CrossRef] [PubMed]

C. Tang, L. Han, H. Ren, T. Gao, Z. Wang, and K. Tang, “The oriented-couple partial differential equations for filtering in wrapped phase patterns,” Opt. Express. 17, 5606–5617(2009).
[CrossRef] [PubMed]

Ghiglia, D. C.

D. C. Ghiglia and L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A. 11, 107–117 (1994).
[CrossRef]

González, E.

J. Villa, R. R. Vera, J. A. Quiroga, I. de la Rosa, and E. González, “Anisotropic phase-map denoising using a regularized cost-function with complex-valued Markov-random-fields,” Opt. Lasers Eng. 48, 650–656 (2010).
[CrossRef]

Govindaraju, V.

S. Chikkerur, A. N. Cartwright, V. Govindaraju, “Fingerprint enhancement using STFT analysis,” Pattern Recogn. 40, 198–211 (2007).
[CrossRef]

Gu, X.

Han, L.

C. Tang, L. Han, H. Ren, T. Gao, Z. Wang, and K. Tang, “The oriented-couple partial differential equations for filtering in wrapped phase patterns,” Opt. Express. 17, 5606–5617(2009).
[CrossRef] [PubMed]

Hong, C. K.

Kao, C. C.

Kaufmann, G. H.

G. H. Kaufmann, A. Davila, and D. Kerr, “Speckle noise reduction in TV holography,” Proc. SPIE. 2730, 96–100 (1996).
[CrossRef]

Kemao, Q.

Kerr, D.

Kmet’, A. B.

T. I. Voronyak, A. B. Kmet’, and O. V. Lychak, “Single-step phase-shifting speckle interferometry,” Mater Sci. 43, 554–567 (2007).
[CrossRef]

Lee, C. K.

Lee, S. S.

Lim, H. C.

Lychak, O. V.

T. I. Voronyak, A. B. Kmet’, and O. V. Lychak, “Single-step phase-shifting speckle interferometry,” Mater Sci. 43, 554–567 (2007).
[CrossRef]

Malacara, D.

Marroguin, J. L.

Mendoza Santoyo, F.

Nakadate, S.

Nam, L. T. H.

Perkins, S.

R. Fisher, S. Perkins, A. Walker, and E. Wolfart, “Frequency filter,” http://homepages.inf.ed.ac.uk/rbf/HIPR2/freqfilt.htm.

Quiroga, J. A.

J. Villa, R. R. Vera, J. A. Quiroga, I. de la Rosa, and E. González, “Anisotropic phase-map denoising using a regularized cost-function with complex-valued Markov-random-fields,” Opt. Lasers Eng. 48, 650–656 (2010).
[CrossRef]

Ren, H.

C. Tang, L. Han, H. Ren, T. Gao, Z. Wang, and K. Tang, “The oriented-couple partial differential equations for filtering in wrapped phase patterns,” Opt. Express. 17, 5606–5617(2009).
[CrossRef] [PubMed]

Robinson, D. W.

D. W. Robinson and D. C. Williams, “Digital phase stepping speckle interferometry,” Opt. Commun. 57, 26–30(1986).
[CrossRef]

Rodriguez-Vera, R.

Romero, L. A.

D. C. Ghiglia and L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A. 11, 107–117 (1994).
[CrossRef]

Ryu, H. S.

Saito, H.

Servin, M.

Soon, S. H.

Tang, C.

C. Tang, T. Gao, S. Yan, L. Wang, and J. Wu, “The oriented spatial filter masks for electronic speckle pattern interferometry phase patterns,” Opt. Express. 18, 8942–8947 (2010).
[CrossRef] [PubMed]

C. Tang, L. Han, H. Ren, T. Gao, Z. Wang, and K. Tang, “The oriented-couple partial differential equations for filtering in wrapped phase patterns,” Opt. Express. 17, 5606–5617(2009).
[CrossRef] [PubMed]

C. Tang, W. Wang, H. Yan, and X. Gu, “Tangent least-squares fitting filtering method for electrical speckle pattern interferometry phase fringe patterns,” Appl. Opt. 46, 2907–2913 (2007).
[CrossRef] [PubMed]

Tang, K.

C. Tang, L. Han, H. Ren, T. Gao, Z. Wang, and K. Tang, “The oriented-couple partial differential equations for filtering in wrapped phase patterns,” Opt. Express. 17, 5606–5617(2009).
[CrossRef] [PubMed]

Tyrer, J. R.

Vera, R. R.

J. Villa, R. R. Vera, J. A. Quiroga, I. de la Rosa, and E. González, “Anisotropic phase-map denoising using a regularized cost-function with complex-valued Markov-random-fields,” Opt. Lasers Eng. 48, 650–656 (2010).
[CrossRef]

Villa, J.

J. Villa, R. R. Vera, J. A. Quiroga, I. de la Rosa, and E. González, “Anisotropic phase-map denoising using a regularized cost-function with complex-valued Markov-random-fields,” Opt. Lasers Eng. 48, 650–656 (2010).
[CrossRef]

Voronyak, T. I.

T. I. Voronyak, A. B. Kmet’, and O. V. Lychak, “Single-step phase-shifting speckle interferometry,” Mater Sci. 43, 554–567 (2007).
[CrossRef]

Waldner, S.

H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999).
[CrossRef]

Walker, A.

R. Fisher, S. Perkins, A. Walker, and E. Wolfart, “Frequency filter,” http://homepages.inf.ed.ac.uk/rbf/HIPR2/freqfilt.htm.

Wang, L.

C. Tang, T. Gao, S. Yan, L. Wang, and J. Wu, “The oriented spatial filter masks for electronic speckle pattern interferometry phase patterns,” Opt. Express. 18, 8942–8947 (2010).
[CrossRef] [PubMed]

Wang, W.

Wang, Z.

C. Tang, L. Han, H. Ren, T. Gao, Z. Wang, and K. Tang, “The oriented-couple partial differential equations for filtering in wrapped phase patterns,” Opt. Express. 17, 5606–5617(2009).
[CrossRef] [PubMed]

Williams, D. C.

D. W. Robinson and D. C. Williams, “Digital phase stepping speckle interferometry,” Opt. Commun. 57, 26–30(1986).
[CrossRef]

Wolfart, E.

R. Fisher, S. Perkins, A. Walker, and E. Wolfart, “Frequency filter,” http://homepages.inf.ed.ac.uk/rbf/HIPR2/freqfilt.htm.

Wu, J.

C. Tang, T. Gao, S. Yan, L. Wang, and J. Wu, “The oriented spatial filter masks for electronic speckle pattern interferometry phase patterns,” Opt. Express. 18, 8942–8947 (2010).
[CrossRef] [PubMed]

Wu, K. C.

Yan, H.

Yan, S.

C. Tang, T. Gao, S. Yan, L. Wang, and J. Wu, “The oriented spatial filter masks for electronic speckle pattern interferometry phase patterns,” Opt. Express. 18, 8942–8947 (2010).
[CrossRef] [PubMed]

Yang, C. S.

Yeh, G. B.

Appl. Opt. (7)

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. A. (1)

D. C. Ghiglia and L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A. 11, 107–117 (1994).
[CrossRef]

Mater Sci. (1)

T. I. Voronyak, A. B. Kmet’, and O. V. Lychak, “Single-step phase-shifting speckle interferometry,” Mater Sci. 43, 554–567 (2007).
[CrossRef]

Opt. Commun. (2)

D. W. Robinson and D. C. Williams, “Digital phase stepping speckle interferometry,” Opt. Commun. 57, 26–30(1986).
[CrossRef]

H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999).
[CrossRef]

Opt. Express. (2)

C. Tang, L. Han, H. Ren, T. Gao, Z. Wang, and K. Tang, “The oriented-couple partial differential equations for filtering in wrapped phase patterns,” Opt. Express. 17, 5606–5617(2009).
[CrossRef] [PubMed]

C. Tang, T. Gao, S. Yan, L. Wang, and J. Wu, “The oriented spatial filter masks for electronic speckle pattern interferometry phase patterns,” Opt. Express. 18, 8942–8947 (2010).
[CrossRef] [PubMed]

Opt. Lasers Eng. (2)

J. Villa, R. R. Vera, J. A. Quiroga, I. de la Rosa, and E. González, “Anisotropic phase-map denoising using a regularized cost-function with complex-valued Markov-random-fields,” Opt. Lasers Eng. 48, 650–656 (2010).
[CrossRef]

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304–317 (2007).
[CrossRef]

Opt. Lett. (1)

Pattern Recogn. (1)

S. Chikkerur, A. N. Cartwright, V. Govindaraju, “Fingerprint enhancement using STFT analysis,” Pattern Recogn. 40, 198–211 (2007).
[CrossRef]

Proc. SPIE. (1)

G. H. Kaufmann, A. Davila, and D. Kerr, “Speckle noise reduction in TV holography,” Proc. SPIE. 2730, 96–100 (1996).
[CrossRef]

Other (1)

R. Fisher, S. Perkins, A. Walker, and E. Wolfart, “Frequency filter,” http://homepages.inf.ed.ac.uk/rbf/HIPR2/freqfilt.htm.

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Figures (4)

Fig. 1
Fig. 1

Computer-simulated phase pattern and its processed images. (a) Initial image; (b) filtered image of (a) by the RFM; (c) filtered image of (a) by the WFF; (d) filtered image of (a) by our method; (e) exact three-dimensional phase; (f) evaluated three-dimensional unwrapped phase based on (b); (g) evaluated three-dimensional unwrapped phase based on Fig. 1c; and (h) evaluated three-dimensional unwrapped phase based on (d).

Fig. 2
Fig. 2

Another computer-simulated phase pattern and its processed results. (a) Initial image; (b) filtered image of (a) by the RFM; (c) filtered image of (a) by the WFF; (d) filtered image of (a) by our method; (e) exact three-dimensional phase; (f) evaluated three- dimensional unwrapped phase based on (b); (g) evaluated three-dimensional unwrapped phase based on (c); and (h) evaluated three- dimensional unwrapped phase based on (d).

Fig. 3
Fig. 3

Experimentally obtained original ESPI phase pattern and filtered images. (a) Initial image; (b) filtered image of (a) by the RFM; (c) filtered image of (a) by the WFF; (d) filtered image of (a) by our method; (e) evaluated three-dimensional unwrapped phase based on (b); (f) evaluated three-dimensional unwrapped phase based on (c); and (g) evaluated three-dimensional unwrapped phase based on (d).

Fig. 4
Fig. 4

Experimental setup for Fig. 3a.

Tables (1)

Tables Icon

Table 1 MSE of the Unwrapped Phase Values and the Computational Times

Equations (29)

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

G ( ξ , η ) = F ( ξ , η ) H ( ξ , η ) ,
H ( ξ , η ) = { 1 if     ξ 2 + η 2 < D 0 0 if     ξ 2 + η 2 > D 0 .
I = I average ( I ) .
F ( ξ , η ) = F F T ( I ( x , y ) ) .
H ( ξ , η ) = | F ( ξ , η ) | α .
G ( ξ , η ) = F ( ξ , η ) | F ( ξ , η ) | α .
I enh ( x , y ) = IFFT [ G ( ξ , η ) ] .
SF ( u , v , ξ , η ) = [ I ( u , v ) g 0 , 0 , ξ , η ( u , v ) ] exp ( - i ξ u - i η v ) ,
I ( x , y ) = 1 4 π 2 η l η h ξ l ξ h [ I ( x , y ) g 0 , 0 , ξ , η ( x , y ) ] g 0 , 0 , ξ , η ( x , y ) ,
g u , v ; ξ , η ( x u , y v ) = g ( x u , y v ) · exp ( i ξ x + i η y ) ,
g ( x , y ) = exp ( x 2 2 σ x 2 y 2 2 σ y 2 ) ,
H ( u , v , ξ , η ) = { 1 if     | SF ( u , v , ξ , η ) | thr 0 if     | S F ( u , v , ξ , η ) | < thr ,
G ( u , v , ξ , η ) = SF ( u , v , ξ , η ) H ( u , v , ξ , η ) .
I ( x , y ) = G σ ( x , y ) * I ( x , y ) ,
G σ ( x , y ) = 1 π σ exp [ ( x 2 + y 2 ) σ ] .
FFT ( I ( x , y ) ) = FFT [ G σ ( x , y ) * I ( x , y ) ] = FFT [ I ( x , y ) ] · FFT [ G σ ( x , y ) ] .
FFT ( G σ ( x , y ) ) = 1 σ π + + exp [ ( x 2 + y 2 ) σ ] · exp [ i 2 π ( ξ x + η y ) ] d x d y = exp [ π 2 σ ( ξ 2 + η 2 ) ] .
FFT [ I ( x , y ) ] = exp [ π 2 σ ( ξ 2 + η 2 ) ] FFT [ I ( x , y ) ] .
H ( ξ , η ) = | FFT [ I ( x , y ) ] | α = | exp [ π 2 σ ( ξ 2 + η 2 ) ] FFT [ I ( x , y ) ] | α .
G ( ξ , η ) = FFT [ I ( x , y ) ] | FFT [ I ( x , y ) ] · exp [ π 2 σ ( ξ 2 + η 2 ) ] | α .
I enh ( x , y ) = IFFT [ G ( ξ , η ) ] .
MSE = 1 M × N i = 1 M j = 1 N [ ψ ( i , j ) ψ 0 ( i , j ) ] 2 i = 1 M j = 1 N [ ψ 0 ( i , j ) ] 2 ,
I 1 , i , j = I o , i , j + I r , i , j + 2 I o , i , j I r , i , j cos φ i , j + n 0 , i , j ,
I 2 , i , j = I o , i , j + I r , i , j + 2 I o , i , j I r , i , j cos ( φ i , j + π 2 ) + n 0 , i , j ,
I 3 , i , j = I o , i , j + I r , i , j + 2 I o , i , j I r , i , j cos ( φ i , j + ψ i , j ) + n 0 , i , j ,
I 4 , i , j = I o , i , j + I r , i , j + 2 I o , i , j I r , i , j cos ( φ i , j + ψ i , j + π 2 ) + n 0 , i , j ,
ψ i , j = 2 tan 1 ( I 4 , i , j I 2 , i , j + I 3 , i , j I 1 , i , j I 4 , i , j + I 2 , i , j I 3 , i , j I 1 , i , j ) .
ψ i , j = 3 π { [ 7.5 × ( i M 2 ) M ] 2 + [ 7.5 × ( j N 2 ) N ] 2 } .
ψ i , j = 100 { exp [ ( 2 i M ) 2 + ( 2 j 3 N 2 ) 2 20000 ] + exp [ ( 2 i M ) 2 + ( 2 j N 2 ) 2 30000 ] } + 15 { [ 3 ( i M 2 ) M ] 2 + [ 3 ( j N 3 ) N ] 2 } .

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