Abstract

In this paper, we introduce the radial basis function (RBF) interpolation method to electronic speckle pattern interferometry (ESPI) and propose the RBF interpolation method to obtain unwrapped phase values based on a skeleton map. Because of the excellent approximation properties of the RBF interpolation, the proposed method can extract accurate phase values from a single fringe pattern effectively, even using a simple 3×3 mean filter as preprocessing. Using our method, both special filtering methods for ESPI fringes as preprocessing and postprocessing, including a dilatation and erosion algorithm for pruning and connecting and the smooth algorithm for improving the phase values are not needed. We test our method on a computer-simulated and two experimentally obtained poor-quality fringe patterns. The results have demonstrated that our RBF interpolation method works well even under a seriously disconnected skeleton map where it is impossible to apply the widely used, Matlab function grid data interpolation or the backpropagation neural networks method [Appl. Opt. 46, 7475 (2007)].

© 2011 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
  6. F. J. Cuevas, J. H. Sossa-Azuela, and M. Servin, “A parametric method applied to phase recovery from a fringe pattern based on a genetic algorithm,” Opt. Commun. 203, 213–223(2002).
    [CrossRef]
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    [CrossRef]
  8. C. Quan, C. J. Tay, F. Yang, and X. He, “Phase extraction from a single fringe pattern based on guidance of an extreme map,” Appl. Opt. 44, 4814–4821 (2005).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  22. C. Tang, L. Wang, S. Yan, J. Wu, L. Cheng, and C. Li, “Displacement field analysis based on the combination digital speckle correlation method with radial basis function interpolation,” Appl. Opt. 49, 4545–4553 (2010).
    [CrossRef] [PubMed]
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    [CrossRef]
  24. C. B. Barber, D. P. Dobkin, and H. T. Huhdanpaa, “The quickhull algorithm for convex hulls,” ACM Trans. Math. Softw. 22, 469–483 (1996).
    [CrossRef]

2010 (3)

2009 (2)

2008 (1)

2007 (1)

2006 (1)

2005 (1)

2004 (1)

2003 (1)

Q. Yu, X. Sun, X. Liu, X. Ding, and Z. Qiu, “Removing speckle noise and extracting the skeletons from a single speckle fringe pattern by spin filtering with curved-surface windows,” Opt. Eng. 42, 68–74 (2003).
[CrossRef]

2002 (1)

F. J. Cuevas, J. H. Sossa-Azuela, and M. Servin, “A parametric method applied to phase recovery from a fringe pattern based on a genetic algorithm,” Opt. Commun. 203, 213–223(2002).
[CrossRef]

2001 (1)

1996 (2)

G. H. Kaufmann and G. E. Galizzi, “Speckle noise reduction in television holography fringes using wavelet thresholding,” Opt. Eng. 35, 9–14 (1996).
[CrossRef]

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

1994 (2)

1993 (1)

Z. Huang, “Fringe skeleton extraction using adaptive refining,” Opt. Lasers Eng. 18, 281–295 (1993).
[CrossRef]

1986 (1)

1984 (1)

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).

Andresen, K.

Barber, C. B.

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

Bernal, F.

M. Kindelan, F. Bernal, P. González-Rodríguez, and M. Moscoso, “Application of the RBF meshless method to the solution of the radiative transport equation,” J. Comput. Phys. 229, 1897–1908 (2010).
[CrossRef]

Buhmann, M. D.

M. D. Buhmann, Radial Basis Functions: Theory and Implementations (Cambridge University, 2003).
[CrossRef]

M. D. Buhmann, “Radial basis functions,” in Acta Numerica (Cambridge University, 2000), pp. 1–38.

Chang, Y.

Chen, S.

Cheng, L.

Cuevas, F. J.

F. J. Cuevas, J. H. Sossa-Azuela, and M. Servin, “A parametric method applied to phase recovery from a fringe pattern based on a genetic algorithm,” Opt. Commun. 203, 213–223(2002).
[CrossRef]

M. Servin, J. L. Marroquin, and F. J. Cuevas, “Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms,” J. Opt. Soc. Am. A 18, 689–695 (2001).
[CrossRef]

Cui, X.

Ding, X.

Q. Yu, X. Sun, X. Liu, X. Ding, and Z. Qiu, “Removing speckle noise and extracting the skeletons from a single speckle fringe pattern by spin filtering with curved-surface windows,” Opt. Eng. 42, 68–74 (2003).
[CrossRef]

Dobkin, D. P.

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

Galizzi, G. E.

G. H. Kaufmann and G. E. Galizzi, “Speckle noise reduction in television holography fringes using wavelet thresholding,” Opt. Eng. 35, 9–14 (1996).
[CrossRef]

Gao, T.

Gao, W.

González-Rodríguez, P.

M. Kindelan, F. Bernal, P. González-Rodríguez, and M. Moscoso, “Application of the RBF meshless method to the solution of the radiative transport equation,” J. Comput. Phys. 229, 1897–1908 (2010).
[CrossRef]

Han, L.

Han, Lin

He, X.

Huang, Z.

Z. Huang, “Fringe skeleton extraction using adaptive refining,” Opt. Lasers Eng. 18, 281–295 (1993).
[CrossRef]

Huhdanpaa, H. T.

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

Kaufmann, G. H.

G. H. Kaufmann and G. E. Galizzi, “Speckle noise reduction in television holography fringes using wavelet thresholding,” Opt. Eng. 35, 9–14 (1996).
[CrossRef]

Kemao, Q.

Kindelan, M.

M. Kindelan, F. Bernal, P. González-Rodríguez, and M. Moscoso, “Application of the RBF meshless method to the solution of the radiative transport equation,” J. Comput. Phys. 229, 1897–1908 (2010).
[CrossRef]

Kreis, T.

Li, B.

Li, C.

Lin, F.

Liu, X.

Q. Yu, X. Sun, X. Liu, X. Ding, and Z. Qiu, “Removing speckle noise and extracting the skeletons from a single speckle fringe pattern by spin filtering with curved-surface windows,” Opt. Eng. 42, 68–74 (2003).
[CrossRef]

Q. Yu, X. Liu, and K. Andresen, “New spin filters for interferometric fringe patterns and grating patterns,” Appl. Opt. 33, 3705–3711 (1994).
[CrossRef] [PubMed]

Lu, W.

Malacara, D.

Malacara, Z.

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (CRC, 2005).
[CrossRef]

Marroquin, J. L.

Moscoso, M.

M. Kindelan, F. Bernal, P. González-Rodríguez, and M. Moscoso, “Application of the RBF meshless method to the solution of the radiative transport equation,” J. Comput. Phys. 229, 1897–1908 (2010).
[CrossRef]

Neimark, A.

A. Neimark, “Investigating filtering methods for the denoising and contrast enhancement, and evenual skeletonization, of ESPI images,” http://mesoscopic.mines.edu/mediawiki/images/d/de/Project_Report_Neimark.pdf (2009).

Qiu, Z.

Q. Yu, X. Sun, X. Liu, X. Ding, and Z. Qiu, “Removing speckle noise and extracting the skeletons from a single speckle fringe pattern by spin filtering with curved-surface windows,” Opt. Eng. 42, 68–74 (2003).
[CrossRef]

Quan, C.

Ren, H.

Rodriguez-Vera, R.

Servin, M.

Servín, M.

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (CRC, 2005).
[CrossRef]

Soon, S. H.

Sossa-Azuela, J. H.

F. J. Cuevas, J. H. Sossa-Azuela, and M. Servin, “A parametric method applied to phase recovery from a fringe pattern based on a genetic algorithm,” Opt. Commun. 203, 213–223(2002).
[CrossRef]

Sun, X.

Q. Yu, X. Sun, X. Liu, X. Ding, and Z. Qiu, “Removing speckle noise and extracting the skeletons from a single speckle fringe pattern by spin filtering with curved-surface windows,” Opt. Eng. 42, 68–74 (2003).
[CrossRef]

Tang, C.

Tang, K.

Tay, C. J.

Wang, H.

Wang, L.

Wang, W.

Wang, X.

Wang, Z.

Womack, K. H.

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).

Wu, J.

Yan, H.

Yan, S.

Yang, F.

Yu, Q.

Q. Yu, X. Sun, X. Liu, X. Ding, and Z. Qiu, “Removing speckle noise and extracting the skeletons from a single speckle fringe pattern by spin filtering with curved-surface windows,” Opt. Eng. 42, 68–74 (2003).
[CrossRef]

Q. Yu, X. Liu, and K. Andresen, “New spin filters for interferometric fringe patterns and grating patterns,” Appl. Opt. 33, 3705–3711 (1994).
[CrossRef] [PubMed]

Zhang, F.

Zhang, Z.

Zhou, D.

ACM Trans. Math. Softw. (1)

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

Appl. Opt. (8)

C. Tang, L. Wang, S. Yan, J. Wu, L. Cheng, and C. Li, “Displacement field analysis based on the combination digital speckle correlation method with radial basis function interpolation,” Appl. Opt. 49, 4545–4553 (2010).
[CrossRef] [PubMed]

C. Tang, Z. Wang, L. Wang, J. Wu, T. Gao, and S. Yan, “Estimation of fringe orientation for optical fringe patterns with poor quality based on Fourier transform,” Appl. Opt. 49, 554–561 (2010).
[CrossRef] [PubMed]

C. Quan, C. J. Tay, F. Yang, and X. He, “Phase extraction from a single fringe pattern based on guidance of an extreme map,” Appl. Opt. 44, 4814–4821 (2005).
[CrossRef] [PubMed]

C. Tang, F. Zhang, B. Li, and H. Yan, “Performance evaluation of partial differential equation models in electronic speckle pattern interferometry and δ-mollification method of phase map,” Appl. Opt. 45, 7392–7400 (2006).
[CrossRef] [PubMed]

C. Tang, W. Lu, S. Chen, Z. Zhang, B. Li, W. Wang, and Lin Han, “Denoising by coupled partial differential equations and extracting phase by backpropagation neural networks for electronic speckle pattern interferometry,” Appl. Opt. 46, 7475–7484 (2007).
[CrossRef] [PubMed]

M. Servin, D. Malacara, and R. Rodriguez-Vera, “Phase-locked-loop interferometry applied to aspheric testing with a computer-stored compensator,” Appl. Opt. 33, 2589–2595(1994).
[CrossRef] [PubMed]

Q. Yu, X. Liu, and K. Andresen, “New spin filters for interferometric fringe patterns and grating patterns,” Appl. Opt. 33, 3705–3711 (1994).
[CrossRef] [PubMed]

Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43, 2695–2702 (2004).
[CrossRef] [PubMed]

J. Comput. Phys. (1)

M. Kindelan, F. Bernal, P. González-Rodríguez, and M. Moscoso, “Application of the RBF meshless method to the solution of the radiative transport equation,” J. Comput. Phys. 229, 1897–1908 (2010).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

F. J. Cuevas, J. H. Sossa-Azuela, and M. Servin, “A parametric method applied to phase recovery from a fringe pattern based on a genetic algorithm,” Opt. Commun. 203, 213–223(2002).
[CrossRef]

Opt. Eng. (3)

Q. Yu, X. Sun, X. Liu, X. Ding, and Z. Qiu, “Removing speckle noise and extracting the skeletons from a single speckle fringe pattern by spin filtering with curved-surface windows,” Opt. Eng. 42, 68–74 (2003).
[CrossRef]

G. H. Kaufmann and G. E. Galizzi, “Speckle noise reduction in television holography fringes using wavelet thresholding,” Opt. Eng. 35, 9–14 (1996).
[CrossRef]

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).

Opt. Express (1)

Opt. Lasers Eng. (1)

Z. Huang, “Fringe skeleton extraction using adaptive refining,” Opt. Lasers Eng. 18, 281–295 (1993).
[CrossRef]

Opt. Lett. (2)

Other (4)

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (CRC, 2005).
[CrossRef]

A. Neimark, “Investigating filtering methods for the denoising and contrast enhancement, and evenual skeletonization, of ESPI images,” http://mesoscopic.mines.edu/mediawiki/images/d/de/Project_Report_Neimark.pdf (2009).

M. D. Buhmann, “Radial basis functions,” in Acta Numerica (Cambridge University, 2000), pp. 1–38.

M. D. Buhmann, Radial Basis Functions: Theory and Implementations (Cambridge University, 2003).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Two experimental ESPI fringe patterns. Initial image that depicts (a) out-of-plane displacement and (b) derivative of the out-of-plane displacement.

Fig. 2
Fig. 2

Filtered results of Figs. 1a, 1b by various methods. (a) Filtered results of Fig. 1a by the AW filter and MF for two times, (b) filtered results of Fig. 1b by the AW filter and MF for five times, (c) filtered results of Fig. 1a by the PDE–ODE method and the MF for eight times, (d) filtered results of Fig. 1b by the PDE–ODE method and the MF for ten times, (e) filtered results of Fig. 1a by the SOPDE filter and MF for two times, (f) filtered results of Fig. 1b by the SOPDE filter and MF for five times.

Fig. 3
Fig. 3

Skeletons of Fig. 2 by the binarization and Hilditch thinning method. (a) Skeletons of dark fringes of Fig. 2a, (b) skeletons of bright fringes of Fig. 2b, (c) skeletons of dark fringes of Fig. 2c, (d) skeletons of bright fringes of Fig. 2d, (e) skeletons of dark fringes of Fig. 2e, (f) skeletons of bright fringes of Fig. 2f.

Fig. 4
Fig. 4

3 × 3 mask window with the center pixel p 4 .

Fig. 5
Fig. 5

Computer-simulated ESPI fringe pattern and its processed results. (a) Initial ESPI fringe image, (b) skeleton map of bright fringes of Fig. 5a, (c-1) exact three-dimensional phase distribution, (c-2) gray image of exact phase, (d-1) evaluated three-dimensional phase obtained by the Matlab function grid data, (d-2) gray image of evaluated phase obtained by the Matlab function grid data, (e-1) evaluated three-dimensional phase obtained by the BPNN method, (e-2) gray image of evaluated phase obtained by the BPNN method, (f-1) evaluated three-dimensional phase obtained by our method, (f-2) gray image of evaluated phase obtained by our method.

Fig. 6
Fig. 6

Processed results of Fig. 1a by our method. (a) Skeleton map of bright fringes of Fig. 1a, (b) gray image of evaluated phase obtained by our method, (c) evaluated three-dimensional phase by our method.

Fig. 7
Fig. 7

Processed results of Fig. 1b by our method. (a) Skeleton map of bright fringes of Fig. 1b, (b) gray image of evaluated phase obtained by our method, (c) evaluated three-dimensional phase by our method.

Equations (15)

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

F ( p 4 ) = k = 0 8 g ( p k ) exp [ G ( k ) ] k = 0 8 exp [ G ( k ) ] ,
u t = { [ M × N M × N × u / L ] η 0 } + C g ( | u | ) | u | div ( u | u | ) , u ( x , y , 0 ) = I ( x , y ) ,
η 0 ( x , y ) = A [ ( v , w ) : u ( v , w , t ) u ( x , y , t ) ] , 1 x , v M , 1 y , w N ,
u t = u x x cos 2 θ + u y y sin 2 θ + 2 u x y sin θ cos θ , u ( x , y , 0 ) = I ( x , y ) ,
s ( X ) = i = 1 N w i ϕ ( X X i ) ,
s ( X i ) = f i , i = 1 , 2 , , N ,
[ A ] [ W ] = [ F ] ,
s ( X ) = i = 1 N w i ϕ ( X X i ) + k = 1 M c k p k ( X ) , X R d ,
j = 1 N c j p k ( X j ) = 0 , k = 1 , 2 , , M .
[ A P P T 0 ] [ W C ] = [ F 0 ] ,
[ ϕ 11 ϕ 12 ϕ 1 N 1 x 1 y 1 ϕ 21 ϕ 22 ϕ 2 N 1 x 2 y 2 ϕ N 1 ϕ N 2 ϕ N N 1 x N y N 1 1 1 0 0 0 x 1 x 2 x N 0 0 0 y 1 y 2 y N 0 0 0 ] [ w 1 w 2 w N c 1 c 2 c 3 ] = [ f 1 f 2 f N 0 0 0 ] .
( p 4 > p 3 ) & ( p 4 > p 5 ) & ( p 4 > p 1 ) & ( p 4 > p 7 ) & ( p 4 > p 0 ) & ( p 4 > p 8 ) & ( p 4 > p 2 ) & ( p 4 > p 6 ) ,
( p 4 < p 3 ) & ( p 4 < p 5 ) & ( p 4 < p 1 ) & ( p 4 < p 7 ) & ( p 4 < p 0 ) & ( p 4 < p 8 ) & ( p 4 < p 2 ) & ( p 4 < p 6 ) ,
I sub = | 4 I o I r sin ( ϕ r ϕ o + ψ 2 ) sin ( ψ 2 ) | ,
ψ ( x , y ) = 20 x 0 2 50 x 0 y 0 + 10 x 0 2 y 0 + 20 x 0 y 0 2 10 y 0 2 + y 0 3 + x 0 2 y 0 2 10 x 0 4 ,

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