Abstract

We extend and refine previous work [Appl. Opt. 46, 2907 (2007)]. Combining the coupled nonlinear partial differential equations (PDEs) denoising model with the ordinary differential equations enhancement method, we propose the new denoising and enhancing model for electronic speckle pattern interferometry (ESPI) fringe patterns. Meanwhile, we propose the backpropagation neural networks (BPNN) method to obtain unwrapped phase values based on a skeleton map instead of traditional interpolations. We test the introduced methods on the computer-simulated speckle ESPI fringe patterns and experimentally obtained fringe pattern, respectively. The experimental results show that the coupled nonlinear PDEs denoising model is capable of effectively removing noise, and the unwrapped phase values obtained by the BPNN method are much more accurate than those obtained by the well-known traditional interpolation. In addition, the accuracy of the BPNN method is adjustable by changing the parameters of networks such as the number of neurons.

© 2007 Optical Society of America

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  1. 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]
  2. A. P. Witkin, "Scale-space filtering," in the International Joint Conference on Artificial Intelligence (Karlsruhe, 1983), pp. 1019-1022.
  3. P. Perona and J. Malik, "Scale-space and edge detection using anisotropic diffusion," IEEE Trans. Pattern Anal. Mach. Intell. 12, 629-639 (1990).
    [CrossRef]
  4. F. Catté, P.-L. Lions, J.-M. Morel, and T. Coll, "Image selective smoothing and edge detection by nonlinear diffusion," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 29, 182-193 (1992).
  5. L. Alvarez, P.-L. Lions, and J.-M. Morel, "Image selective smoothing and edge detection by nonlinear diffusion," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 29, 845-866 (1992).
  6. G. Sapiro and V. Caselles, "Contrast enhancement via image evolution flows," Graph. Models Image Process. 59, 407-416 (1997).
    [CrossRef]
  7. C. Tang, F. Zhang, and Z. Chen, "Contrast enhancement for electronic speckle pattern interferometry fringes by the differential equation enhancement method," Appl. Opt. 45, 2287-2294 (2006).
    [CrossRef] [PubMed]
  8. D. Mumford and J. Shah, "Optimal approximations by piecewise smooth functions and associated variational problems," Commun. Pure Appl. Math. 17, 577-685 (1989).
    [CrossRef]
  9. J. M. Morel and S. Solimini, Variational Methods in Image Segmentation (Birkhaäuser, 1995).
  10. Y. Chen, C. A. Z. Barcelos, and B. A. Mairz, "Smoothing and edge detection by time-varying coupled nonlinear diffusion equations," Comput. Vision Image Understand. 82, 85-100 (2001).
    [CrossRef]
  11. C. Tang, F. Zhang, H. Yan, and Z. Chen, "Denoising in electronic speckle pattern interferometry fringes by the filtering method based on partial differential equations," Opt. Commun. 260, 91-96 (2006).
    [CrossRef]
  12. "Neural Networks," http://www.statsoft.com/textbook/stneunet.html.
  13. J. Principe, N. Euliano, and W. Lefebvre, Neural and Adaptive System--Fundamentals Through Simulations (Wiley, 2000).
  14. J. Nazari and O. K. Ersoy, "Implementation of back-propagation neural networks with MatLab," http://docs.lib.purdue.edu/ecetr/275.
  15. N. A. Ochoa, F. M. Santoyo, A. J. Moore, and C. P. López, "Contrast enhancement of electronic speckle pattern interferometry addition fringes," Appl. Opt. 36, 2783-2787 (1997).
    [CrossRef] [PubMed]

2006 (3)

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, F. Zhang, H. Yan, and Z. Chen, "Denoising in electronic speckle pattern interferometry fringes by the filtering method based on partial differential equations," Opt. Commun. 260, 91-96 (2006).
[CrossRef]

C. Tang, F. Zhang, and Z. Chen, "Contrast enhancement for electronic speckle pattern interferometry fringes by the differential equation enhancement method," Appl. Opt. 45, 2287-2294 (2006).
[CrossRef] [PubMed]

2001 (1)

Y. Chen, C. A. Z. Barcelos, and B. A. Mairz, "Smoothing and edge detection by time-varying coupled nonlinear diffusion equations," Comput. Vision Image Understand. 82, 85-100 (2001).
[CrossRef]

1997 (2)

1992 (2)

F. Catté, P.-L. Lions, J.-M. Morel, and T. Coll, "Image selective smoothing and edge detection by nonlinear diffusion," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 29, 182-193 (1992).

L. Alvarez, P.-L. Lions, and J.-M. Morel, "Image selective smoothing and edge detection by nonlinear diffusion," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 29, 845-866 (1992).

1990 (1)

P. Perona and J. Malik, "Scale-space and edge detection using anisotropic diffusion," IEEE Trans. Pattern Anal. Mach. Intell. 12, 629-639 (1990).
[CrossRef]

1989 (1)

D. Mumford and J. Shah, "Optimal approximations by piecewise smooth functions and associated variational problems," Commun. Pure Appl. Math. 17, 577-685 (1989).
[CrossRef]

Alvarez, L.

L. Alvarez, P.-L. Lions, and J.-M. Morel, "Image selective smoothing and edge detection by nonlinear diffusion," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 29, 845-866 (1992).

Barcelos, C. A. Z.

Y. Chen, C. A. Z. Barcelos, and B. A. Mairz, "Smoothing and edge detection by time-varying coupled nonlinear diffusion equations," Comput. Vision Image Understand. 82, 85-100 (2001).
[CrossRef]

Caselles, V.

G. Sapiro and V. Caselles, "Contrast enhancement via image evolution flows," Graph. Models Image Process. 59, 407-416 (1997).
[CrossRef]

Catté, F.

F. Catté, P.-L. Lions, J.-M. Morel, and T. Coll, "Image selective smoothing and edge detection by nonlinear diffusion," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 29, 182-193 (1992).

Chen, Y.

Y. Chen, C. A. Z. Barcelos, and B. A. Mairz, "Smoothing and edge detection by time-varying coupled nonlinear diffusion equations," Comput. Vision Image Understand. 82, 85-100 (2001).
[CrossRef]

Chen, Z.

C. Tang, F. Zhang, and Z. Chen, "Contrast enhancement for electronic speckle pattern interferometry fringes by the differential equation enhancement method," Appl. Opt. 45, 2287-2294 (2006).
[CrossRef] [PubMed]

C. Tang, F. Zhang, H. Yan, and Z. Chen, "Denoising in electronic speckle pattern interferometry fringes by the filtering method based on partial differential equations," Opt. Commun. 260, 91-96 (2006).
[CrossRef]

Coll, T.

F. Catté, P.-L. Lions, J.-M. Morel, and T. Coll, "Image selective smoothing and edge detection by nonlinear diffusion," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 29, 182-193 (1992).

Ersoy, O. K.

J. Nazari and O. K. Ersoy, "Implementation of back-propagation neural networks with MatLab," http://docs.lib.purdue.edu/ecetr/275.

Euliano, N.

J. Principe, N. Euliano, and W. Lefebvre, Neural and Adaptive System--Fundamentals Through Simulations (Wiley, 2000).

Lefebvre, W.

J. Principe, N. Euliano, and W. Lefebvre, Neural and Adaptive System--Fundamentals Through Simulations (Wiley, 2000).

Li, B.

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]

Lions, P.-L.

L. Alvarez, P.-L. Lions, and J.-M. Morel, "Image selective smoothing and edge detection by nonlinear diffusion," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 29, 845-866 (1992).

F. Catté, P.-L. Lions, J.-M. Morel, and T. Coll, "Image selective smoothing and edge detection by nonlinear diffusion," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 29, 182-193 (1992).

López, C. P.

Mairz, B. A.

Y. Chen, C. A. Z. Barcelos, and B. A. Mairz, "Smoothing and edge detection by time-varying coupled nonlinear diffusion equations," Comput. Vision Image Understand. 82, 85-100 (2001).
[CrossRef]

Malik, J.

P. Perona and J. Malik, "Scale-space and edge detection using anisotropic diffusion," IEEE Trans. Pattern Anal. Mach. Intell. 12, 629-639 (1990).
[CrossRef]

Moore, A. J.

Morel, J. M.

J. M. Morel and S. Solimini, Variational Methods in Image Segmentation (Birkhaäuser, 1995).

Morel, J.-M.

L. Alvarez, P.-L. Lions, and J.-M. Morel, "Image selective smoothing and edge detection by nonlinear diffusion," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 29, 845-866 (1992).

F. Catté, P.-L. Lions, J.-M. Morel, and T. Coll, "Image selective smoothing and edge detection by nonlinear diffusion," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 29, 182-193 (1992).

Mumford, D.

D. Mumford and J. Shah, "Optimal approximations by piecewise smooth functions and associated variational problems," Commun. Pure Appl. Math. 17, 577-685 (1989).
[CrossRef]

Nazari, J.

J. Nazari and O. K. Ersoy, "Implementation of back-propagation neural networks with MatLab," http://docs.lib.purdue.edu/ecetr/275.

Ochoa, N. A.

Perona, P.

P. Perona and J. Malik, "Scale-space and edge detection using anisotropic diffusion," IEEE Trans. Pattern Anal. Mach. Intell. 12, 629-639 (1990).
[CrossRef]

Principe, J.

J. Principe, N. Euliano, and W. Lefebvre, Neural and Adaptive System--Fundamentals Through Simulations (Wiley, 2000).

Santoyo, F. M.

Sapiro, G.

G. Sapiro and V. Caselles, "Contrast enhancement via image evolution flows," Graph. Models Image Process. 59, 407-416 (1997).
[CrossRef]

Shah, J.

D. Mumford and J. Shah, "Optimal approximations by piecewise smooth functions and associated variational problems," Commun. Pure Appl. Math. 17, 577-685 (1989).
[CrossRef]

Solimini, S.

J. M. Morel and S. Solimini, Variational Methods in Image Segmentation (Birkhaäuser, 1995).

Tang, C.

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, F. Zhang, H. Yan, and Z. Chen, "Denoising in electronic speckle pattern interferometry fringes by the filtering method based on partial differential equations," Opt. Commun. 260, 91-96 (2006).
[CrossRef]

C. Tang, F. Zhang, and Z. Chen, "Contrast enhancement for electronic speckle pattern interferometry fringes by the differential equation enhancement method," Appl. Opt. 45, 2287-2294 (2006).
[CrossRef] [PubMed]

Witkin, A. P.

A. P. Witkin, "Scale-space filtering," in the International Joint Conference on Artificial Intelligence (Karlsruhe, 1983), pp. 1019-1022.

Yan, H.

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, F. Zhang, H. Yan, and Z. Chen, "Denoising in electronic speckle pattern interferometry fringes by the filtering method based on partial differential equations," Opt. Commun. 260, 91-96 (2006).
[CrossRef]

Zhang, F.

C. Tang, F. Zhang, H. Yan, and Z. Chen, "Denoising in electronic speckle pattern interferometry fringes by the filtering method based on partial differential equations," Opt. Commun. 260, 91-96 (2006).
[CrossRef]

C. Tang, F. Zhang, and Z. Chen, "Contrast enhancement for electronic speckle pattern interferometry fringes by the differential equation enhancement method," Appl. Opt. 45, 2287-2294 (2006).
[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]

Appl. Opt. (3)

Commun. Pure Appl. Math. (1)

D. Mumford and J. Shah, "Optimal approximations by piecewise smooth functions and associated variational problems," Commun. Pure Appl. Math. 17, 577-685 (1989).
[CrossRef]

Comput. Vision Image Understand. (1)

Y. Chen, C. A. Z. Barcelos, and B. A. Mairz, "Smoothing and edge detection by time-varying coupled nonlinear diffusion equations," Comput. Vision Image Understand. 82, 85-100 (2001).
[CrossRef]

Graph. Models Image Process. (1)

G. Sapiro and V. Caselles, "Contrast enhancement via image evolution flows," Graph. Models Image Process. 59, 407-416 (1997).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

P. Perona and J. Malik, "Scale-space and edge detection using anisotropic diffusion," IEEE Trans. Pattern Anal. Mach. Intell. 12, 629-639 (1990).
[CrossRef]

Opt. Commun. (1)

C. Tang, F. Zhang, H. Yan, and Z. Chen, "Denoising in electronic speckle pattern interferometry fringes by the filtering method based on partial differential equations," Opt. Commun. 260, 91-96 (2006).
[CrossRef]

SIAM (2)

F. Catté, P.-L. Lions, J.-M. Morel, and T. Coll, "Image selective smoothing and edge detection by nonlinear diffusion," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 29, 182-193 (1992).

L. Alvarez, P.-L. Lions, and J.-M. Morel, "Image selective smoothing and edge detection by nonlinear diffusion," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 29, 845-866 (1992).

Other (5)

"Neural Networks," http://www.statsoft.com/textbook/stneunet.html.

J. Principe, N. Euliano, and W. Lefebvre, Neural and Adaptive System--Fundamentals Through Simulations (Wiley, 2000).

J. Nazari and O. K. Ersoy, "Implementation of back-propagation neural networks with MatLab," http://docs.lib.purdue.edu/ecetr/275.

J. M. Morel and S. Solimini, Variational Methods in Image Segmentation (Birkhaäuser, 1995).

A. P. Witkin, "Scale-space filtering," in the International Joint Conference on Artificial Intelligence (Karlsruhe, 1983), pp. 1019-1022.

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Figures (11)

Fig. 1
Fig. 1

Four-layer BPNN used for phase extraction from a single ESPI fringe pattern.

Fig. 2
Fig. 2

Computer-simulated ESPI fringe pattern corresponding to Eq. (21): (a) initial image and (b) ideal three-dimensional phase.

Fig. 3
Fig. 3

(a) Filtered and enhanced image by the model (1), (b) filtered and enhanced image by the new model (9), (c) gray-distribution curve of (a) for the cross section ( row = 120 ) , (d) gray-distribution curve of (b) for the cross section ( row = 120 ) .

Fig. 4
Fig. 4

(a) Skeletons of Fig. 3(b), (b) three-dimensional unwrapped phase map obtained by the C-spline interpolation method based on (a), (c) three-dimensional unwrapped phase map obtained by our BPNN method based on (a).

Fig. 5
Fig. 5

(a) Skeletons of white fringes in Fig. 2(b), (b) three-dimensional unwrapped phase map obtained by the C-spline interpolation method based on (a), (c) three-dimensional unwrapped phase map obtained by our BPNN method based on (a).

Fig. 6
Fig. 6

Computer-simulated ESPI fringe pattern corresponding to Eq. (22): (a) initial image and (b) ideal three-dimensional phase.

Fig. 7
Fig. 7

(a) Filtered and enhanced images by the model (1), (b) filtered and enhanced images by the new model (9), (c) gray-distribution curve of (a) for the cross section ( row = 200 ) , (d) gray-distribution curve of (b) for the cross section ( row = 200 ) .

Fig. 8
Fig. 8

(a) Skeletons of Fig. 7(b), (b) three-dimensional unwrapped phase map obtained by the C-spline interpolation method based on (a), (c) three-dimensional unwrapped phase map obtained by our BPNN method based on (a).

Fig. 9
Fig. 9

(a) Skeletons of white fringes in 7(b), (b) three-dimensional unwrapped phase map obtained by the C-spline interpolation method based on (a), (c) three-dimensional unwrapped phase map obtained by our BPNN method based on (a).

Fig. 10
Fig. 10

Experimentally obtained ESPI fringe pattern and its results, (a) initial image, (b) filtered and enhanced image by the model (1), (c) filtered and enhanced image by the new model (9), (d) skeletons of (c).

Fig. 11
Fig. 11

(a) Three-dimensional unwrapped phase map obtained by the C-spline interpolation method based on Fig. 10(d), (b) three-dimensional unwrapped phase map obtained by our BPNN method based on Fig. 10(d).

Equations (28)

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

u t = C g ( | G σ u | ) | u | div ( u | u | ) + { [ M × N M × N × u / L ] η 0 } .
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 ,
E ( u ) = Ω f ( | u | ) d x d y ,
u t = α g ( | v | ) | u | div ( u | u | ) + α ( g ( | v | ) ) u β ( u I ) | u | ,
E u ( v ) = Ω [ a | v | + ( b / 2 ) | v u | 2 ] d x d y .
v t = a ( t ) div ( v | v | ) b ( v u ) ,
u ( x , y , 0 ) = I ( x , y ) ,   v ( x , y , 0 ) = I ( x , y ) ,   x , y Ω .
u t = C 1 { α g ( | v | ) | u | div ( u | u | ) + α ( g ( | v | ) ) u β ( u I ) | u | } + γ ( t ) { [ M × N M × N × u / L ] η 0 } ,
v t = C 2 { a ( t ) div ( v | v | ) b ( v u ) } ,
( u t ) i , j n = u i , j n + 1 u i , j n Δ t .
( d 1 ) i , j n = | u ( i , j , t n ) | ( div ( u ( i , j , t n ) | u ( i , j , t n ) | ) ) ,
( d 2 ) i , j n = div ( v ( i , j , t n ) | v ( i , j , t n ) | ) .
( d 3 ) i , j n = | v i , j n | .
g i , j n = 1 ( 1 + k ( ( d 3 ) i , j n ) 2 ) .
( d 4 ) i , j n = ( g u ) i , j n .
( d 5 ) i , j n = | u ( i , j , t n ) | = ( ( ( u x ) i , j n ) 2 + ( ( u y ) i , j n ) 2 ) 1 / 2 .
u i , j n + 1 = u i , j n + Δ t × C 1 × { α × ( g i , j n ) × ( d 1 ) i , j n + α × ( d 4 ) i , j n β × ( u i , j n I i , j ) × ( d 5 ) i , j n } + γ n × { [ M × N M × N × u i , j n / L ] ( η 0 ) i , j } ,
v i , j n + 1 = v i , j n + Δ t × C 2 × { a × ( d 2 ) i , j n b × ( v i , j n u i , j n ) } ,
f ( x ) = 1 e x 1 + e x .
a n = { 35 , 1 n 9 a n 1 0.7 , 10 n 55 1 2 a n 1 , n 55 ,   γ n = { 1 , n N 0 0 , n > N 0 ,
u i , j n + 1 = u i , j n + Δ t C ( g i , j n ) ( d 1 ) i , j n + { [ M × N M × N × u i , j n / L ] ( η 0 ) i , j } ,
I s u b = | 4 I o I r   sin ( ϕ r ϕ o + ψ 2 ) sin ( ψ 2 ) | ,
ψ i , j = 25 × ( exp ( ( 1 .4 i 78 ) 2 ( j 78 ) 2 45,000 ) + exp ( ( 1 .4 i 128 ) 2 ( j 192 ) 2 45,000 ) ) .
ψ i , j = ψ i , j 1 + ψ i , j 2 ,
ψ i , j 1 = 80 [ exp ( ( i 0 M 0 ) 2 + ( j 0 N 0 ) 2 30,000 ) + exp ( ( i 0 3 M 0 ) 2 + ( j 0 3 N 0 ) 2 30,000 ) ] ,
ψ i , j 2 = 50 i 0 2 50 i 0 j 0 + 10 i 0 2 j 0 + 20 i 0 j 0 2 10 j 0 2 + j 0 3 + i 0 2 j 0 2 10 i 0 4 ,

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