Abstract

The ordinary differential equation (ODE) and partial differential equation (PDE) image- processing methods have been applied to reduce noise and enhance the contrast of electronic speckle pattern interferometry fringe patterns. We evaluate the performance of a few representative PDE denoising models quantitatively with two parameters called image fidelity and speckle index, and then we choose a good denoising model. Combining this denoising model with the ODE enhancement method, we make it possible to perform contrast enhancement and denoising simultaneously. Second, we introduce the δ-mollification method to smooth the unwrapped phase map. Finally, based on PDE image processing, δ mollification and some traditional techniques, an approach of phase extraction from a single fringe pattern is tested for computer-simulated and experimentally obtained fringe patterns. The method works well under a high noise level and limited visibility and can extract accurate phase values.

© 2006 Optical Society of America

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  1. S.-H. Baik, S.-K. Park, C.-J. Kim, and S.-Y. Kim, "Two-channel spatial phase shifting electronic speckle pattern interferometer," Opt. Commun. 192, 205-211 (2001).
    [CrossRef]
  2. 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]
  3. X. Su and W. Chen, "Fourier transform profilometry: a review," Opt. Lasers Eng. 35, 263-284 (2001).
    [CrossRef]
  4. K. H. Womack, "Interferometric phase measurement using spatial synchronous detection," Opt. Eng. 23, 391-395 (1984).
  5. 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]
  6. 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]
  7. B. V. Dorrío and J. L. Fernández, "Phase-evaluation methods in whole-field optical measurement techniques," Meas. Sci. Technol. 10, R33-R55 (1999).
    [CrossRef]
  8. A. P. Witkin, "Scale-space filtering," in Proceedings of International Joint Conference on Artificial Intelligence (Karlsruhe, 1983), pp. 1019-1021.
  9. P. Perona and J. Malik, "Scale-space and edge detection using anisotropic diffusion," IEEE Trans. Pattern Anal. Mach. Intell. 12, 629-639 (1990).
    [CrossRef]
  10. 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).
  11. 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).
  12. G. Sapiro, "Geometric partial differential equations in image analysis: past, present, and future," in Proceedings of IEEE Second International Conference on Image Processing 3 (IEEE, 1995), pp. 1-4.
    [CrossRef]
  13. G. Sapiro and V. Caselles, "Contrast enhancement via image evolution flows," Graph. Models Image Process. 59, 407-416 (1997).
    [CrossRef]
  14. 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]
  15. 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]
  16. A. Savitzky and M. J. E. Golay, "Smoothing and differentiation of data by simplified least squares procedures," Anal. Chem. 36, 1627-1639 (1964).
    [CrossRef]
  17. A. Dávila, G. H. Kaufmann, and D. Kerr, "Scale-space filter for smoothing electronic speckle pattern interferometry fringes," Opt. Eng. 35, 3549-3554 (1996).
    [CrossRef]
  18. 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]
  19. C. E. Mejía and D. A. Murio, "Numerical solution of generalized IHCP by discrete mollification," Comput. Math. Appl. 32, 33-50 (1996).

2006

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]

2005

2001

X. Su and W. Chen, "Fourier transform profilometry: a review," Opt. Lasers Eng. 35, 263-284 (2001).
[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]

S.-H. Baik, S.-K. Park, C.-J. Kim, and S.-Y. Kim, "Two-channel spatial phase shifting electronic speckle pattern interferometer," Opt. Commun. 192, 205-211 (2001).
[CrossRef]

1999

B. V. Dorrío and J. L. Fernández, "Phase-evaluation methods in whole-field optical measurement techniques," Meas. Sci. Technol. 10, R33-R55 (1999).
[CrossRef]

1997

1996

C. E. Mejía and D. A. Murio, "Numerical solution of generalized IHCP by discrete mollification," Comput. Math. Appl. 32, 33-50 (1996).

A. Dávila, G. H. Kaufmann, and D. Kerr, "Scale-space filter for smoothing electronic speckle pattern interferometry fringes," Opt. Eng. 35, 3549-3554 (1996).
[CrossRef]

1994

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).

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

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

1984

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

1964

A. Savitzky and M. J. E. Golay, "Smoothing and differentiation of data by simplified least squares procedures," Anal. Chem. 36, 1627-1639 (1964).
[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).

Baik, S.-H.

S.-H. Baik, S.-K. Park, C.-J. Kim, and S.-Y. Kim, "Two-channel spatial phase shifting electronic speckle pattern interferometer," Opt. Commun. 192, 205-211 (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, W.

X. Su and W. Chen, "Fourier transform profilometry: a review," Opt. Lasers Eng. 35, 263-284 (2001).
[CrossRef]

Chen, Z.

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]

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).

Cuevas, F. J.

Dávila, A.

A. Dávila, G. H. Kaufmann, and D. Kerr, "Scale-space filter for smoothing electronic speckle pattern interferometry fringes," Opt. Eng. 35, 3549-3554 (1996).
[CrossRef]

Dorrío, B. V.

B. V. Dorrío and J. L. Fernández, "Phase-evaluation methods in whole-field optical measurement techniques," Meas. Sci. Technol. 10, R33-R55 (1999).
[CrossRef]

Fernández, J. L.

B. V. Dorrío and J. L. Fernández, "Phase-evaluation methods in whole-field optical measurement techniques," Meas. Sci. Technol. 10, R33-R55 (1999).
[CrossRef]

Golay, M. J. E.

A. Savitzky and M. J. E. Golay, "Smoothing and differentiation of data by simplified least squares procedures," Anal. Chem. 36, 1627-1639 (1964).
[CrossRef]

He, X.

Kaufmann, G. H.

A. Dávila, G. H. Kaufmann, and D. Kerr, "Scale-space filter for smoothing electronic speckle pattern interferometry fringes," Opt. Eng. 35, 3549-3554 (1996).
[CrossRef]

Kerr, D.

A. Dávila, G. H. Kaufmann, and D. Kerr, "Scale-space filter for smoothing electronic speckle pattern interferometry fringes," Opt. Eng. 35, 3549-3554 (1996).
[CrossRef]

Kim, C.-J.

S.-H. Baik, S.-K. Park, C.-J. Kim, and S.-Y. Kim, "Two-channel spatial phase shifting electronic speckle pattern interferometer," Opt. Commun. 192, 205-211 (2001).
[CrossRef]

Kim, S.-Y.

S.-H. Baik, S.-K. Park, C.-J. Kim, and S.-Y. Kim, "Two-channel spatial phase shifting electronic speckle pattern interferometer," Opt. Commun. 192, 205-211 (2001).
[CrossRef]

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.

Malacara, D.

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]

Marroquin, J. L.

Moore, A. J.

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).

Ochoa, N. A.

Park, S.-K.

S.-H. Baik, S.-K. Park, C.-J. Kim, and S.-Y. Kim, "Two-channel spatial phase shifting electronic speckle pattern interferometer," Opt. Commun. 192, 205-211 (2001).
[CrossRef]

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]

Quan, C.

Rodriguez-Vera, R.

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]

G. Sapiro, "Geometric partial differential equations in image analysis: past, present, and future," in Proceedings of IEEE Second International Conference on Image Processing 3 (IEEE, 1995), pp. 1-4.
[CrossRef]

Savitzky, A.

A. Savitzky and M. J. E. Golay, "Smoothing and differentiation of data by simplified least squares procedures," Anal. Chem. 36, 1627-1639 (1964).
[CrossRef]

Servin, M.

Su, X.

X. Su and W. Chen, "Fourier transform profilometry: a review," Opt. Lasers Eng. 35, 263-284 (2001).
[CrossRef]

Tang, C.

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]

Tay, C. J.

Witkin, A. P.

A. P. Witkin, "Scale-space filtering," in Proceedings of International Joint Conference on Artificial Intelligence (Karlsruhe, 1983), pp. 1019-1021.

Womack, K. H.

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

Yan, H.

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]

Yang, F.

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]

Anal. Chem.

A. Savitzky and M. J. E. Golay, "Smoothing and differentiation of data by simplified least squares procedures," Anal. Chem. 36, 1627-1639 (1964).
[CrossRef]

Appl. Opt.

Comput. Math. Appl.

C. E. Mejía and D. A. Murio, "Numerical solution of generalized IHCP by discrete mollification," Comput. Math. Appl. 32, 33-50 (1996).

Graph. Models Image Process.

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.

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

J. Opt. Soc. Am. A

Meas. Sci. Technol.

B. V. Dorrío and J. L. Fernández, "Phase-evaluation methods in whole-field optical measurement techniques," Meas. Sci. Technol. 10, R33-R55 (1999).
[CrossRef]

Opt. Commun.

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]

S.-H. Baik, S.-K. Park, C.-J. Kim, and S.-Y. Kim, "Two-channel spatial phase shifting electronic speckle pattern interferometer," Opt. Commun. 192, 205-211 (2001).
[CrossRef]

Opt. Eng.

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

A. Dávila, G. H. Kaufmann, and D. Kerr, "Scale-space filter for smoothing electronic speckle pattern interferometry fringes," Opt. Eng. 35, 3549-3554 (1996).
[CrossRef]

Opt. Lasers Eng.

X. Su and W. Chen, "Fourier transform profilometry: a review," Opt. Lasers Eng. 35, 263-284 (2001).
[CrossRef]

SIAM

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

G. Sapiro, "Geometric partial differential equations in image analysis: past, present, and future," in Proceedings of IEEE Second International Conference on Image Processing 3 (IEEE, 1995), pp. 1-4.
[CrossRef]

A. P. Witkin, "Scale-space filtering," in Proceedings of International Joint Conference on Artificial Intelligence (Karlsruhe, 1983), pp. 1019-1021.

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Figures (7)

Fig. 1
Fig. 1

Computer-simulated fringe pattern and its filtered images with different PDE filterings and one mean filtering: (a) initial image, (b) filtered image by Eq. (2), (c) filtered image by Eq. (3), (d) filtered image by Eq. (5), (e) filtered image by Eq. (7), (f) filtered image by Eq. (8).

Fig. 2
Fig. 2

Skeletons obtained by binarization and thinning algorithms: (a) skeleton of Fig. 1(d) and (b) skeleton of Fig. 1(f).

Fig. 3
Fig. 3

Comparing δ mollification with a Savitzky–Golay filter operator. Top, a computer-simulated noisy sine signal; middle, filtered signal obtained by the Savitzky–Golay filter operator; bottom, filtered signal obtained by δ mollification.

Fig. 4
Fig. 4

Computer-simulated fringe pattern and its processed results: (a) initial image, (b) ideal 3D phase, (c) PDE-processed image by Eq. (17), (d) skeleton map of (c). Three-dimensional unwrapped phase distribution (e) before and (f) after δ mollification.

Fig. 5
Fig. 5

Computer-simulated fringe pattern and its processed results: (a) initial image, (b) ideal 3D phase distribution; (c) PDE-processed image by Eq. (17), (d) skeleton map of (c). Three-dimensional unwrapped phase distribution (e) before and (f) after δ mollification.

Fig. 6
Fig. 6

Computer-simulated fringe pattern and its processed results: (a) initial image, (b) ideal 3D phase distribution, (c) PDE-processed image by Eq. (17), (d) skeleton map of (c). Three-dimensional unwrapped phase distribution (e) before and (f) after δ mollification.

Fig. 7
Fig. 7

Experimental fringe pattern and its processed results: (a) initial image, (b) enhanced and filtered image of (a) by Eq. (17), (c) skeleton map of (b). Three-dimensional unwrapped phase distribution (d) before and (e) after δ mollification.

Tables (1)

Tables Icon

Table 1 Performance Evaluation Results for the Various PDE Denoising Models Based on Fig. 1 Fringe Patterns

Equations (27)

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

u t = F [ u ( x , y , t ) ] , u ( x , y , 0 ) = I ( x , y ) ,
u t = 2 u , u ( x , y , 0 ) = I ( x , y ) .
u t = div [ g ( | u | ) u ] , u ( x , y , 0 ) = I ( x , y ) ,
g k ( s ) = ( 1 + ks 2 ) - 1 ,
u t = div [ g ( | G σ u | ) u ] , u ( x , y , 0 ) = I ( x , y ) ,
G ( x , y , t ) = C t - 1 exp ( - ( x 2 + y 2 ) / 4 t ) .
u t = | u | div ( u | u | ) , u ( x , y , 0 ) = I ( x , y ) .
u t = g ( | G σ 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 ( x , y , t ) t = { M × N - [ M × N × u ( x , y , t ) ] L } - η 0 ( x , y ) ,
u ( x , y , 0 ) = I ( x , y ) ,
f = 1 - ( I 0 - I ) 2 I 0 2 ,
I sub = | 4 I o I r sin ( ϕ r - ϕ o + ψ 2 ) sin ( ψ 2 ) | ,
ψ ( x , y ) = 4.5 π [ - 0.02 x + ( x - M / 2 M / 2 ) 2 + ( y - N / 2 N / 2 ) 2 ] .
u t = F 1 [ u ( x , y , t ) ] ,
u t = F 2 [ u ( x , y , t ) ] ,
u t = F 1 [ u ( x , y , t ) ] + C F 2 [ u ( x , y , t ) ] ,
t u = { [ M × N - M × N × u L ] η 0 } + C g ( | G σ u | ) × | u | div ( u | u | ) .
A p = [ - p p exp ( s 2 ) d s ] - 1 .
ρ δ , p ( t ) = { A p δ - 1 exp ( - t 2 δ 2 ) , 0 , | t | < p δ | t | p δ .
- p δ p δ ρ δ , p ( t ) d t = 1 .
J δ f ( t ) = 0 1 ρ δ , p ( t - s ) f ( s ) d s ,
ψ ( x , y ) = 40 [ exp ( - ( x 0 - M 0 / 2 ) 2 + y 0 2 12000 ) - exp ( - ( x 0 - M 0 / 2 ) 2 + ( y 0 - N 0 ) 2 12000 ) ] ,
ψ ( x , y ) = 40 x 0 2 - 20 x 0 y 0 + 10 x 0 y 0 2 - 10 y 0 2 + 20 y 0 3 ,
ψ ( x , y ) = 60 [ exp ( - ( x 0 - M 0 ) 2 + ( y 0 - N 0 ) 2 15000 ) + exp ( - ( x 0 - 3 M 0 ) 2 + ( y 0 - 3 N 0 ) 2 15000 ) ] ,

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