Abstract

In this paper, we first present the general description for partial differential equations (PDEs) based image processing methods, including the basic idea, the main advantages and disadvantages, a few representative PDE models, and the derivation of PDE models. Then we review our contributions on PDE-based anisotropic filtering methods for electronic speckle pattern interferometry, including the second-order, fourth-order, and coupled nonoriented PDE filtering models and the second-order and coupled nonlinear oriented PDE filtering models. We have summarized the features of each model.

© 2012 Optical Society of America

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  1. M. Aslan, “Toward the development of high-speed microscopic ESPI system for monitoring laser heating/drilling of alumina Al2O3 substrates,” Ph.D. dissertation (Pennsylvania State University, 2000).
  2. X. Su and W. Chen, “Fourier transform profilometry: a review,” Opt. Lasers Eng. 35, 263–284 (2001).
    [CrossRef]
  3. K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
  4. S. Nakadate and H. Saito, “Fringe scanning speckle-pattern interferometry,” Appl. Opt. 24, 2172–2180 (1985).
    [CrossRef]
  5. 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]
  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. D. Kerr, F. M. 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]
  8. C. Tang, L. Han, H. Ren, D. Zhou, Y. Chang, X. Wang, and X. Cui, “Second-order oriented partial-differential equations for denoising in electronic-speckle-pattern interferometry fringes,” Opt. Lett. 33, 2179–2181 (2008).
    [CrossRef]
  9. J. Weikert, Anisotropic Diffusion in Image Processing (Teubner Verlag, 1998).
  10. 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]
  11. W. Lv, C. Tang, and W. Wang, “Noise reduction in electronic speckle pattern interferometry fringes by fourth-order partial differential equations,” Proc. SPIE 6279, (2007).
    [CrossRef]
  12. Y. You and M. Kaveh, “Fourth-order partial differential equations for noise removal,” IEEE Trans. Image Process. 9, 1723–1730 (2000).
    [CrossRef]
  13. A. P. Witkin, “Scale-space filtering,” in Proceedings of International Joint Conference on Artificial Intelligence (Morgan Kaufmann, 1983), pp. 1019–1021.
  14. P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Machine Intell. 12, 629–639 (1990).
    [CrossRef]
  15. F. Catté, P.-L. Lions, J.-M. Morel, and T. Coll, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Numer. Anal. 29, 182–193 (1992).
    [CrossRef]
  16. Y. Chen, C. A. Z. Barcelos, and B. A. Mairz, “Smoothing and edge detection by time-varying coupled nonlinear diffusion equations,” Comput. Vis. Image Understand. 82, 85–100 (2001).
    [CrossRef]
  17. J. Wu and C. Tang, “PDE-based random-valued impulse noise removal based on new class of controlling functions,” IEEE Trans. Image Process. 20, 2428–2438 (2011).
    [CrossRef]
  18. L. Alvarez, P.-L. Lions, and J.-M. Morel, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Numer. Anal. 29, 845–866 (1992).
    [CrossRef]
  19. L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica 60, 259–268 (1992).
    [CrossRef]
  20. M. Lysaker, A. Lundervold, and X.-C. Tai, “Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time,” IEEE Trans. Image Process 12, 1579–1590(2003).
    [CrossRef]
  21. G. Sapiro and D. L. Ringach, “Anisotropic diffusion of multivalued images with applications to color filtering,” IEEE Trans. Image Process 5, 1582–1586 (1996).
    [CrossRef]
  22. Wikipedia, “Euler-Lagrange equation,” http://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation .
  23. G. Sapiro, Geometric Partial Differential Equations and Image Analysis (Cambridge University, 2001).
  24. 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]
  25. 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]
  26. L. Cheng, C. Tang, S. Yan, X. Chen, L. Wang, and B. Wang, “New fourth-order partial differential equations for filtering in electronic speckle pattern interferometry fringes,” Opt. Commun. 284, 5549–5555 (2011).
    [CrossRef]
  27. S. K. Weeratunga and C. Kamath, “A comparison of PDE based non-linear anisotropic diffusion techniques for image denoising,” Proc. SPIE 5014, 151493 (2003).
    [CrossRef]
  28. H. M. Salinas and D. C. Fernández, “Comparison of PDE-based nonlinear diffusion approaches for image enhancement and denoising in optical coherence tomography,” IEEE Trans. Med. Imaging 26, 761–771 (2007).
    [CrossRef]
  29. 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]
  30. H. A. Aebischery and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999).
    [CrossRef]
  31. H. Wang, Q. Kemao, W. Gao, F. Lin, and H. S. Seah, “Fringe pattern denoising using coherence enhancing diffusion,” Opt. Lett. 34, 1141–1143 (2009).
    [CrossRef]

2011

J. Wu and C. Tang, “PDE-based random-valued impulse noise removal based on new class of controlling functions,” IEEE Trans. Image Process. 20, 2428–2438 (2011).
[CrossRef]

L. Cheng, C. Tang, S. Yan, X. Chen, L. Wang, and B. Wang, “New fourth-order partial differential equations for filtering in electronic speckle pattern interferometry fringes,” Opt. Commun. 284, 5549–5555 (2011).
[CrossRef]

2009

2008

2007

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]

H. M. Salinas and D. C. Fernández, “Comparison of PDE-based nonlinear diffusion approaches for image enhancement and denoising in optical coherence tomography,” IEEE Trans. Med. Imaging 26, 761–771 (2007).
[CrossRef]

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

W. Lv, C. Tang, and W. Wang, “Noise reduction in electronic speckle pattern interferometry fringes by fourth-order partial differential equations,” Proc. SPIE 6279, (2007).
[CrossRef]

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]

2003

S. K. Weeratunga and C. Kamath, “A comparison of PDE based non-linear anisotropic diffusion techniques for image denoising,” Proc. SPIE 5014, 151493 (2003).
[CrossRef]

M. Lysaker, A. Lundervold, and X.-C. Tai, “Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time,” IEEE Trans. Image Process 12, 1579–1590(2003).
[CrossRef]

2002

2001

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

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

2000

Y. You and M. Kaveh, “Fourth-order partial differential equations for noise removal,” IEEE Trans. Image Process. 9, 1723–1730 (2000).
[CrossRef]

1999

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

1996

G. Sapiro and D. L. Ringach, “Anisotropic diffusion of multivalued images with applications to color filtering,” IEEE Trans. Image Process 5, 1582–1586 (1996).
[CrossRef]

1992

L. Alvarez, P.-L. Lions, and J.-M. Morel, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Numer. Anal. 29, 845–866 (1992).
[CrossRef]

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica 60, 259–268 (1992).
[CrossRef]

F. Catté, P.-L. Lions, J.-M. Morel, and T. Coll, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Numer. Anal. 29, 182–193 (1992).
[CrossRef]

1990

1985

1984

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

Aebischery, H. A.

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

Alvarez, L.

L. Alvarez, P.-L. Lions, and J.-M. Morel, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Numer. Anal. 29, 845–866 (1992).
[CrossRef]

Aslan, M.

M. Aslan, “Toward the development of high-speed microscopic ESPI system for monitoring laser heating/drilling of alumina Al2O3 substrates,” Ph.D. dissertation (Pennsylvania State University, 2000).

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. Vis. Image Understand. 82, 85–100 (2001).
[CrossRef]

Catté, F.

F. Catté, P.-L. Lions, J.-M. Morel, and T. Coll, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Numer. Anal. 29, 182–193 (1992).
[CrossRef]

Chang, Y.

Chen, S.

Chen, W.

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

Chen, X.

L. Cheng, C. Tang, S. Yan, X. Chen, L. Wang, and B. Wang, “New fourth-order partial differential equations for filtering in electronic speckle pattern interferometry fringes,” Opt. Commun. 284, 5549–5555 (2011).
[CrossRef]

Chen, Y.

Y. Chen, C. A. Z. Barcelos, and B. A. Mairz, “Smoothing and edge detection by time-varying coupled nonlinear diffusion equations,” Comput. Vis. Image Understand. 82, 85–100 (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]

Cheng, L.

L. Cheng, C. Tang, S. Yan, X. Chen, L. Wang, and B. Wang, “New fourth-order partial differential equations for filtering in electronic speckle pattern interferometry fringes,” Opt. Commun. 284, 5549–5555 (2011).
[CrossRef]

Coll, T.

F. Catté, P.-L. Lions, J.-M. Morel, and T. Coll, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Numer. Anal. 29, 182–193 (1992).
[CrossRef]

Cui, X.

Fatemi, E.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica 60, 259–268 (1992).
[CrossRef]

Fernández, D. C.

H. M. Salinas and D. C. Fernández, “Comparison of PDE-based nonlinear diffusion approaches for image enhancement and denoising in optical coherence tomography,” IEEE Trans. Med. Imaging 26, 761–771 (2007).
[CrossRef]

Gao, T.

Gao, W.

Han, L.

Han, Lin

Kamath, C.

S. K. Weeratunga and C. Kamath, “A comparison of PDE based non-linear anisotropic diffusion techniques for image denoising,” Proc. SPIE 5014, 151493 (2003).
[CrossRef]

Kao, C. C.

Kaveh, M.

Y. You and M. Kaveh, “Fourth-order partial differential equations for noise removal,” IEEE Trans. Image Process. 9, 1723–1730 (2000).
[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.

Li, B.

Lin, F.

Lions, P.-L.

L. Alvarez, P.-L. Lions, and J.-M. Morel, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Numer. Anal. 29, 845–866 (1992).
[CrossRef]

F. Catté, P.-L. Lions, J.-M. Morel, and T. Coll, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Numer. Anal. 29, 182–193 (1992).
[CrossRef]

Lu, W.

Lundervold, A.

M. Lysaker, A. Lundervold, and X.-C. Tai, “Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time,” IEEE Trans. Image Process 12, 1579–1590(2003).
[CrossRef]

Lv, W.

W. Lv, C. Tang, and W. Wang, “Noise reduction in electronic speckle pattern interferometry fringes by fourth-order partial differential equations,” Proc. SPIE 6279, (2007).
[CrossRef]

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]

Lysaker, M.

M. Lysaker, A. Lundervold, and X.-C. Tai, “Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time,” IEEE Trans. Image Process 12, 1579–1590(2003).
[CrossRef]

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. Vis. 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. Machine Intell. 12, 629–639 (1990).
[CrossRef]

Morel, J.-M.

L. Alvarez, P.-L. Lions, and J.-M. Morel, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Numer. Anal. 29, 845–866 (1992).
[CrossRef]

F. Catté, P.-L. Lions, J.-M. Morel, and T. Coll, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Numer. Anal. 29, 182–193 (1992).
[CrossRef]

Nakadate, S.

Osher, S.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica 60, 259–268 (1992).
[CrossRef]

Perona, P.

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

Ren, H.

Ringach, D. L.

G. Sapiro and D. L. Ringach, “Anisotropic diffusion of multivalued images with applications to color filtering,” IEEE Trans. Image Process 5, 1582–1586 (1996).
[CrossRef]

Rudin, L. I.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica 60, 259–268 (1992).
[CrossRef]

Saito, H.

Salinas, H. M.

H. M. Salinas and D. C. Fernández, “Comparison of PDE-based nonlinear diffusion approaches for image enhancement and denoising in optical coherence tomography,” IEEE Trans. Med. Imaging 26, 761–771 (2007).
[CrossRef]

Santoyo, F. M.

Sapiro, G.

G. Sapiro and D. L. Ringach, “Anisotropic diffusion of multivalued images with applications to color filtering,” IEEE Trans. Image Process 5, 1582–1586 (1996).
[CrossRef]

G. Sapiro, Geometric Partial Differential Equations and Image Analysis (Cambridge University, 2001).

Seah, H. S.

Su, X.

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

Tai, X.-C.

M. Lysaker, A. Lundervold, and X.-C. Tai, “Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time,” IEEE Trans. Image Process 12, 1579–1590(2003).
[CrossRef]

Tang, C.

L. Cheng, C. Tang, S. Yan, X. Chen, L. Wang, and B. Wang, “New fourth-order partial differential equations for filtering in electronic speckle pattern interferometry fringes,” Opt. Commun. 284, 5549–5555 (2011).
[CrossRef]

J. Wu and C. Tang, “PDE-based random-valued impulse noise removal based on new class of controlling functions,” IEEE Trans. Image Process. 20, 2428–2438 (2011).
[CrossRef]

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]

C. Tang, L. Han, H. Ren, D. Zhou, Y. Chang, X. Wang, and X. Cui, “Second-order oriented partial-differential equations for denoising in electronic-speckle-pattern interferometry fringes,” Opt. Lett. 33, 2179–2181 (2008).
[CrossRef]

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]

W. Lv, C. Tang, and W. Wang, “Noise reduction in electronic speckle pattern interferometry fringes by fourth-order partial differential equations,” Proc. SPIE 6279, (2007).
[CrossRef]

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]

Tang, K.

Tyrer, J. R.

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

Wang, B.

L. Cheng, C. Tang, S. Yan, X. Chen, L. Wang, and B. Wang, “New fourth-order partial differential equations for filtering in electronic speckle pattern interferometry fringes,” Opt. Commun. 284, 5549–5555 (2011).
[CrossRef]

Wang, H.

Wang, L.

L. Cheng, C. Tang, S. Yan, X. Chen, L. Wang, and B. Wang, “New fourth-order partial differential equations for filtering in electronic speckle pattern interferometry fringes,” Opt. Commun. 284, 5549–5555 (2011).
[CrossRef]

Wang, W.

Wang, X.

Wang, Z.

Weeratunga, S. K.

S. K. Weeratunga and C. Kamath, “A comparison of PDE based non-linear anisotropic diffusion techniques for image denoising,” Proc. SPIE 5014, 151493 (2003).
[CrossRef]

Weikert, J.

J. Weikert, Anisotropic Diffusion in Image Processing (Teubner Verlag, 1998).

Witkin, A. P.

A. P. Witkin, “Scale-space filtering,” in Proceedings of International Joint Conference on Artificial Intelligence (Morgan Kaufmann, 1983), pp. 1019–1021.

Womack, K. H.

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

Wu, J.

J. Wu and C. Tang, “PDE-based random-valued impulse noise removal based on new class of controlling functions,” IEEE Trans. Image Process. 20, 2428–2438 (2011).
[CrossRef]

Wu, K. C.

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]

Yan, S.

L. Cheng, C. Tang, S. Yan, X. Chen, L. Wang, and B. Wang, “New fourth-order partial differential equations for filtering in electronic speckle pattern interferometry fringes,” Opt. Commun. 284, 5549–5555 (2011).
[CrossRef]

Yang, C. S.

Yeh, G. B.

You, Y.

Y. You and M. Kaveh, “Fourth-order partial differential equations for noise removal,” IEEE Trans. Image Process. 9, 1723–1730 (2000).
[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]

Zhang, Z.

Zhou, D.

Appl. Opt.

Comput. Vis. Image Understand.

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

IEEE Trans. Image Process

M. Lysaker, A. Lundervold, and X.-C. Tai, “Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time,” IEEE Trans. Image Process 12, 1579–1590(2003).
[CrossRef]

G. Sapiro and D. L. Ringach, “Anisotropic diffusion of multivalued images with applications to color filtering,” IEEE Trans. Image Process 5, 1582–1586 (1996).
[CrossRef]

IEEE Trans. Image Process.

J. Wu and C. Tang, “PDE-based random-valued impulse noise removal based on new class of controlling functions,” IEEE Trans. Image Process. 20, 2428–2438 (2011).
[CrossRef]

Y. You and M. Kaveh, “Fourth-order partial differential equations for noise removal,” IEEE Trans. Image Process. 9, 1723–1730 (2000).
[CrossRef]

IEEE Trans. Med. Imaging

H. M. Salinas and D. C. Fernández, “Comparison of PDE-based nonlinear diffusion approaches for image enhancement and denoising in optical coherence tomography,” IEEE Trans. Med. Imaging 26, 761–771 (2007).
[CrossRef]

IEEE Trans. Pattern Anal. Machine Intell.

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

J. Opt. Soc. Am. A

Mater. Sci.

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.

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

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]

L. Cheng, C. Tang, S. Yan, X. Chen, L. Wang, and B. Wang, “New fourth-order partial differential equations for filtering in electronic speckle pattern interferometry fringes,” Opt. Commun. 284, 5549–5555 (2011).
[CrossRef]

Opt. Eng.

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

Opt. Express

Opt. Lasers Eng.

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

Opt. Lett.

Physica

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica 60, 259–268 (1992).
[CrossRef]

Proc. SPIE

S. K. Weeratunga and C. Kamath, “A comparison of PDE based non-linear anisotropic diffusion techniques for image denoising,” Proc. SPIE 5014, 151493 (2003).
[CrossRef]

W. Lv, C. Tang, and W. Wang, “Noise reduction in electronic speckle pattern interferometry fringes by fourth-order partial differential equations,” Proc. SPIE 6279, (2007).
[CrossRef]

SIAM J. Numer. Anal.

F. Catté, P.-L. Lions, J.-M. Morel, and T. Coll, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Numer. Anal. 29, 182–193 (1992).
[CrossRef]

L. Alvarez, P.-L. Lions, and J.-M. Morel, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Numer. Anal. 29, 845–866 (1992).
[CrossRef]

Other

Wikipedia, “Euler-Lagrange equation,” http://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation .

G. Sapiro, Geometric Partial Differential Equations and Image Analysis (Cambridge University, 2001).

A. P. Witkin, “Scale-space filtering,” in Proceedings of International Joint Conference on Artificial Intelligence (Morgan Kaufmann, 1983), pp. 1019–1021.

M. Aslan, “Toward the development of high-speed microscopic ESPI system for monitoring laser heating/drilling of alumina Al2O3 substrates,” Ph.D. dissertation (Pennsylvania State University, 2000).

J. Weikert, Anisotropic Diffusion in Image Processing (Teubner Verlag, 1998).

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Figures (7)

Fig. 1.
Fig. 1.

Our main PDE filtering models used in ESPI.

Fig. 2.
Fig. 2.

A computer-simulated fringe pattern, its filtered results, and skeletons. (a-1): Initial image; (a-2) the ideal skeletons of the light fringes; (b-1), (c-1), (d-1), (e-1), (f-1), and (g-1): filtered images by SOPDE, COPDE, PFOPDE_1, PFOPDE_2, PFOPDE_3, and NFOPDE, respectively; (b-2), (c-2), (d-2), (e-2), (f-2), and (g-2): the corresponding skeletons of light fringes, respectively.

Fig. 3.
Fig. 3.

The improved results by using mean filter. (a-1), (b-1), (c-1) and (d-1): The filtered images of Fig. 2(b-1), 2(c-1), 2(d-1), and 2(e-1) by using mean filter; (a-2), (b-2), (c-2), and (d-2): the corresponding skeletons of light fringes.

Fig. 4.
Fig. 4.

An experimentally obtained original ESPI fringe image with high density and its filtered results. (a) Initial image; (b) the filtered image by SOPDE; (c) the filtered image by OSOPDE; (d) the filtered image by COPDE; (e) the filtered image by OCOPDE.

Fig. 5.
Fig. 5.

A computer-simulated phase fringe pattern and its filtered results. (a) Initial image; (b-1) and (b-2): the filtered images by SOPDE with n=10, 20, respectively; (c-1) and (c-2): the filtered images by OSOPDE with n=50, 100, respectively; (d-1) and (d-2): the filtered images by COPDE with n=20, 30, respectively; (e-1) and (e-2): the filtered images by OCOPDE with n=100, 120, respectively.

Fig. 6.
Fig. 6.

A computer-simulated fringe pattern with variable density and its filtered results. (a) Initial image; (b) the filtered image of Fig. 6(a) by OSOPDE with n=50; (c) the filtered image of Fig. 6(b) by using mean filter three times; (d) the filtered image of Fig. 6(a) by OSOPDE with n=150; (e) the filtered image of Fig. 6(a) by OSOPDE with n=200.

Fig. 7.
Fig. 7.

An experimentally obtained original ESPI fringe image with variable fringe density and its filtered results. (a) Initial image; (b) the filtered image of Fig. 7(a) by OSOPDE with n=300; (c) the filtered image of Fig. 7(b) by using mean filter eight times; (d) the filtered image of Fig. 7(a) by OSOPDE with n=1000; (e) the filtered image of Fig. 7(a) by OSOPDE with n=1500.

Equations (30)

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ut=F[u(x,y,t)],u(x,y,0)=I(x,y),
ut=div(g(|u|)u),
g(|u|)=11+(|u|/k)2,
g(|u|)=exp((|u|/k)2),
ut=2[g(|2u|)2u].
ut=αg(|v|)|u|div(u|u|)+α(g(|v|))uβ(uI)|u|,
vt=a(t)div(v|v|)b(vu),
ut=|u|div(u|u|).
ut=div(u|u|)+λ(t)(u0u),
ut=(uxx|uxx|)xx(uyy|uyy|)yyλ(uu0),
ut=(uxx|D2u|)xx(uxy|D2u|)yx(uyx|D2u|)xy(uyy|D2u|)yyλ(uu0),
λ=1σ2Ω(uxx|uxx|(uu0)xx+uyy|uyy|(uu0)yy)dxdy,σ2uu02,|D2u|=(|uxx|2+|uxy|2+|uyx|2+|uyy|2).
ut=g(|u|)|u|div(u|u|).
ut=g(λ+,λ)2uθ2.
minE(u).
E(u)=ΩF(x,y,u,ux,uy,uxx,uxy,uyy,uxxx,uxy)dxdy,
Fux(Fux)y(Fuy)+2x2(Fuxx)+2xy(Fuxy)+2y2(Fuyy)++(1)nnyn(Fuyyy)=0.
E(u)=Ω(|u|+λ2(uI)2)dxdy.
Fux(Fux)y(Fuy)=0.
Fu=λ(uI),Fux=uxux2+uy2,Fuy=uyux2+uy2.
λ(Iu)div(u|u|)=0.
ut=g(|u|)(uxxxx+uyyyy+uxyxy).
Isub=|4IoIrsin(φrφo+ψ2)sin(ψ2)|,
ψi,j=45[exp((2im)2+(2j3n/2)230000)+exp((2im)2+(2jn/2)235000)]+10[(3(im/2)m)2+(3(jn/3)n)2],
{Δt=0.5,k=0.00001,n=20forSOPDEΔt=0.8,α=0.27,b=0.02,k=0.0001,β=0.0005,n=100forCOPDEΔt=0.25,k=0.00001,n=500forPFOPDE_1Δt=0.25,n=500forPFOPDE_2Δt=0.25,n=500forPFOPDE_3Δt=0.06,k=0.00001,n=500forNFOPDE.
E(u)=Ω12|uρ|2dxdy,
ut=uxxcos2θ+uyysin2θ+2uxysinθcosθ,
ut=g(|u|)(uxxcos2θ+uyysin2θ+2uxysinθcosθ).
{ut=αg(|v|)(uxxcos2θ+uyysin2θ+2uxysinθcosθ)β(uI)|u|vt=a(t)|v|(vxxcos2θ+vyysin2θ+2vxysinθcosθ)b(vu).
{Δt=0.5forSOPDE and OSOPDEΔt=0.8,α=0.27,b=0.02,β=0.0005forCOPDEΔt=0.8,α=0.40,b=0.02,β=0.0005forOCOPDE.

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