Abstract

In this paper, we are concerned with denoising in experimentally obtained electronic speckle pattern interferometry (ESPI) speckle fringe patterns with poor quality. We extend the application of two existing oriented partial differential equation (PDE) filters, including the second-order single oriented PDE filter and the double oriented PDE filter, to two experimentally obtained ESPI speckle fringe patterns with very poor quality, and compare them with other efficient filtering methods, including the adaptive weighted filter, the improved nonlinear complex diffusion PDE, and the windowed Fourier transform method. All of the five filters have been illustrated to be efficient denoising methods through previous comparative analyses in published papers. The experimental results have demonstrated that the two oriented PDE models are applicable to low-quality ESPI speckle fringe patterns. Then for solving the main shortcoming of the two oriented PDE models, we develop the numerically fast algorithms based on Gauss–Seidel strategy for the two oriented PDE models. The proposed numerical algorithms are capable of accelerating the convergence greatly, and perform significantly better in terms of computational efficiency. Our numerically fast algorithms are extended automatically to some other PDE filtering models.

© 2013 Optical Society of America

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References

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  1. 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]
  2. C. Tang, W. Lu, Y. Cai, L. Han, and G. Wang, “Nearly preprocessing-free method for skeletonization of gray-scale electronic speckle pattern interferometry fringe patterns via partial differential equations,” Opt. Lett. 33, 183–185 (2008).
    [CrossRef]
  3. G. Wang, Y. Li, and H. Zhou, “Application of the radial basis function interpolation to phase extraction from a single electronic speckle pattern interferometric fringe,” Appl. Opt. 50, 3110–3117 (2011).
    [CrossRef]
  4. 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]
  5. 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]
  6. 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]
  7. G. H. Kaufmann, A. Davila, and D. Kerr, “Speckle noise reduction in TV holography,” Proc. SPIE 2730, 96–100 (1995).
    [CrossRef]
  8. Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43, 2695–2702 (2004).
    [CrossRef]
  9. Q. Yu, X. Sun, and X. Liu, “Spin filtering with curve windows for interferometric fringes,” Appl. Opt. 41, 2650–2654 (2002).
    [CrossRef]
  10. 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]
  11. 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]
  12. H. Wang, Q. Kemao, W. Gao, Feng Lin, and H. S. Seah, “Fringe pattern denoising using coherence enhancing diffusion,” Opt. Lett. 34, 1141–1143 (2009).
    [CrossRef]
  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, 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]
  15. A. Neimark, “Investigating filtering methods for the denoising and contrast enhancement, and eventual skeletonization, of ESPI images,” http://mesoscopic.mines.edu/mediawiki/images/d/de/Project_Report_Neimark.pdf 2009 .
  16. K. Qian, L. Nam, L. Feng, and S. Soon, “Comparative analysis on some filters for wrapped phase maps,” Appl. Opt. 46, 7412–7418 (2007).
    [CrossRef]
  17. H. Wang and K. Qian, “Comparative analysis on some spatial-domain filters for fringe pattern denoising,” Appl. Opt. 50, 1687–1696 (2011).
    [CrossRef]
  18. 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]
  19. C. Tang, L. Wang, H. Yan, and C. Li, “Comparison on performance of some representative and recent filtering methods in electronic speckle pattern interferometry,” Opt. Lasers Eng. 50, 1036–1051 (2012).
    [CrossRef]
  20. 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]
  21. H. Y. Yun, C. K. Hong, and S. W. Chang, “Least-square phase estimation with multiple parameters in phase-shifting electronic speckle pattern interferometry,” J. Opt. Soc. Am. A 20, 240–247 (2003).
    [CrossRef]
  22. Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304–317 (2007).
    [CrossRef]
  23. E. Trouve, M. Caramma, and H. Maitre, “Fringe detection in noisy complex interferograms,” Appl. Opt. 35, 3799–3806 (1996).
    [CrossRef]
  24. J. M. Huntley, “Random phase measurement errors in digital speckle pattern interferometry,” Opt. Lasers Eng. 26, 131–150 (1997).
    [CrossRef]
  25. R. Bernardes, C. Maduro, P. Serranho, A. Araújo, S. Barbeiro, and J. Cunha-Vaz, “Improved adaptive complex diffusion despeckling filter,” Opt. Express 18, 24048–24059 (2010).
    [CrossRef]
  26. G. Gilboa, N. Sochen, and Y. Y. Zeevi, “Image enhancement and denoising by complex diffusion processes,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 1020–1036 (2004).
    [CrossRef]
  27. L. Wang, G. Leedham, and D. Cho, “Minutiae feature analysis for infrared hand vein pattern biometrics,” Pattern Recogn. 41, 920–929 (2008).
    [CrossRef]
  28. 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]
  29. J. Vargas, J. Quiroga, and T. Belenguer, “Local fringe density determination by adaptive filtering,” Opt. Lett. 36, 70–72 (2011).
    [CrossRef]
  30. Y. You and M. Kaveh, “Fourth-order partial differential equation for noise removal,” IEEE Trans. Image Process 9, 1723–1730 (2000).
    [CrossRef]
  31. M. Lysaker, A. Lundervold, and X. 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]

2012 (1)

C. Tang, L. Wang, H. Yan, and C. Li, “Comparison on performance of some representative and recent filtering methods in electronic speckle pattern interferometry,” Opt. Lasers Eng. 50, 1036–1051 (2012).
[CrossRef]

2011 (3)

2010 (4)

2009 (2)

2008 (3)

2007 (3)

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

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

2006 (1)

2005 (1)

2004 (2)

G. Gilboa, N. Sochen, and Y. Y. Zeevi, “Image enhancement and denoising by complex diffusion processes,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 1020–1036 (2004).
[CrossRef]

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

2003 (2)

H. Y. Yun, C. K. Hong, and S. W. Chang, “Least-square phase estimation with multiple parameters in phase-shifting electronic speckle pattern interferometry,” J. Opt. Soc. Am. A 20, 240–247 (2003).
[CrossRef]

M. Lysaker, A. Lundervold, and X. 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 (1)

2000 (1)

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

1999 (2)

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]

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]

1997 (1)

J. M. Huntley, “Random phase measurement errors in digital speckle pattern interferometry,” Opt. Lasers Eng. 26, 131–150 (1997).
[CrossRef]

1996 (1)

1995 (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]

Araújo, A.

Barbeiro, S.

Belenguer, T.

Bernardes, R.

Cai, Y.

Caramma, M.

Chang, S. W.

Chang, Y.

Cho, D.

L. Wang, G. Leedham, and D. Cho, “Minutiae feature analysis for infrared hand vein pattern biometrics,” Pattern Recogn. 41, 920–929 (2008).
[CrossRef]

Cuevas, F. J.

Cui, X.

Cunha-Vaz, J.

Davila, A.

G. H. Kaufmann, A. Davila, and D. Kerr, “Speckle noise reduction in TV holography,” Proc. SPIE 2730, 96–100 (1995).
[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.

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.

Gilboa, G.

G. Gilboa, N. Sochen, and Y. Y. Zeevi, “Image enhancement and denoising by complex diffusion processes,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 1020–1036 (2004).
[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]

Han, L.

He, X.

Hong, C. K.

Huntley, J. M.

J. M. Huntley, “Random phase measurement errors in digital speckle pattern interferometry,” Opt. Lasers Eng. 26, 131–150 (1997).
[CrossRef]

Kaufmann, G. H.

G. H. Kaufmann, A. Davila, and D. Kerr, “Speckle noise reduction in TV holography,” Proc. SPIE 2730, 96–100 (1995).
[CrossRef]

Kaveh, M.

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

Kemao, Q.

Kerr, D.

G. H. Kaufmann, A. Davila, and D. Kerr, “Speckle noise reduction in TV holography,” Proc. SPIE 2730, 96–100 (1995).
[CrossRef]

Leedham, G.

L. Wang, G. Leedham, and D. Cho, “Minutiae feature analysis for infrared hand vein pattern biometrics,” Pattern Recogn. 41, 920–929 (2008).
[CrossRef]

Li, B.

Li, C.

C. Tang, L. Wang, H. Yan, and C. Li, “Comparison on performance of some representative and recent filtering methods in electronic speckle pattern interferometry,” Opt. Lasers Eng. 50, 1036–1051 (2012).
[CrossRef]

Li, Y.

Lim, H. C.

Lin, Feng

Liu, X.

Lu, W.

Lundervold, A.

M. Lysaker, A. Lundervold, and X. 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]

Lysaker, M.

M. Lysaker, A. Lundervold, and X. 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]

Maduro, C.

Maitre, H.

Malacara, D.

Marroguin, J. L.

Nam, L.

Qian, K.

Quan, C.

Quiroga, J.

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.

Rodriguez-Vera, R.

Ryu, H. S.

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]

Seah, H. S.

Serranho, P.

Servin, M.

Sochen, N.

G. Gilboa, N. Sochen, and Y. Y. Zeevi, “Image enhancement and denoising by complex diffusion processes,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 1020–1036 (2004).
[CrossRef]

Soon, S.

Sun, X.

Tai, X.

M. Lysaker, A. Lundervold, and X. 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.

C. Tang, L. Wang, H. Yan, and C. Li, “Comparison on performance of some representative and recent filtering methods in electronic speckle pattern interferometry,” Opt. Lasers Eng. 50, 1036–1051 (2012).
[CrossRef]

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]

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]

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, W. Lu, Y. Cai, L. Han, and G. Wang, “Nearly preprocessing-free method for skeletonization of gray-scale electronic speckle pattern interferometry fringe patterns via partial differential equations,” Opt. Lett. 33, 183–185 (2008).
[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, 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]

Tang, K.

Tay, C. J.

Trouve, E.

Vargas, J.

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]

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]

Wang, G.

Wang, H.

Wang, L.

C. Tang, L. Wang, H. Yan, and C. Li, “Comparison on performance of some representative and recent filtering methods in electronic speckle pattern interferometry,” Opt. Lasers Eng. 50, 1036–1051 (2012).
[CrossRef]

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]

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]

L. Wang, G. Leedham, and D. Cho, “Minutiae feature analysis for infrared hand vein pattern biometrics,” Pattern Recogn. 41, 920–929 (2008).
[CrossRef]

Wang, X.

Wang, Z.

Wu, J.

Yan, H.

C. Tang, L. Wang, H. Yan, and C. Li, “Comparison on performance of some representative and recent filtering methods in electronic speckle pattern interferometry,” Opt. Lasers Eng. 50, 1036–1051 (2012).
[CrossRef]

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]

Yan, S.

Yang, F.

You, Y.

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

Yu, Q.

Yun, H. Y.

Zeevi, Y. Y.

G. Gilboa, N. Sochen, and Y. Y. Zeevi, “Image enhancement and denoising by complex diffusion processes,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 1020–1036 (2004).
[CrossRef]

Zhang, F.

Zhou, D.

Zhou, H.

Appl. Opt. (10)

E. Trouve, M. Caramma, and H. Maitre, “Fringe detection in noisy complex interferograms,” Appl. Opt. 35, 3799–3806 (1996).
[CrossRef]

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]

Q. Yu, X. Sun, and X. Liu, “Spin filtering with curve windows for interferometric fringes,” Appl. Opt. 41, 2650–2654 (2002).
[CrossRef]

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

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]

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]

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

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]

H. Wang and K. Qian, “Comparative analysis on some spatial-domain filters for fringe pattern denoising,” Appl. Opt. 50, 1687–1696 (2011).
[CrossRef]

G. Wang, Y. Li, and H. Zhou, “Application of the radial basis function interpolation to phase extraction from a single electronic speckle pattern interferometric fringe,” Appl. Opt. 50, 3110–3117 (2011).
[CrossRef]

IEEE Trans. Image Process (2)

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

M. Lysaker, A. Lundervold, and X. 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]

IEEE Trans. Med. Imaging (1)

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. Mach. Intell. (1)

G. Gilboa, N. Sochen, and Y. Y. Zeevi, “Image enhancement and denoising by complex diffusion processes,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 1020–1036 (2004).
[CrossRef]

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

Opt. Commun. (1)

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 (3)

Opt. Lasers Eng. (4)

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]

C. Tang, L. Wang, H. Yan, and C. Li, “Comparison on performance of some representative and recent filtering methods in electronic speckle pattern interferometry,” Opt. Lasers Eng. 50, 1036–1051 (2012).
[CrossRef]

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

J. M. Huntley, “Random phase measurement errors in digital speckle pattern interferometry,” Opt. Lasers Eng. 26, 131–150 (1997).
[CrossRef]

Opt. Lett. (5)

Pattern Recogn. (1)

L. Wang, G. Leedham, and D. Cho, “Minutiae feature analysis for infrared hand vein pattern biometrics,” Pattern Recogn. 41, 920–929 (2008).
[CrossRef]

Proc. SPIE (1)

G. H. Kaufmann, A. Davila, and D. Kerr, “Speckle noise reduction in TV holography,” Proc. SPIE 2730, 96–100 (1995).
[CrossRef]

Other (1)

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

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Figures (5)

Fig. 1.
Fig. 1.

Two experimentally obtained low-quality original ESPI fringe images.

Fig. 2.
Fig. 2.

Filtered results of Figs. 1(a) and 1(b) by the five filters. (a-1), (b-1), (c-1), (d-1), and (e-1): filtered results of Fig. 1(a) by AW, ICDPDE, WFF, SOOPDE, and CEDPDE models. (a-2), (b-2), (c-2), (d-2), and (e-2): filtered results of Fig. 1(b) by AW, ICDPDE, WFF, SOOPDE, and CEDPDE models.

Fig. 3.
Fig. 3.

Skeletons of filtered results. (a-1), (b-1), (c-1), (d-1), and (e-1): skeletons of Figs. 2(a-1), 2(b-1), 2(c-1), 2(d-1), and 2(e-1). (a-2), (b-2), (c-2), (d-2), and (e-2): skeletons of Figs. 2(a-2), 2(b-2), 2(c-2), 2(d-2), and 2(e-2).

Fig. 4.
Fig. 4.

Computer-simulated ESPI fringe pattern, and PSNR values. (a) Initial image, (b-1) PSNR values by SOOPDE_GD and SOOPDE_GS for various iteration times, and (b-2) PSNR values by CEDPDE_GD and CEDPDE_GS for various iteration times.

Fig. 5.
Fig. 5.

Filtered results of Fig. 1 and their skeletons. (a-1) and (a-2): filtered resutls by SOOPDE_GS. (b-1) and (b-2): filtered results by CEDPDE_GS. (c-1), (c-2), (d-1), and (d-2): corresponding skeletons.

Tables (2)

Tables Icon

Table 1. Algorithm Parameters and the Computational Time Used in Fig. 2

Tables Icon

Table 2. Algorithm Parameters and the Computational Time used in Fig. 5

Equations (28)

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 ( x , y , t ) t = · ( D ( Im ( u ) ) u ) ,
D ( Im ( u ) ) = exp ( i θ ) 1 + ( Im ( u ) k θ ) 2 ,
k = K max + ( K min K max ) g min ( g ) max ( g ) min ( g ) ,
g = G N , σ * Re ( u ) ,
Δ t ( n ) = 1 α [ a + b exp ( max ( | Re ( u ( n ) t ) | / Re ( u ( n ) ) ) ) ] ,
G ( ξ , η ) = F ( ξ , η ) H ( 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 | SF ( u , v , ξ , η ) | < thr ,
G ( u , v , ξ , η ) = SF ( u , v , ξ , η ) H ( u , v , ξ , η ) .
u t = 2 u ρ 2 = u x x cos 2 θ + u y y sin 2 θ + 2 u x y sin θ cos θ ,
u t = 2 u ρ 2 + λ 1 2 u ρ 2 = ( u x x cos 2 θ + u y y sin 2 θ + 2 u x y sin θ cos θ ) + λ 1 ( u x x sin 2 θ + u y y cos 2 θ 2 u x y sin θ cos θ ) ,
λ 1 = { α n N 0 max ( α , ( ( d thr ) / max ( d ) ) ) N 0 < n N ,
u ( i , j ) = { = 1 if u ( i , j ) ( μ ( i , j ) T g ) = 0 otherwise ,
E ( u ) = Ω 1 2 | u ρ | 2 d x d y ,
u x x cos 2 θ + u y y sin 2 θ + 2 u x y sin θ cos θ = 0 .
u i , j n + 1 = u i , j n + Δ t ( ( u x x ) i , j n cos 2 ( θ i , j ) + ( u y y ) i , j n sin 2 ( θ i , j ) + 2 ( u x y ) i , j n cos ( θ i , j ) sin ( θ i , j ) ) ,
E ( u ) = 1 2 Ω ( λ 1 | u ρ | 2 + λ 2 | u ρ | 2 ) d x d y ,
λ 1 ( u x x sin 2 θ + u y y cos 2 θ 2 u x y sin θ cos θ ) + λ 2 ( u x x cos 2 θ + u y y sin 2 θ + 2 u x y sin θ cos θ ) = 0 .
u i , j n + 1 = u i , j n + Δ t ( λ 1 ( ( u y y ) i , j n cos 2 θ i , j + ( u x x ) i , j n sin 2 θ i , j 2 ( u x y ) i , j n cos θ i , j sin θ i , j ) + λ 2 ( ( u x x ) i , j n cos 2 θ i , j + ( u y y ) i , j n sin 2 θ i , j + 2 ( u x y ) i , j n cos θ i , j sin θ i , j ) ) .
( u i + 1 , j 2 u i , j + u i 1 , j ) cos 2 θ i , j + ( u i , j + 1 2 u i , j + u i , j 1 ) sin 2 θ i , j + 2 ( u i + 1 , j + 1 u i + 1 , j 1 u i 1 , j + 1 + u i 1 , j 1 ) 4 sin θ i , j cos θ i , j = 0 .
u i , j n + 1 = 1 2 ( ( u i + 1 , j n + u i 1 , j n ) cos 2 θ i , j + ( u i , j + 1 n + u i , j 1 n ) sin 2 θ i , j + ( u i + 1 , j + 1 n u i + 1 , j 1 n u i 1 , j + 1 n + u i 1 , j 1 n ) 2 sin θ i , j cos θ i , j ) .
u i , j n + 1 = 1 2 ( ( u i + 1 , j n + u i 1 , j n + 1 ) cos 2 θ i , j + ( u i , j + 1 n + u i , j 1 n + 1 ) sin 2 θ i , j + ( u i + 1 , j + 1 n u i + 1 , j 1 n u i 1 , j + 1 n + u i 1 , j 1 n + 1 ) 2 sin θ i , j cos θ i , j ) .
u i , j n + 1 = λ 2 λ 1 2 ( λ 2 + λ 1 ) ( ( u i + 1 , j n + u i 1 , j n + 1 ) cos 2 θ i , j + ( u i , j + 1 n + u i , j 1 n + 1 ) sin 2 θ i , j + ( u i + 1 , j + 1 n u i + 1 , j 1 n u i 1 , j + 1 n + u i 1 , j 1 n + 1 ) 2 sin θ i , j cos θ i , j ) + λ 1 2 ( λ 2 + λ 1 ) ( u i + 1 , j n u i + 1 , j n + 1 u i , j + 1 n + u i , j 1 n + 1 ) .
PSNR = 20 log 10 ( 255 1 M N i = 1 M j = 1 N ( u i , j u ¯ i , j ) 2 ) ,

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