Abstract

We derive the second-order oriented partial-differential equations (PDEs) for denoising in electronic-speckle-pattern interferometry fringe patterns from two points of view. The first is based on variational methods, and the second is based on controlling diffusion direction. Our oriented PDE models make the diffusion along only the fringe orientation. The main advantage of our filtering method, based on oriented PDE models, is that it is very easy to implement compared with the published filtering methods along the fringe orientation. We demonstrate the performance of our oriented PDE models via application to two computer-simulated and experimentally obtained speckle fringes and compare with related PDE models.

© 2008 Optical Society of America

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References

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  1. C. Tang, F. Zhang, and Z. Chen, Appl. Opt. 45, 2287 (2006).
    [Crossref] [PubMed]
  2. C. Tang, F. Zhang, B. Li, and H. Yan, Appl. Opt. 45, 7392 (2006).
    [Crossref] [PubMed]
  3. A. P. Witkin, in Proceedings of the International Joint Conferences on Artificial Intelligence (Karlsruhe, 1983), pp. 1019-1021.
  4. P. Perona and J. Malik, IEEE Trans. Pattern Anal. Mach. Intell. 12, 629 (1990).
    [Crossref]
  5. F. Catté, P.-L. Lions, J.-M. Morel, and T. Coll, SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 29, 182 (1992).
  6. L. Alvarez, P.-L. Lions, and J.-M. Morel, SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 29, 845 (1992).
  7. C. Tang, F. Zhang, H. Yan, and Z. Chen, Opt. Commun. 260, 91 (2006).
    [Crossref]
  8. C. Tang, W. Lu, S. Chen, Z. Zhang, B. Li, W. Wang, and Lin Han, Appl. Opt. 46, 7475 (2007).
    [Crossref] [PubMed]
  9. Y. L. You and M. Kaveh, IEEE Trans. Image Process. 9, 1723 (2000).
    [Crossref]
  10. G. Aubert and K. Pierre, Mathematical Problems in Image Processing (Springer, 2002).
  11. L. Hong, Y. Wan, and A. Jain, IEEE Trans. Pattern Anal. Mach. Intell. 20, 777 (1998).
    [Crossref]

2007 (1)

2006 (3)

2000 (1)

Y. L. You and M. Kaveh, IEEE Trans. Image Process. 9, 1723 (2000).
[Crossref]

1998 (1)

L. Hong, Y. Wan, and A. Jain, IEEE Trans. Pattern Anal. Mach. Intell. 20, 777 (1998).
[Crossref]

1992 (2)

F. Catté, P.-L. Lions, J.-M. Morel, and T. Coll, SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 29, 182 (1992).

L. Alvarez, P.-L. Lions, and J.-M. Morel, SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 29, 845 (1992).

1990 (1)

P. Perona and J. Malik, IEEE Trans. Pattern Anal. Mach. Intell. 12, 629 (1990).
[Crossref]

Alvarez, L.

L. Alvarez, P.-L. Lions, and J.-M. Morel, SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 29, 845 (1992).

Aubert, G.

G. Aubert and K. Pierre, Mathematical Problems in Image Processing (Springer, 2002).

Catté, F.

F. Catté, P.-L. Lions, J.-M. Morel, and T. Coll, SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 29, 182 (1992).

Chen, S.

Chen, Z.

C. Tang, F. Zhang, and Z. Chen, Appl. Opt. 45, 2287 (2006).
[Crossref] [PubMed]

C. Tang, F. Zhang, H. Yan, and Z. Chen, Opt. Commun. 260, 91 (2006).
[Crossref]

Coll, T.

F. Catté, P.-L. Lions, J.-M. Morel, and T. Coll, SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 29, 182 (1992).

Han, Lin

Hong, L.

L. Hong, Y. Wan, and A. Jain, IEEE Trans. Pattern Anal. Mach. Intell. 20, 777 (1998).
[Crossref]

Jain, A.

L. Hong, Y. Wan, and A. Jain, IEEE Trans. Pattern Anal. Mach. Intell. 20, 777 (1998).
[Crossref]

Kaveh, M.

Y. L. You and M. Kaveh, IEEE Trans. Image Process. 9, 1723 (2000).
[Crossref]

Li, B.

Lions, P.-L.

F. Catté, P.-L. Lions, J.-M. Morel, and T. Coll, SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 29, 182 (1992).

L. Alvarez, P.-L. Lions, and J.-M. Morel, SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 29, 845 (1992).

Lu, W.

Malik, J.

P. Perona and J. Malik, IEEE Trans. Pattern Anal. Mach. Intell. 12, 629 (1990).
[Crossref]

Morel, J.-M.

L. Alvarez, P.-L. Lions, and J.-M. Morel, SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 29, 845 (1992).

F. Catté, P.-L. Lions, J.-M. Morel, and T. Coll, SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 29, 182 (1992).

Perona, P.

P. Perona and J. Malik, IEEE Trans. Pattern Anal. Mach. Intell. 12, 629 (1990).
[Crossref]

Pierre, K.

G. Aubert and K. Pierre, Mathematical Problems in Image Processing (Springer, 2002).

Tang, C.

Wan, Y.

L. Hong, Y. Wan, and A. Jain, IEEE Trans. Pattern Anal. Mach. Intell. 20, 777 (1998).
[Crossref]

Wang, W.

Witkin, A. P.

A. P. Witkin, in Proceedings of the International Joint Conferences on Artificial Intelligence (Karlsruhe, 1983), pp. 1019-1021.

Yan, H.

C. Tang, F. Zhang, H. Yan, and Z. Chen, Opt. Commun. 260, 91 (2006).
[Crossref]

C. Tang, F. Zhang, B. Li, and H. Yan, Appl. Opt. 45, 7392 (2006).
[Crossref] [PubMed]

You, Y. L.

Y. L. You and M. Kaveh, IEEE Trans. Image Process. 9, 1723 (2000).
[Crossref]

Zhang, F.

Zhang, Z.

Appl. Opt. (3)

IEEE Trans. Image Process. (1)

Y. L. You and M. Kaveh, IEEE Trans. Image Process. 9, 1723 (2000).
[Crossref]

IEEE Trans. Pattern Anal. Mach. Intell. (2)

L. Hong, Y. Wan, and A. Jain, IEEE Trans. Pattern Anal. Mach. Intell. 20, 777 (1998).
[Crossref]

P. Perona and J. Malik, IEEE Trans. Pattern Anal. Mach. Intell. 12, 629 (1990).
[Crossref]

Opt. Commun. (1)

C. Tang, F. Zhang, H. Yan, and Z. Chen, Opt. Commun. 260, 91 (2006).
[Crossref]

SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. (2)

F. Catté, P.-L. Lions, J.-M. Morel, and T. Coll, SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 29, 182 (1992).

L. Alvarez, P.-L. Lions, and J.-M. Morel, SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 29, 845 (1992).

Other (2)

A. P. Witkin, in Proceedings of the International Joint Conferences on Artificial Intelligence (Karlsruhe, 1983), pp. 1019-1021.

G. Aubert and K. Pierre, Mathematical Problems in Image Processing (Springer, 2002).

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Figures (3)

Fig. 1
Fig. 1

Computer-simulated fringe pattern and its filtered images: (a) the initial image, (b) the filtered image by the model in Eq. (1), (c) the filtered image by the model in Eq. (6).

Fig. 2
Fig. 2

Computer-simulated fringe pattern and its filtered images: (a) the initial image, (b) the filtered image by the model in Eq. (2), (c) the filtered image by the model in Eq. (12).

Fig. 3
Fig. 3

Experimentally obtained ESPI fringe and its filtered images: (a) the initial image, (b) the filtered image by the model in Eq. (2), (c) the filtered image by the model in Eq. (12).

Equations (15)

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u t = u d i v ( u u ) .
u t = g ( ) u div ( u u ) .
E ( u ) = Ω 1 2 u ρ 2 d x d y .
f u x ( f u x ) y ( f u y ) = 0.
f = 1 2 u ρ 2 = 1 2 ( u x cos θ + u y sin θ ) 2 ,
u t = u x x cos 2 θ + u y y sin 2 θ + 2 u x y sin θ cos θ .
u t = 1 u x 2 + u y 2 ( u y 2 u x x 2 u x u y u x y + u x 2 u y y ) .
2 u T 2 = 1 u x 2 + u y 2 ( u y 2 u x x 2 u x u y u x y + u x 2 u y y ) .
u t = 2 u T 2 ,
u t = 2 u ρ 2 .
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 = g ( ) ( u x x cos 2 θ + u y y sin 2 θ + 2 u x y sin θ cos θ ) .
θ i , j = 1 2 tan 1 k , l 2 u x ( k , l ) u y ( k , l ) k , l ( u x 2 ( k , l ) u y 2 ( k , l ) ) ,
u i , j n + 1 = u i , j n + Δ t g i , j n [ ( 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 ) ] ,
g i , j n = 1 1 + k ( ( ( u x ) i , j n ) 2 + ( ( u y ) i , j n ) 2 ) ,

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