Abstract

We derive the new oriented-couple partial differential equation (PDE) models based on the variational methods for filtering in electronic speckle pattern interferometry phase fringe patterns. In the filtering methods based on the oriented PDE models, filtering along fringe orientation for the entire image is simply realized through solving the PDEs numerically, without having to laboriously establish the small filtering window along the fringe orientation and move this filtering window over each pixel in an image. We test the proposed models on two computer-simulated speckle phase fringe patterns and an experimentally obtained phase fringe pattern, respectively, in which the fringe density is variable, and compare our models with related PDE models. Further, we quantitatively evaluate the performance of these PDE models with a comparative parameter, named the image fidelity. We also compare the computational time of our method with that of a traditional filtering method along the fringe orientation. The experimental results demonstrate the performance of our new oriented PDE models.

© 2009 Optical Society of America

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    [CrossRef] [PubMed]
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    [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).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  8. A. P. Witkin, "Scale-space filtering," in Proceedings of IJCAI, (Karlsruhe, 1983), pp. 1019-1021.
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  18. L. Hong, Y. Wan, and A. Jain, "Fingerprint Image Enhancement: Algorithm and Performance Evaluation," IEEE Transactions on pattern analysis and machine intelligence,  20, 777- 789 (1998).
    [CrossRef]
  19. U. Schnars and W. P. O. Jueptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002)
    [CrossRef]

2008

2007

2006

2002

QifengYu , X. Sun, and X. Liu, "Spin filtering with curve windows for interferometric fringe patterns," Appl. Opt. 412650-2654 (2002).
[CrossRef] [PubMed]

U. Schnars and W. P. O. Jueptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002)
[CrossRef]

2001

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

1998

L. Hong, Y. Wan, and A. Jain, "Fingerprint Image Enhancement: Algorithm and Performance Evaluation," IEEE Transactions on pattern analysis and machine intelligence,  20, 777- 789 (1998).
[CrossRef]

1995

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]

1990

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

1986

D. W. Robinson and D. C. Williams, "Digital phase stepping speckle interferometry," Opt. Commun. 57,26-30 (1986).
[CrossRef]

1985

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]

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]

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," Computer Vision Image Understand. 82,85-100 (2001).
[CrossRef]

Cai, Y.

Chang, Y.

Chen, S.

Chen, Y.

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

Chen, Z.

Creath, K.

Cuevas, F. J.

Cui, X.

Han, L.

Hong, C. K.

Hong, L.

L. Hong, Y. Wan, and A. Jain, "Fingerprint Image Enhancement: Algorithm and Performance Evaluation," IEEE Transactions on pattern analysis and machine intelligence,  20, 777- 789 (1998).
[CrossRef]

Jain, A.

L. Hong, Y. Wan, and A. Jain, "Fingerprint Image Enhancement: Algorithm and Performance Evaluation," IEEE Transactions on pattern analysis and machine intelligence,  20, 777- 789 (1998).
[CrossRef]

Jueptner, W. P. O.

U. Schnars and W. P. O. Jueptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002)
[CrossRef]

Kaveh, M.

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

Li, B.

Lim, H. C.

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]

Liu, X.

Lu, W.

Mairz, B. A.

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

Malacara, D.

Malik, J.

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

Marroguin, J. L.

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]

Nakadate, S.

Perona, P.

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

Qifeng,

Ren, H.

Robinson, D. W.

D. W. Robinson and D. C. Williams, "Digital phase stepping speckle interferometry," Opt. Commun. 57,26-30 (1986).
[CrossRef]

Rodriguez-Vera, R.

Ryu, H. S.

Saito, H.

Schnars, U.

U. Schnars and W. P. O. Jueptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002)
[CrossRef]

Servin, M.

Sun, X.

Tang, C.

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]

Wan, Y.

L. Hong, Y. Wan, and A. Jain, "Fingerprint Image Enhancement: Algorithm and Performance Evaluation," IEEE Transactions on pattern analysis and machine intelligence,  20, 777- 789 (1998).
[CrossRef]

Wang, G.

Wang, W.

Wang, X.

Williams, D. C.

D. W. Robinson and D. C. Williams, "Digital phase stepping speckle interferometry," Opt. Commun. 57,26-30 (1986).
[CrossRef]

Yan, H.

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.

Zhang, Z.

Zhou, D.

Appl. Opt.

Computer Vision Image Understand.

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

IEEE TPAMI.

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

IEEE Trans. Image Process.

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

IEEE Transactions on pattern analysis and machine intelligence

L. Hong, Y. Wan, and A. Jain, "Fingerprint Image Enhancement: Algorithm and Performance Evaluation," IEEE Transactions on pattern analysis and machine intelligence,  20, 777- 789 (1998).
[CrossRef]

Meas. Sci. Technol.

U. Schnars and W. P. O. Jueptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002)
[CrossRef]

Opt. Commun.

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]

D. W. Robinson and D. C. Williams, "Digital phase stepping speckle interferometry," Opt. Commun. 57,26-30 (1986).
[CrossRef]

Opt. Lett.

SIAM J. Numer. Anal.

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

A. P. Witkin, "Scale-space filtering," in Proceedings of IJCAI, (Karlsruhe, 1983), pp. 1019-1021.

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Figures (3)

Fig. 1.
Fig. 1.

A computer-simulated phase fringe pattern and its filtered images. (a) Initial image.(b) The second-order oriented PDE model. (c) The conventional coupled PDE models.(d) Our oriented-couple PDE models.

Fig. 2.
Fig. 2.

A computer-simulated phase fringe pattern and its filtered images. (a) Initial image. (b) The second-order oriented PDE model. (c) The conventional coupled PDE models.(d) Our oriented-couple PDE models.

Fig. 3.
Fig. 3.

An experimentally obtained ESPI phase fringe pattern and its filtered images. (a) Initial image. (b) The second-order oriented PDE model. (c) The conventional coupled PDE models. (d) Our oriented-couple PDE models.

Tables (2)

Tables Icon

Table 1. Performance evaluation results for the various PDE filtering models

Tables Icon

Table 2. Comparison the computational time of our method and the traditional method

Equations (36)

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

{ u t = α g ( v ) u div ( u u ) + α ( g ( v ) ) u β ( u I ) u v t = a ( t ) div ( v v ) b ( v u )
u t = g ( u ) ( u xx cos 2 θ + u yy sin 2 θ + 2 u xy sin θ cos θ )
E ( u ) = Ω { 1 2 g ( v ) u ρ 2 + 1 2 β ( u I ) 2 } dxdy
= Ω { 1 2 g ( v ) ( u x cos θ + u y sin θ ) 2 + 1 2 β ( u I ) 2 } dxdy
f u x ( f u x ) y ( f u y ) = 0
f = 1 2 g ( v ) ( u x cos θ + u y sin θ ) 2 + 1 2 β ( u I ) 2
f u = β ( u I )
x ( f u x ) = g cos θ ( u xx cos θ + u xy sin θ )
y ( f u y ) = g sin θ ( u yy sin θ + u xy cos θ )
αg ( v ) ( u xx cos 2 θ + u yy sin 2 θ + 2 u xy sin θ cos θ ) + β ( u I ) u = 0
u t = αg ( v ) ( u xx cos 2 θ + u yy sin 2 θ + 2 u xy sin θ cos θ ) β ( u I ) u
v t = a ( t ) v v div ( v v ) b ( v u )
v t = a ( t ) v ( v xx cos 2 θ + v yy sin 2 θ + 2 v xy sin θ cos θ ) b ( v u )
u x y 0 = I x y , v x y 0 = I x y
{ u i , j n + 1 = u i , j n + Δ t g i , j n [ ( u xx ) i , j n cos 2 ( θ i , j ) + ( u yy ) i , j n sin 2 ( θ i , j ) + 2 ( u xy ) i , j n cos ( θ i , j ) sin ( θ i , j ) ] β ( u i , j n I ) ( u ) i , j n v i , j n + 1 = v i , j n + Δ t a ( t ) ( v ) i , j n [ ( u xx ) i , j n cos 2 ( θ i , j ) + ( u yy ) i , j n sin 2 ( θ i , j ) + 2 ( u xy ) i , j n cos ( θ i , j ) sin ( θ i , j ) ] b ( v i , j n u i , j n )
g i , j n = 1 ( 1 + k ( v i , j n ) 2 )
v i , j n = ( ( max ( Δ i v i , j n , 0 ) ) 2 + ( min ( Δ i + v i , j n , 0 ) ) 2 + ( max ( Δ j + v i , j n , 0 ) ) 2 + ( min ( Δ j v i , j n , 0 ) ) 2 ) 1 2
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 )
Φ x i j = cos ( 2 θ i j )
Φ y i j = sin ( 2 θ i j )
Φ x i j = k = w 2 1 2 w 2 1 2 l = w 2 1 2 w 2 1 2 F k l Φ x i + k j + l
Φ y i j = k = w 2 1 2 w 2 1 2 l = w 2 1 2 w 2 1 2 F k l Φ y i + k j + l
θ i j = 1 2 tan 1 Φ y i j Φ x i j
u x y = a + bx + cy
b = u x = x , y u x y x x , y x 2
c = u y = x , y u x y y x , y y 2
I 1 , i , j = I o , i , j + I r , i , j + 2 I o , i , j I r , i , j cos φ i , j + n 0 , i , j
I 2 , i , j = I o , i , j + I r , i , j + 2 I o , i , j I r , i , j cos ( φ i , j + π 2 ) + n 0 , i , j
I 3 , i , j = I o , i , j + I r , i , j + 2 I o , i , j I r , i , j cos ( φ i , j + ψ i , j ) + n 0 , i , j
I 4 , i , j = I o , i , j + I r , i , j + 2 I o , i , j I r , i , j cos ( φ i , j + ψ i , j + π 2 ) + n 0 , i , j
ψ i , j = 2 tan 1 ( I 4 , i , j I 2 , i , j + I 3 , i , j I 1 , i , j I 4 , i , j + I 2 , i , j I 3 , i , j I 1 , i , j )
ψ i , j = 3 π [ ( 6 × i 150 300 ) 2 + ( 6 × j 150 300 ) 2 ]
ψ i , j = ψ i , j 1 + ψ i , j 2
ψ i , j 1 = 80 [ exp ( ( 1 85 ) 2 + ( j 85 ) 2 3500 ) + exp ( ( i 295 ) 2 + ( j 295 ) 2 3500 ) ]
ψ i , j 2 = 100 ( i 190 222 ) 2 10 ( i 190 222 ) ( j 190 222 ) + 40 ( j 190 222 ) 2
f = 1 ( I 0 I ) 2 I 0 2

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