Abstract

The skeletonization of optical fringes with high density and high noise has been an open problem. We describe a skeletonization process for gray-scale optical fringe patterns with high density and high noise based on the gradient vector fields (GVFs). We derive the new oriented couple governing partial differential equations (PDEs) for calculating the GVFs of dense, noisy optical fringes based on the variational methods, in which the fringe orientation is taken into account fully. We test the proposed PDEs on a computer simulation and experimentally obtained fringe patterns, in which the fringes contain obvious high density regions and heavy noise, and compare them with related PDEs and the fringe extreme tracking method, respectively. The experimental results demonstrate that the new governing PDEs perform favorably.

© 2010 Optical Society of America

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  1. M. Aslan, “Toward the development of high-speed microscpoic ESPI system for monitoring laser heating/drilling of alumina Al2O3 substrates,” Ph.D. dissertation (Pennsylvania State University, 2000).
  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. L. Wang, G. Leedham, and D. S. Y. Cho, “Minutiae feature analysis for infrared hand vein pattern biometrics,” Pattern Recogn. 41, 920–929 (2008).
    [CrossRef]
  4. Z. Yu and C. Bajaj, “A segmentation-free approach for skeletonization of gray-scale images via anisotropic vector diffusion,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR’04) (2004), pp. 415–420.
  5. 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] [PubMed]
  6. C. Xu and J. L. Prince, “Snakes, shapes, and gradient vector flow,” IEEE Trans. Image Process. 7, 359–369 (1998).
    [CrossRef]
  7. C. Tang, L. Han, H. I. Ren, T. Gao, Z. Wang, and Ke Tang, “The oriented-couple partial differential equations for filtering in wrapped phase patterns,” Opt. Express 175606–5617 (2009).
    [CrossRef] [PubMed]
  8. J. Villa, J. A. Quiroga, and I. D. L. Rosa, “Regularized quadratic cost function for oriented fringe-pattern filtering,” Opt. Lett. 34,1741–1743 (2009).
    [CrossRef] [PubMed]
  9. Y. Chen, C. A. Z. Barcelos, and B. A. Mair, “Smoothing and edge detection by time-varying coupled nonlinear diffusion equations,” Comput. Vision Image Understand. 82, 85–100(2001).
    [CrossRef]
  10. 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] [PubMed]

2010

2009

2008

2005

2001

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

1998

C. Xu and J. L. Prince, “Snakes, shapes, and gradient vector flow,” IEEE Trans. Image Process. 7, 359–369 (1998).
[CrossRef]

Aslan, M.

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

Bajaj, C.

Z. Yu and C. Bajaj, “A segmentation-free approach for skeletonization of gray-scale images via anisotropic vector diffusion,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR’04) (2004), pp. 415–420.

Barcelos, C. A. Z.

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

Cai, Y.

Chen, Y.

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

Cho, D. S. Y.

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

Gao, T.

Han, L.

He, X.

Leedham, G.

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

Lu, W.

Mair, B. A.

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

Prince, J. L.

C. Xu and J. L. Prince, “Snakes, shapes, and gradient vector flow,” IEEE Trans. Image Process. 7, 359–369 (1998).
[CrossRef]

Quan, C.

Quiroga, J. A.

Ren, H. I.

Rosa, I. D. L.

Tang, C.

Tang, Ke

Tay, C. J.

Villa, J.

Wang, G.

Wang, L.

Wang, Z.

Wu, J.

Xu, C.

C. Xu and J. L. Prince, “Snakes, shapes, and gradient vector flow,” IEEE Trans. Image Process. 7, 359–369 (1998).
[CrossRef]

Yan, S.

Yang, F.

Yu, Z.

Z. Yu and C. Bajaj, “A segmentation-free approach for skeletonization of gray-scale images via anisotropic vector diffusion,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR’04) (2004), pp. 415–420.

Appl. Opt.

Comput. Vision Image Understand.

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

IEEE Trans. Image Process.

C. Xu and J. L. Prince, “Snakes, shapes, and gradient vector flow,” IEEE Trans. Image Process. 7, 359–369 (1998).
[CrossRef]

Opt. Express

Opt. Lett.

Pattern Recogn.

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

Other

Z. Yu and C. Bajaj, “A segmentation-free approach for skeletonization of gray-scale images via anisotropic vector diffusion,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR’04) (2004), pp. 415–420.

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

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Figures (2)

Fig. 1
Fig. 1

Skeletonization of an experimentally obtained fringe image by various methods: (a) initial image, (b) skeletons obtained by the fringe extreme tracking method, (c) skeletons obtained by PDEs (1), (d) skeletons obtained by our PDEs (4).

Fig. 2
Fig. 2

Skeletonization of a computer-simulated ESPI fringe image by various methods: (a) initial image, (b) skeletons obtained by the fringe extreme tracking method, (c) skeletons obtained by PDEs (1), (d) skeletons obtained by our PDEs (4).

Equations (36)

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u ( x , y , t ) t = α g ( | u | ) | u | div ( u | u | ) + α ( g ( | u | ) ) u β ( u I x ) | u | + γ div ( g ( | u | ) u ) ,
v ( x , y , t ) t = α g ( | v | ) | v | div ( v | v | ) + α ( g ( | v | ) ) v β ( v I y ) | v | + γ div ( g ( | v | ) v ) ,
u ( x , y , t ) t = a ( t ) div ( u | u | ) b ( u u ) ,
v ( x , y , t ) t = a ( t ) div ( v | v | ) b ( v v ) ,
E 1 = Ω { [ α g ( V ) | V | + β ( V I ) 2 ] + γ f ( | V | ) } d x d y .
E 1 = Ω { 1 2 α g ( | V | ) | V ρ | 2 + 1 2 β ( V I ) 2 } d x d y = Ω { 1 2 α g ( | V | ) ( u ρ 2 + v ρ 2 ) + 1 2 β ( V I ) 2 } d x d y ,
u t = α g ( | u | ) ( u x x cos 2 θ + u y y sin 2 θ + 2 u x y sin θ cos θ ) β ( u I ) | u | ,
v t = α g ( | v | ) ( v x x cos 2 θ + v y y sin 2 θ + 2 v x y sin θ cos θ ) β ( v I ) | v | ,
u t = a ( t ) g 1 ( | u | ) ( u x x cos 2 θ + u y y sin 2 θ + 2 u x y sin θ cos θ ) b ( u u ) ,
v t = a ( t ) g 1 ( | v | ) ( v x x cos 2 θ + v y y sin 2 θ + 2 v x y sin θ cos θ ) b ( v v ) ,
u ( x , y , 0 ) = I x , v ( x , y , 0 ) = I y , u ( x , y , 0 ) = I x , v ( x , y , 0 ) = I y .
u x , y N = u u 2 + v 2 , v x , y N = v u 2 + v 2 .
S x , y = ( u x , y N x ) 2 + ( v x , y N y ) 2 max ( ( u x , y N x ) 2 + ( v x , y N y ) 2 ) .
f FS ( x , y ) = { 1 , skeleton pixel if     S x , y > T ¯ 0 , non-skeleton pixel otherwise ,
u i , j n + 1 = u i , j n + Δ t α 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 ) ] Δ t β ( u i , j n I ) ( | u | ) i , j n ,
v i , j n + 1 = v i , j n + Δ t α g i , j n [ ( v x x ) i , j n cos 2 ( θ i , j ) + ( v y y ) i , j n sin 2 ( θ i , j ) + 2 ( v x y ) i , j n cos ( θ i , j ) sin ( θ i , j ) ] Δ t β ( v i , j n I ) ( | v | ) i , j n ,
u i , j n + 1 = u i , j n + Δ t a ( t ) ( g 1 ) 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 ) ] Δ t b ( u i , j n u i , j n ) ,
v i , j n + 1 = v i , j n + Δ t a ( t ) ( g 1 ) i , j n [ ( v x x ) i , j n cos 2 ( θ i , j ) + ( v y y ) i , j n sin 2 ( θ i , j ) + 2 ( v x y ) i , j n cos ( θ i , j ) sin ( θ i , j ) ] Δ t b ( v i , j n v i , j n ) ,
( u x x ) i , j n = u i + 1 , j n 2 u i , j n + u i 1 , j n .
θ i , j = 1 2 tan 1 ( k l E ( ω k , ω l ) sin ( 2 θ k , l ) k l E ( ω k , ω l ) cos ( 2 θ k , l ) ) ,
I sub = | 4 I o I r sin ( ϕ r ϕ o + ψ 2 ) sin ( ψ 2 ) | ,
ψ x , y = 110 × [ exp ( ( x 0.5 M ) 2 + y 2 15000 ) exp ( ( x 0.5 M ) 2 + ( y N ) 2 15000 ) ] .
u ρ = u x cos θ + u y sin θ ,
v ρ = v x cos θ + v y sin θ ,
E 1 = Ω { 1 2 α g ( | V | ) ( ( u x cos θ + u y sin θ ) 2 + ( v x cos θ + v y sin θ ) 2 ) + 1 2 β ( V I ) 2 } d x d y .
f 1 u x ( f 1 u x ) y ( f 1 u y ) = 0 ,
f 2 v x ( f 2 v x ) y ( f 2 v y ) = 0 ,
f 1 = 1 2 α g ( | u | ) ( ( u x cos θ + u y sin θ ) 2 + ( v x cos θ + v y sin θ ) 2 ) + 1 2 β ( u I ) 2 ,
f 2 = 1 2 α g ( | v | ) ( ( u x cos θ + u y sin θ ) 2 + ( v x cos θ + v y sin θ ) 2 ) + 1 2 β ( v I ) 2 .
{ α g ( | u | ) ( u x x cos 2 θ + u y y sin 2 θ + 2 u x y sin θ cos θ ) + β ( u I ) | u | = 0 α g ( | v | ) ( v x x cos 2 θ + v y y sin 2 θ + 2 v x y sin θ cos θ ) + β ( v I ) | v | = 0 .
u t = α g ( | u | ) ( u x x cos 2 θ + u y y sin 2 θ + 2 u x y sin θ cos θ ) β ( u I ) | u | ,
v t = α g ( | v | ) ( v x x cos 2 θ + v y y sin 2 θ + 2 v x y sin θ cos θ ) β ( v I ) | v | .
E 2 = Ω { 1 2 a ( t ) g 1 ( | V | ) | V ρ | 2 + ( b / 2 ) | V V | 2 } d x d y ,
E 2 = Ω [ 1 2 a ( t ) g 1 ( | V | ) ( u ρ 2 + v ρ 2 ) + ( b / 2 ) | V V | 2 ] d x d y = Ω [ 1 2 a ( t ) g 1 ( | V | ) ( ( u x cos θ + u y sin θ ) 2 + ( v x cos θ + v y sin θ ) 2 ) + b 2 | V V | 2 ] d x d y .
u t = a ( t ) g 1 ( | u | ) ( u x x cos 2 θ + u y y sin 2 θ + 2 u x y sin θ cos θ ) b ( u u ) ,
v t = a ( t ) g 1 ( | v | ) ( v x x cos 2 θ + v y y sin 2 θ + 2 v x y sin θ cos θ ) b ( v v ) .

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