Abstract

A finite element formulation for a digital image correlation method is presented that will determine directly the complete, two-dimensional displacement field during the image correlation process on digital images. The entire interested image area is discretized into finite elements that are involved in the common image correlation process by use of our algorithms. This image correlation method with finite element formulation has an advantage over subset-based image correlation methods because it satisfies the requirements of displacement continuity and derivative continuity among elements on images. Numerical studies and a real experiment are used to verify the proposed formulation. Results have shown that the image correlation with the finite element formulation is computationally efficient, accurate, and robust.

© 2005 Optical Society of America

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References

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  1. W. H. Peters, W. F. Ranson, “Digital imaging techniques in experimental stress analysis,” Opt. Eng. 21, 427–432 (1982).
    [CrossRef]
  2. M. A. Sutton, W. J. Wolters, W. H. Peters, W. F. Ranson, S. R. McNel, “Determination of displacements using an improved digital correlation method,” Image Vision Comput. 1, 133–139 (1983).
    [CrossRef]
  3. T. C. Chu, W. F. Ranson, M. A. Sutton, W. H. Peters, “Applications of digital image correlation techniques to experimental mechanics,” Exp. Mech. 25, 232–244 (1985).
    [CrossRef]
  4. H. A. Bruck, S. R. McNeill, M. A. Sutton, W. H. Peters, “Digital image correlation using Newton–Raphson method of partial differential correction,” Exp. Mech. 29, 262–267 (1989).
    [CrossRef]
  5. H. Lu, G. Vendroux, W. G. Knauss, “Surface deformation measurements of a cylindrical specimen by digital image correlation,” Exp. Mech. 37, 433–439 (1997).
    [CrossRef]
  6. C. C.-B. Wang, J. M. Deng, G. A. Ateshian, C. T. Hung, “An automated approach for direct measurement of two-dimensional strain distributions within articular cartilage under unconfined compression,” J. Biomech. Eng. 124, 557–567 (2002).
    [CrossRef] [PubMed]
  7. J. S. Lyons, J. Liu, M. A. Sutton, “High-temperature deformation measurements using digital-image correlation,” Exp. Mech. 36, 64–70 (1996).
    [CrossRef]
  8. H. Lu, “Applications of digital speckle correlation to microscopic strain measurement and materials property characterization,” J. Electron. Packaging 120, 275–279 (1998).
    [CrossRef]
  9. D. Vogel, R. Kühnert, M. Dost, B. Michel, “Determination of packaging material properties utilizing image correlation techniques,” J. Electron. Packaging 124, 345–351 (2002).
    [CrossRef]
  10. U. Eisaku, I. Tadashi, “Measurement of deformation of epoxy resin plates with an embedded SMA wire using digital image correlation,” Int. J. Mod. Phys. B 17, 1750–1755 (2003).
    [CrossRef]
  11. D. J. Segalman, D. B. Woyak, R. E. Rowlands, “Smooth spline-like finite-element differentiation of full-field experimental data over arbitrary geometry,” Exp. Mech. 19, 429–437 (1979).
    [CrossRef]
  12. Z. Feng, R. E. Rowlands, “Continuous full-field representation and differentiation of three-dimensional experimental vector data,” Comput. Struct. 26, 979–990 (1987).
    [CrossRef]
  13. G. Vendroux, W. G. Knauss, “Submicron deformation field measurements: Part 2. Improved digital image correlation,” Exp. Mech. 38, 86–92 (1998).
    [CrossRef]
  14. H. Lu, P. D. Cary, “Deformation measurements by digital image correlation: implementation of a second-order displacement gradient,” Exp. Mech. 40, 393–400 (2000).
    [CrossRef]
  15. O. C. Zienkiewicz, R. L. Taylor, “The Finite Element Method,” 5th ed. (Butterworth-Heinemann, 2000).
  16. Wang Xucheng, Shao Min, “The Basis of Finite Element Method and Numerical Method,” 2nd ed. (Tsinghua University Publishing House, 1997), in Chinese.
  17. J. H. L. Pang, Shi Xunqing, Zhang Xueren, Liu Qinjun, “Application of digital speckle correlation to micro-deformation measurement of a flip chip assembly,” in Proceedings of IEEE Conference on Electronic Components and Technology (Institute of Electrical and Electronics Engineers, 2003), pp. 926–932.

2003 (1)

U. Eisaku, I. Tadashi, “Measurement of deformation of epoxy resin plates with an embedded SMA wire using digital image correlation,” Int. J. Mod. Phys. B 17, 1750–1755 (2003).
[CrossRef]

2002 (2)

D. Vogel, R. Kühnert, M. Dost, B. Michel, “Determination of packaging material properties utilizing image correlation techniques,” J. Electron. Packaging 124, 345–351 (2002).
[CrossRef]

C. C.-B. Wang, J. M. Deng, G. A. Ateshian, C. T. Hung, “An automated approach for direct measurement of two-dimensional strain distributions within articular cartilage under unconfined compression,” J. Biomech. Eng. 124, 557–567 (2002).
[CrossRef] [PubMed]

2000 (1)

H. Lu, P. D. Cary, “Deformation measurements by digital image correlation: implementation of a second-order displacement gradient,” Exp. Mech. 40, 393–400 (2000).
[CrossRef]

1998 (2)

G. Vendroux, W. G. Knauss, “Submicron deformation field measurements: Part 2. Improved digital image correlation,” Exp. Mech. 38, 86–92 (1998).
[CrossRef]

H. Lu, “Applications of digital speckle correlation to microscopic strain measurement and materials property characterization,” J. Electron. Packaging 120, 275–279 (1998).
[CrossRef]

1997 (1)

H. Lu, G. Vendroux, W. G. Knauss, “Surface deformation measurements of a cylindrical specimen by digital image correlation,” Exp. Mech. 37, 433–439 (1997).
[CrossRef]

1996 (1)

J. S. Lyons, J. Liu, M. A. Sutton, “High-temperature deformation measurements using digital-image correlation,” Exp. Mech. 36, 64–70 (1996).
[CrossRef]

1989 (1)

H. A. Bruck, S. R. McNeill, M. A. Sutton, W. H. Peters, “Digital image correlation using Newton–Raphson method of partial differential correction,” Exp. Mech. 29, 262–267 (1989).
[CrossRef]

1987 (1)

Z. Feng, R. E. Rowlands, “Continuous full-field representation and differentiation of three-dimensional experimental vector data,” Comput. Struct. 26, 979–990 (1987).
[CrossRef]

1985 (1)

T. C. Chu, W. F. Ranson, M. A. Sutton, W. H. Peters, “Applications of digital image correlation techniques to experimental mechanics,” Exp. Mech. 25, 232–244 (1985).
[CrossRef]

1983 (1)

M. A. Sutton, W. J. Wolters, W. H. Peters, W. F. Ranson, S. R. McNel, “Determination of displacements using an improved digital correlation method,” Image Vision Comput. 1, 133–139 (1983).
[CrossRef]

1982 (1)

W. H. Peters, W. F. Ranson, “Digital imaging techniques in experimental stress analysis,” Opt. Eng. 21, 427–432 (1982).
[CrossRef]

1979 (1)

D. J. Segalman, D. B. Woyak, R. E. Rowlands, “Smooth spline-like finite-element differentiation of full-field experimental data over arbitrary geometry,” Exp. Mech. 19, 429–437 (1979).
[CrossRef]

Ateshian, G. A.

C. C.-B. Wang, J. M. Deng, G. A. Ateshian, C. T. Hung, “An automated approach for direct measurement of two-dimensional strain distributions within articular cartilage under unconfined compression,” J. Biomech. Eng. 124, 557–567 (2002).
[CrossRef] [PubMed]

Bruck, H. A.

H. A. Bruck, S. R. McNeill, M. A. Sutton, W. H. Peters, “Digital image correlation using Newton–Raphson method of partial differential correction,” Exp. Mech. 29, 262–267 (1989).
[CrossRef]

Cary, P. D.

H. Lu, P. D. Cary, “Deformation measurements by digital image correlation: implementation of a second-order displacement gradient,” Exp. Mech. 40, 393–400 (2000).
[CrossRef]

Chu, T. C.

T. C. Chu, W. F. Ranson, M. A. Sutton, W. H. Peters, “Applications of digital image correlation techniques to experimental mechanics,” Exp. Mech. 25, 232–244 (1985).
[CrossRef]

Deng, J. M.

C. C.-B. Wang, J. M. Deng, G. A. Ateshian, C. T. Hung, “An automated approach for direct measurement of two-dimensional strain distributions within articular cartilage under unconfined compression,” J. Biomech. Eng. 124, 557–567 (2002).
[CrossRef] [PubMed]

Dost, M.

D. Vogel, R. Kühnert, M. Dost, B. Michel, “Determination of packaging material properties utilizing image correlation techniques,” J. Electron. Packaging 124, 345–351 (2002).
[CrossRef]

Eisaku, U.

U. Eisaku, I. Tadashi, “Measurement of deformation of epoxy resin plates with an embedded SMA wire using digital image correlation,” Int. J. Mod. Phys. B 17, 1750–1755 (2003).
[CrossRef]

Feng, Z.

Z. Feng, R. E. Rowlands, “Continuous full-field representation and differentiation of three-dimensional experimental vector data,” Comput. Struct. 26, 979–990 (1987).
[CrossRef]

Hung, C. T.

C. C.-B. Wang, J. M. Deng, G. A. Ateshian, C. T. Hung, “An automated approach for direct measurement of two-dimensional strain distributions within articular cartilage under unconfined compression,” J. Biomech. Eng. 124, 557–567 (2002).
[CrossRef] [PubMed]

Knauss, W. G.

G. Vendroux, W. G. Knauss, “Submicron deformation field measurements: Part 2. Improved digital image correlation,” Exp. Mech. 38, 86–92 (1998).
[CrossRef]

H. Lu, G. Vendroux, W. G. Knauss, “Surface deformation measurements of a cylindrical specimen by digital image correlation,” Exp. Mech. 37, 433–439 (1997).
[CrossRef]

Kühnert, R.

D. Vogel, R. Kühnert, M. Dost, B. Michel, “Determination of packaging material properties utilizing image correlation techniques,” J. Electron. Packaging 124, 345–351 (2002).
[CrossRef]

Liu, J.

J. S. Lyons, J. Liu, M. A. Sutton, “High-temperature deformation measurements using digital-image correlation,” Exp. Mech. 36, 64–70 (1996).
[CrossRef]

Lu, H.

H. Lu, P. D. Cary, “Deformation measurements by digital image correlation: implementation of a second-order displacement gradient,” Exp. Mech. 40, 393–400 (2000).
[CrossRef]

H. Lu, “Applications of digital speckle correlation to microscopic strain measurement and materials property characterization,” J. Electron. Packaging 120, 275–279 (1998).
[CrossRef]

H. Lu, G. Vendroux, W. G. Knauss, “Surface deformation measurements of a cylindrical specimen by digital image correlation,” Exp. Mech. 37, 433–439 (1997).
[CrossRef]

Lyons, J. S.

J. S. Lyons, J. Liu, M. A. Sutton, “High-temperature deformation measurements using digital-image correlation,” Exp. Mech. 36, 64–70 (1996).
[CrossRef]

McNeill, S. R.

H. A. Bruck, S. R. McNeill, M. A. Sutton, W. H. Peters, “Digital image correlation using Newton–Raphson method of partial differential correction,” Exp. Mech. 29, 262–267 (1989).
[CrossRef]

McNel, S. R.

M. A. Sutton, W. J. Wolters, W. H. Peters, W. F. Ranson, S. R. McNel, “Determination of displacements using an improved digital correlation method,” Image Vision Comput. 1, 133–139 (1983).
[CrossRef]

Michel, B.

D. Vogel, R. Kühnert, M. Dost, B. Michel, “Determination of packaging material properties utilizing image correlation techniques,” J. Electron. Packaging 124, 345–351 (2002).
[CrossRef]

Min, Shao

Wang Xucheng, Shao Min, “The Basis of Finite Element Method and Numerical Method,” 2nd ed. (Tsinghua University Publishing House, 1997), in Chinese.

Pang, J. H. L.

J. H. L. Pang, Shi Xunqing, Zhang Xueren, Liu Qinjun, “Application of digital speckle correlation to micro-deformation measurement of a flip chip assembly,” in Proceedings of IEEE Conference on Electronic Components and Technology (Institute of Electrical and Electronics Engineers, 2003), pp. 926–932.

Peters, W. H.

H. A. Bruck, S. R. McNeill, M. A. Sutton, W. H. Peters, “Digital image correlation using Newton–Raphson method of partial differential correction,” Exp. Mech. 29, 262–267 (1989).
[CrossRef]

T. C. Chu, W. F. Ranson, M. A. Sutton, W. H. Peters, “Applications of digital image correlation techniques to experimental mechanics,” Exp. Mech. 25, 232–244 (1985).
[CrossRef]

M. A. Sutton, W. J. Wolters, W. H. Peters, W. F. Ranson, S. R. McNel, “Determination of displacements using an improved digital correlation method,” Image Vision Comput. 1, 133–139 (1983).
[CrossRef]

W. H. Peters, W. F. Ranson, “Digital imaging techniques in experimental stress analysis,” Opt. Eng. 21, 427–432 (1982).
[CrossRef]

Qinjun, Liu

J. H. L. Pang, Shi Xunqing, Zhang Xueren, Liu Qinjun, “Application of digital speckle correlation to micro-deformation measurement of a flip chip assembly,” in Proceedings of IEEE Conference on Electronic Components and Technology (Institute of Electrical and Electronics Engineers, 2003), pp. 926–932.

Ranson, W. F.

T. C. Chu, W. F. Ranson, M. A. Sutton, W. H. Peters, “Applications of digital image correlation techniques to experimental mechanics,” Exp. Mech. 25, 232–244 (1985).
[CrossRef]

M. A. Sutton, W. J. Wolters, W. H. Peters, W. F. Ranson, S. R. McNel, “Determination of displacements using an improved digital correlation method,” Image Vision Comput. 1, 133–139 (1983).
[CrossRef]

W. H. Peters, W. F. Ranson, “Digital imaging techniques in experimental stress analysis,” Opt. Eng. 21, 427–432 (1982).
[CrossRef]

Rowlands, R. E.

Z. Feng, R. E. Rowlands, “Continuous full-field representation and differentiation of three-dimensional experimental vector data,” Comput. Struct. 26, 979–990 (1987).
[CrossRef]

D. J. Segalman, D. B. Woyak, R. E. Rowlands, “Smooth spline-like finite-element differentiation of full-field experimental data over arbitrary geometry,” Exp. Mech. 19, 429–437 (1979).
[CrossRef]

Segalman, D. J.

D. J. Segalman, D. B. Woyak, R. E. Rowlands, “Smooth spline-like finite-element differentiation of full-field experimental data over arbitrary geometry,” Exp. Mech. 19, 429–437 (1979).
[CrossRef]

Sutton, M. A.

J. S. Lyons, J. Liu, M. A. Sutton, “High-temperature deformation measurements using digital-image correlation,” Exp. Mech. 36, 64–70 (1996).
[CrossRef]

H. A. Bruck, S. R. McNeill, M. A. Sutton, W. H. Peters, “Digital image correlation using Newton–Raphson method of partial differential correction,” Exp. Mech. 29, 262–267 (1989).
[CrossRef]

T. C. Chu, W. F. Ranson, M. A. Sutton, W. H. Peters, “Applications of digital image correlation techniques to experimental mechanics,” Exp. Mech. 25, 232–244 (1985).
[CrossRef]

M. A. Sutton, W. J. Wolters, W. H. Peters, W. F. Ranson, S. R. McNel, “Determination of displacements using an improved digital correlation method,” Image Vision Comput. 1, 133–139 (1983).
[CrossRef]

Tadashi, I.

U. Eisaku, I. Tadashi, “Measurement of deformation of epoxy resin plates with an embedded SMA wire using digital image correlation,” Int. J. Mod. Phys. B 17, 1750–1755 (2003).
[CrossRef]

Taylor, R. L.

O. C. Zienkiewicz, R. L. Taylor, “The Finite Element Method,” 5th ed. (Butterworth-Heinemann, 2000).

Vendroux, G.

G. Vendroux, W. G. Knauss, “Submicron deformation field measurements: Part 2. Improved digital image correlation,” Exp. Mech. 38, 86–92 (1998).
[CrossRef]

H. Lu, G. Vendroux, W. G. Knauss, “Surface deformation measurements of a cylindrical specimen by digital image correlation,” Exp. Mech. 37, 433–439 (1997).
[CrossRef]

Vogel, D.

D. Vogel, R. Kühnert, M. Dost, B. Michel, “Determination of packaging material properties utilizing image correlation techniques,” J. Electron. Packaging 124, 345–351 (2002).
[CrossRef]

Wang, C. C.-B.

C. C.-B. Wang, J. M. Deng, G. A. Ateshian, C. T. Hung, “An automated approach for direct measurement of two-dimensional strain distributions within articular cartilage under unconfined compression,” J. Biomech. Eng. 124, 557–567 (2002).
[CrossRef] [PubMed]

Wolters, W. J.

M. A. Sutton, W. J. Wolters, W. H. Peters, W. F. Ranson, S. R. McNel, “Determination of displacements using an improved digital correlation method,” Image Vision Comput. 1, 133–139 (1983).
[CrossRef]

Woyak, D. B.

D. J. Segalman, D. B. Woyak, R. E. Rowlands, “Smooth spline-like finite-element differentiation of full-field experimental data over arbitrary geometry,” Exp. Mech. 19, 429–437 (1979).
[CrossRef]

Xucheng, Wang

Wang Xucheng, Shao Min, “The Basis of Finite Element Method and Numerical Method,” 2nd ed. (Tsinghua University Publishing House, 1997), in Chinese.

Xueren, Zhang

J. H. L. Pang, Shi Xunqing, Zhang Xueren, Liu Qinjun, “Application of digital speckle correlation to micro-deformation measurement of a flip chip assembly,” in Proceedings of IEEE Conference on Electronic Components and Technology (Institute of Electrical and Electronics Engineers, 2003), pp. 926–932.

Xunqing, Shi

J. H. L. Pang, Shi Xunqing, Zhang Xueren, Liu Qinjun, “Application of digital speckle correlation to micro-deformation measurement of a flip chip assembly,” in Proceedings of IEEE Conference on Electronic Components and Technology (Institute of Electrical and Electronics Engineers, 2003), pp. 926–932.

Zienkiewicz, O. C.

O. C. Zienkiewicz, R. L. Taylor, “The Finite Element Method,” 5th ed. (Butterworth-Heinemann, 2000).

Comput. Struct. (1)

Z. Feng, R. E. Rowlands, “Continuous full-field representation and differentiation of three-dimensional experimental vector data,” Comput. Struct. 26, 979–990 (1987).
[CrossRef]

Exp. Mech. (7)

G. Vendroux, W. G. Knauss, “Submicron deformation field measurements: Part 2. Improved digital image correlation,” Exp. Mech. 38, 86–92 (1998).
[CrossRef]

H. Lu, P. D. Cary, “Deformation measurements by digital image correlation: implementation of a second-order displacement gradient,” Exp. Mech. 40, 393–400 (2000).
[CrossRef]

D. J. Segalman, D. B. Woyak, R. E. Rowlands, “Smooth spline-like finite-element differentiation of full-field experimental data over arbitrary geometry,” Exp. Mech. 19, 429–437 (1979).
[CrossRef]

T. C. Chu, W. F. Ranson, M. A. Sutton, W. H. Peters, “Applications of digital image correlation techniques to experimental mechanics,” Exp. Mech. 25, 232–244 (1985).
[CrossRef]

H. A. Bruck, S. R. McNeill, M. A. Sutton, W. H. Peters, “Digital image correlation using Newton–Raphson method of partial differential correction,” Exp. Mech. 29, 262–267 (1989).
[CrossRef]

H. Lu, G. Vendroux, W. G. Knauss, “Surface deformation measurements of a cylindrical specimen by digital image correlation,” Exp. Mech. 37, 433–439 (1997).
[CrossRef]

J. S. Lyons, J. Liu, M. A. Sutton, “High-temperature deformation measurements using digital-image correlation,” Exp. Mech. 36, 64–70 (1996).
[CrossRef]

Image Vision Comput. (1)

M. A. Sutton, W. J. Wolters, W. H. Peters, W. F. Ranson, S. R. McNel, “Determination of displacements using an improved digital correlation method,” Image Vision Comput. 1, 133–139 (1983).
[CrossRef]

Int. J. Mod. Phys. B (1)

U. Eisaku, I. Tadashi, “Measurement of deformation of epoxy resin plates with an embedded SMA wire using digital image correlation,” Int. J. Mod. Phys. B 17, 1750–1755 (2003).
[CrossRef]

J. Biomech. Eng. (1)

C. C.-B. Wang, J. M. Deng, G. A. Ateshian, C. T. Hung, “An automated approach for direct measurement of two-dimensional strain distributions within articular cartilage under unconfined compression,” J. Biomech. Eng. 124, 557–567 (2002).
[CrossRef] [PubMed]

J. Electron. Packaging (2)

H. Lu, “Applications of digital speckle correlation to microscopic strain measurement and materials property characterization,” J. Electron. Packaging 120, 275–279 (1998).
[CrossRef]

D. Vogel, R. Kühnert, M. Dost, B. Michel, “Determination of packaging material properties utilizing image correlation techniques,” J. Electron. Packaging 124, 345–351 (2002).
[CrossRef]

Opt. Eng. (1)

W. H. Peters, W. F. Ranson, “Digital imaging techniques in experimental stress analysis,” Opt. Eng. 21, 427–432 (1982).
[CrossRef]

Other (3)

O. C. Zienkiewicz, R. L. Taylor, “The Finite Element Method,” 5th ed. (Butterworth-Heinemann, 2000).

Wang Xucheng, Shao Min, “The Basis of Finite Element Method and Numerical Method,” 2nd ed. (Tsinghua University Publishing House, 1997), in Chinese.

J. H. L. Pang, Shi Xunqing, Zhang Xueren, Liu Qinjun, “Application of digital speckle correlation to micro-deformation measurement of a flip chip assembly,” in Proceedings of IEEE Conference on Electronic Components and Technology (Institute of Electrical and Electronics Engineers, 2003), pp. 926–932.

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Figures (9)

Fig. 1
Fig. 1

Schematic of subset-based image correlation: (a) undeformed and (b) deformed images.

Fig. 2
Fig. 2

Wrong deformation data in the displacement vector field.

Fig. 3
Fig. 3

Schematic of element-based DIC: (a) undeformed and (b) deformed images.

Fig. 4
Fig. 4

Determination of nodal assembly matrices by use of the correspondence between global and element nodes.

Fig. 5
Fig. 5

Reference image.

Fig. 6
Fig. 6

Horizontal skew image.

Fig. 7
Fig. 7

Variation in strain εxx from the element-based method.

Fig. 8
Fig. 8

Variation in strain εxx from the subset-based method.

Fig. 9
Fig. 9

DIC specimen mounted on a high-resolution rotation stage.

Tables (4)

Tables Icon

Table 1 Results of Simulated Rigid Body Translations

Tables Icon

Table 2 Results of Simulated Uniform Deformations

Tables Icon

Table 3 Results of Simulated Rigid Body Rotation

Tables Icon

Table 4 Results of Rigid Body Rotation Experiment

Equations (22)

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

C = S p S { f ( x , y ) g [ x + u ( x , y , a ) , y + v ( x , y , a ) ] } 2 S p S f 2 ( x , y ) ,
u ( x , y , a ) = a 0 + a 2 ( x x 0 ) + a 4 ( y y 0 ) , v ( x , y , a ) = a 1 + a 3 ( x x 0 ) + a 5 ( y y 0 ) .
a [ u ( x 0 , y 0 ) , v ( x 0 , y 0 ) , u x | x 0 , y 0 , v x | x 0 , y 0 , u y | x 0 , y 0 , v y | x 0 , y 0 ]
C ( a 0 ) ( a a 0 ) = C ( a 0 ) .
ɛ x x = u x + 1 2 [ ( u x ) 2 + ( v x ) 2 ] , ɛ y y = v y + 1 2 [ ( u y ) 2 + ( v y ) 2 ] , ɛ x y = 1 2 ( u y + v x ) + 1 2 ( u x u y + v x v y ) .
θ 1 2 ( v x u y ) .
g ( x , ) = i = 0 3 j = 0 3 α i j x i j .
C = e S S p e { f ( x , y ) g [ x + u ( x , y , p e ) , y + v ( x , y , p e ) ] } 2 S p S f 2 ( x , y ) ,
x = x + u ( x , y , p e ) , = y + v ( x , y , p e ) .
u ( x , y , p e ) = i = 1 m N i ( x , y ) u i , v ( x , y , p e ) = i = 1 m N i ( x , y ) v i ,
C = F e S S p e [ f ( S p ) g ( S p , p e ) ] 2 .
C = ( C p i ) i = 1 , 2 n = 2 F e S G T { S p e [ f ( S p ) g ( S p , p e ) ] g ( S p , p e ) p j e } j = 1 , 2 m = 0 .
p e = G p .
G = [ 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 ] .
C ( p 0 ) ( p p 0 ) = C ( p 0 ) ,
C = ( 2 C p i p j ) i = 1 , 2 n j = 1 , 2 n = e S G T { 2 F S p e [ f ( S p ) g ( S p , p e ) ] 2 g ( S p , p e ) p h e p k e + 2 F S p e g ( S p , p e ) p h e g ( S p , p e ) p k e } h = 1 , 2 m k = 1 , 2 m G .
2 C p i p j 2 F e S G T S p e g ( S p , p e ) p h e g ( S p , p e ) p k e G .
g ( S p , p e ) p k e = g ( x , , p e ) x x p k e + g ( x , , p e ) p k e .
v ¯ v 0 ( pixels )
ɛ ¯ x x ɛ x x 0
ɛ ¯ y y ɛ y y 0
ɛ ¯ x y ɛ x y 0

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