Abstract

Electronic speckle pattern interferometry fringe patterns usually have poor contrast so it is important to enhance fringe contrast for the extraction of phase from a single fringe pattern. We present new enhancement methods based on differential equations (called DE enhancement methods) to electronic speckle pattern interferometry fringes. The DE enhancement methods transform the image processing to solve differential equations. With the proposed methods, the visibility of the correlation speckle fringe patterns can be improved significantly. We tested the proposed methods on computer-simulated speckle correlation fringes and experimentally obtained fringes, and we compared the new method with other contrast enhancement techniques. The experimental results illustrate the performance of this approach.

© 2006 Optical Society of America

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  1. S.-H. Baik, S.-K. Park, C.-J. Kim, and S.-Y. Kim, "Two-channel spatial phase shifting electronic speckle pattern interferometer," Opt. Commun. 192, 205-211 (2001).
    [CrossRef]
  2. C. J. Tay, C. Quan, L. Chen, and Y. Fu, "Phase extraction from electronic speckle patterns by statistical analysis," Opt. Commun. 236, 259-269 (2004).
    [CrossRef]
  3. 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]
  4. X. Su and W. Chen, "Fourier transform profilometry: a review," Opt. Laser Eng. 35, 263-284 (2001).
    [CrossRef]
  5. K. H. Womack, "Interferometric phase measurement using spatial synchronous detection," Opt. Eng. 23, 391-395 (1984).
  6. M. Servin, D. Malacara, and R. Rodriguez-Vera, "Phase-locked-loop interferometry applied to aspheric testing with a computer-stored compensator," Appl. Opt. 33, 2589-2595 (1994).
    [CrossRef] [PubMed]
  7. F. J. Cuevas, M. Servin, O. N. Stavroudis, and R. Rodriguez-Vera, "Multi-layer neural network applied to phase and depth recovery from fringe patterns," Opt. Commun. 181, 239-259 (2000).
    [CrossRef]
  8. F. J. Cuevas, J. H. Sossa-Azuela, and M. Servin, "A parametric method applied to phase recovery from a fringe pattern based on a genetic algorithm," Opt. Commun. 203, 213-223 (2002).
    [CrossRef]
  9. M. Servin, J. L. Marroquin, and F. J. Cuevas, "Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms," J. Opt. Soc. Am. A. 18, 689-695 (2001).
    [CrossRef]
  10. A. Davila, D. Kerr, and G. H. Kaufmann, "Digital processing of electronic speckle pattern interferometry addition fringes," Appl. Opt. 33, 5964-5969 (1994).
    [CrossRef] [PubMed]
  11. N. A. Ochoa, F. M. Santoyo, A. J. Moore, and C. P. López, "Contrast enhancement of electronic speckle pattern interferometry addition fringes," Appl. Opt. 36, 2783-2787 (1997).
    [CrossRef] [PubMed]
  12. T. Singh and C. S. Vikram, "Histogram equalization of Young's fringes in speckle photography," in Interferometry XI: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 4777, 27-30 (2002).
    [CrossRef]
  13. T. Singh and C. S. Vikram, "Histogram equalization of vibration fringes in holography," Optik (Stuttgart) 113, 499-503 (2002).
    [CrossRef]
  14. F. Catté, P. L. Lions, J. M. Morel, and T. Coll, "Image selective smoothing and edge detection by nonlinear diffusion," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 29, 182-193 (1992).
  15. G. Sapiro, "Geometric partial differential equations in image processing: past, present, and future," in Proceedings of the Second IEEE International Conference on Image Processing 3 (Institute of Electrical and Electronics Engineers, 1995), pp. 1-4.
  16. 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]
  17. G. Sapiro and V. Caselles, "Contrast enhancement via image evolution flows," Graph. Models Image Process. 59, 407-416 (1997).
    [CrossRef]
  18. R. C. Gonzalez and R. E. Woods, Digital Image Processing, 2nd ed. (Prentice-Hall, 2002), pp. 91-97.

2005 (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] [PubMed]

2004 (1)

C. J. Tay, C. Quan, L. Chen, and Y. Fu, "Phase extraction from electronic speckle patterns by statistical analysis," Opt. Commun. 236, 259-269 (2004).
[CrossRef]

2002 (4)

F. J. Cuevas, J. H. Sossa-Azuela, and M. Servin, "A parametric method applied to phase recovery from a fringe pattern based on a genetic algorithm," Opt. Commun. 203, 213-223 (2002).
[CrossRef]

T. Singh and C. S. Vikram, "Histogram equalization of Young's fringes in speckle photography," in Interferometry XI: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 4777, 27-30 (2002).
[CrossRef]

T. Singh and C. S. Vikram, "Histogram equalization of vibration fringes in holography," Optik (Stuttgart) 113, 499-503 (2002).
[CrossRef]

R. C. Gonzalez and R. E. Woods, Digital Image Processing, 2nd ed. (Prentice-Hall, 2002), pp. 91-97.

2001 (4)

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]

M. Servin, J. L. Marroquin, and F. J. Cuevas, "Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms," J. Opt. Soc. Am. A. 18, 689-695 (2001).
[CrossRef]

S.-H. Baik, S.-K. Park, C.-J. Kim, and S.-Y. Kim, "Two-channel spatial phase shifting electronic speckle pattern interferometer," Opt. Commun. 192, 205-211 (2001).
[CrossRef]

X. Su and W. Chen, "Fourier transform profilometry: a review," Opt. Laser Eng. 35, 263-284 (2001).
[CrossRef]

2000 (1)

F. J. Cuevas, M. Servin, O. N. Stavroudis, and R. Rodriguez-Vera, "Multi-layer neural network applied to phase and depth recovery from fringe patterns," Opt. Commun. 181, 239-259 (2000).
[CrossRef]

1997 (2)

G. Sapiro and V. Caselles, "Contrast enhancement via image evolution flows," Graph. Models Image Process. 59, 407-416 (1997).
[CrossRef]

N. A. Ochoa, F. M. Santoyo, A. J. Moore, and C. P. López, "Contrast enhancement of electronic speckle pattern interferometry addition fringes," Appl. Opt. 36, 2783-2787 (1997).
[CrossRef] [PubMed]

1995 (1)

G. Sapiro, "Geometric partial differential equations in image processing: past, present, and future," in Proceedings of the Second IEEE International Conference on Image Processing 3 (Institute of Electrical and Electronics Engineers, 1995), pp. 1-4.

1994 (2)

1992 (1)

F. Catté, P. L. Lions, J. M. Morel, and T. Coll, "Image selective smoothing and edge detection by nonlinear diffusion," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 29, 182-193 (1992).

1984 (1)

K. H. Womack, "Interferometric phase measurement using spatial synchronous detection," Opt. Eng. 23, 391-395 (1984).

Baik, S.-H.

S.-H. Baik, S.-K. Park, C.-J. Kim, and S.-Y. Kim, "Two-channel spatial phase shifting electronic speckle pattern interferometer," Opt. Commun. 192, 205-211 (2001).
[CrossRef]

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]

Caselles, V.

G. Sapiro and V. Caselles, "Contrast enhancement via image evolution flows," Graph. Models Image Process. 59, 407-416 (1997).
[CrossRef]

Catté, F.

F. Catté, P. L. Lions, J. M. Morel, and T. Coll, "Image selective smoothing and edge detection by nonlinear diffusion," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 29, 182-193 (1992).

Chen, L.

C. J. Tay, C. Quan, L. Chen, and Y. Fu, "Phase extraction from electronic speckle patterns by statistical analysis," Opt. Commun. 236, 259-269 (2004).
[CrossRef]

Chen, W.

X. Su and W. Chen, "Fourier transform profilometry: a review," Opt. Laser Eng. 35, 263-284 (2001).
[CrossRef]

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]

Coll, T.

F. Catté, P. L. Lions, J. M. Morel, and T. Coll, "Image selective smoothing and edge detection by nonlinear diffusion," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 29, 182-193 (1992).

Cuevas, F. J.

F. J. Cuevas, J. H. Sossa-Azuela, and M. Servin, "A parametric method applied to phase recovery from a fringe pattern based on a genetic algorithm," Opt. Commun. 203, 213-223 (2002).
[CrossRef]

M. Servin, J. L. Marroquin, and F. J. Cuevas, "Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms," J. Opt. Soc. Am. A. 18, 689-695 (2001).
[CrossRef]

F. J. Cuevas, M. Servin, O. N. Stavroudis, and R. Rodriguez-Vera, "Multi-layer neural network applied to phase and depth recovery from fringe patterns," Opt. Commun. 181, 239-259 (2000).
[CrossRef]

Davila, A.

Fu, Y.

C. J. Tay, C. Quan, L. Chen, and Y. Fu, "Phase extraction from electronic speckle patterns by statistical analysis," Opt. Commun. 236, 259-269 (2004).
[CrossRef]

Gonzalez, R. C.

R. C. Gonzalez and R. E. Woods, Digital Image Processing, 2nd ed. (Prentice-Hall, 2002), pp. 91-97.

He, X.

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]

Kaufmann, G. H.

Kerr, D.

Kim, C.-J.

S.-H. Baik, S.-K. Park, C.-J. Kim, and S.-Y. Kim, "Two-channel spatial phase shifting electronic speckle pattern interferometer," Opt. Commun. 192, 205-211 (2001).
[CrossRef]

Kim, S.-Y.

S.-H. Baik, S.-K. Park, C.-J. Kim, and S.-Y. Kim, "Two-channel spatial phase shifting electronic speckle pattern interferometer," Opt. Commun. 192, 205-211 (2001).
[CrossRef]

Lions, P. L.

F. Catté, P. L. Lions, J. M. Morel, and T. Coll, "Image selective smoothing and edge detection by nonlinear diffusion," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 29, 182-193 (1992).

López, C. P.

N. A. Ochoa, F. M. Santoyo, A. J. Moore, and C. P. López, "Contrast enhancement of electronic speckle pattern interferometry addition fringes," Appl. Opt. 36, 2783-2787 (1997).
[CrossRef] [PubMed]

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]

Malacara, D.

Marroquin, J. L.

M. Servin, J. L. Marroquin, and F. J. Cuevas, "Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms," J. Opt. Soc. Am. A. 18, 689-695 (2001).
[CrossRef]

Moore, A. J.

N. A. Ochoa, F. M. Santoyo, A. J. Moore, and C. P. López, "Contrast enhancement of electronic speckle pattern interferometry addition fringes," Appl. Opt. 36, 2783-2787 (1997).
[CrossRef] [PubMed]

Morel, J. M.

F. Catté, P. L. Lions, J. M. Morel, and T. Coll, "Image selective smoothing and edge detection by nonlinear diffusion," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 29, 182-193 (1992).

Ochoa, N. A.

N. A. Ochoa, F. M. Santoyo, A. J. Moore, and C. P. López, "Contrast enhancement of electronic speckle pattern interferometry addition fringes," Appl. Opt. 36, 2783-2787 (1997).
[CrossRef] [PubMed]

Park, S.-K.

S.-H. Baik, S.-K. Park, C.-J. Kim, and S.-Y. Kim, "Two-channel spatial phase shifting electronic speckle pattern interferometer," Opt. Commun. 192, 205-211 (2001).
[CrossRef]

Quan, C.

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]

C. J. Tay, C. Quan, L. Chen, and Y. Fu, "Phase extraction from electronic speckle patterns by statistical analysis," Opt. Commun. 236, 259-269 (2004).
[CrossRef]

Rodriguez-Vera, R.

F. J. Cuevas, M. Servin, O. N. Stavroudis, and R. Rodriguez-Vera, "Multi-layer neural network applied to phase and depth recovery from fringe patterns," Opt. Commun. 181, 239-259 (2000).
[CrossRef]

M. Servin, D. Malacara, and R. Rodriguez-Vera, "Phase-locked-loop interferometry applied to aspheric testing with a computer-stored compensator," Appl. Opt. 33, 2589-2595 (1994).
[CrossRef] [PubMed]

Santoyo, F. M.

N. A. Ochoa, F. M. Santoyo, A. J. Moore, and C. P. López, "Contrast enhancement of electronic speckle pattern interferometry addition fringes," Appl. Opt. 36, 2783-2787 (1997).
[CrossRef] [PubMed]

Sapiro, G.

G. Sapiro and V. Caselles, "Contrast enhancement via image evolution flows," Graph. Models Image Process. 59, 407-416 (1997).
[CrossRef]

G. Sapiro, "Geometric partial differential equations in image processing: past, present, and future," in Proceedings of the Second IEEE International Conference on Image Processing 3 (Institute of Electrical and Electronics Engineers, 1995), pp. 1-4.

Servin, M.

F. J. Cuevas, J. H. Sossa-Azuela, and M. Servin, "A parametric method applied to phase recovery from a fringe pattern based on a genetic algorithm," Opt. Commun. 203, 213-223 (2002).
[CrossRef]

M. Servin, J. L. Marroquin, and F. J. Cuevas, "Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms," J. Opt. Soc. Am. A. 18, 689-695 (2001).
[CrossRef]

F. J. Cuevas, M. Servin, O. N. Stavroudis, and R. Rodriguez-Vera, "Multi-layer neural network applied to phase and depth recovery from fringe patterns," Opt. Commun. 181, 239-259 (2000).
[CrossRef]

M. Servin, D. Malacara, and R. Rodriguez-Vera, "Phase-locked-loop interferometry applied to aspheric testing with a computer-stored compensator," Appl. Opt. 33, 2589-2595 (1994).
[CrossRef] [PubMed]

Singh, T.

T. Singh and C. S. Vikram, "Histogram equalization of Young's fringes in speckle photography," in Interferometry XI: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 4777, 27-30 (2002).
[CrossRef]

T. Singh and C. S. Vikram, "Histogram equalization of vibration fringes in holography," Optik (Stuttgart) 113, 499-503 (2002).
[CrossRef]

Sossa-Azuela, J. H.

F. J. Cuevas, J. H. Sossa-Azuela, and M. Servin, "A parametric method applied to phase recovery from a fringe pattern based on a genetic algorithm," Opt. Commun. 203, 213-223 (2002).
[CrossRef]

Stavroudis, O. N.

F. J. Cuevas, M. Servin, O. N. Stavroudis, and R. Rodriguez-Vera, "Multi-layer neural network applied to phase and depth recovery from fringe patterns," Opt. Commun. 181, 239-259 (2000).
[CrossRef]

Su, X.

X. Su and W. Chen, "Fourier transform profilometry: a review," Opt. Laser Eng. 35, 263-284 (2001).
[CrossRef]

Tay, C. J.

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]

C. J. Tay, C. Quan, L. Chen, and Y. Fu, "Phase extraction from electronic speckle patterns by statistical analysis," Opt. Commun. 236, 259-269 (2004).
[CrossRef]

Vikram, C. S.

T. Singh and C. S. Vikram, "Histogram equalization of vibration fringes in holography," Optik (Stuttgart) 113, 499-503 (2002).
[CrossRef]

T. Singh and C. S. Vikram, "Histogram equalization of Young's fringes in speckle photography," in Interferometry XI: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 4777, 27-30 (2002).
[CrossRef]

Womack, K. H.

K. H. Womack, "Interferometric phase measurement using spatial synchronous detection," Opt. Eng. 23, 391-395 (1984).

Woods, R. E.

R. C. Gonzalez and R. E. Woods, Digital Image Processing, 2nd ed. (Prentice-Hall, 2002), pp. 91-97.

Yang, F.

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]

Graph. Models Image Process. (1)

G. Sapiro and V. Caselles, "Contrast enhancement via image evolution flows," Graph. Models Image Process. 59, 407-416 (1997).
[CrossRef]

Appl. Opt. (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] [PubMed]

Appl. Opt. (3)

Comput. Vision Image Understand. (1)

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]

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

M. Servin, J. L. Marroquin, and F. J. Cuevas, "Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms," J. Opt. Soc. Am. A. 18, 689-695 (2001).
[CrossRef]

Opt. Commun. (2)

S.-H. Baik, S.-K. Park, C.-J. Kim, and S.-Y. Kim, "Two-channel spatial phase shifting electronic speckle pattern interferometer," Opt. Commun. 192, 205-211 (2001).
[CrossRef]

C. J. Tay, C. Quan, L. Chen, and Y. Fu, "Phase extraction from electronic speckle patterns by statistical analysis," Opt. Commun. 236, 259-269 (2004).
[CrossRef]

Opt. Laser Eng. (1)

X. Su and W. Chen, "Fourier transform profilometry: a review," Opt. Laser Eng. 35, 263-284 (2001).
[CrossRef]

Opt. Commun. (2)

F. J. Cuevas, M. Servin, O. N. Stavroudis, and R. Rodriguez-Vera, "Multi-layer neural network applied to phase and depth recovery from fringe patterns," Opt. Commun. 181, 239-259 (2000).
[CrossRef]

F. J. Cuevas, J. H. Sossa-Azuela, and M. Servin, "A parametric method applied to phase recovery from a fringe pattern based on a genetic algorithm," Opt. Commun. 203, 213-223 (2002).
[CrossRef]

Opt. Eng. (1)

K. H. Womack, "Interferometric phase measurement using spatial synchronous detection," Opt. Eng. 23, 391-395 (1984).

Optik (Stuttgart) (1)

T. Singh and C. S. Vikram, "Histogram equalization of vibration fringes in holography," Optik (Stuttgart) 113, 499-503 (2002).
[CrossRef]

Proc. SPIE (1)

T. Singh and C. S. Vikram, "Histogram equalization of Young's fringes in speckle photography," in Interferometry XI: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 4777, 27-30 (2002).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. (1)

F. Catté, P. L. Lions, J. M. Morel, and T. Coll, "Image selective smoothing and edge detection by nonlinear diffusion," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 29, 182-193 (1992).

Other (2)

G. Sapiro, "Geometric partial differential equations in image processing: past, present, and future," in Proceedings of the Second IEEE International Conference on Image Processing 3 (Institute of Electrical and Electronics Engineers, 1995), pp. 1-4.

R. C. Gonzalez and R. E. Woods, Digital Image Processing, 2nd ed. (Prentice-Hall, 2002), pp. 91-97.

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Figures (3)

Fig. 1
Fig. 1

Computer-simulated addition fringe pattern and its enhanced images: (a) initial image, (b) Sobel operator, (c) normalized variance, (d) traditional histogram specification for 20 iterations, (e) DE enhancement with Δ t = 0.001 and n = 20 (i.e., 20 iterations) in Eq. (12).

Fig. 2
Fig. 2

Experimental subtraction fringe pattern and its enhanced images: (a) subtraction fringe pattern, (b) Sobel operator, (c) normalized variance, (d) traditional histogram equalization, (e) DE enhancement with Δ t = 0.001 and n = 1 in Eq. (6), (f) DE enhancement with Δ t = 0.0015 and n = 1 in Eq. (6).

Fig. 3
Fig. 3

Experimental subtraction fringe pattern and its enhanced images: (a) subtraction fringe pattern, (b) Sobel operator, (c) normalized variance, (d) traditional histogram equalization, (e) DE enhancement with Δ t = 0.001 and n = 1 in Eq. (6), (f) DE enhancement with Δ t = 0.0005 and n = 2 in Eq. (6).

Equations (26)

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

p ( r ) = K r K ,         0 r L ,
s ( r ) = 0 r p ( ω ) d ω ,         0 r L .
s ( r ) = 0 r p ( ω ) d ω ,         0 r L ,
v ( z ) = 0 z h ( t ) d t ,         0 z L ,
s ( r ) = v ( z ) .
η 0 ( x , y ) = Α [ ( v , w ) : I ( v , w , t ) I ( x , y , t ) ] ,
1 x , v M ,         1 y , w N ,
I ( x , y , t ) t = [ M N M N I ( x , y , t ) / L ] η 0 ( x , y ) ,
I ( x , y , 0 ) = I 0 ( x , y ) ,
Α [ ( v , w ) : I ( v , w ) I ( x , y ) ] = [ M N M N I ( x , y ) / L ] ,
Α [ ( v , w ) : b I ( v , w ) a ] = ( M N / L ) ( b a ) ,
I ( x , y , t ) t = { M N H [ I ( x , y , t ) ] } η 0 ( x , y ) ,
I ( x , y , 0 ) = I 0 ( x , y ),
Α [ ( v , w ) : I ( v , w ) I ( x , y ) ] = { M N H [ I ( x , y ) ] } .
Α [ ( v , w ) : I ( x , y ) I ( v , w ) I ( x , y ) + δ ] = H [ I ( x , y ) + δ ] H [ I ( x , y ) ] ,
I ( x , y , t ) = I ( x , y , t ) I ( t ) , I ( x , y , 0 ) = I 0 ( x , y ) ,
η 0 ( x , y ) = Α [ ( v , w ) : I ( v , w , t ) I ( x , y , t ) ] ,
H [ I ( x , y , t ) ] = 0 I ( x , y , t ) h ( ξ ) d ξ ,
I ( x , y , t ) t = { M N H [ I ( x , y , t ) ] } η 0 ( x , y ) ,
( I t ) i , j n = I i , j n + 1 I i , j n Δ t ,
( I t ) i , j n = ( I ) i , j n + 1 ( I ) i , j n Δ t .
I i , j n + 1 = I i , j n + Δ t { [ M N M N I i , j n / L ] ( η 0 ) i , j } ,
( I ) i , j n + 1 = ( I ) i , j n + Δ t { [ M N H [ ( I ) i , j n ] ( η 0 ) i , j } ,
I a d d = 2 ( I o + I r ) + 4 I o I r cos ( ϕ r ϕ o + φ 2 ) cos ( φ 2 ) ,
φ ( x , y ) = 4 π [ ( 3 x 2 ) 2 + ( 3 y 2 ) 2 ] ,     | x | < 1 ,     | y | < 1.
h ( s ) = { cos ( s 50 π 2 ) ,               0 s < 50 0 ,                       50 s 205 cos ( s 205 50 * π 2 + 3 π 2 ) ,       205 < s 255 .

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