Abstract

Electronic speckle pattern interferometry is useful for the qualitative depiction of the deformation profile of harmonically vibrating objects. However, extending the process to achieve quantitative results requires unwrapping the phase in the interferogram, which contains significant noise due to the speckle. Two methods to achieve accurate phase information from time-averaged speckle pattern interferograms are presented. The first is based on a direct inverse of the regions within corresponding phase intervals, and the second is based on optimization of four independent parameters. The optimization method requires less time than more commonly used algorithms and shows higher precision of the resulting surface displacement.

© 2016 Optical Society of America

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References

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    [Crossref]
  3. C. Tang, J. Zhang, C. Sun, Y. Su, and K. L. Su, “Electronic speckle pattern interferometry for fracture expansion in nuclear graphite based on PDE image processing methods,” Proc. SPIE 9525, 952521 (2015).
    [Crossref]
  4. C. Song, M. V. Matham, and K. H. K. Chan, “Speckle referencing: digital speckle pattern interferometry (SR-DSPI) for imaging of non-diffusive surfaces,” Proc. SPIE 9524, 952421 (2015).
    [Crossref]
  5. J. N. Petzing, “Vibration analysis using fast speckle metrology,” Proc. SPIE 3745, 134–140 (1999).
  6. W. Wang and C. Hwang, The Development and Applications of Amplitude Fluctuation Electronic Speckle Pattern Interferometry Method, Recent Advances in Mechanics (Springer, 2011).
  7. X. Dai, X. Shao, Z. Geng, F. Yang, Y. Jiang, and X. He, “Vibration measurement based on electronic speckle pattern interferometry and radial basis function,” Opt. Commun. 355, 33–43 (2015).
    [Crossref]
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    [Crossref]
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    [Crossref]
  11. T. R. Moore and J. J. Skubal, “Time-averaged electronic speckle pattern interferometry in the presence of ambient motion. Part I. Theory and experiments,” Appl. Opt. 47, 4640–4648 (2008).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  23. C. Li, C. Tang, H. Yan, L. Wang, and H. Zhang, “Localized Fourier transform filter for noise removal in electronic speckle pattern interferometry wrapped phase patterns,” Appl. Opt. 50, 4903–4911 (2011).
    [Crossref]
  24. H. H. Rosenbrock, “An automatic method for finding the greatest or least value of a function,” Comput. J. 3, 175–184 (1960).
    [Crossref]

2015 (3)

C. Tang, J. Zhang, C. Sun, Y. Su, and K. L. Su, “Electronic speckle pattern interferometry for fracture expansion in nuclear graphite based on PDE image processing methods,” Proc. SPIE 9525, 952521 (2015).
[Crossref]

C. Song, M. V. Matham, and K. H. K. Chan, “Speckle referencing: digital speckle pattern interferometry (SR-DSPI) for imaging of non-diffusive surfaces,” Proc. SPIE 9524, 952421 (2015).
[Crossref]

X. Dai, X. Shao, Z. Geng, F. Yang, Y. Jiang, and X. He, “Vibration measurement based on electronic speckle pattern interferometry and radial basis function,” Opt. Commun. 355, 33–43 (2015).
[Crossref]

2012 (2)

L. Krzemien and M. Lukomski, “Algorithm for automated analysis of surface vibrations using time-averaged DSPI,” Appl. Opt. 51, 5154–5160 (2012).
[Crossref]

C. Tang, L. Wang, H. Yan, and C. Li, “Comparison on performance of some representative and recent filtering methods in electronic speckle pattern interferometry,” Opt. Lasers Eng. 50, 1036–1051 (2012).
[Crossref]

2011 (2)

2009 (1)

2008 (2)

M. J. Huang and J. K. Liou, “Retrieving ESPI map of discontinuous objects via a novel phase unwrapping algorithm,” Strain 44, 239–247 (2008).
[Crossref]

T. R. Moore and J. J. Skubal, “Time-averaged electronic speckle pattern interferometry in the presence of ambient motion. Part I. Theory and experiments,” Appl. Opt. 47, 4640–4648 (2008).
[Crossref]

2007 (2)

L. Ernesto, M. Espinosa, H. M. Carpio Valadez, and F. J. Cuevas, “Demodulation of interferograms of closed fringes by Zernike polynomials using a technique of soft computing,” Eng. Lett. 15, 99–104 (2007).

J. M. Bioucas-Dias and G. Valadão, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16, 698–709 (2007).
[Crossref]

2006 (2)

T. R. Moore and S. A. Zietlow, “Interferometric studies of a piano soundboard,” Acoust. Soc. Am. 119, 1783–1793 (2006).
[Crossref]

D. N. Borza, “Full-field vibration amplitude recovery from high-resolution time-averaged speckle interferograms and digital holograms by regional inverting of the Bessel function,” Opt. Lasers Eng. 44, 747–770 (2006).
[Crossref]

1999 (1)

J. N. Petzing, “Vibration analysis using fast speckle metrology,” Proc. SPIE 3745, 134–140 (1999).

1996 (1)

1993 (1)

B. Pouet, T. Chatters, and S. Krishnaswamy, “Synchronized reference updating technique for electronic speckle interferometry,” J. Nondestr. Eval. 12, 133–138 (1993).
[Crossref]

1990 (1)

1985 (1)

1960 (1)

H. H. Rosenbrock, “An automatic method for finding the greatest or least value of a function,” Comput. J. 3, 175–184 (1960).
[Crossref]

Bioucas-Dias, J. M.

J. M. Bioucas-Dias and G. Valadão, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16, 698–709 (2007).
[Crossref]

Borza, D. N.

D. N. Borza, “Full-field vibration amplitude recovery from high-resolution time-averaged speckle interferograms and digital holograms by regional inverting of the Bessel function,” Opt. Lasers Eng. 44, 747–770 (2006).
[Crossref]

Carpio Valadez, H. M.

L. Ernesto, M. Espinosa, H. M. Carpio Valadez, and F. J. Cuevas, “Demodulation of interferograms of closed fringes by Zernike polynomials using a technique of soft computing,” Eng. Lett. 15, 99–104 (2007).

Casillas-Rodríguez, F. J.

Chan, K. H. K.

C. Song, M. V. Matham, and K. H. K. Chan, “Speckle referencing: digital speckle pattern interferometry (SR-DSPI) for imaging of non-diffusive surfaces,” Proc. SPIE 9524, 952421 (2015).
[Crossref]

Chatters, T.

B. Pouet, T. Chatters, and S. Krishnaswamy, “Synchronized reference updating technique for electronic speckle interferometry,” J. Nondestr. Eval. 12, 133–138 (1993).
[Crossref]

Chatziioannou, V.

T. Statsenko, V. Chatziioannou, and W. Kausel, “Interferometric studies of the Brazilian Cuíca,” in Proceedings of the Third Vienna Talk on Music Acoustics (2015), pp. 153–157.

Creath, K.

Cuevas, F. J.

L. Ernesto, M. Espinosa, H. M. Carpio Valadez, and F. J. Cuevas, “Demodulation of interferograms of closed fringes by Zernike polynomials using a technique of soft computing,” Eng. Lett. 15, 99–104 (2007).

Dai, X.

X. Dai, X. Shao, Z. Geng, F. Yang, Y. Jiang, and X. He, “Vibration measurement based on electronic speckle pattern interferometry and radial basis function,” Opt. Commun. 355, 33–43 (2015).
[Crossref]

de la Torre-Ibarra, M.

Deán, J. L.

Doval, A. F.

Duran-Ramírez, V. M.

Ernesto, L.

L. Ernesto, M. Espinosa, H. M. Carpio Valadez, and F. J. Cuevas, “Demodulation of interferograms of closed fringes by Zernike polynomials using a technique of soft computing,” Eng. Lett. 15, 99–104 (2007).

Espinosa, M.

L. Ernesto, M. Espinosa, H. M. Carpio Valadez, and F. J. Cuevas, “Demodulation of interferograms of closed fringes by Zernike polynomials using a technique of soft computing,” Eng. Lett. 15, 99–104 (2007).

Geng, Z.

X. Dai, X. Shao, Z. Geng, F. Yang, Y. Jiang, and X. He, “Vibration measurement based on electronic speckle pattern interferometry and radial basis function,” Opt. Commun. 355, 33–43 (2015).
[Crossref]

Gómez-Rosas, G.

Goodman, J. W.

J. W. Goodman, Laser Speckle and Related Phenomena (Springer, 1975).

He, X.

X. Dai, X. Shao, Z. Geng, F. Yang, Y. Jiang, and X. He, “Vibration measurement based on electronic speckle pattern interferometry and radial basis function,” Opt. Commun. 355, 33–43 (2015).
[Crossref]

Huang, M. J.

M. J. Huang and J. K. Liou, “Retrieving ESPI map of discontinuous objects via a novel phase unwrapping algorithm,” Strain 44, 239–247 (2008).
[Crossref]

Hwang, C.

W. Wang and C. Hwang, The Development and Applications of Amplitude Fluctuation Electronic Speckle Pattern Interferometry Method, Recent Advances in Mechanics (Springer, 2011).

Hwang, C. H.

Jiang, Y.

X. Dai, X. Shao, Z. Geng, F. Yang, Y. Jiang, and X. He, “Vibration measurement based on electronic speckle pattern interferometry and radial basis function,” Opt. Commun. 355, 33–43 (2015).
[Crossref]

Jones, R.

R. Jones and C. Wykes, Holographic and Speckle Interferometry, 2nd ed. (Cambridge University, 2001).

Kausel, W.

T. Statsenko, V. Chatziioannou, and W. Kausel, “Interferometric studies of the Brazilian Cuíca,” in Proceedings of the Third Vienna Talk on Music Acoustics (2015), pp. 153–157.

Krishnaswamy, S.

B. Pouet, T. Chatters, and S. Krishnaswamy, “Synchronized reference updating technique for electronic speckle interferometry,” J. Nondestr. Eval. 12, 133–138 (1993).
[Crossref]

Krzemien, L.

Li, C.

C. Tang, L. Wang, H. Yan, and C. Li, “Comparison on performance of some representative and recent filtering methods in electronic speckle pattern interferometry,” Opt. Lasers Eng. 50, 1036–1051 (2012).
[Crossref]

C. Li, C. Tang, H. Yan, L. Wang, and H. Zhang, “Localized Fourier transform filter for noise removal in electronic speckle pattern interferometry wrapped phase patterns,” Appl. Opt. 50, 4903–4911 (2011).
[Crossref]

Lin, S. Y.

Liou, J. K.

M. J. Huang and J. K. Liou, “Retrieving ESPI map of discontinuous objects via a novel phase unwrapping algorithm,” Strain 44, 239–247 (2008).
[Crossref]

Lukomski, M.

Matham, M. V.

C. Song, M. V. Matham, and K. H. K. Chan, “Speckle referencing: digital speckle pattern interferometry (SR-DSPI) for imaging of non-diffusive surfaces,” Proc. SPIE 9524, 952421 (2015).
[Crossref]

Mendoza-Santoyo, F.

Moore, T. R.

Mora-González, M.

Muñoz Maciel, J.

Peña Lecona, F. G.

Pérez-López, C.

Petzing, J. N.

J. N. Petzing, “Vibration analysis using fast speckle metrology,” Proc. SPIE 3745, 134–140 (1999).

Pouet, B.

B. Pouet, T. Chatters, and S. Krishnaswamy, “Synchronized reference updating technique for electronic speckle interferometry,” J. Nondestr. Eval. 12, 133–138 (1993).
[Crossref]

Rosenbrock, H. H.

H. H. Rosenbrock, “An automatic method for finding the greatest or least value of a function,” Comput. J. 3, 175–184 (1960).
[Crossref]

Shao, X.

X. Dai, X. Shao, Z. Geng, F. Yang, Y. Jiang, and X. He, “Vibration measurement based on electronic speckle pattern interferometry and radial basis function,” Opt. Commun. 355, 33–43 (2015).
[Crossref]

Skubal, J. J.

Song, C.

C. Song, M. V. Matham, and K. H. K. Chan, “Speckle referencing: digital speckle pattern interferometry (SR-DSPI) for imaging of non-diffusive surfaces,” Proc. SPIE 9524, 952421 (2015).
[Crossref]

Statsenko, T.

T. Statsenko, V. Chatziioannou, and W. Kausel, “Interferometric studies of the Brazilian Cuíca,” in Proceedings of the Third Vienna Talk on Music Acoustics (2015), pp. 153–157.

Su, K. L.

C. Tang, J. Zhang, C. Sun, Y. Su, and K. L. Su, “Electronic speckle pattern interferometry for fracture expansion in nuclear graphite based on PDE image processing methods,” Proc. SPIE 9525, 952521 (2015).
[Crossref]

Su, Y.

C. Tang, J. Zhang, C. Sun, Y. Su, and K. L. Su, “Electronic speckle pattern interferometry for fracture expansion in nuclear graphite based on PDE image processing methods,” Proc. SPIE 9525, 952521 (2015).
[Crossref]

Sun, C.

C. Tang, J. Zhang, C. Sun, Y. Su, and K. L. Su, “Electronic speckle pattern interferometry for fracture expansion in nuclear graphite based on PDE image processing methods,” Proc. SPIE 9525, 952521 (2015).
[Crossref]

Tang, C.

C. Tang, J. Zhang, C. Sun, Y. Su, and K. L. Su, “Electronic speckle pattern interferometry for fracture expansion in nuclear graphite based on PDE image processing methods,” Proc. SPIE 9525, 952521 (2015).
[Crossref]

C. Tang, L. Wang, H. Yan, and C. Li, “Comparison on performance of some representative and recent filtering methods in electronic speckle pattern interferometry,” Opt. Lasers Eng. 50, 1036–1051 (2012).
[Crossref]

C. Li, C. Tang, H. Yan, L. Wang, and H. Zhang, “Localized Fourier transform filter for noise removal in electronic speckle pattern interferometry wrapped phase patterns,” Appl. Opt. 50, 4903–4911 (2011).
[Crossref]

Trillo, C.

Valadão, G.

J. M. Bioucas-Dias and G. Valadão, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16, 698–709 (2007).
[Crossref]

Vikhagen, E.

Wang, L.

C. Tang, L. Wang, H. Yan, and C. Li, “Comparison on performance of some representative and recent filtering methods in electronic speckle pattern interferometry,” Opt. Lasers Eng. 50, 1036–1051 (2012).
[Crossref]

C. Li, C. Tang, H. Yan, L. Wang, and H. Zhang, “Localized Fourier transform filter for noise removal in electronic speckle pattern interferometry wrapped phase patterns,” Appl. Opt. 50, 4903–4911 (2011).
[Crossref]

Wang, W.

W. Wang and C. Hwang, The Development and Applications of Amplitude Fluctuation Electronic Speckle Pattern Interferometry Method, Recent Advances in Mechanics (Springer, 2011).

Wang, W. C.

Wykes, C.

R. Jones and C. Wykes, Holographic and Speckle Interferometry, 2nd ed. (Cambridge University, 2001).

Yan, H.

C. Tang, L. Wang, H. Yan, and C. Li, “Comparison on performance of some representative and recent filtering methods in electronic speckle pattern interferometry,” Opt. Lasers Eng. 50, 1036–1051 (2012).
[Crossref]

C. Li, C. Tang, H. Yan, L. Wang, and H. Zhang, “Localized Fourier transform filter for noise removal in electronic speckle pattern interferometry wrapped phase patterns,” Appl. Opt. 50, 4903–4911 (2011).
[Crossref]

Yang, F.

X. Dai, X. Shao, Z. Geng, F. Yang, Y. Jiang, and X. He, “Vibration measurement based on electronic speckle pattern interferometry and radial basis function,” Opt. Commun. 355, 33–43 (2015).
[Crossref]

Zhang, H.

Zhang, J.

C. Tang, J. Zhang, C. Sun, Y. Su, and K. L. Su, “Electronic speckle pattern interferometry for fracture expansion in nuclear graphite based on PDE image processing methods,” Proc. SPIE 9525, 952521 (2015).
[Crossref]

Zietlow, S. A.

T. R. Moore and S. A. Zietlow, “Interferometric studies of a piano soundboard,” Acoust. Soc. Am. 119, 1783–1793 (2006).
[Crossref]

Acoust. Soc. Am. (1)

T. R. Moore and S. A. Zietlow, “Interferometric studies of a piano soundboard,” Acoust. Soc. Am. 119, 1783–1793 (2006).
[Crossref]

Appl. Opt. (7)

Comput. J. (1)

H. H. Rosenbrock, “An automatic method for finding the greatest or least value of a function,” Comput. J. 3, 175–184 (1960).
[Crossref]

Eng. Lett. (1)

L. Ernesto, M. Espinosa, H. M. Carpio Valadez, and F. J. Cuevas, “Demodulation of interferograms of closed fringes by Zernike polynomials using a technique of soft computing,” Eng. Lett. 15, 99–104 (2007).

IEEE Trans. Image Process. (1)

J. M. Bioucas-Dias and G. Valadão, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16, 698–709 (2007).
[Crossref]

J. Nondestr. Eval. (1)

B. Pouet, T. Chatters, and S. Krishnaswamy, “Synchronized reference updating technique for electronic speckle interferometry,” J. Nondestr. Eval. 12, 133–138 (1993).
[Crossref]

Opt. Commun. (1)

X. Dai, X. Shao, Z. Geng, F. Yang, Y. Jiang, and X. He, “Vibration measurement based on electronic speckle pattern interferometry and radial basis function,” Opt. Commun. 355, 33–43 (2015).
[Crossref]

Opt. Express (1)

Opt. Lasers Eng. (2)

D. N. Borza, “Full-field vibration amplitude recovery from high-resolution time-averaged speckle interferograms and digital holograms by regional inverting of the Bessel function,” Opt. Lasers Eng. 44, 747–770 (2006).
[Crossref]

C. Tang, L. Wang, H. Yan, and C. Li, “Comparison on performance of some representative and recent filtering methods in electronic speckle pattern interferometry,” Opt. Lasers Eng. 50, 1036–1051 (2012).
[Crossref]

Proc. SPIE (3)

C. Tang, J. Zhang, C. Sun, Y. Su, and K. L. Su, “Electronic speckle pattern interferometry for fracture expansion in nuclear graphite based on PDE image processing methods,” Proc. SPIE 9525, 952521 (2015).
[Crossref]

C. Song, M. V. Matham, and K. H. K. Chan, “Speckle referencing: digital speckle pattern interferometry (SR-DSPI) for imaging of non-diffusive surfaces,” Proc. SPIE 9524, 952421 (2015).
[Crossref]

J. N. Petzing, “Vibration analysis using fast speckle metrology,” Proc. SPIE 3745, 134–140 (1999).

Strain (1)

M. J. Huang and J. K. Liou, “Retrieving ESPI map of discontinuous objects via a novel phase unwrapping algorithm,” Strain 44, 239–247 (2008).
[Crossref]

Other (4)

J. W. Goodman, Laser Speckle and Related Phenomena (Springer, 1975).

W. Wang and C. Hwang, The Development and Applications of Amplitude Fluctuation Electronic Speckle Pattern Interferometry Method, Recent Advances in Mechanics (Springer, 2011).

R. Jones and C. Wykes, Holographic and Speckle Interferometry, 2nd ed. (Cambridge University, 2001).

T. Statsenko, V. Chatziioannou, and W. Kausel, “Interferometric studies of the Brazilian Cuíca,” in Proceedings of the Third Vienna Talk on Music Acoustics (2015), pp. 153–157.

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Figures (11)

Fig. 1.
Fig. 1. Arrangement for the ESPI measurements with a speckled reference beam. Light from the laser is divided into two beams by a polarizing beam splitter ( P B S ). Half-wave plates ( λ / 2 ) are added to control the intensity of the both beams. The beams are expanded by lenses ( L ). A PZT-driven mirror is introduced in the reference arm in order to produce a changeable optical path difference. The reference beam passes through the ground glass ( G ), and then two beams are combined by a beam splitter ( B S ) and directed to the camera.
Fig. 2.
Fig. 2. Plot of the maximum value of the subtraction intensity without the multiplicative constant versus environmental phase amplitude γ t cam . The gray vertical line shows that the intensity ranges from zero to its maximum value for different φ n with a single value of γ .
Fig. 3.
Fig. 3. Brazilian Cuíca, produced by Meinl Percussion. The photo is taken from the manufacturer’s website.
Fig. 4.
Fig. 4. Image processing for a part of the membrane of the Brazilian drum, vibrating at 1971 Hz. The normalization of the differential frame on the reference frame is followed by the Fourier filtering. (a) Temporally averaged reference frame, (b) temporally averaged differential frame, (c) normalized intensity, and (d) Fourier-filtered normalized intensity.
Fig. 5.
Fig. 5. Assignment of the values for bright and dark fringes. Odd numbers for the bright fringes are based on the local maximum of the region, even numbers follow from the neighboring fringes.
Fig. 6.
Fig. 6. Image with assigned phase interval orders.
Fig. 7.
Fig. 7. Unwrapping via the peak direct inverse method. (a) Cloud of the recovered phase points. (b) Interpolated phase map.
Fig. 8.
Fig. 8. Resulting 3D vibration amplitude and corresponding intensity pattern for the peak direct inverse method. (a) 3D displacement. (b) Reconstructed interferogram.
Fig. 9.
Fig. 9. Unwrapping via optimization algorithm. (a) Grid of the recovered phase points. (b) Interpolated phase map.
Fig. 10.
Fig. 10. Resulting 3D vibration amplitude and corresponding intensity pattern for the optimization method. (a) 3D displacement. (b) Reconstructed interferogram.
Fig. 11.
Fig. 11. Profiles of the experimentally obtained interferometric image and reconstructed interferometric patterns, taken as the absolute value of the Bessel function of zeroth order of the phase maps, obtained using two unwrapping methods.

Equations (21)

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

I rest = I o + I r + 2 I o I r cos ( ϕ 0 ) ,
ϕ = ϕ 0 + ξ sin ( ω 0 t ) ,
ξ = 4 π A λ .
I = I o + I r + β cos ( ϕ 0 + ξ sin ( ω 0 t ) ) ,
I = I o + I r + β cos ( ϕ 0 ) J 0 ( ξ ) .
ϕ = ϕ 0 + γ t + ξ sin ( ω 0 t ) ,
I n = I o + I r + β cos ( ϕ n + γ t + ξ sin ω 0 t ) ,
I sub ( ξ ) = | I n + 1 I n | = β | cos ( ϕ n + γ t + ξ sin ω 0 t ) cos ( ϕ n + γ t + ξ sin ω 0 t ) | ,
I sub ( ξ ) = β | 1 t ex 0 t ex cos [ ϕ n + γ ( t cam + t ) + ξ sin ω 0 ( t cam + t ) ] d t 1 t ex 0 t ex cos [ ϕ n + γ t + ξ sin ω 0 t ] d t | ,
I sub ( ξ ) = β | M J 0 ( ξ ) + N H 0 ( ξ ) | ,
M = I sub ( 0 ) / β = | 1 t ex 0 t ex cos ( γ t ) d t 1 t ex 0 t ex cos ( γ ( t cam + t ) ) d t | = 4 γ t ex | sin ( γ 2 t cam ) sin ( γ 2 t ex ) sin ( γ 2 ( t ex + t cam ) ) | .
M = 4 γ t ex | sin ( γ 2 t cam ) sin ( γ 2 t ex ) sin ( γ 2 ( t ex + t cam ) + ϕ n ) | .
I sub ( ξ ) = | β M J 0 ( ξ ) | ,
P inv = I P ,
F { P inv } | F ( P inv ) | a = F { P filt } ,
F obj = C 1 E 1 + C 2 E 2 + C 3 E 3 + C 4 E 4 ,
E 1 = [ i = 1 m j = 1 n ( I c i , j I s i , j ) 2 ] 2 ,
I c i , j = | J 0 ( Φ i , j ) | .
E 2 = i = 1 m j = 1 n ( Θ ( Φ p i , j r i , j ) Θ ( Φ p i , j l i , j ) ) ,
E 3 = [ { i = 3 m 2 j = 3 n 2 ( Φ xx i , j Φ xx i + 1 , j ) 2 + ( Φ xx i , j Φ xx i 1 , j ) 2 + ( Φ xx i , j Φ xx i , j + 1 ) 2 + ( Φ xx i , j Φ xx i , j 1 ) 2 } / ( Δ x ) 2 ] 2 ,
E 4 = [ { i = 3 m 2 j = 3 n 2 ( Φ yy i , j Φ yy i + 1 , j ) 2 + ( Φ yy i , j Φ yy i 1 , j ) 2 + ( Φ yy i , j Φ yy i , j + 1 ) 2 + ( Φ yy i , j Φ yy i , j 1 ) 2 } / ( Δ y ) 2 ] 2 ,

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