Abstract

We present a method to determine micro and nano in-plane displacements based on the phase singularities generated by application of directional wavelet transforms to speckle pattern images. The spatial distribution of the obtained phase singularities by the wavelet transform configures a network, which is characterized by two quasi-orthogonal directions. The displacement value is determined by identifying the intersection points of the network before and after the displacement produced by the tested object. The performance of this method is evaluated using simulated speckle patterns and experimental data. The proposed approach is compared with the optical vortex metrology and digital image correlation methods in terms of performance and noise robustness, and the advantages and limitations associated to each method are also discussed.

© 2013 Optical Society of America

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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]

2009

B. Pan, K. Qian, H. Xie, and A. Asundi, “Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review,” Meas. Sci. Technol. 20, 062001 (2009).
[CrossRef]

Y. Qiao, W. Wang, N. Minematsu, J. Liu, M. Takeda, and X. Tang, “A theory of phase singularities for image representation and its applications to object tracking and image matching,” IEEE Trans. Image Process. 18, 2153–2166 (2009).
[CrossRef]

A. Federico and G. H. Kaufmann, “Robust phase recovery in temporal speckle pattern interferometry using a 3D directional wavelet transform,” Opt. Lett. 34, 2336–2338 (2009).
[CrossRef]

2008

2007

2006

2005

W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement,” Opt. Commun. 248, 59–68 (2005).
[CrossRef]

2000

M. V. Berry and M. R. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. A 456, 2059–2079 (2000).
[CrossRef]

1997

1993

1983

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

Antoine, J. P.

J. P. Antoine, R. Murenzi, P. Vandergheynst, and S. Twareque Ali, Two-Dimensional Wavelets and their Relatives (Cambridge University, 2004).

Asundi, A.

B. Pan, K. Qian, H. Xie, and A. Asundi, “Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review,” Meas. Sci. Technol. 20, 062001 (2009).
[CrossRef]

Benckert, L. R.

Berry, M. V.

M. V. Berry and M. R. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. A 456, 2059–2079 (2000).
[CrossRef]

Bo-qin, X.

B. Pan, X. Hui-min, X. Bo-qin, and D. Fu-long, “Performance of sub-pixel registration algorithms in digital image correlation,” Meas. Sci. Technol. 17, 1615–1621 (2006).
[CrossRef]

Chen, D. J.

Chiang, F. P.

Dennis, M. R.

Don, H. S.

Equis, S.

S. Equis and P. Jacquot, “Simulation of speckle complex amplitude: advocating the linear model,” Proc. SPIE 6341, 381–386 (2006).
[CrossRef]

Federico, A.

Fu-long, D.

B. Pan, X. Hui-min, X. Bo-qin, and D. Fu-long, “Performance of sub-pixel registration algorithms in digital image correlation,” Meas. Sci. Technol. 17, 1615–1621 (2006).
[CrossRef]

Hanson, S. G.

Hui-min, X.

B. Pan, X. Hui-min, X. Bo-qin, and D. Fu-long, “Performance of sub-pixel registration algorithms in digital image correlation,” Meas. Sci. Technol. 17, 1615–1621 (2006).
[CrossRef]

Ishii, N.

W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement,” Opt. Commun. 248, 59–68 (2005).
[CrossRef]

Ishijima, R.

Jacquot, P.

S. Equis and P. Jacquot, “Simulation of speckle complex amplitude: advocating the linear model,” Proc. SPIE 6341, 381–386 (2006).
[CrossRef]

Kaufmann, G. H.

Liu, J.

Y. Qiao, W. Wang, N. Minematsu, J. Liu, M. Takeda, and X. Tang, “A theory of phase singularities for image representation and its applications to object tracking and image matching,” IEEE Trans. Image Process. 18, 2153–2166 (2009).
[CrossRef]

Matsuda, A.

McNeill, S. R.

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

Minematsu, N.

Y. Qiao, W. Wang, N. Minematsu, J. Liu, M. Takeda, and X. Tang, “A theory of phase singularities for image representation and its applications to object tracking and image matching,” IEEE Trans. Image Process. 18, 2153–2166 (2009).
[CrossRef]

Miyamoto, Y.

W. Wang, T. Yokozeki, R. Ishijima, A. Wada, Y. Miyamoto, M. Takeda, and S. G. Hanson, “Optical vortex metrology for nanometric speckle displacement measurement,” Opt. Express 14, 10195–10206 (2006).
[CrossRef]

W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement,” Opt. Commun. 248, 59–68 (2005).
[CrossRef]

Murenzi, R.

J. P. Antoine, R. Murenzi, P. Vandergheynst, and S. Twareque Ali, Two-Dimensional Wavelets and their Relatives (Cambridge University, 2004).

Pan, B.

B. Pan, K. Qian, H. Xie, and A. Asundi, “Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review,” Meas. Sci. Technol. 20, 062001 (2009).
[CrossRef]

B. Pan, X. Hui-min, X. Bo-qin, and D. Fu-long, “Performance of sub-pixel registration algorithms in digital image correlation,” Meas. Sci. Technol. 17, 1615–1621 (2006).
[CrossRef]

Peters, W. H.

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

Qian, K.

B. Pan, K. Qian, H. Xie, and A. Asundi, “Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review,” Meas. Sci. Technol. 20, 062001 (2009).
[CrossRef]

Qiao, Y.

Y. Qiao, W. Wang, N. Minematsu, J. Liu, M. Takeda, and X. Tang, “A theory of phase singularities for image representation and its applications to object tracking and image matching,” IEEE Trans. Image Process. 18, 2153–2166 (2009).
[CrossRef]

Ranson, W. F.

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

Sjödahl, M.

Sutton, M. A.

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

Takeda, M.

Tan, Y. S.

Tang, X.

Y. Qiao, W. Wang, N. Minematsu, J. Liu, M. Takeda, and X. Tang, “A theory of phase singularities for image representation and its applications to object tracking and image matching,” IEEE Trans. Image Process. 18, 2153–2166 (2009).
[CrossRef]

Twareque Ali, S.

J. P. Antoine, R. Murenzi, P. Vandergheynst, and S. Twareque Ali, Two-Dimensional Wavelets and their Relatives (Cambridge University, 2004).

Vandergheynst, P.

J. P. Antoine, R. Murenzi, P. Vandergheynst, and S. Twareque Ali, Two-Dimensional Wavelets and their Relatives (Cambridge University, 2004).

Wada, A.

Wang, W.

Wolters, W. J.

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

Xie, H.

B. Pan, K. Qian, H. Xie, and A. Asundi, “Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review,” Meas. Sci. Technol. 20, 062001 (2009).
[CrossRef]

Yokozeki, T.

Appl. Opt.

IEEE Trans. Image Process.

Y. Qiao, W. Wang, N. Minematsu, J. Liu, M. Takeda, and X. Tang, “A theory of phase singularities for image representation and its applications to object tracking and image matching,” IEEE Trans. Image Process. 18, 2153–2166 (2009).
[CrossRef]

Image Vis. Comput.

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

Meas. Sci. Technol.

B. Pan, K. Qian, H. Xie, and A. Asundi, “Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review,” Meas. Sci. Technol. 20, 062001 (2009).
[CrossRef]

B. Pan, X. Hui-min, X. Bo-qin, and D. Fu-long, “Performance of sub-pixel registration algorithms in digital image correlation,” Meas. Sci. Technol. 17, 1615–1621 (2006).
[CrossRef]

Opt. Commun.

W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement,” Opt. Commun. 248, 59–68 (2005).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. R. Soc. A

M. V. Berry and M. R. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. A 456, 2059–2079 (2000).
[CrossRef]

Proc. SPIE

S. Equis and P. Jacquot, “Simulation of speckle complex amplitude: advocating the linear model,” Proc. SPIE 6341, 381–386 (2006).
[CrossRef]

Other

J. P. Antoine, R. Murenzi, P. Vandergheynst, and S. Twareque Ali, Two-Dimensional Wavelets and their Relatives (Cambridge University, 2004).

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Figures (7)

Fig. 1.
Fig. 1.

(a) Simulated intensity image of a speckle pattern with its corresponding optical vortices marked in white solid circles for real (light cyan) and imaginary (dark red) contours of level zero of the associated electric field; (b) the same SI as in (a) with phase singularities (white squares) detected by the OVM method with real (light cyan) and imaginary (dark red) contours of level zero corresponding to the Laguerre–Gauss filtering values; (c) spatial positions of the optical vortices (black solid circles) and OVM phase singularities (red squares) simultaneously shown.

Fig. 2.
Fig. 2.

Illustration of the mechanism involved in the movement of the positions of the DCWT phase singularities for a given in-plane translation. (a) Simulated SI; (b) phase of the DCWT of (a); (c) spatial positions of the wavelet phase singularities from (b); (d) SI shown in (a) displaced 10 pixels to the right; (e) phase of the DCWT of (d); (f) spatial positions of the wavelet phase singularities obtained from (b) (light cyan) and (e) (dark red).

Fig. 3.
Fig. 3.

(a) Amplitude of ψ ( r ) , (b) phase of ψ ( r ) , (c)  ψ ^ ( k ) , (d) schematic of ψ ^ ( k ) with modulated frequency k 0 and radius r .

Fig. 4.
Fig. 4.

PSCP and OVM analysis of an experimental SI. (a) Phase of the coefficients of the directional wavelet transform of the speckle pattern with θ = 0 ; (b) idem to (a) with θ = π / 2 ; (c) result of the Canny edge detector applied to image (a); (d) result of the Canny edge detector applied to image (b); (e) overlapping of (c) and (d), which defines the PSCP; (f) phase singularities obtained by the OVM technique (solid circles).

Fig. 5.
Fig. 5.

100 × 100 pixels region of detected pairs before (solid circles) and after (empty circles) the introduction of an in-plane translation of three pixels to the right direction and two pixels up using (a) the PSCP approach and (b) the OVM method.

Fig. 6.
Fig. 6.

Phase singularities determined before (solid circles) and after (empty circles) the localized displacement over 40 × 40 pixels by using (a) the PSCP method and (b) OVM method.

Fig. 7.
Fig. 7.

Optical setup consisting of a linearly polarized laser beam, a Leica microscope objective, a polarization beam splitter (BS), and a CCD Camera.

Tables (3)

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Table 1. Results Obtained for a Translation Movementa

Tables Icon

Table 2. Results Obtained for a Rotation Movementa

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Table 3. Measurement of Experimental In-Plane Displacements

Equations (7)

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DCWT [ I ( r ) ] ( a , b , θ ) = 1 a 2 R 2 d 2 r ψ * [ a 1 R θ ( r b ) ] I ( r ) = R 2 d 2 k e i k · b ψ ^ * [ a R θ ( k ) ] I ^ ( k ) ,
ψ ( r ) = ( 2 | r / σ i σ k 0 | 2 σ 2 ) exp ( i k 0 · r ) exp ( | r | 2 / 2 σ 2 ) ,
ψ ^ ( k ) = σ 2 | k | 2 exp [ σ 2 2 | k k 0 | 2 ] ,
MAD ( X , Y ) = 1 n i = 1 n ( | Δ x i mode ( X ) | , | Δ y i mode ( Y ) | ) ,
x c = ( i b i 2 ) ( i a i c i ) ( i a i b i ) ( i b i c i ) ( i a i 2 ) ( i b i 2 ) ( i a i b i ) 2 ,
y c = ( i a i 2 ) ( i b i c i ) ( i a i b i ) ( i a i c i ) ( i a i 2 ) ( i b i 2 ) ( i a i b i ) 2 ,
I = | U ( 1 α d [ 0 , 1 ] ) + α d [ 0 , 1 ] U | 2 ,

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