Abstract

The concentric fringe patterns created by features in holograms may be associated with a complex-valued orientational order field. Convolution with an orientational alignment operator then identifies centers of symmetry that correspond to the two-dimensional positions of the features. Feature identification through orientational alignment is reminiscent of voting algorithms such as Hough transforms, but may be implemented with fast convolution methods, and so can be orders of magnitude faster.

© 2014 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  21. J. Rubinstein, J. Segman, Y. Zeevi, “Recognition of distorted patterns by invariance kernels,” Pattern Recogn. 24, 959–967 (1991).
    [CrossRef]
  22. T. J. Atherton, D. J. Kerbyson, “Size invariant circle detection,” Image Vision Comput. 17, 795–803 (1999).
    [CrossRef]
  23. A. Savitzky, M. J. E. Golay, “Smoothing and differentionation of data by simplified least squares procedures,” Acta Crystallog. 36, 1627–1639 (1964).
  24. D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
    [CrossRef] [PubMed]
  25. T. Savin, P. S. Doyle, “Role of finite exposure time on measuring an elastic modulus using microrheology,” Phys. Rev. E 71, 041106 (2005).
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  26. T. Savin, P. S. Doyle, “Static and dynamic errors in particle tracking microrheology,” Biophys. J. 88, 623–638 (2005).
    [CrossRef]
  27. B. J. Krishnatreya, A. Colen-Landy, P. Hasebe, B. A. Bell, J. R. Jones, A. Sunda-Meya, D. G. Grier, “Measuring Boltzmann’s constant through holographic video microscopy of a single sphere,” Am. J. Phys. 82, 23–31 (2014).
    [CrossRef]

2014 (1)

B. J. Krishnatreya, A. Colen-Landy, P. Hasebe, B. A. Bell, J. R. Jones, A. Sunda-Meya, D. G. Grier, “Measuring Boltzmann’s constant through holographic video microscopy of a single sphere,” Am. J. Phys. 82, 23–31 (2014).
[CrossRef]

2013 (2)

J. Fung, V. N. Manoharan, “Holographic measurements of anisotropic three-dimensional diffusion of colloidal clusters,” Phys. Rev. E 88, 020302 (2013).
[CrossRef]

C. Hollitt, “A convolution approach to the circle Hough transform for arbitrary radius,” Mach. Vision Appl. 24, 683–694 (2013).
[CrossRef]

2012 (2)

H. Shpaisman, B. J. Krishnatreya, D. G. Grier, “Holographic microrefractometer,” Appl. Phys. Lett. 101, 091102 (2012).
[CrossRef]

R. Parthasarathy, “Rapid, accurate particle tracking by calculation of radial symmetry centers,” Nature Methods 9, 724–726 (2012).
[CrossRef] [PubMed]

2011 (4)

2010 (1)

K. Xiao, D. G. Grier, “Multidimensional optical fractionation with holographic verification,” Phys. Rev. Lett. 104, 028302 (2010).
[CrossRef]

2009 (3)

F. C. Cheong, B. Sun, R. Dreyfus, J. Amato-Grill, K. Xiao, L. Dixon, D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express 17, 13071–13079 (2009).
[CrossRef] [PubMed]

F. C. Cheong, K. Xiao, D. G. Grier, “Characterization of individual milk fat globules with holographic video microscopy,” J. Dairy Sci. 92, 95–99 (2009).
[CrossRef]

F. C. Cheong, S. Duarte, S.-H. Lee, D. G. Grier, “Holographic microrheology of polysaccharides from Streptococcus mutans biofilms,” Rheol. Acta 48, 109–115 (2009).
[CrossRef]

2008 (1)

Y. Roichman, B. Sun, A. Stolarski, D. G. Grier, “Influence of non-conservative optical forces on the dynamics of optically trapped colloidal spheres: The fountain of probability,” Phys. Rev. Lett. 101, 128301 (2008).
[CrossRef]

2007 (2)

2006 (1)

2005 (2)

T. Savin, P. S. Doyle, “Role of finite exposure time on measuring an elastic modulus using microrheology,” Phys. Rev. E 71, 041106 (2005).
[CrossRef]

T. Savin, P. S. Doyle, “Static and dynamic errors in particle tracking microrheology,” Biophys. J. 88, 623–638 (2005).
[CrossRef]

2003 (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[CrossRef] [PubMed]

1999 (1)

T. J. Atherton, D. J. Kerbyson, “Size invariant circle detection,” Image Vision Comput. 17, 795–803 (1999).
[CrossRef]

1996 (1)

J. C. Crocker, D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. 179, 298–310 (1996).
[CrossRef]

1991 (1)

J. Rubinstein, J. Segman, Y. Zeevi, “Recognition of distorted patterns by invariance kernels,” Pattern Recogn. 24, 959–967 (1991).
[CrossRef]

1981 (1)

D. H. Ballard, “Generalizing the Hough transform to detect arbitrary shapes,” Pattern Recogn. 13, 111–122 (1981).
[CrossRef]

1979 (1)

D. R. Nelson, B. I. Halperin, “Dislocation-mediated melting in two dimensions,” Phys. Rev. B 19(5), 2457–2484 (1979).
[CrossRef]

1978 (1)

B. I. Halperin, D. R. Nelson, “Theory of two-dimensional melting,” Phys. Rev. Lett. 41(2), 121–124 (1978).
[CrossRef]

1964 (1)

A. Savitzky, M. J. E. Golay, “Smoothing and differentionation of data by simplified least squares procedures,” Acta Crystallog. 36, 1627–1639 (1964).

Amato-Grill, J.

Atherton, T. J.

T. J. Atherton, D. J. Kerbyson, “Size invariant circle detection,” Image Vision Comput. 17, 795–803 (1999).
[CrossRef]

Ballard, D. H.

D. H. Ballard, “Generalizing the Hough transform to detect arbitrary shapes,” Pattern Recogn. 13, 111–122 (1981).
[CrossRef]

Bell, B. A.

B. J. Krishnatreya, A. Colen-Landy, P. Hasebe, B. A. Bell, J. R. Jones, A. Sunda-Meya, D. G. Grier, “Measuring Boltzmann’s constant through holographic video microscopy of a single sphere,” Am. J. Phys. 82, 23–31 (2014).
[CrossRef]

Bianchi, S.

Bolognesi, G.

Cheong, F. C.

F. C. Cheong, K. Xiao, D. J. Pine, D. G. Grier, “Holographic characterization of individual colloidal spheres’ porosities,” Soft Matter 7, 6816–6819 (2011).
[CrossRef]

L. Dixon, F. C. Cheong, D. G. Grier, “Holographic particle-streak velocimetry,” Opt. Express 19, 4393–4398 (2011).
[CrossRef] [PubMed]

F. C. Cheong, S. Duarte, S.-H. Lee, D. G. Grier, “Holographic microrheology of polysaccharides from Streptococcus mutans biofilms,” Rheol. Acta 48, 109–115 (2009).
[CrossRef]

F. C. Cheong, B. Sun, R. Dreyfus, J. Amato-Grill, K. Xiao, L. Dixon, D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express 17, 13071–13079 (2009).
[CrossRef] [PubMed]

F. C. Cheong, K. Xiao, D. G. Grier, “Characterization of individual milk fat globules with holographic video microscopy,” J. Dairy Sci. 92, 95–99 (2009).
[CrossRef]

Colen-Landy, A.

B. J. Krishnatreya, A. Colen-Landy, P. Hasebe, B. A. Bell, J. R. Jones, A. Sunda-Meya, D. G. Grier, “Measuring Boltzmann’s constant through holographic video microscopy of a single sphere,” Am. J. Phys. 82, 23–31 (2014).
[CrossRef]

Crocker, J. C.

J. C. Crocker, D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. 179, 298–310 (1996).
[CrossRef]

Di Leonardo, R.

Dixon, L.

Doyle, P. S.

T. Savin, P. S. Doyle, “Static and dynamic errors in particle tracking microrheology,” Biophys. J. 88, 623–638 (2005).
[CrossRef]

T. Savin, P. S. Doyle, “Role of finite exposure time on measuring an elastic modulus using microrheology,” Phys. Rev. E 71, 041106 (2005).
[CrossRef]

Dreyfus, R.

Duarte, S.

F. C. Cheong, S. Duarte, S.-H. Lee, D. G. Grier, “Holographic microrheology of polysaccharides from Streptococcus mutans biofilms,” Rheol. Acta 48, 109–115 (2009).
[CrossRef]

Fung, J.

Golay, M. J. E.

A. Savitzky, M. J. E. Golay, “Smoothing and differentionation of data by simplified least squares procedures,” Acta Crystallog. 36, 1627–1639 (1964).

Grier, D. G.

B. J. Krishnatreya, A. Colen-Landy, P. Hasebe, B. A. Bell, J. R. Jones, A. Sunda-Meya, D. G. Grier, “Measuring Boltzmann’s constant through holographic video microscopy of a single sphere,” Am. J. Phys. 82, 23–31 (2014).
[CrossRef]

H. Shpaisman, B. J. Krishnatreya, D. G. Grier, “Holographic microrefractometer,” Appl. Phys. Lett. 101, 091102 (2012).
[CrossRef]

F. C. Cheong, K. Xiao, D. J. Pine, D. G. Grier, “Holographic characterization of individual colloidal spheres’ porosities,” Soft Matter 7, 6816–6819 (2011).
[CrossRef]

L. Dixon, F. C. Cheong, D. G. Grier, “Holographic particle-streak velocimetry,” Opt. Express 19, 4393–4398 (2011).
[CrossRef] [PubMed]

K. Xiao, D. G. Grier, “Multidimensional optical fractionation with holographic verification,” Phys. Rev. Lett. 104, 028302 (2010).
[CrossRef]

F. C. Cheong, B. Sun, R. Dreyfus, J. Amato-Grill, K. Xiao, L. Dixon, D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express 17, 13071–13079 (2009).
[CrossRef] [PubMed]

F. C. Cheong, S. Duarte, S.-H. Lee, D. G. Grier, “Holographic microrheology of polysaccharides from Streptococcus mutans biofilms,” Rheol. Acta 48, 109–115 (2009).
[CrossRef]

F. C. Cheong, K. Xiao, D. G. Grier, “Characterization of individual milk fat globules with holographic video microscopy,” J. Dairy Sci. 92, 95–99 (2009).
[CrossRef]

Y. Roichman, B. Sun, A. Stolarski, D. G. Grier, “Influence of non-conservative optical forces on the dynamics of optically trapped colloidal spheres: The fountain of probability,” Phys. Rev. Lett. 101, 128301 (2008).
[CrossRef]

S.-H. Lee, Y. Roichman, G.-R. Yi, S.-H. Kim, S.-M. Yang, A. van Blaaderen, P. van Oostrum, D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. Express 15, 18275–18282 (2007).
[CrossRef]

S.-H. Lee, D. G. Grier, “Holographic microscopy of holographically trapped three-dimensional structures,” Opt. Express 15, 1505–1512 (2007).
[CrossRef] [PubMed]

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[CrossRef] [PubMed]

J. C. Crocker, D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. 179, 298–310 (1996).
[CrossRef]

Halperin, B. I.

D. R. Nelson, B. I. Halperin, “Dislocation-mediated melting in two dimensions,” Phys. Rev. B 19(5), 2457–2484 (1979).
[CrossRef]

B. I. Halperin, D. R. Nelson, “Theory of two-dimensional melting,” Phys. Rev. Lett. 41(2), 121–124 (1978).
[CrossRef]

Hasebe, P.

B. J. Krishnatreya, A. Colen-Landy, P. Hasebe, B. A. Bell, J. R. Jones, A. Sunda-Meya, D. G. Grier, “Measuring Boltzmann’s constant through holographic video microscopy of a single sphere,” Am. J. Phys. 82, 23–31 (2014).
[CrossRef]

Hollitt, C.

C. Hollitt, “A convolution approach to the circle Hough transform for arbitrary radius,” Mach. Vision Appl. 24, 683–694 (2013).
[CrossRef]

Jones, J. R.

B. J. Krishnatreya, A. Colen-Landy, P. Hasebe, B. A. Bell, J. R. Jones, A. Sunda-Meya, D. G. Grier, “Measuring Boltzmann’s constant through holographic video microscopy of a single sphere,” Am. J. Phys. 82, 23–31 (2014).
[CrossRef]

Katz, J.

Kaz, D. M.

Kerbyson, D. J.

T. J. Atherton, D. J. Kerbyson, “Size invariant circle detection,” Image Vision Comput. 17, 795–803 (1999).
[CrossRef]

Kim, S.-H.

Krishnatreya, B. J.

B. J. Krishnatreya, A. Colen-Landy, P. Hasebe, B. A. Bell, J. R. Jones, A. Sunda-Meya, D. G. Grier, “Measuring Boltzmann’s constant through holographic video microscopy of a single sphere,” Am. J. Phys. 82, 23–31 (2014).
[CrossRef]

H. Shpaisman, B. J. Krishnatreya, D. G. Grier, “Holographic microrefractometer,” Appl. Phys. Lett. 101, 091102 (2012).
[CrossRef]

Lee, S.-H.

Malkiel, E.

Manoharan, V. N.

Martin, K. E.

McGorty, R.

Nelson, D. R.

D. R. Nelson, B. I. Halperin, “Dislocation-mediated melting in two dimensions,” Phys. Rev. B 19(5), 2457–2484 (1979).
[CrossRef]

B. I. Halperin, D. R. Nelson, “Theory of two-dimensional melting,” Phys. Rev. Lett. 41(2), 121–124 (1978).
[CrossRef]

Parthasarathy, R.

R. Parthasarathy, “Rapid, accurate particle tracking by calculation of radial symmetry centers,” Nature Methods 9, 724–726 (2012).
[CrossRef] [PubMed]

Perry, R. W.

Pine, D. J.

F. C. Cheong, K. Xiao, D. J. Pine, D. G. Grier, “Holographic characterization of individual colloidal spheres’ porosities,” Soft Matter 7, 6816–6819 (2011).
[CrossRef]

Roichman, Y.

Y. Roichman, B. Sun, A. Stolarski, D. G. Grier, “Influence of non-conservative optical forces on the dynamics of optically trapped colloidal spheres: The fountain of probability,” Phys. Rev. Lett. 101, 128301 (2008).
[CrossRef]

S.-H. Lee, Y. Roichman, G.-R. Yi, S.-H. Kim, S.-M. Yang, A. van Blaaderen, P. van Oostrum, D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. Express 15, 18275–18282 (2007).
[CrossRef]

Rubinstein, J.

J. Rubinstein, J. Segman, Y. Zeevi, “Recognition of distorted patterns by invariance kernels,” Pattern Recogn. 24, 959–967 (1991).
[CrossRef]

Savin, T.

T. Savin, P. S. Doyle, “Role of finite exposure time on measuring an elastic modulus using microrheology,” Phys. Rev. E 71, 041106 (2005).
[CrossRef]

T. Savin, P. S. Doyle, “Static and dynamic errors in particle tracking microrheology,” Biophys. J. 88, 623–638 (2005).
[CrossRef]

Savitzky, A.

A. Savitzky, M. J. E. Golay, “Smoothing and differentionation of data by simplified least squares procedures,” Acta Crystallog. 36, 1627–1639 (1964).

Segman, J.

J. Rubinstein, J. Segman, Y. Zeevi, “Recognition of distorted patterns by invariance kernels,” Pattern Recogn. 24, 959–967 (1991).
[CrossRef]

Sheng, J.

Shpaisman, H.

H. Shpaisman, B. J. Krishnatreya, D. G. Grier, “Holographic microrefractometer,” Appl. Phys. Lett. 101, 091102 (2012).
[CrossRef]

Stolarski, A.

Y. Roichman, B. Sun, A. Stolarski, D. G. Grier, “Influence of non-conservative optical forces on the dynamics of optically trapped colloidal spheres: The fountain of probability,” Phys. Rev. Lett. 101, 128301 (2008).
[CrossRef]

Sun, B.

F. C. Cheong, B. Sun, R. Dreyfus, J. Amato-Grill, K. Xiao, L. Dixon, D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express 17, 13071–13079 (2009).
[CrossRef] [PubMed]

Y. Roichman, B. Sun, A. Stolarski, D. G. Grier, “Influence of non-conservative optical forces on the dynamics of optically trapped colloidal spheres: The fountain of probability,” Phys. Rev. Lett. 101, 128301 (2008).
[CrossRef]

Sunda-Meya, A.

B. J. Krishnatreya, A. Colen-Landy, P. Hasebe, B. A. Bell, J. R. Jones, A. Sunda-Meya, D. G. Grier, “Measuring Boltzmann’s constant through holographic video microscopy of a single sphere,” Am. J. Phys. 82, 23–31 (2014).
[CrossRef]

van Blaaderen, A.

van Oostrum, P.

Xiao, K.

F. C. Cheong, K. Xiao, D. J. Pine, D. G. Grier, “Holographic characterization of individual colloidal spheres’ porosities,” Soft Matter 7, 6816–6819 (2011).
[CrossRef]

K. Xiao, D. G. Grier, “Multidimensional optical fractionation with holographic verification,” Phys. Rev. Lett. 104, 028302 (2010).
[CrossRef]

F. C. Cheong, B. Sun, R. Dreyfus, J. Amato-Grill, K. Xiao, L. Dixon, D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express 17, 13071–13079 (2009).
[CrossRef] [PubMed]

F. C. Cheong, K. Xiao, D. G. Grier, “Characterization of individual milk fat globules with holographic video microscopy,” J. Dairy Sci. 92, 95–99 (2009).
[CrossRef]

Yang, S.-M.

Yi, G.-R.

Zeevi, Y.

J. Rubinstein, J. Segman, Y. Zeevi, “Recognition of distorted patterns by invariance kernels,” Pattern Recogn. 24, 959–967 (1991).
[CrossRef]

Acta Crystallog. (1)

A. Savitzky, M. J. E. Golay, “Smoothing and differentionation of data by simplified least squares procedures,” Acta Crystallog. 36, 1627–1639 (1964).

Am. J. Phys. (1)

B. J. Krishnatreya, A. Colen-Landy, P. Hasebe, B. A. Bell, J. R. Jones, A. Sunda-Meya, D. G. Grier, “Measuring Boltzmann’s constant through holographic video microscopy of a single sphere,” Am. J. Phys. 82, 23–31 (2014).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

H. Shpaisman, B. J. Krishnatreya, D. G. Grier, “Holographic microrefractometer,” Appl. Phys. Lett. 101, 091102 (2012).
[CrossRef]

Biophys. J. (1)

T. Savin, P. S. Doyle, “Static and dynamic errors in particle tracking microrheology,” Biophys. J. 88, 623–638 (2005).
[CrossRef]

Image Vision Comput. (1)

T. J. Atherton, D. J. Kerbyson, “Size invariant circle detection,” Image Vision Comput. 17, 795–803 (1999).
[CrossRef]

J. Colloid Interface Sci. (1)

J. C. Crocker, D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. 179, 298–310 (1996).
[CrossRef]

J. Dairy Sci. (1)

F. C. Cheong, K. Xiao, D. G. Grier, “Characterization of individual milk fat globules with holographic video microscopy,” J. Dairy Sci. 92, 95–99 (2009).
[CrossRef]

Mach. Vision Appl. (1)

C. Hollitt, “A convolution approach to the circle Hough transform for arbitrary radius,” Mach. Vision Appl. 24, 683–694 (2013).
[CrossRef]

Nature (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[CrossRef] [PubMed]

Nature Methods (1)

R. Parthasarathy, “Rapid, accurate particle tracking by calculation of radial symmetry centers,” Nature Methods 9, 724–726 (2012).
[CrossRef] [PubMed]

Opt. Express (6)

Pattern Recogn. (2)

D. H. Ballard, “Generalizing the Hough transform to detect arbitrary shapes,” Pattern Recogn. 13, 111–122 (1981).
[CrossRef]

J. Rubinstein, J. Segman, Y. Zeevi, “Recognition of distorted patterns by invariance kernels,” Pattern Recogn. 24, 959–967 (1991).
[CrossRef]

Phys. Rev. B (1)

D. R. Nelson, B. I. Halperin, “Dislocation-mediated melting in two dimensions,” Phys. Rev. B 19(5), 2457–2484 (1979).
[CrossRef]

Phys. Rev. E (2)

T. Savin, P. S. Doyle, “Role of finite exposure time on measuring an elastic modulus using microrheology,” Phys. Rev. E 71, 041106 (2005).
[CrossRef]

J. Fung, V. N. Manoharan, “Holographic measurements of anisotropic three-dimensional diffusion of colloidal clusters,” Phys. Rev. E 88, 020302 (2013).
[CrossRef]

Phys. Rev. Lett. (3)

B. I. Halperin, D. R. Nelson, “Theory of two-dimensional melting,” Phys. Rev. Lett. 41(2), 121–124 (1978).
[CrossRef]

Y. Roichman, B. Sun, A. Stolarski, D. G. Grier, “Influence of non-conservative optical forces on the dynamics of optically trapped colloidal spheres: The fountain of probability,” Phys. Rev. Lett. 101, 128301 (2008).
[CrossRef]

K. Xiao, D. G. Grier, “Multidimensional optical fractionation with holographic verification,” Phys. Rev. Lett. 104, 028302 (2010).
[CrossRef]

Rheol. Acta (1)

F. C. Cheong, S. Duarte, S.-H. Lee, D. G. Grier, “Holographic microrheology of polysaccharides from Streptococcus mutans biofilms,” Rheol. Acta 48, 109–115 (2009).
[CrossRef]

Soft Matter (1)

F. C. Cheong, K. Xiao, D. J. Pine, D. G. Grier, “Holographic characterization of individual colloidal spheres’ porosities,” Soft Matter 7, 6816–6819 (2011).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Feature detection by orientation alignment. (a) Normalized hologram b(r) of a 0.8 μm-radius polystyrene sphere in water. (b) Magnitude |∇b(r)| of the gradient of the image in (a). (c) The orientation, 2ϕ(r), of the gradients. Inset: phase angle of the orientation alignment convolution kernel, (d) Orientation alignment transform of the image in (a). Inset: Schematic representation of how three pixels (colored red) contribute to the real part of the transform. Blue lobes represent real-valued contributions to Ψ(r).

Fig. 2
Fig. 2

Feature identification in a multi-particle hologram. The greyscale hologram b(r) of 12 colloidal spheres is transformed by the orientation alignment transform into sharply resolved peaks in B(r) whose centers are plotted as crosses. The scale bar indicates 10 μm.

Fig. 3
Fig. 3

(a) Trajectory r(t) of a colloidal sphere obtained by analyzing a holographic video with the orientation alignment transform, colored by time. (b) The mean-squared displacement along and ŷ computed from r(t), together fits to Eq. (8), plotted as dashed curves.

Equations (8)

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ϕ ( r ) = tan 1 ( y b ( r ) x b ( r ) ) ,
ψ ( r ) = | b ( r ) | 2 e 2 i ϕ ( r ) .
K ( r ) = 1 r e 2 i θ ,
Ψ ( r ) = K ( r r ) ψ ( r ) d 2 r .
Ψ ˜ ( k ) = K ˜ ( k ) ψ ˜ ( k ) ,
K ˜ ( k ) = 1 k e 2 i θ
Δ r j 2 ( τ ) = [ r j ( t + τ ) r j ( t ) ] 2
Δ r j 2 ( τ ) = 2 D j τ + 2 ε j 2 ,

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