Abstract

Computerized image-analysis routines deployed widely to locate and track the positions of particles in microscope images include several steps where images are convolved with kernels to remove noise. In many common implementations, some kernels are rotationally asymmetric. Here we show that the use of these asymmetric kernels creates significant artifacts, distorting apparent particle positions in a way that gives the artificial appearance of orientational crystalline order, even in such fully-disordered isotropic systems as simple fluids of hard-sphere-like colloids. We rectify this problem by replacing all asymmetric kernels with rotationally-symmetric kernels, which does not impact code performance. We show that these corrected codes locate particle positions properly, restoring measured isotropy to colloidal fluids. We also investigate rapidly-formed colloidal sediments, and with the corrected codes show that these sediments, often thought to be amorphous, may exhibit strong orientational correlations among bonds between neighboring colloidal particles.

© 2013 Optical Society of America

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    [CrossRef] [PubMed]
  4. Z. Zhang, N. Xu, D. T. N. Chen, P. Yunker, A. M. Alsayed, K. B. Aptowicz, P. Habdas, A. J. Liu, S. R. Nagel, and A. G. Yodh, “Thermal vestige of the zero-temperature jamming transition,” Nature459, 230–233 (2009).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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  24. C. P. Royall, M. E. Leunissen, A.-P. Hynninen, M. Dijkstra, and A. van Blaaderen, “Re-entrant melting and freezing in a model system of charged colloids,” J. Chem. Phys.124, 244706 (2006).
    [CrossRef] [PubMed]
  25. A. P. Cohen, E. Janai, E. Mogilko, A. B. Schofield, and E. Sloutskin, “Fluid suspensions of colloidal ellipsoids: Direct structural measurements,” Phys. Rev. Lett.107, 238301 (2011).
    [CrossRef] [PubMed]
  26. A. P. Cohen, E. Janai, D. C. Rapaport, A. B. Schofield, and E. Sloutskin, “Structure and interactions in fluids of prolate colloidal ellipsoids: Comparison between experiment, theory, and simulation,” J. Chem. Phys.137, 184505 (2012).
    [CrossRef] [PubMed]

2013 (1)

S. R. Liber, S. Borohovich, A. V. Butenko, A. B. Schofield, and E. Sloutskin, “Dense colloidal fluids form denser amorphous sediments,” Proc. Nat. Acad. Sci. U. S. A.110, 5769–5773 (2013).
[CrossRef]

2012 (2)

A. P. Cohen, E. Janai, D. C. Rapaport, A. B. Schofield, and E. Sloutskin, “Structure and interactions in fluids of prolate colloidal ellipsoids: Comparison between experiment, theory, and simulation,” J. Chem. Phys.137, 184505 (2012).
[CrossRef] [PubMed]

G. L. Hunter and E. R. Weeks, “The physics of the colloidal glass transition,” Rep. Prog. Phys.75, 066501 (2012).
[CrossRef] [PubMed]

2011 (3)

Z. Zheng, F. Wang, and Y. Han, “Glass transitions in quasi-two-dimensional suspensions of colloidal ellipsoids,” Phys. Rev. Lett.107, 065702 (2011).
[CrossRef] [PubMed]

A. P. Cohen, E. Janai, E. Mogilko, A. B. Schofield, and E. Sloutskin, “Fluid suspensions of colloidal ellipsoids: Direct structural measurements,” Phys. Rev. Lett.107, 238301 (2011).
[CrossRef] [PubMed]

X. Cheng, J. H. McCoy, J. N. Israelachvili, and I. Cohen, “Imaging the microscopic structure of shear thinning and thickening colloidal suspensions,” Science333, 1276–1279 (2011).
[CrossRef] [PubMed]

2009 (3)

U. Gasser, “Crystallization in three- and two-dimensional colloidal suspensions,” J. Phys. Condens. Mat.21, 203101 (2009).
[CrossRef]

Z. Zhang, N. Xu, D. T. N. Chen, P. Yunker, A. M. Alsayed, K. B. Aptowicz, P. Habdas, A. J. Liu, S. R. Nagel, and A. G. Yodh, “Thermal vestige of the zero-temperature jamming transition,” Nature459, 230–233 (2009).
[CrossRef] [PubMed]

Y. Gao and M. L. Kilfoil, “Accurate detection and complete tracking of large populations of features in three dimensions,” Opt. Express17, 4685–4704 (2009).
[CrossRef] [PubMed]

2008 (2)

M. Jerkins, M. Schröter, H. L. Swinney, T. J. Senden, M. Saadatfar, and T. Aste, “Onset of mechanical stability in random packings of frictional particles,” Phys. Rev. Lett.101, 018301 (2008).
[CrossRef]

P. J. Lu, E. Zaccarelli, F. Ciulla, A. B. Schofield, F. Sciortino, and D. A. Weitz, “Gelation of particles with short-range attraction,” Nature453, 499–503 (2008).
[CrossRef] [PubMed]

2007 (1)

2006 (1)

C. P. Royall, M. E. Leunissen, A.-P. Hynninen, M. Dijkstra, and A. van Blaaderen, “Re-entrant melting and freezing in a model system of charged colloids,” J. Chem. Phys.124, 244706 (2006).
[CrossRef] [PubMed]

2005 (1)

T. Aste, M. Saadatfar, and T. J. Senden, “Geometrical structure of disordered sphere packings,” Phys. Rev. E71, 061302 (2005).
[CrossRef]

2001 (1)

U. Gasser, E. R. Weeks, A. B. Schofield, P. N. Pusey, and D. A. Weitz, “Real-space imaging of nucleation and growth in colloidal crystallization,” Science292, 258–262 (2001).
[CrossRef] [PubMed]

2000 (1)

E. R. Weeks, J. C. Crocker, A. C. Levitt, A. Schofield, and D. A. Weitz, “Three-dimensional direct imaging of structural relaxation near the colloidal glass transition,” Science287, 627–631 (2000).
[CrossRef] [PubMed]

1996 (1)

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

1995 (1)

A. van Blaaderen and P. Wiltzius, “Real-space structure of colloidal hard-sphere glasses,” Science270, 1177–1179 (1995).
[CrossRef]

1986 (1)

D. Frenkel, R. J. Vos, C. G. de Kruif, and A. Vrij, “Structure factors of polydisperse systems of hard spheres: A comparison of Monte Carlo simulations and Percus-Yevick theory,” J. Chem. Phys.84, 4625–4630 (1986).
[CrossRef]

1982 (1)

D. R. Wilkinson and S. F. Edwards, “The use of stereology to determine the partial two-body correlation functions for hard sphere ensembles,” J. Phys. D Appl. Phys.15, 551–562 (1982).
[CrossRef]

Alsayed, A. M.

Z. Zhang, N. Xu, D. T. N. Chen, P. Yunker, A. M. Alsayed, K. B. Aptowicz, P. Habdas, A. J. Liu, S. R. Nagel, and A. G. Yodh, “Thermal vestige of the zero-temperature jamming transition,” Nature459, 230–233 (2009).
[CrossRef] [PubMed]

Aptowicz, K. B.

Z. Zhang, N. Xu, D. T. N. Chen, P. Yunker, A. M. Alsayed, K. B. Aptowicz, P. Habdas, A. J. Liu, S. R. Nagel, and A. G. Yodh, “Thermal vestige of the zero-temperature jamming transition,” Nature459, 230–233 (2009).
[CrossRef] [PubMed]

Aste, T.

M. Jerkins, M. Schröter, H. L. Swinney, T. J. Senden, M. Saadatfar, and T. Aste, “Onset of mechanical stability in random packings of frictional particles,” Phys. Rev. Lett.101, 018301 (2008).
[CrossRef]

T. Aste, M. Saadatfar, and T. J. Senden, “Geometrical structure of disordered sphere packings,” Phys. Rev. E71, 061302 (2005).
[CrossRef]

Borohovich, S.

S. R. Liber, S. Borohovich, A. V. Butenko, A. B. Schofield, and E. Sloutskin, “Dense colloidal fluids form denser amorphous sediments,” Proc. Nat. Acad. Sci. U. S. A.110, 5769–5773 (2013).
[CrossRef]

Butenko, A. V.

S. R. Liber, S. Borohovich, A. V. Butenko, A. B. Schofield, and E. Sloutskin, “Dense colloidal fluids form denser amorphous sediments,” Proc. Nat. Acad. Sci. U. S. A.110, 5769–5773 (2013).
[CrossRef]

Chen, D. T. N.

Z. Zhang, N. Xu, D. T. N. Chen, P. Yunker, A. M. Alsayed, K. B. Aptowicz, P. Habdas, A. J. Liu, S. R. Nagel, and A. G. Yodh, “Thermal vestige of the zero-temperature jamming transition,” Nature459, 230–233 (2009).
[CrossRef] [PubMed]

Cheng, X.

X. Cheng, J. H. McCoy, J. N. Israelachvili, and I. Cohen, “Imaging the microscopic structure of shear thinning and thickening colloidal suspensions,” Science333, 1276–1279 (2011).
[CrossRef] [PubMed]

Cheong, O.

M. de Berg, O. Cheong, M. van Kreveld, and M. Overmars, Computational Geometry: Algorithms and Applications (Springer, 2008).

Ciulla, F.

P. J. Lu, E. Zaccarelli, F. Ciulla, A. B. Schofield, F. Sciortino, and D. A. Weitz, “Gelation of particles with short-range attraction,” Nature453, 499–503 (2008).
[CrossRef] [PubMed]

Cohen, A. P.

A. P. Cohen, E. Janai, D. C. Rapaport, A. B. Schofield, and E. Sloutskin, “Structure and interactions in fluids of prolate colloidal ellipsoids: Comparison between experiment, theory, and simulation,” J. Chem. Phys.137, 184505 (2012).
[CrossRef] [PubMed]

A. P. Cohen, E. Janai, E. Mogilko, A. B. Schofield, and E. Sloutskin, “Fluid suspensions of colloidal ellipsoids: Direct structural measurements,” Phys. Rev. Lett.107, 238301 (2011).
[CrossRef] [PubMed]

Cohen, I.

X. Cheng, J. H. McCoy, J. N. Israelachvili, and I. Cohen, “Imaging the microscopic structure of shear thinning and thickening colloidal suspensions,” Science333, 1276–1279 (2011).
[CrossRef] [PubMed]

Crocker, J. C.

E. R. Weeks, J. C. Crocker, A. C. Levitt, A. Schofield, and D. A. Weitz, “Three-dimensional direct imaging of structural relaxation near the colloidal glass transition,” Science287, 627–631 (2000).
[CrossRef] [PubMed]

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

D. G. Grier and J. C. Crocker, personal communication.

de Berg, M.

M. de Berg, O. Cheong, M. van Kreveld, and M. Overmars, Computational Geometry: Algorithms and Applications (Springer, 2008).

de Kruif, C. G.

D. Frenkel, R. J. Vos, C. G. de Kruif, and A. Vrij, “Structure factors of polydisperse systems of hard spheres: A comparison of Monte Carlo simulations and Percus-Yevick theory,” J. Chem. Phys.84, 4625–4630 (1986).
[CrossRef]

Dijkstra, M.

C. P. Royall, M. E. Leunissen, A.-P. Hynninen, M. Dijkstra, and A. van Blaaderen, “Re-entrant melting and freezing in a model system of charged colloids,” J. Chem. Phys.124, 244706 (2006).
[CrossRef] [PubMed]

Edwards, S. F.

D. R. Wilkinson and S. F. Edwards, “The use of stereology to determine the partial two-body correlation functions for hard sphere ensembles,” J. Phys. D Appl. Phys.15, 551–562 (1982).
[CrossRef]

Frenkel, D.

D. Frenkel, R. J. Vos, C. G. de Kruif, and A. Vrij, “Structure factors of polydisperse systems of hard spheres: A comparison of Monte Carlo simulations and Percus-Yevick theory,” J. Chem. Phys.84, 4625–4630 (1986).
[CrossRef]

Gao, Y.

Gasser, U.

U. Gasser, “Crystallization in three- and two-dimensional colloidal suspensions,” J. Phys. Condens. Mat.21, 203101 (2009).
[CrossRef]

U. Gasser, E. R. Weeks, A. B. Schofield, P. N. Pusey, and D. A. Weitz, “Real-space imaging of nucleation and growth in colloidal crystallization,” Science292, 258–262 (2001).
[CrossRef] [PubMed]

Grier, D. G.

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

D. G. Grier and J. C. Crocker, personal communication.

Habdas, P.

Z. Zhang, N. Xu, D. T. N. Chen, P. Yunker, A. M. Alsayed, K. B. Aptowicz, P. Habdas, A. J. Liu, S. R. Nagel, and A. G. Yodh, “Thermal vestige of the zero-temperature jamming transition,” Nature459, 230–233 (2009).
[CrossRef] [PubMed]

Han, Y.

Z. Zheng, F. Wang, and Y. Han, “Glass transitions in quasi-two-dimensional suspensions of colloidal ellipsoids,” Phys. Rev. Lett.107, 065702 (2011).
[CrossRef] [PubMed]

Hansen, J.-P.

J.-P. Hansen and I. R. McDonald, Theory of Simple Liquids (Elsevier, 2006).

Hunter, G. L.

G. L. Hunter and E. R. Weeks, “The physics of the colloidal glass transition,” Rep. Prog. Phys.75, 066501 (2012).
[CrossRef] [PubMed]

Hynninen, A.-P.

C. P. Royall, M. E. Leunissen, A.-P. Hynninen, M. Dijkstra, and A. van Blaaderen, “Re-entrant melting and freezing in a model system of charged colloids,” J. Chem. Phys.124, 244706 (2006).
[CrossRef] [PubMed]

Israelachvili, J. N.

X. Cheng, J. H. McCoy, J. N. Israelachvili, and I. Cohen, “Imaging the microscopic structure of shear thinning and thickening colloidal suspensions,” Science333, 1276–1279 (2011).
[CrossRef] [PubMed]

Janai, E.

A. P. Cohen, E. Janai, D. C. Rapaport, A. B. Schofield, and E. Sloutskin, “Structure and interactions in fluids of prolate colloidal ellipsoids: Comparison between experiment, theory, and simulation,” J. Chem. Phys.137, 184505 (2012).
[CrossRef] [PubMed]

A. P. Cohen, E. Janai, E. Mogilko, A. B. Schofield, and E. Sloutskin, “Fluid suspensions of colloidal ellipsoids: Direct structural measurements,” Phys. Rev. Lett.107, 238301 (2011).
[CrossRef] [PubMed]

Jerkins, M.

M. Jerkins, M. Schröter, H. L. Swinney, T. J. Senden, M. Saadatfar, and T. Aste, “Onset of mechanical stability in random packings of frictional particles,” Phys. Rev. Lett.101, 018301 (2008).
[CrossRef]

Kilfoil, M. L.

Leunissen, M. E.

C. P. Royall, M. E. Leunissen, A.-P. Hynninen, M. Dijkstra, and A. van Blaaderen, “Re-entrant melting and freezing in a model system of charged colloids,” J. Chem. Phys.124, 244706 (2006).
[CrossRef] [PubMed]

Levitt, A. C.

E. R. Weeks, J. C. Crocker, A. C. Levitt, A. Schofield, and D. A. Weitz, “Three-dimensional direct imaging of structural relaxation near the colloidal glass transition,” Science287, 627–631 (2000).
[CrossRef] [PubMed]

Liber, S. R.

S. R. Liber, S. Borohovich, A. V. Butenko, A. B. Schofield, and E. Sloutskin, “Dense colloidal fluids form denser amorphous sediments,” Proc. Nat. Acad. Sci. U. S. A.110, 5769–5773 (2013).
[CrossRef]

Liu, A. J.

Z. Zhang, N. Xu, D. T. N. Chen, P. Yunker, A. M. Alsayed, K. B. Aptowicz, P. Habdas, A. J. Liu, S. R. Nagel, and A. G. Yodh, “Thermal vestige of the zero-temperature jamming transition,” Nature459, 230–233 (2009).
[CrossRef] [PubMed]

Lu, P. J.

P. J. Lu, E. Zaccarelli, F. Ciulla, A. B. Schofield, F. Sciortino, and D. A. Weitz, “Gelation of particles with short-range attraction,” Nature453, 499–503 (2008).
[CrossRef] [PubMed]

P. J. Lu, P. A. Sims, H. Oki, J. B. Macarthur, and D. A. Weitz, “Target-locking acquisition with real-time confocal (TARC) microscopy,” Opt. Express15, 8702–8712 (2007).
[CrossRef] [PubMed]

Macarthur, J. B.

McCoy, J. H.

X. Cheng, J. H. McCoy, J. N. Israelachvili, and I. Cohen, “Imaging the microscopic structure of shear thinning and thickening colloidal suspensions,” Science333, 1276–1279 (2011).
[CrossRef] [PubMed]

McDonald, I. R.

J.-P. Hansen and I. R. McDonald, Theory of Simple Liquids (Elsevier, 2006).

Mogilko, E.

A. P. Cohen, E. Janai, E. Mogilko, A. B. Schofield, and E. Sloutskin, “Fluid suspensions of colloidal ellipsoids: Direct structural measurements,” Phys. Rev. Lett.107, 238301 (2011).
[CrossRef] [PubMed]

Nagel, S. R.

Z. Zhang, N. Xu, D. T. N. Chen, P. Yunker, A. M. Alsayed, K. B. Aptowicz, P. Habdas, A. J. Liu, S. R. Nagel, and A. G. Yodh, “Thermal vestige of the zero-temperature jamming transition,” Nature459, 230–233 (2009).
[CrossRef] [PubMed]

Oki, H.

Overmars, M.

M. de Berg, O. Cheong, M. van Kreveld, and M. Overmars, Computational Geometry: Algorithms and Applications (Springer, 2008).

Pusey, P. N.

U. Gasser, E. R. Weeks, A. B. Schofield, P. N. Pusey, and D. A. Weitz, “Real-space imaging of nucleation and growth in colloidal crystallization,” Science292, 258–262 (2001).
[CrossRef] [PubMed]

Rapaport, D. C.

A. P. Cohen, E. Janai, D. C. Rapaport, A. B. Schofield, and E. Sloutskin, “Structure and interactions in fluids of prolate colloidal ellipsoids: Comparison between experiment, theory, and simulation,” J. Chem. Phys.137, 184505 (2012).
[CrossRef] [PubMed]

Royall, C. P.

C. P. Royall, M. E. Leunissen, A.-P. Hynninen, M. Dijkstra, and A. van Blaaderen, “Re-entrant melting and freezing in a model system of charged colloids,” J. Chem. Phys.124, 244706 (2006).
[CrossRef] [PubMed]

Saadatfar, M.

M. Jerkins, M. Schröter, H. L. Swinney, T. J. Senden, M. Saadatfar, and T. Aste, “Onset of mechanical stability in random packings of frictional particles,” Phys. Rev. Lett.101, 018301 (2008).
[CrossRef]

T. Aste, M. Saadatfar, and T. J. Senden, “Geometrical structure of disordered sphere packings,” Phys. Rev. E71, 061302 (2005).
[CrossRef]

Schofield, A.

E. R. Weeks, J. C. Crocker, A. C. Levitt, A. Schofield, and D. A. Weitz, “Three-dimensional direct imaging of structural relaxation near the colloidal glass transition,” Science287, 627–631 (2000).
[CrossRef] [PubMed]

Schofield, A. B.

S. R. Liber, S. Borohovich, A. V. Butenko, A. B. Schofield, and E. Sloutskin, “Dense colloidal fluids form denser amorphous sediments,” Proc. Nat. Acad. Sci. U. S. A.110, 5769–5773 (2013).
[CrossRef]

A. P. Cohen, E. Janai, D. C. Rapaport, A. B. Schofield, and E. Sloutskin, “Structure and interactions in fluids of prolate colloidal ellipsoids: Comparison between experiment, theory, and simulation,” J. Chem. Phys.137, 184505 (2012).
[CrossRef] [PubMed]

A. P. Cohen, E. Janai, E. Mogilko, A. B. Schofield, and E. Sloutskin, “Fluid suspensions of colloidal ellipsoids: Direct structural measurements,” Phys. Rev. Lett.107, 238301 (2011).
[CrossRef] [PubMed]

P. J. Lu, E. Zaccarelli, F. Ciulla, A. B. Schofield, F. Sciortino, and D. A. Weitz, “Gelation of particles with short-range attraction,” Nature453, 499–503 (2008).
[CrossRef] [PubMed]

U. Gasser, E. R. Weeks, A. B. Schofield, P. N. Pusey, and D. A. Weitz, “Real-space imaging of nucleation and growth in colloidal crystallization,” Science292, 258–262 (2001).
[CrossRef] [PubMed]

Schröter, M.

M. Jerkins, M. Schröter, H. L. Swinney, T. J. Senden, M. Saadatfar, and T. Aste, “Onset of mechanical stability in random packings of frictional particles,” Phys. Rev. Lett.101, 018301 (2008).
[CrossRef]

Sciortino, F.

P. J. Lu, E. Zaccarelli, F. Ciulla, A. B. Schofield, F. Sciortino, and D. A. Weitz, “Gelation of particles with short-range attraction,” Nature453, 499–503 (2008).
[CrossRef] [PubMed]

Senden, T. J.

M. Jerkins, M. Schröter, H. L. Swinney, T. J. Senden, M. Saadatfar, and T. Aste, “Onset of mechanical stability in random packings of frictional particles,” Phys. Rev. Lett.101, 018301 (2008).
[CrossRef]

T. Aste, M. Saadatfar, and T. J. Senden, “Geometrical structure of disordered sphere packings,” Phys. Rev. E71, 061302 (2005).
[CrossRef]

Sims, P. A.

Sloutskin, E.

S. R. Liber, S. Borohovich, A. V. Butenko, A. B. Schofield, and E. Sloutskin, “Dense colloidal fluids form denser amorphous sediments,” Proc. Nat. Acad. Sci. U. S. A.110, 5769–5773 (2013).
[CrossRef]

A. P. Cohen, E. Janai, D. C. Rapaport, A. B. Schofield, and E. Sloutskin, “Structure and interactions in fluids of prolate colloidal ellipsoids: Comparison between experiment, theory, and simulation,” J. Chem. Phys.137, 184505 (2012).
[CrossRef] [PubMed]

A. P. Cohen, E. Janai, E. Mogilko, A. B. Schofield, and E. Sloutskin, “Fluid suspensions of colloidal ellipsoids: Direct structural measurements,” Phys. Rev. Lett.107, 238301 (2011).
[CrossRef] [PubMed]

Smith, S. W.

S. W. Smith, The Scientist & Engineer’s Guide to Digital Signal Processing (California Technical, 1997).

Swinney, H. L.

M. Jerkins, M. Schröter, H. L. Swinney, T. J. Senden, M. Saadatfar, and T. Aste, “Onset of mechanical stability in random packings of frictional particles,” Phys. Rev. Lett.101, 018301 (2008).
[CrossRef]

van Blaaderen, A.

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G. L. Hunter and E. R. Weeks, “The physics of the colloidal glass transition,” Rep. Prog. Phys.75, 066501 (2012).
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U. Gasser, E. R. Weeks, A. B. Schofield, P. N. Pusey, and D. A. Weitz, “Real-space imaging of nucleation and growth in colloidal crystallization,” Science292, 258–262 (2001).
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http://tacaswell.github.io/tracking/html/

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Figures (4)

Fig. 1
Fig. 1

(a) Unprocessed confocal microscope image of a dense fluid at ϕ = 0.39 of colloidal spheres (σ =2.4 μm; 19 pixels). (b)–(d) Images filtered via convolution with several kernels, whose intensity maps are inset (not to scale): (b) square Tsq (28 pixels), (c) circular pillbox Tcir (28 pixels) and (d) circular gaussian Tgau (24 pixels). After filtering, the same particle-location algorithm is applied; final particle positions are marked with white dots.

Fig. 2
Fig. 2

(a) g(r) derived from particle positions, such as in Fig. 1(b), using Tsq of size 2.3 μm (green squares). Theoretical g(r) calculated for a fluid of hard spheres at the same ϕ (solid grey curve) is completely different; changing the size of Tsq (blue and pink squares) does not improve the agreement. Inset: 2D density distribution function is highly anisotropic and shows an unphysical square symmetry. (b) By contrast, filtering with a rotationally-symmetric kernel, either Tcir (red circles) or Tgau (blue circles) leads to very close agreement for r > σ. Inset: 2D density distribution function is circularly symmetric.

Fig. 3
Fig. 3

(a) P(θ) of disordered particles in the Tsq-filtered image in Fig. 1(b) has a cross-like shape, independent of sample orientation, either along the -axis of the confocal scanner (χ = 0°, red dashed curve) or physically rotated by χ = 45° (black curve). P(θ) is insensitive to a 45°-image rotations before Tsq filtering (blue dotted curve), yet rotates with the image when this rotation follows Tsq filtering (green curve). (b) P(θ) after refining particle positions with fracshift [15] (purple curve) has the same artifact; anisotropy is less pronounced for lower ϕ =0.05 (orange circles). (c) P(θ) after filtration with rotationally-symmetric kernels, either Tcir (black circles) or Tgau (white circles), is correctly isotropic. (a)–(c) are at the same scale. (d) P(θ) for a crystalline sample, filtered with Tsq (red squares) vs. Tcir (blue circles) and Tgau (blue triangles), show smaller differences.

Fig. 4
Fig. 4

(a) Confocal microscope image of a colloidal sediment prepared in a capillary with a centrifuge; effective gravity direction marked by the arrow. (b) P(θ) for this sediment demonstrates preferential bond orientation along the direction of gravity (θ = 0) and at integer multiples of 60° with respect to this direction; data points (blue circles) have uncertainty of ±0.015 and B-spline (solid curve) is included as a guide to the eye. Physically rotating the sample by χ = 30° on the microscope stage, then rotating the resulting P(θ) by −30° (red squares and curve), follows the original data closely.

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