Morphological Analysis of Brownian Motion for Physical Measurements

  • Élodie Puybareau
  • Hugues Talbot
  • Noha Gaber
  • Tarik Bourouina
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10225)

Abstract

Brownian motion is a well-known, apparently chaotic motion affecting microscopic objects in fluid media. The mathematical and physical basis of Brownian motion have been well studied but not often exploited. In this article we propose a particle tracking methodology based on mathematical morphology, suitable for Brownian motion analysis, which can provide difficult physical measurements such as the local temperature and viscosity. We illustrate our methodology on simulation and real data, showing that interesting phenomena and good precision can be achieved.

Keywords

Random walk Particle suspension Particle tracking Segmentation 

References

  1. 1.
    Brown, R.: A brief account of microscopical observations made in the months of June, July and August, 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies. Philos. Mag. 4(21), 161–173 (1827)Google Scholar
  2. 2.
    Einstein, A.: Über die von der molekularkinetischen theorie der wärme geforderte bewegung von in ruhenden flüssigkeiten suspendierten teilchen. Ann. Phys. 322(8), 549–560 (1905)CrossRefMATHGoogle Scholar
  3. 3.
    Perrin, J.: Mouvement brownien et réalité moléculaire. Ann. Chim. Phys. 18(8), 5–114 (1909)Google Scholar
  4. 4.
    Bachelier, L.: Théorie de la spéculation. Ann. Sci. l’École Normale Supér. 3(17), 21–86 (1900)CrossRefMATHGoogle Scholar
  5. 5.
    Skellam, J.G.: Random dispersal in theoretical populations. Biometrika 38(1/2), 196–218 (1951)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Colding, E., et al.: Random walk models in biology. J. R. Soc. Interface 5, 813–834 (2008)CrossRefGoogle Scholar
  7. 7.
    De Gennes, P.G.: Scaling Concepts in Polymer Physics. Cornell University Press, Ithaca (1979)Google Scholar
  8. 8.
    Grady, L.: Random walks for image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 28(11), 1768–1783 (2006)CrossRefGoogle Scholar
  9. 9.
    Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E.: Equation of state calculations by fast computing machines. J. Chem. Phys. 21(6), 1087–1092 (1953)CrossRefGoogle Scholar
  10. 10.
    Hastings, W.: Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57(1), 97–109 (1970)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Geman, S., Geman, D.: Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intell. 6, 721–741 (1984)CrossRefMATHGoogle Scholar
  12. 12.
    Bertsimas, D., Vempala, S.: Solving convex programs by random walks. J. ACM (JACM) 51(4), 540–556 (2004)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Park, J., Choi, C., Kihm, K.: Temperature measurement for a nanoparticle suspension by detecting the Brownian motion using optical serial sectioning microscopy (OSSM). Meas. Sci. Technol. 16(7), 1418 (2005)CrossRefGoogle Scholar
  14. 14.
    Donsker, M.: An invariance principle for certain probability limit theorems. Mem. Am. Math. Soc. 6 (1951)Google Scholar
  15. 15.
    Pólya, G.: Über eine aufgabe betreffend die irrfahrt im strassennetz. Math. Ann. 84, 149–160 (1921)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Nordlund, K.: Basics of Monte Carlo simulations. http://www.acclab.helsinki.fi/~knordlun/mc/mc5nc.pdf
  17. 17.
    Gaber, N., Malak, M., Marty, F., Angelescu, D.E., Richalot, E., Bourouina, T.: Optical trapping and binding of particles in an optofluidic stable fabry-pérot resonator with single-sided injection. Lab Chip 14(13), 2259–2265 (2014)CrossRefGoogle Scholar
  18. 18.
    Allan, D., et al.: Trackpy: fast, flexible particle-tracking toolkit. http://soft-matter.github.io/trackpy
  19. 19.
    Najman, L., Talbot, H. (eds.): Mathematical Morphology: from Theory to Applications. ISTE-Wiley, London, September 2010. ISBN 978-1848212152Google Scholar
  20. 20.
    Vincent, L.: Morphological grayscale reconstruction in image analysis: applications and efficient algorithms. IEEE Trans. Image Process. 2(2), 176–201 (1993)CrossRefGoogle Scholar
  21. 21.
    Vincent, L.: Grayscale area openings and closings, their efficient implementation and applications. In: Proceedings of the Conference on Mathematical Morphology and Its Applications to Signal Processing, Barcelona, Spain, pp. 22–27, May 1993Google Scholar
  22. 22.
    Meijster, A., Wilkinson, H.: A comparison of algorithms for connected set openings and closings. IEEE Trans. Pattern Anal. Mach. Intell. 24(4), 484–494 (2002)CrossRefGoogle Scholar
  23. 23.
    Géraud, T., Talbot, H., Vandroogenbroeck, M.: Algorithms for mathematical morphology. In: [19] Chap. 12, pp. 323–354. ISBN 978-1848212152Google Scholar
  24. 24.
    Bertrand, G., Couprie, M.: Transformations topologiques discretes. In: Coeurjolly, D., Montanvert, A., Chassery, J. (eds.) Géométrie discrète et images numériques, pp. 187–209. Hermès, Mumbai (2007)Google Scholar
  25. 25.
    Matheron, G.: The Theory of Regionalized Variables and Its Applications, vol. 5. École national supérieure des mines, Paris (1971)Google Scholar
  26. 26.
    Olea, R.A.: Optimal contour mapping using universal kriging. J. Geophys. Res. 79(5), 695–702 (1974)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Élodie Puybareau
    • 1
    • 2
  • Hugues Talbot
    • 2
  • Noha Gaber
    • 3
  • Tarik Bourouina
    • 2
  1. 1.EPITA Research and DevelopmentLe Kremlin-BicetreFrance
  2. 2.Université Paris-Est/ESIEENoisy-le-GrandFrance
  3. 3.National Oceanic and Atmospheric AdministrationWashington, D.C.USA

Personalised recommendations