Skip to main content

Estimation of Velocity Fields and Propagation on Non-Euclidian Domains: Application to the Exploration of Cortical Spatiotemporal Dynamics

  • 1610 Accesses

Part of the Lecture Notes in Mathematics book series (LNMBIOS,volume 1983)

Abstract

Better understanding of the interrelationship between the brain’s structural architecture and functional processing is one of the leading questions in today’s integrative neuroscience. Non-invasive imaging techniques have revealed as major contributing tools to this endeavor, which obviously requires the cooperation of space and time-resolved experimental evidences. Electromagnetic brain mapping using magneto- and electro-encephalography (M/EEG) source estimation is so far the imaging method with the best trade-off between spatial and temporal resolution (∼1 cm and <1ms respectively, [4,5]). Combined with individual anatomical information from Magnetic Resonance Imaging (MRI) and statistical inference techniques [35], M/EEG source estimation has now reached considerable maturity and may indeed be considered as a true functional brain imaging technique.

With or without considering the estimation of M/EEG generators as a priority, the analysis of M/EEG data is classically motivated by the detection of salient features in the time course of surface measures either/both at the sensor or/and cortical levels. These features of interest may be extracted from waveform peaks and/or their related time latencies, band-specific oscillatory patterns surging from a time-frequency decomposition of the data or regional activation blobs at the cortical level.

By nature, the extraction of such features usually results from an extremely reductive – though pragmatic – point of view on the spatio-temporal dynamics of brain responses. It is pragmatic because it responds to a need for the reduction in the information mass from the original data. It is reductive though because most studies report on either/both the localization or/and the dynamical properties of brain events as defined according to the investigator, hence with an uncontrolled level of subjectivity.

Keywords

  • Optical Flow
  • Angular Error
  • Advection Equation
  • Global Intensity
  • Arbitrary Surface

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. G. Allaire, Analyse numérique et optimisation, Editions de l’Ecole Polytechnique, 2005.

    Google Scholar 

  2. G. Aubert, R. Deriche, and P. Kornprobst, Computing optical flow via variational techniques, SIAM J. Appl. Math., 1999, 60(1), 156–182.

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. P. Azerad and G. Pousin, Inégalite de Poincaré courbe pour le traitement variationnel de l’équation de transport, CR Acad. Sci. Paris, 1996, 721–727.

    Google Scholar 

  4. S. Baillet, J.C. Mosher, and R.M. Leahy, Electromagnetic brain mapping, IEEE Signal Processing Magazine, November, 2001.

    Google Scholar 

  5. J.M. Barrie, W.J. Freeman, and M.D. Lenhart, Spatiotemporal Analysis of Prepyriform, Visual, Auditory, and Somesthetic Surface EEGs in Trained Rabbits, Journal of Neurophysiology, 1996, 76, 520–539.

    Google Scholar 

  6. J.L. Barron, D.J. Fleet, and S.S. Beauchemin, Performance of optical flow techniques, International Journal of Computer Vision, 1994, 12, 43–77.

    CrossRef  Google Scholar 

  7. J.L. Barron and A. Liptay, Measuring 3-D plant growth using optical flow, Bioimaging, 1997, 5, 82–86.

    CrossRef  Google Scholar 

  8. S.S. Beauchemin and J.L. Barron, The computation of optical flow, ACM Computing Surveys, 1995, 27(3), 433–467.

    CrossRef  Google Scholar 

  9. D. Bereziat, I. Herlin, and L. Younes, A Generalized Optical Flow Constraint and its Physical Interpretation, Proc. Conf. Computer Vision and Pattern Recognition, 2000, 02, 2487.

    Google Scholar 

  10. O. Besson and G. De Montmollin, Space-time integrated least squares: a time-marching approach, International Journal for Numerical Methods in Fluids, 2004, 44(5), 525–543.

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. D.L. Book, J.P. Boris, and K. Hain, Flux-Corrected Transport. II – Generalizations of the method, Journal of Computational Physics, 1975, 18, 248–283.

    CrossRef  MATH  Google Scholar 

  12. A. Bossavit, Whitney forms: a class of finite elements for three-dimensional computations in electromagnetism, IEE Proceedings, 1988, 135, Part A, 493–500, 8, November.

    Google Scholar 

  13. N.E.H. Bowler, C.E. Pierce, and A. Seed, Development of a precipitation nowcasting algorithm based upon optical flow techniques, Journal of Hydrology, 2004, 288, 74–91.

    CrossRef  Google Scholar 

  14. U. Clarenz, M. Rumpf, and A. Telea, Finite elements on point based surfaces, Proc. of Symp. on Point-Based Graphics, 2004, 201–211.

    Google Scholar 

  15. T. Corpetti, D. Heitz, G. Arroyo, E. Mémin, and A. Santa-Cruz, Fluid experimental flow estimation based on an optical-flow scheme, Experiments in fluids, 2006, 40, 80–97.

    CrossRef  Google Scholar 

  16. T. Corpetti, E. Mémin, and P. Pérez, Dense Estimation of Fluid Flows, IEEE Transactions on Pattern Analysis and Machine Intelligence, 2002, 24 (3), 365–380.

    CrossRef  Google Scholar 

  17. D. Cosmelli, O. David, J.P. Lachaux, J. Martinerie, L. Garnero, B. Renault, and F. Varela, Waves of consciousness: ongoing cortical patterns during binocular rivalry, Neuroimage, 2004, 23, 128–140.

    CrossRef  Google Scholar 

  18. U. Diewald, T. Preusser, and M. Rumpf, Anisotropic Diffusion in Vector Field Visualization on Euclidean Domains and Surfaces, IEEE Transactions on Visualization and Computer Graphics, 2000, 6, 139–149.

    CrossRef  Google Scholar 

  19. M.P. Do Carmo, Riemannian Geometry, Birkhuser, 1993.

    Google Scholar 

  20. O. Druet, E. Hebey, and F. Robert, Blow-up Theory for Elliptic PDEs in Riemannian Geometry, Princeton University Press, Princeton, N.J., 2004.

    MATH  Google Scholar 

  21. A. Giachetti, M. Campani, and V. Torre, The use of optical flow for road navigation, IEEE Trans. Robotics and Automation, 1998, 14, 34–48.

    CrossRef  Google Scholar 

  22. J.L. Guermond, A finite element technique for solving first order PDE’s in L1, SIAM J. Numer. Anal., 2004, 42(2), 714–737.

    CrossRef  MathSciNet  MATH  Google Scholar 

  23. B.K.P. Horn and B.G. Schunck, Determining optical flow, Artificial Intelligence, 1981, 17, 185–204.

    CrossRef  Google Scholar 

  24. A. Imiya, H. Sugaya, A. Torii, and Y. Mochizuki, Variational Analysis of Spherical Images, Proc. Computer Analysis of Images and Patterns, 2005.

    Google Scholar 

  25. T. Inouye, K. Shinosaki, A. Iyama, Y. Matsumoto, S. Toi, and T. Ishthara, Potential flow of frontal midline theta activity during a mental task in the human electroencephalogram, Neurosci. Lett., 1994, 169, 145–148.

    Google Scholar 

  26. T. Inouye, K. Shinosaki, S. Toi, Y. Matsumoto, and N. Hosaka, Potential flow of alpha- activity in the human electroencephalogram, Neurosci. Lett., 1995, 187, 29–32.

    Google Scholar 

  27. V.K. Jirsa, K.J. Jantzen, A. Fuchs, and J.A.S. Kelso, Spatiotemporal forward solution of the EEG and MEG using networkmodeling, IEEE Trans. Medical Imaging, 2002, 21, 493–504.

    CrossRef  Google Scholar 

  28. C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, New York, 1987.

    MATH  Google Scholar 

  29. J. Lefèvre, G. Obozinski, and S. Baillet, Imaging Brain Activation Streams from Optical Flow Computation on 2-Riemannian Manifold, Proc. Information Processing in Medical Imaging, 2007, 470–481.

    Google Scholar 

  30. H. Liu, T. Hong, M. Herman, T. Camus, and R. Chellapa, Accuracy vs. efficiency trade-off in optical flow algorithms, Comput. Vision Image Understand., 1998, 72, 271–286.

    CrossRef  Google Scholar 

  31. L.M. Lui, Y. Wang, and T.F. Chan, Solving PDEs on Manifold using Global Conformal Parameterization, Proc. Variational, Geometric, and Level Set Methods in Computer Vision, 2005, 307–319.

    Google Scholar 

  32. F. Mémoli, G. Sapiro, and S. Osher, Solving variational problems and partial differential equations mapping into general target manifolds, J. Comput. Phys., 2004, 195, 263–292.

    CrossRef  MathSciNet  MATH  Google Scholar 

  33. H.H. Nagel, On the estimation of optical flow: relations between different approaches and some new results., Artificial Intelligence, 1987, 33, 299–324.

    Google Scholar 

  34. P.L. Nunez, Toward a quantitative description of large-scale neocortical dynamic function and EEG, Behavioral and Brain Sciences, 2000, 23, 371–398.

    CrossRef  Google Scholar 

  35. D. Pantazis, T.E. Nichols, S. Baillet, and R.M. Leahy, A comparison of random field theory and permutation methods for the statistical analysis of MEG data., Neuroimage, 2005, 25, 383–394.

    CrossRef  Google Scholar 

  36. P. Perrochet and P. Azérad, Space-time integrated least-squares: Solving a pure advection equation with a pure diffusion operator., J.Computer.Phys., 1995, 117, 183–193.

    CrossRef  MATH  Google Scholar 

  37. W. Rekik, D. Bereziat, and S. Dubuisson, Optical flow computation and visualization in spherical context. Application on 3D+ t bio-cellular sequences., Engineering in Medicine and Biology Society, 2006. EMBS’06. 28th Annual International Conference of the IEEE, 2006, 1645–1648.

    Google Scholar 

  38. J.A. Rossmanith, D.S. Bale, and R.J. LeVeque, A wave propagation algorithm for hyperbolic systems on curved manifolds, J. Comput. Phys., 2004, 199, 631–662.

    CrossRef  MathSciNet  MATH  Google Scholar 

  39. C. Schnörr, Determining optical flow for irregular domains by minimizing quadratic functionals of a certain class., Int. J. Computer Vision, 1991, 6(1), 25–38.

    CrossRef  Google Scholar 

  40. N. Sochen, R. Deriche, and L. Lopez Perez, The Beltrami Flow over Implicit Manifolds, Proc. of the Ninth IEEE International Conference on Computer Vision, 2003, 832.

    Google Scholar 

  41. A. Spira and R. Kimmel, Geometric curve flows on parametric manifolds, J. Comput. Phys., 2007, 223, 235–249.

    CrossRef  MathSciNet  MATH  Google Scholar 

  42. Z. Sun, G. Bebis, and R. Miller, On-Road Vehicle Detection: A Review, IEEE Trans. Pattern Anal. Mach. Intell., 2006, 28, 694–711.

    CrossRef  Google Scholar 

  43. A. Torii, A. Imiya, H. Sugaya, and Y. Mochizuki, Optical Flow Computation for Compound Eyes: Variational Analysis of Omni-Directional Views, Proc. Brain, Vision and Artificial Intelligence, 2005.

    Google Scholar 

  44. K. Wang, Weiwei, Y. Tong, M. Desbrun, and P. Schroder, Edge subdivision schemes and the construction of smooth vector fields, ACM Trans. Graph., 2006, 25, 1041–1048.

    Google Scholar 

  45. J. Weickert and C. Schnörr, A theoretical framework for convex regularizers in PDE-based computation of image motion, International Journal of Computer Vision, 2001, 45(3), 245–264.

    CrossRef  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Sylvain Baillet .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Lefèvre, J., Baillet, S. (2009). Estimation of Velocity Fields and Propagation on Non-Euclidian Domains: Application to the Exploration of Cortical Spatiotemporal Dynamics. In: Ammari, H. (eds) Mathematical Modeling in Biomedical Imaging I. Lecture Notes in Mathematics(), vol 1983. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03444-2_5

Download citation