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Finslerian Diffusion and the Bloch–Torrey Equation

  • T. C. J. Dela HaijeEmail author
  • A. Fuster
  • L. M. J. Florack
Conference paper
Part of the Mathematics and Visualization book series (MATHVISUAL)

Abstract

By analyzing stochastic processes on a Riemannian manifold, in particular Brownian motion, one can deduce the metric structure of the space. This fact is implicitly used in diffusion tensor imaging of the brain when cast into a Riemannian framework. When modeling the brain white matter as a Riemannian manifold one finds (under some provisions) that the metric tensor is proportional to the inverse of the diffusion tensor, and this opens up a range of geometric analysis techniques. Unfortunately a number of these methods have limited applicability, as the Riemannian framework is not rich enough to capture key aspects of the tissue structure, such as fiber crossings.An extension of the Riemannian framework to the more general Finsler manifolds has been proposed in the literature as a possible alternative. The main contribution of this work is the conclusion that simply considering Brownian motion on the Finsler base manifold does not reproduce the signal model proposed in the Finslerian framework, nor lead to a model that allows the extraction of the Finslerian metric structure from the signal.

Keywords

Brownian Motion Riemannian Manifold Diffusion Tensor Image Diffusion Tensor Beltrami Operator 
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.

Notes

Acknowledgements

Tom Dela Haije gratefully acknowledges The Netherlands Organisation for Scientific Research (NWO) for financial support. The authors would like to thank Thomas Schultz and Remco Duits for their input regarding the quadratic scaling assumption. Data were provided by the Human Connectome Project, WU-Minn Consortium (Principal Investigators: David Van Essen and Kamil Ugurbil; 1U54MH091657) funded by the 16 NIH Institutes and Centers that support the NIH Blueprint for Neuroscience Research; and by the McDonnell Center for Systems Neuroscience at Washington University.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • T. C. J. Dela Haije
    • 1
    Email author
  • A. Fuster
    • 1
  • L. M. J. Florack
    • 1
  1. 1.Imaging Science and Technology Eindhoven (IST/e)Eindhoven University of TechnologyEindhovenThe Netherlands

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