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Visualization and Processing of Higher Order Descriptors for Multi-Valued Data pp 21–35Cite as

Finslerian Diffusion and the Bloch–Torrey Equation

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Part of the book series: Mathematics and Visualization ((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.

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Notes

  1. 1.

    We only consider reversible Finsler functions, meaning that for all \((\mathbf{x},\mathbf{y}) \in \mathit{TM}\), we have \(F(\mathbf{x},-\mathbf{y}) = F(\mathbf{x},\mathbf{y})\).

  2. 2.

    Different gradient sequences can be used to find the coefficients {D ij}, based on modified (but similar) versions of Eq. (8).

  3. 3.

    In conformity with diffusion MRI literature we omit the diacritical mark above the covectors \(\mathbf{q}\) and \(\mathbf{G}\).

  4. 4.

    There are two diffusion MRI models called generalized DTI, one by Özarslan et al. [34] and one by Liu et al. [27]. Whenever we refer to GDTI we mean the former.

  5. 5.

    Since we only report results of a single experiment we provide only the b-value. A more extensive analysis should consider the influence of the different parameters δ, Δ, and \(\|\mathbf{G}\|\) separately.

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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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  1. Imaging Science and Technology Eindhoven (IST/e), Eindhoven University of Technology, 5612 AZ, Eindhoven, The Netherlands

    T. C. J. Dela Haije, A. Fuster & L. M. J. Florack

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  1. T. C. J. Dela Haije
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  2. A. Fuster
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  3. L. M. J. Florack
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Correspondence to T. C. J. Dela Haije .

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  1. Linköping University, Norrköping, Sweden

    Ingrid Hotz

  2. Visualization and Medical Image Analysis, University of Bonn, Bonn, Germany

    Thomas Schultz

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Dela Haije, T.C.J., Fuster, A., Florack, L.M.J. (2015). Finslerian Diffusion and the Bloch–Torrey Equation. In: Hotz, I., Schultz, T. (eds) Visualization and Processing of Higher Order Descriptors for Multi-Valued Data. Mathematics and Visualization. Springer, Cham. https://doi.org/10.1007/978-3-319-15090-1_2

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