Riemann-Finsler Geometry for Diffusion Weighted Magnetic Resonance Imaging

Florack, Luc; Fuster, Andrea

doi:10.1007/978-3-642-54301-2_8

Cited by 9 publications

(5 citation statements)

References 56 publications

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“…However, higher order diffusion models may benefit from our approach as well, provided one can define a proper metrical distance. For example, the framework proposed in [14] stipulates a Finsler metric for geodesic tractography in HARDI, which can in principle be adapted in a similar way to our modification of the Riemannian metric in DTI.…”

Section: Resultsmentioning

confidence: 99%

Adjugate Diffusion Tensors for Geodesic Tractography in White Matter

et al. 2015

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One of the approaches in diffusion tensor imaging is to consider a Riemannian metric given by the inverse diffusion tensor. Such a metric is used for geodesic tractography and connectivity analysis in white matter. We propose a metric tensor given by the adjugate rather than the previously proposed inverse diffusion tensor. The adjugate metric can also be employed in the sharpening framework. Tractography experiments on synthetic and real brain diffusion data show improvement for high-curvature tracts and in the vicinity of isotropic diffusion regions relative to most results for inverse (sharpened) diffusion tensors, and especially on real data. In addition, adjugate tensors are shown to be more robust to noise.

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Section: Resultsmentioning

confidence: 99%

Adjugate Diffusion Tensors for Geodesic Tractography in White Matter

et al. 2015

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“…See e.g. [15] where in a highly anisotropic situation (diffusion weighted magnetic resonance imaging of brain), the measurement returned a Finsler metric which is very close to a Riemannian metric.…”

Section: Limit Diffusion As a Riemannian Brownian Motion With Driftmentioning

confidence: 99%

Geodesic Random Walks, Diffusion Processes and Brownian Motion on Finsler Manifolds

Matveev

Pavlyukevich

2021

J Geom Anal

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“…Although Riemannian geodesic tractography does not cover traditional "streamline tractography", the latter may be viewed as a singular limit that can be inferred from a sequence of Riemannian metrics. In turn, Finsler geodesic tractography may be related to an orientation-parametrized family of Riemannian metrics (Astola, 2010;Florack and Fuster, 2014;Florack et al, 2015;Dela Haije, 2017;Dela Haije et al, 2019). In this sense Riemannian geodesic tractography plays an intermediate role, being a generalisation of the original streamline case and a special instance of the Finslerian case.…”

Section: Theorymentioning

confidence: 99%

Geodesic Uncertainty in Diffusion MRI

Sengers

Florack

Fuster

2021

Front. Comput. Sci.

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We study theoretical and operational issues of geodesic tractography, a geometric methodology for retrieving biologically plausible neural fibers in the brain from diffusion weighted magnetic resonance imaging. The premise is that true positives are geodesics in a suitably constructed metric space, but unlike traditional first order methods these are not a priori constrained to connect nongeneric points on subdimensional manifolds, such as the characteristics in traditional streamline methods. By virtue of the Hopf-Rinow theorem geodesic tractography furnishes a huge amount of redundancy, ensuring the a priori existence of at least one tentative fiber between any two points and permitting additional tractometric and data-extrinsic constraints for (fuzzy or crisp) classification of true and false positives. In our feasibility study we consider a hybrid paradigm that unifies existing ideas on tractography, combining deterministic and probabilistic elements in a way naturally supported by metric geometry. Particular attention is paid to an analytical prediction of geodesic deviation on numerically computed geodesics, a ‘tidal’ effect induced by small perturbations resulting from data noise. Taking these effects into account clarifies the inherent uncertainty of geodesics, while simultaneosuly offering a dimensionality reduction of the tractography problem.

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Riemann-Finsler Geometry for Diffusion Weighted Magnetic Resonance Imaging

Cited by 9 publications

References 56 publications

Adjugate Diffusion Tensors for Geodesic Tractography in White Matter

Adjugate Diffusion Tensors for Geodesic Tractography in White Matter

Geodesic Random Walks, Diffusion Processes and Brownian Motion on Finsler Manifolds

Geodesic Uncertainty in Diffusion MRI

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