A Random Riemannian Metric for Probabilistic Shortest-Path Tractography

Hauberg, Søren; Schober, Michael; Liptrot, Matthew G.; Hennig, Philipp; Feragen, Aasa

doi:10.1007/978-3-319-24553-9_73

Cited by 7 publications

(7 citation statements)

References 20 publications

(33 reference statements)

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“…This strategy offers novel ways to quantify and visualize uncertainty arising from the numerical computation and allow marginalization over a space of feasible solutions. Hauberg et al [40] extended this work and incorporated data uncertainty in DTI by sub-sampling the diffusion gradients and solving the noisy ODE. Several other studies using the shortest path algorithms in fiber tracking can be found in the literature [39,63].…”

Section: Methodsmentioning

confidence: 99%

“…Color coding the fiber tracts according to their seed points [22] does not suffice to minimize the complexity of the visualization. Schober et al [88] and Hauberg et al [40] used wobbly spaghetti plot that emphasize the fact that the individual resulting paths cannot be considered as real fibers in the brain which is a common misinterpretation in spaghetti plot. Instead, they are uncertain estimates of fibers.…”

Section: Global Uncertainty Visualizationmentioning

confidence: 99%

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Uncertainty in the DTI Visualization Pipeline

Siddiqui

Höllt

Vilanova

2021

Mathematics and Visualization

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Diffusion-Weighted Magnetic Resonance Imaging (DWI) enables the in-vivo visualization of fibrous tissues such as white matter in the brain. Diffusion-Tensor Imaging (DTI) specifically models the DWI diffusion measurements as a second order-tensor. The processing pipeline to visualize this data, from image acquisition to the final rendering, is rather complex. It involves a considerable amount of measurements, parameters and model assumptions, all of which generate uncertainties in the final result which typically are not shown to the analyst in the visualization. In recent years, there has been a considerable amount of work on the visualization of uncertainty in DWI, and specifically DTI. In this chapter, we primarily focus on DTI given its simplicity and applicability, however, several aspects presented are valid for DWI as a whole. We explore the various sources of uncertainties involved, approaches for modeling those uncertainties, and, finally, we survey different strategies to visually represent them. We also look at several related methods of uncertainty visualization that have been applied outside DTI and discuss how these techniques can be adopted to the DTI domain. We conclude our discussion with an overview of potential research directions.

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Section: Methodsmentioning

confidence: 99%

Section: Global Uncertainty Visualizationmentioning

confidence: 99%

Uncertainty in the DTI Visualization Pipeline

Siddiqui

Höllt

Vilanova

2021

Mathematics and Visualization

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“…The most successful area of research to date has been on the development of Bayesian methods for global optimisation [Snoek et al, 2012], which have become standard to the point of being embedded into commercial software [The MathWorks Inc.] and deployed in realistic [Acerbi, 2018, Paul et al, 2018 and indeed high-profile [Chen et al, 2018] applications. Other numerical tasks have yet to experience the same level of practical interest, though we note applications of probabilistic methods for cubature in computer graphics [Marques et al, 2013] and tracking [Prüher et al, 2018], as well as applications of probabilistic numerical methods in medical tractography [Hauberg et al, 2015] and nonlinear state estimation in an industrial context.…”

Section: Killer Appsmentioning

confidence: 99%

A modern retrospective on probabilistic numerics

Oates

Sullivan

2019

Stat Comput

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This article attempts to place the emergence of probabilistic numerics as a mathematical-statistical research field within its historical context and to explore how its gradual development can be related both to applications and to a modern formal treatment. We highlight in particular the parallel contributions of Sul din and Larkin in the 1960s and how their pioneering early ideas have reached a degree of maturity in the intervening period, mediated by paradigms such as average-case analysis and information-based complexity. We provide a subjective assessment of the state of research in probabilistic numerics and highlight some difficulties to be addressed by future works.

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“…Finally, second order tensors are in 1-1 correspondence with covariance matrices, and any metric on second order tensors therefore also defines a metric on centered multivariate normal distributions. Statistics on probability distributions have many possible applications, from population statistics on uncertain tractography results represented as Gaussian Processes [27,20] via evolutionary algorithms for optimization [19], to information geometry [3].…”

Section: Why Are Second Order Tensors Still Interesting?mentioning

confidence: 99%

Geometries and Interpolations for Symmetric Positive Definite Matrices

Feragen

Fuster

2017

Mathematics and Visualization

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In this survey we review classical and recently proposed Riemannian metrics and interpolation schemes on the space of symmetric positive definite (SPD) matrices. We perform simulations that illustrate the problem of tensor fattening not only in the usually avoided Frobenius metric, but also in other classical metrics on SPD matrices such as the Wasserstein metric, the affine invariant / Fisher Rao metric, and the log Euclidean metric. For comparison, we perform the same simulations on several recently proposed frameworks for SPD matrices that decompose tensors into shape and orientation. In light of the simulation results, we discuss the mathematical and qualitative properties of these new metrics in comparison with the classical ones. Finally, we explore the nonlinear variation of properties such as shape and scale throughout principal geodesics in different metrics, which affects the visualization of scale and shape variation in tensorial data. With the paper, we will release a software package with Matlab scripts for computing the interpolations and statistics used for the experiments in the paper. 1

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A Random Riemannian Metric for Probabilistic Shortest-Path Tractography

Cited by 7 publications

References 20 publications

Uncertainty in the DTI Visualization Pipeline

Uncertainty in the DTI Visualization Pipeline

A modern retrospective on probabilistic numerics

Geometries and Interpolations for Symmetric Positive Definite Matrices

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