A Random Riemannian Metric for Probabilistic Shortest-Path Tractography
نویسندگان
چکیده
Shortest-path tractography (SPT) algorithms solve global optimization problems defined from local distance functions. As diffusion MRI data is inherently noisy, so are the voxelwise tensors from which local distances are derived. We extend Riemannian SPT by modeling the stochasticity of the diffusion tensor as a “random Riemannian metric”, where a geodesic is a distribution over tracts. We approximate this distribution with a Gaussian process and present a probabilistic numerics algorithm for computing the geodesic distribution. We demonstrate SPT improvements on data from the Human Connectome Project.
منابع مشابه
Improved segmentation of white matter tracts with adaptive Riemannian metrics
We present a novel geodesic approach to segmentation of white matter tracts from diffusion tensor imaging (DTI). Compared to deterministic and stochastic tractography, geodesic approaches treat the geometry of the brain white matter as a manifold, often using the inverse tensor field as a Riemannian metric. The white matter pathways are then inferred from the resulting geodesics, which have the...
متن کاملShortest Path through Random Points
Let (M, g1) be a compact d-dimensional Riemannian manifold. Let Xn be a set of n sample points in M drawn randomly from a Lebesgue density f . Define two points x, y ∈ M . We prove that the normalized length of the power-weighted shortest path between x, y passing through Xn converges to the Riemannian distance between x, y under the metric gp = f g1, where p > 1 is the power parameter. The res...
متن کاملShortest Path Embeddings of Graphs on Surfaces
The classical theorem of Fáry states that every planar graph can be represented by an embedding in which every edge is represented by a straight line segment. We consider generalizations of Fáry’s theorem to surfaces equipped with Riemannian metrics. In this setting, we require that every edge is drawn as a shortest path between its two endpoints and we call an embedding with this property a sh...
متن کاملProbabilistic Shortest Path Tractography in DTI Using Gaussian Process ODE Solvers
Tractography in diffusion tensor imaging estimates connectivity in the brain through observations of local diffusivity. These observations are noisy and of low resolution and, as a consequence, connections cannot be found with high precision. We use probabilistic numerics to estimate connectivity between regions of interest and contribute a Gaussian Process tractography algorithm which allows f...
متن کاملPhysarum Can Solve the Shortest Path Problem on Riemannian Surface Mathematically Rigorously
Abstract: In this paper we report a mathematical theory inspired by P. polycephalum. It is an amoeba-like organism that exhibits the shortest path-finding behavior in a labyrinth. Physarum solver is a model equation of this interesting behavior of the creature. In this paper, we prove mathematically rigorously it on graph included in two dimensional manifold with Riemannian metric, that is, Rie...
متن کامل