This paper presents new conditions under which sub-Riemannian distance can be measured by means of a C 1 sub-Riemannian geodesic.

This paper presents a novel method for 2D and 3D shape matching that is insensitive to articulation. It uses the eccentricity transform, which is based on the computation of geodesic distances. Geodesic distances computed over a 2D or 3D shape are articulation insensitive. The eccentricity transform considers the length of the longest geodesics. Histograms of the eccentricity transform characte...

A geodesic is a parameterized curve on a Riemannian manifold governed by a second order partial differential equation. Geodesics are notoriously unstable: small perturbations of the underlying manifold may lead to dramatic changes of the course of a geodesic. Such instability makes it difficult to use geodesics in many applications, in particular in the world of discrete geometry. In this paper...

Intra-shape deformations complicate 3D shape recognition and therefore need proper modeling. Thereto, an isometric deformation model is used in this paper. The method proposed does not need explicit point correspondences for the comparison of 3D shapes. The geodesic distance matrix is used as an isometry-invariant shape representation. Two approaches are described to arrive at a sampling order ...

We propose data structures for answering a geodesic-distance query between two query points in a two-dimensional or three-dimensional dynamic environment, in which obstacles are deforming continuously. Each obstacle in the environment is modeled as the convex hull of a continuously deforming point cloud. The key to our approach is to avoid maintaining the convex hull of each point cloud explici...

We propose a novel algorithm to compute Voronoi diagrams of order k in arbitrary 2D and 3D domains. The algorithm is based on a fast ordered propagation distance transformation called occlusion points propagation geodesic distance transformation (OPPGDT) which is robust and linear in the domain size, and has higher accuracy than other geodesic distance transformations published before. Our appr...

The radial basis function (RBF) collocation techniques for the numerical solution of partial differential equation problems are increasingly popular in recent years thanks to their striking merits being inherently meshless, integration-free, and highly accurate. However, the RBF-based methods have markedly been limited to handle isotropic problems due to the use of the isotropic Euclidean dista...

We present a novel data smoothing and analysis framework for cortical thickness data defined on the brain cortical manifold. Gaussian kernel smoothing, which weights neighboring observations according to their 3D Euclidean distance, has been widely used in 3D brain images to increase the signal-to-noise ratio. When the observations lie on a convoluted brain surface, however, it is more natural ...

We present a new framework for multidimensional shape analysis. The proposed framework represents solid objects as points on an infinite-dimensional Riemannian manifold and distances between objects as minimal length geodesic paths. Intershape distance forms the foundation for shape-based statistical analysis. The proposed method incorporates a metric that naturally prevents self-intersections ...

Geodesic distance matrices can reveal shape properties that are largely invariant to non-rigid deformations, and thus are often used to analyze and represent 3-D shapes. However, these matrices grow quadratically with the number of points. Thus for large point sets it is common to use a low-rank approximation to the distance matrix, which fits in memory and can be efficiently analyzed using met...