The geodesic distance (path length) and effective resistance are both metrics defined on the vertices of a graph. The effective resistance is a more refined measure of connectivity than the geodesic distance. For example if there are k edge disjoint paths of geodesic distance d between two vertices, then the effective resistance is no more than d k . Thus, the more paths, the closer the vertice...

Shape deformations preserving the intrinsic properties of a surface are called isometries. An isometry deforms a surface without tearing or stretching it, and preserves geodesic distances. We present a technique for matching point set surfaces, which is invariant with respect to isometries. A set of reference points, evenly distributed on the point set surface, is sampled by farthest point samp...

We present the first algorithm to compute the geodesic Fréchet distance between two polygonal curves in a plane with polygonal obstacles, where distances between points are measured as the length of a shortest path between them. Using shortest path structures that we call dynamic and static spotlights, we efficiently construct and propagate reachability information through the free space diagra...

We propose a new image compression method based on geodesic Delaunay triangulations. Triangulations are generated by a progressive geodesic meshing algorithm which exploits the anisotropy of images through a farthest point sampling strategy. This seeding is performed according to anisotropic geodesic distances which force the anisotropic Delaunay triangles to follow the geometry of the image. G...

For manifold learning, it is assumed that high-dimensional sample/data points are on an embedded low-dimensional manifold. Usually, distances among samples are computed to represent the underlying data structure, for a specified distance measure such as the Euclidean distance or geodesic distance. For manifold learning, here we propose a metric according to the angular change along a geodesic l...

This paper reviews both the theory and practice of the numerical computation of geodesic distances on Riemannian manifolds . The notion of Riemannian manifold allows one to define a local metric (a symmetric positive tensor field) that encodes the information about the problem one wishes to solve. This takes into account a local isotropic cost (whether some point should be avoided or not) and a...

Optical coherence tomography (OCT) is a non-invasive imaging technique that can produce images of the eye at the microscopic level. OCT image segmentation to localise retinal layer boundaries is a fundamental procedure for diagnosing and monitoring the progression of retinal and optical nerve disorders. In this paper, we introduce a novel and accurate geodesic distance method (GDM) for OCT segm...

Locally Linear Embedding (LLE) has recently been proposed as a method for dimensional reduction of high-dimensional nonlinear data sets. In LLE each data point is reconstructed from a linear combination of its n nearest neighbors, which are typically found using the Euclidean Distance. We propose an extension of LLE which consists in performing the search for the neighbors with respect to the g...

In recent years trajectory data has become one of the main types of geographic data, and hence algorithmic tools to handle large quantities of trajectories are essential. A single trajectory is typically represented as a sequence of time-stamped points in the plane. In a collection of trajectories one wants to detect maximal groups of moving entities and their behaviour (merges and splits) over...