We introduce the heat method for computing the shortest geodesic distance to an arbitrary subset of a given domain. The heat method is robust, efficient, and simple to implement since it is based on solving a pair of standard linear elliptic problems. The resulting algorithm represents a significant breakthrough in the practical computation of distance on a wide variety of geometric domains, si...

Let S be a set of n points and let w be a function that assigns non-negative weights to points in S. The additive weighted distance dw(p, q) between two points p, q ∈ S is defined as w(p) + d(p, q) + w(q) if p 6= q and it is zero if p = q. Here, d(p, q) denotes the (geodesic) Euclidean distance between p and q. A graph G(S,E) is called a t-spanner for the additive weighted set S of points if fo...

Abstract. The present paper discusses the question of formulating and solving minimal-distance problems over the group-manifold of real symplectic matrices. In order to tackle the related optimization problem, the real symplectic group is regarded as a pseudo-Riemannian manifold and a metric is chosen that affords the computation of geodesic arcs in closed forms. Then, the considered minimal-di...

We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish space and dN is a geodesic Borel distance which makes (X, dN ) a possibly branching geodesic space. We show that under some assumptions on the transference plan we can reduce the transport problem to transport problems along family of geodesics. We introduce three assumptions on a given dN -monotone trans...

We present an efficient dynamic data structure that supports geodesic nearest neighbor queries for a set S of point sites in a static simple polygon P . Our data structure allows us to insert a new site in S, delete a site from S, and ask for the site in S closest to an arbitrary query point q ∈ P . All distances are measured using the geodesic distance, that is, the length of the shortest path...

We present a new texture discrimination method for textured color images in the wavelet domain. In each wavelet subband, the correlation between the color bands is modeled by a multivariate generalized Gaussian distribution with fixed shape parameter (Gaussian, Laplacian). On the corresponding Riemannian manifold, the shape of texture clusters is characterized by means of principal geodesic ana...

We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish space and dN is a geodesic Borel distance which makes (X, dN ) a possibly branching geodesic space. We show that under some assumptions on the transference plan we can reduce the transport problem to transport problems along family of geodesics. We introduce two assumptions on the transference plan π whic...