In this paper, an optimal coordinated motion planning problem for multiple agents subject to constraints on the admissible formation patterns is formulated. Solutions to the problem are reinterpreted as distance minimizing geodesics on a certain manifold with boundary. A geodesic on this manifold may fail to be a solution for different reasons. In particular, if a geodesic possesses conjugate p...

In this paper, the problem of non-rigid shape recognition is viewed from the perspective of metric geometry, and the applicability of diffusion distances within the Gromov-Hausdorff framework is explored. While the commonly used geodesic distance exploits the shortest path between points on the surface, the diffusion distance averages all paths connecting between the points. The diffusion dista...

A new method for obtaining the optimal path to robot navigation in 2-D environments is presented in this paper. To obtain the optimal path we use mathematical morphology in binary worlds and the geodesic distance. The navigation algorithm is based on the search for a path of minimum cost by using the wave-front of the geodesic distance of the mathematical morphology. The optimal path will be th...

A polygonal domain P with n corners V and h holes is a connected polygonal region of genus h whose boundary consists of h + 1 closed chains of n total line segments. The holes and the outer boundary of P are regarded as obstacles. Then, the geodesic distance d(p, q) between any two points p, q in polygonal domain P is defined to be the (Euclidean) length of a shortest obstacle-avoiding path bet...

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 method for lossy compression of genus-0 surfaces is presented. Geometry, texture and other surface attributes are incorporated in a unified manner. The input surfaces are represented by surfels (surface elements), i.e., by a set of disks with attributes. Each surfel, with its attribute vector, is optimally mapped onto a sphere in the sense of geodesic distance preservation. The resulting sphe...

Over the past few years, symmetric positive definite matrices (SPD) have been receiving considerable attention from computer vision community. Though various distance measures have been proposed in the past for comparing SPD matrices, the two most widely-used measures are affine-invariant distance and log-Euclidean distance. This is because these two measures are true geodesic distances induced...

The concept of a geodesic invex subset of a Riemannian manifold is introduced. Geodesic invex and preinvex functions on a geodesic invex set with respect to particular maps are defined. The relation between geodesic invexity and preinvexity of functions on manifolds is studied. Using proximal subdifferential, certain results concerning extremum points of a non smooth geodesic preinvex function ...