Abstract A conjecture of Burns and Knieper (1991) asks whether a 2-plane with metric without conjugate points, geodesic foliation whose lines are at bounded Hausdorff distance, is necessarily flat. We prove this in two cases: under the hypothesis that plane admits total curvature, visibility some point. Along way, we show all line foliations on Riemannian must be homeomorphic to standard one.

The objective of this work is to show how geodesic propagation techniques known in mathematical morphology can be employed to reconstruct terrain surfaces, and more generally grid data, from contour maps. We present the so-called generalised geodesic distance that was briefly introduced in a previous article. The shortest paths defined by the generalised geodesic distance can be used to linearl...

Let P be a simple polygon of n vertices and let S be a set of n points called sites lying in the interior of P. The geodesic-distance between two sites x and y in P is defined as the length of the shortest polygonal path connecting x and y constrained to lie in P. It is a useful notion in graphics problems concerning visibility between objects, computer vision problems concerned with the descri...

The purpose of this paper is to introduce a form of update based on the minimization of the geodesic distance on a graph. We provide a characterization of this class using settheoretic operators and show that such operators bijectively correspond to geodesic metrics. As distance is generated by distinguishability, our framework is appropriate in contexts where distance is generated by threshold...

Finding shortest paths and shortest distances between points on a surface S in three-dimensional space is a well-studied problem in differential geometry and computational geometry. The shortest path between two points on S is denoted a geodesic path on the surface and the shortest distance between two points on S is denoted a geodesic distance. In this survey, we consider the case where a disc...

Let P be a simple polygon of n vertices and let S be a set of N points lying in the interior of P . A geodesic disk GD(p, r) with center p and radius r is the set of points in P that have a geodesic distance ≤ r from p (where the geodesic distance is the length of the shortest polygonal path connection that lies in P ). In this paper we present an output sensitive algorithm for finding all N ge...

Hashing is a popular solution to Approximate Nearest Neighbor (ANN) problems. Many hashing schemes aim at preserving the Euclidean distance of the original data. However, it is the geodesic distance rather than the Euclidean distance that more accurately characterizes the semantic similarity of data, especially in a high dimensional space. Consequently, manifold based hashing methods have achie...

Spectral clustering is a method of subspace clustering which is suitable for the data of any shape and converges to global optimal solution. By combining concepts of shared nearest neighbors and geodesic distance with spectral clustering, a self-adaptive spectral clustering based on geodesic distance and shared nearest neighbors was proposed. Experiments show that the improved spectral clusteri...