In this paper, we investigate the L1 geodesic farthest neighbors in a simple polygon P , and address several fundamental problems related to farthest neighbors. Given a subset S ⊆ P , an L1 geodesic farthest neighbor of p ∈ P from S is one that maximizes the length of L1 shortest path from p in P . Our list of problems include: computing the diameter, radius, center, farthestneighbor Voronoi di...

The eccentricity transform associates to each point of a shape the geodesic distance to the points farthest away. This can be used to decompose shapes into parts based on the connectivity of the isoheight lines. The adjacency graph of the obtained decomposition is similar to a Reeb graph and characterizes the topology of the object. Graph pyramids are used to efficiently compute the decompositi...

We have measured by tunneling spectroscopy the electronic density of states in a nonsuperconducting wire in good contact with a superconductor, at distances of 200, 300, and 800 nm from the interface. Closest to the interface, the density of states near the Fermi energy is reduced to 55% of its normal value. At the farthest measurement point, this dip has nearly completely disappeared. We compa...

Given a family of k disjoint connected polygonal sites in general position and of total complexity n, we consider the farthest-site Voronoi diagram of these sites, where the distance to a site is the distance to a closest point on it. We show that the complexity of this diagram is O(n), and give an O(n log n) time algorithm to compute it. We also prove a number of structural properties of this ...

We propose a surface segmentation method based on Fast Marching Farthest Point Sampling designed for noisy, visually reconstructed point clouds or laser range data. Adjusting the distance metric between neighboring vertices we obtain robust, edge-preserving segmentations based on local curvature. We formulate a cost function given a segmentation in terms of a description length to be minimized....

The eccentricity transform associates to each point of a shape the shortest distance to the point farthest away from it. It is defined in any dimension, for open and closed manyfolds. Top-down decomposition of the shape can be used to speed up the computation, with some partitions being better suited than others. We study basic convex shapes and their decomposition in the context of the continu...

Let X be a complete CAT(0) space with the geodesic extension property and Alexandrov curvature bounded below. It is shown that if C is a closed subset of X , then the set of points of X which have a unique nearest point in C is Gδ and of the second Baire category inX. If, in addition,C is bounded, then the set of points ofX which have a unique farthest point in C is dense in X. A proximity resu...

We provide an efficient reduction from the problem of querying approximate multiplicatively weighted farthest neighbors in a metric space to the unweighted problem. Combining our techniques with core-sets for approximate unweighted farthest neighbors, we show how to find (1+ ε)-approximate farthest neighbors in time O(logn) per query in D-dimensional Euclidean space for any constants D and ε. A...