In this paper we propose algorithms for solving a variety of geometric optimization problems on a stream of points in R or R. These problems include various extent measures (e.g. diameter, width, smallest enclosing disk), collision detection (penetration depth and distance between polytopes), and shape fitting (minimum width annulus, circle/line fitting). The main contribution of this paper is ...

Shortest Paths in Intersection Graphs of Unit Disks Sergio Cabello Let G be a unit disk graph in the plane defined by n unit disks whose positions are known. We show that in G a shortest path tree from a given source can be computed in O(n log n) time in the unweighted case and in O(n1+ ) time in the weighted case. As an application of this result, we will discuss how to solve the following pro...

This paper presents an optimization approach for estimating the 3D parameters of buildings from multiple images by fitting the observations to the geometric building models. Through multi-image traingulation, the coordinates of the building corners and their covariances are estimated from the located image corresponding points. Then by a constrained optimization the estimated point positions an...

We summarize techniques for optimal geometric estimation from noisy observations for computer vision applications. We first discuss the interpretation of optimality and point out that geometric estimation is different from the standard statistical estimation. We also describe our noise modeling and a theoretical accuracy limit called the KCR lower bound. Then, we formulate estimation techniques...

The techniques and analysis presented in this thesis provide new methods to solve optimization problems posed on Riemannian manifolds. These methods are applied to the subspace tracking problem found in adaptive signal processing and adaptive control. A new point of view is offered for the constrained optimization problem. Some classical optimization techniques on Euclidean space are generalize...

What is the effectiveness of local search algorithms for geometric problems in the plane? We prove that local search with neighborhoods of magnitude 1/ c is an approximation scheme for the following problems in the Euclidean plane: TSP with random inputs, Steiner tree with random inputs, uniform facility location (with worst case inputs), and bicriteria k-median (also with worst case inputs). T...

This thesis deals with a number of geometric optimization problems which are all NP-hard. The first problem we consider is the set cover problem for polytopes in R3. Here, we are given a set of points in R3 and a fixed set of translates of an arbitrary polytope. We would like to select a subset of the given polytopes such that each input point is covered by at least one polytope and the number ...

We overview techniques for optimal geometric estimation from noisy observations for computer vision applications. We first describe estimation techniques based on minimization of given cost functions: least squares (LS), maximum likelihood (ML), which includes reprojection error minimization (Gold Standard) as a special case, and Sampson error minimization. We then formulate estimation techniqu...

AJ1STRACT In this paper we apply Galois theoretic algebraic methods to certain fundamental geometric optimization problems whose recognition versions are not even known to be in the class NP. In particular we show that the classic Weber problem, the Line-restricted' Weber probe lem and the 3-Dimension version of this problem are in general not solvable by radicals over the field of rationals. O...