Results 1  10
of
18
The Quickhull algorithm for convex hulls
 ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE
, 1996
"... The convex hull of a set of points is the smallest convex set that contains the points. This article presents a practical convex hull algorithm that combines the twodimensional Quickhull Algorithm with the generaldimension BeneathBeyond Algorithm. It is similar to the randomized, incremental algo ..."
Abstract

Cited by 452 (0 self)
 Add to MetaCart
The convex hull of a set of points is the smallest convex set that contains the points. This article presents a practical convex hull algorithm that combines the twodimensional Quickhull Algorithm with the generaldimension BeneathBeyond Algorithm. It is similar to the randomized, incremental algorithms for convex hull and Delaunay triangulation. We provide empirical evidence that the algorithm runs faster when the input contains nonextreme points and that it uses less memory. Computational geometry algorithms have traditionally assumed that input sets are well behaved. When an algorithm is implemented with floatingpoint arithmetic, this assumption can lead to serious errors. We briefly describe a solution to this problem when computing the convex hull in two, three, or four dimensions. The output is a set of “thick ” facets that contain all possible exact convex hulls of the input. A variation is effective in five or more dimensions.
Constructing Strongly Convex Hulls Using Exact or Rounded Arithmetic
 Algorithmica
, 1992
"... One useful generalization of the convex hull of a set S of n points is the fflstrongly convex ffihull. It is defined to be a convex polygon P with vertices taken from S such that no point in S lies farther than ffi outside P and such that even if the vertices of P are perturbed by as much as ffl, ..."
Abstract

Cited by 28 (5 self)
 Add to MetaCart
One useful generalization of the convex hull of a set S of n points is the fflstrongly convex ffihull. It is defined to be a convex polygon P with vertices taken from S such that no point in S lies farther than ffi outside P and such that even if the vertices of P are perturbed by as much as ffl, P remains convex. It was an open question 1 as to whether an fflstrongly convex O(ffl)hull existed for all positive ffl. We give here an O(n log n) algorithm for constructing it (which thus proves its existence). This algorithm uses exact rational arithmetic. We also show how to construct an fflstrongly convex O(ffl + ¯)hull in O(n log n) time using rounded arithmetic with rounding unit ¯. This is the first rounded arithmetic convex hull algorithm which guarantees a convex output and which has error independent of n. 1 Introduction Recently, some work in the area of computational geometry has focused on the numerical issues that arise when geometric algorithms are implemented using r...
Delaunay Triangulations of Imprecise Points in Linear Time after Preprocessing
, 2008
"... An assumption of nearly all algorithms in computational geometry is that the input points are given precisely, so it is interesting to ask what is the value of imprecise information about points. We show how to preprocess a set of n disjoint unit disks in the plane in O(n log n) time so that if one ..."
Abstract

Cited by 20 (6 self)
 Add to MetaCart
An assumption of nearly all algorithms in computational geometry is that the input points are given precisely, so it is interesting to ask what is the value of imprecise information about points. We show how to preprocess a set of n disjoint unit disks in the plane in O(n log n) time so that if one point per disk is specified with precise coordinates, the Delaunay triangulation can be computed in linear time. From the Delaunay, one can obtain the Gabriel graph and a Euclidean minimum spanning tree; it is interesting to note the roles that these two structures play in our algorithm to quickly compute the Delaunay.
Largest bounding box, smallest diameter, and related problems on imprecise points
 In Proc. 10th Workshop on Algorithms and Data Structures, LNCS 4619
, 2007
"... We model imprecise points as regions in which one point must be located. We study computing the largest and smallest possible values of various basic geometric measures on sets of imprecise points, such as the diameter, width, closest pair, smallest enclosing circle, and smallest enclosing bounding ..."
Abstract

Cited by 14 (7 self)
 Add to MetaCart
We model imprecise points as regions in which one point must be located. We study computing the largest and smallest possible values of various basic geometric measures on sets of imprecise points, such as the diameter, width, closest pair, smallest enclosing circle, and smallest enclosing bounding box. We give efficient algorithms for most of these problems, and identify the hardness of others. 1
Preprocessing imprecise points and splitting triangulations
 UTRECHT UNIVERSITY
, 2009
"... Traditional algorithms in computational geometry assume that the input points are given precisely. In practice, data is usually imprecise, but information about the imprecision is often available. In this context, we investigate what the value of this information is. We show here how to preprocess a ..."
Abstract

Cited by 12 (4 self)
 Add to MetaCart
Traditional algorithms in computational geometry assume that the input points are given precisely. In practice, data is usually imprecise, but information about the imprecision is often available. In this context, we investigate what the value of this information is. We show here how to preprocess a set of disjoint regions in the plane of total complexity n in O(n log n) time so that if one point per set is specified with precise coordinates, a triangulation of the points can be computed in linear time. In our solution, we solve another problem which we believe to be of independent interest. Given a triangulation with red and blue vertices, we show how to compute a triangulation of only the blue vertices in linear time.
Largest and Smallest Convex Hulls for Imprecise Points
 ALGORITHMICA
, 2008
"... Assume that a set of imprecise points is given, where each point is specified by a region in which the point may lie. We study the problem of computing the smallest and largest possible convex hulls, measured by length and by area. Generally we assume the imprecision region to be a square, but we d ..."
Abstract

Cited by 10 (4 self)
 Add to MetaCart
Assume that a set of imprecise points is given, where each point is specified by a region in which the point may lie. We study the problem of computing the smallest and largest possible convex hulls, measured by length and by area. Generally we assume the imprecision region to be a square, but we discuss the case where it is a segment or circle as well. We give polynomial time algorithms for several variants of this problem, ranging in running time from O(n log n) to O(n^13), and prove NPhardness for some other variants.
Shape Fitting on Point Sets with Probability Distributions
"... Abstract. We consider problems on data sets where each data point has uncertainty described by an individual probability distribution. We develop several frameworks and algorithms for calculating statistics on these uncertain data sets. Our examples focus on geometric shape fitting problems. We prov ..."
Abstract

Cited by 6 (4 self)
 Add to MetaCart
Abstract. We consider problems on data sets where each data point has uncertainty described by an individual probability distribution. We develop several frameworks and algorithms for calculating statistics on these uncertain data sets. Our examples focus on geometric shape fitting problems. We prove approximation guarantees for the algorithms with respect to the full probability distributions. We then empirically demonstrate that our algorithms are simple and practical, solving for a constant hidden by asymptotic analysis so that a user can reliably trade speed and size for accuracy. 1
Computing Convex Hull in Floating Point Arithmetic
 KFC + 01] Shankar
, 1994
"... : We present a numerically stable and time and space complexity optimal algorithm for constructing a convex hull for a set of points on a plane. In contrast to already existing numerically stable algorithms which return only an approximate hull, our algorithm constructs a polygon that is truly conve ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
: We present a numerically stable and time and space complexity optimal algorithm for constructing a convex hull for a set of points on a plane. In contrast to already existing numerically stable algorithms which return only an approximate hull, our algorithm constructs a polygon that is truly convex. The algorithm is simple and easy to implement. 1 Introduction There are many optimal O(n log n) time complexity algorithms for constructing the convex hull of a set of n points in IR 2 (see e.g., [1, 7]). All these algorithms assume that infinite precision arithmetic is used and, in general, they are numerically unstable when a (finite precision) floating point arithmetic is used. In floating point arithmetic, which is widely used for implementing algorithms, the computation of a convex hull is a much less explored problem. Relatively few numerically stable and time complexity optimal algorithms are known; see e.g., [2] for computing a convex hull of a point set, and [5] for computing...
On Detecting Spatial Regularity in Noisy Images
 Information Processing Letters
, 1999
"... Detecting spatial regularity in images arises in computer vision, scene analysis, military applications, and other areas. In this paper we present an O(n 5 2 ) algorithm that reports all maximal equallyspaced collinear subsets. The algorithm is robust in that it can tolerate noise or imprecision t ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
Detecting spatial regularity in images arises in computer vision, scene analysis, military applications, and other areas. In this paper we present an O(n 5 2 ) algorithm that reports all maximal equallyspaced collinear subsets. The algorithm is robust in that it can tolerate noise or imprecision that may be inherent in the measuring process, where the error threshold is a userspecified parameter. Our method also generalizes to higher dimensions. Keywords: Algorithms, combinatorial problems, computational geometry, pattern recognition. 1 Introduction Spatial regularity detection is an important problem in a number of domains such as computer vision, scene analysis, and landmine detection from infrared terrain images [5]. This paper addresses the problem of recognizing equallyspaced collinear subsets of a given pointset, where there may be imprecision in the input data. Kahng and Robins [5] gave an optimal O(n 2 )time algorithm for the exact version of this problem (i.e., where n...