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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 454 (0 self)
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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.
On the convex layers of a planar set
 IEEE Transactions on Information Theory
, 1985
"... AbstractLet S be a set of n points in the Euclidean plane. The convex layers of S are the convex polygons obtained by iterating on the following procedure: compute the convex hull of S and remove its vertices from S. This process of peeling a planar point set is central in the study of robust estim ..."
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Cited by 55 (1 self)
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AbstractLet S be a set of n points in the Euclidean plane. The convex layers of S are the convex polygons obtained by iterating on the following procedure: compute the convex hull of S and remove its vertices from S. This process of peeling a planar point set is central in the study of robust estimators in statistics. It also provides valuable information on the morphology of a set of sites and has proven to be an efficient preconditioning for range search problems. An optimal algorithm is described for computing the convex layers of S. The algorithm runs in O ( n log n) time and requires O(n) space. Also addressed is the problem of determining the depth of a query point within the convex layers of S, i.e., the number of layers that enclose the query point. This is essentially a planar point location problem, for which optimal solutions are therefore known. Taking advantage of structural properties of the problem, however, a much simpler optimal solution is derived. L I.
Statistical detection of independent movement from a moving camera
 Image Vision Comput
, 1993
"... This paper describes the use of a low level, computationally inexpensive closed form motion detector to define regions of interest within an image, based upon statistical measures. The algorithm requires only the first order properties of the image intensities and does not require known camera motio ..."
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Cited by 25 (5 self)
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This paper describes the use of a low level, computationally inexpensive closed form motion detector to define regions of interest within an image, based upon statistical measures. The algorithm requires only the first order properties of the image intensities and does not require known camera motion. It has been tested on a variety of real imagery. A bspline snake is initialised on the occluding contours of this region of interest. 1
Randomized Quick Hull
 Algorithmica
, 1995
"... This paper contains a simple, randomized algorithm for constructing the convex hull of a set of n points in the plane with expected running time O(n log h) where h is the number of points on the convex hull. Introduction Determining the convex hull of a set of points is one of the most basic pro ..."
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Cited by 5 (0 self)
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This paper contains a simple, randomized algorithm for constructing the convex hull of a set of n points in the plane with expected running time O(n log h) where h is the number of points on the convex hull. Introduction Determining the convex hull of a set of points is one of the most basic problems in computational geometry. Ron Graham presented the first O(n log n) algorithm for finding the convex hull of points in the plane in 1972 [12]. The following year, R. Jarvis gave an algorithm whose running time depends on the output size [14]. Jarvis's algorithm runs in O(nh) time where h is the number of points in the convex hull. The next ten years saw many other algorithms for finding convex hulls in the plane most of which run in O(n log n) time [1, 4, 11, 13, 16]. Some very simple algorithms were proposed which have O(n) expected running time for many distributions of points in the plane (such as points with normal density) [10, 3]. During this period, Avis [2] and Yao [20] proved...
HOMOGENEITY ANALYSIS
"... Abstract. The Gifi system of analyzing categorical data through nonlinear varieties of classical multivariate analysis techniques is reviewed. The system is characterized by the optimal scaling of categorical variables which is implemented through alternating least squares algorithms. The main techn ..."
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Cited by 1 (0 self)
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Abstract. The Gifi system of analyzing categorical data through nonlinear varieties of classical multivariate analysis techniques is reviewed. The system is characterized by the optimal scaling of categorical variables which is implemented through alternating least squares algorithms. The main technique of homogeneity analysis is presented, along with its extensions and generalizations leading to nonmetric principal components analysis and canonical correlation analysis. Several examples are used to illustrate the methods. A brief account of stability issues and areas of applications of the techniques is also given. Key words and phrases: Optimal scaling, alternating least squares, multivariate techniques, loss functions, stability.
Extreme Point Detection and the Polar Dual
"... This thesis is concerned with the identification of the extreme points of a given finite set of points denoted by S. In Jibrin, Boneh and Caron[18] the authors explored the advantage of using probabilistic methods applied to the polar dual of S to quickly detect some extreme points. This thesis buil ..."
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This thesis is concerned with the identification of the extreme points of a given finite set of points denoted by S. In Jibrin, Boneh and Caron[18] the authors explored the advantage of using probabilistic methods applied to the polar dual of S to quickly detect some extreme points. This thesis builds on those results with the introduction of a polynomial time procedure that uses the detected extreme points to eliminate redundant points. i Acknowledgments ii
A New Ultimate Convex Hull Algorithm in R²
, 2000
"... We present a very simple algorithm  NEWHULL  to find the convex hull of S = {P 1 , . . . , Pn}, n given points in R 2 . It may be thought of as a variant of Quickhull; however if the hull of S has h vertices, the algorithm runs in time #(n log h), worstcase. On the average it is much faster. Analy ..."
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We present a very simple algorithm  NEWHULL  to find the convex hull of S = {P 1 , . . . , Pn}, n given points in R 2 . It may be thought of as a variant of Quickhull; however if the hull of S has h vertices, the algorithm runs in time #(n log h), worstcase. On the average it is much faster. Analysis suggests that NEWHULL should be twice as fast as the "ultimate" algorithm of Kirkpatrick and Seidel, and experimental evidence bears this out.