Results 11  20
of
31
Minimum Enclosures with Specified Angles
, 1994
"... Given a convex polygon P , an menvelope is a convex msided polygon that contains P . Given any convex polygon P , and any sequence of m 3 angles A = hff 1 ; ff 2 ; : : : ; ff m i, we consider the problem of computing the minimum area menvelope for P whose counterclockwise sequence of exterior an ..."
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Given a convex polygon P , an menvelope is a convex msided polygon that contains P . Given any convex polygon P , and any sequence of m 3 angles A = hff 1 ; ff 2 ; : : : ; ff m i, we consider the problem of computing the minimum area menvelope for P whose counterclockwise sequence of exterior angles is given by A. We show that such envelopes can be computed in O(nm log m) time. The main result on which the correctness of the algorithm rests is a flushness condition stating that for any locally minimum enclosure with specified angles, one of its sides must be collinear with one of the sides of P . The support of the National Science Foundation under Grant CCR8908901, the Bureau of the Census under grant JSA 915, and the University of the District of Columbia under a Faculty Senate Summer Research Grant is gratefully acknowledged, as is the help of Sandy German in preparing this paper. A preliminary version of this paper appeared in Vision Geometry , R.A. Melter and A.Y. Wu, Edit...
Upper and lower bounds on the quality of the PCA bounding boxes
, 2006
"... Principle component analysis (PCA) is commonly used to compute a bounding box of a point set in R d. The popularity of this heuristic lies in its speed, easy implementation and in the fact that usually, PCA bounding boxes quite well approximate the minimumvolume bounding boxes. In this paper we giv ..."
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Principle component analysis (PCA) is commonly used to compute a bounding box of a point set in R d. The popularity of this heuristic lies in its speed, easy implementation and in the fact that usually, PCA bounding boxes quite well approximate the minimumvolume bounding boxes. In this paper we give a lower bound on the approximation factor of PCA bounding boxes of convex polytopes in arbitrary dimension, and an upper bound on the approximation factor of PCA bounding boxes of convex polygons in R².
Bounds on the Quality of the PCA Bounding Boxes
"... Principal component analysis (PCA) is commonly used to compute a bounding box of a point set in R d. The popularity of this heuristic lies in its speed, easy implementation and in the fact that usually, PCA bounding boxes quite well approximate the minimumvolume bounding boxes. We present examples o ..."
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Principal component analysis (PCA) is commonly used to compute a bounding box of a point set in R d. The popularity of this heuristic lies in its speed, easy implementation and in the fact that usually, PCA bounding boxes quite well approximate the minimumvolume bounding boxes. We present examples of discrete points sets in the plane, showing that the worst case ratio of the volume of the PCA bounding box and the volume of the minimumvolume bounding box tends to infinity. Thus, we concentrate our attention on PCA bounding boxes for continuous sets, especially for the convex hull of a point set. Here, we contribute lower bounds on the approximation factor of PCA bounding boxes of convex sets in arbitrary dimension, and upper bounds in R² and R³.
Cutting a convex polyhedron out of a sphere
 Graphs and Combinatorics
"... Abstract Given a convex polyhedron P of n vertices inside a sphere Q, we give an O(n 3)time algorithm that cuts P out of Q by using guillotine cuts and has cutting cost O((log n) 2) times the optimal. 1. ..."
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Abstract Given a convex polyhedron P of n vertices inside a sphere Q, we give an O(n 3)time algorithm that cuts P out of Q by using guillotine cuts and has cutting cost O((log n) 2) times the optimal. 1.
On the Bounding Boxes Obtained by Principal Component Analysis
"... Principle component analysis (PCA) is a commonly used to compute a bounding box of a point set in R d. In this paper we give bounds on the approximation factor of PCA bounding boxes of convex polygons in R² (lower and upper bounds) and convex polyhedra in R³ (lower bound). ..."
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Principle component analysis (PCA) is a commonly used to compute a bounding box of a point set in R d. In this paper we give bounds on the approximation factor of PCA bounding boxes of convex polygons in R² (lower and upper bounds) and convex polyhedra in R³ (lower bound).
Oriented Bounding Box Computation Using Particle Swarm Optimization ∗
"... Abstract. The problem of finding the optimal oriented bounding box (OBB) for a given set of points in R 3, yet simple to state, is computationally challenging. Existing stateoftheart methods dealing with this problem are either exact but slow, or fast but very approximative and unreliable. We pro ..."
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Abstract. The problem of finding the optimal oriented bounding box (OBB) for a given set of points in R 3, yet simple to state, is computationally challenging. Existing stateoftheart methods dealing with this problem are either exact but slow, or fast but very approximative and unreliable. We propose a method based on Particle Swarm Optimization (PSO) to approximate solutions both effectively and accurately. The original PSO algorithm is modified so as to search for optimal solutions over the rotation group SO(3). Particles are defined as 3D rotation matrices and operations are expressed over SO(3) using matrix products, exponentials and logarithms. The symmetry of the problem is also exploited. Numerical experiments show that the proposed algorithm outperforms existing methods, often by far. 1
22.1 Some Geometry
, 2010
"... Isn’t it an artificial, sterilized, didactically pruned world, a mere sham world in which you cravenly vegetate, a world without vices, without passions without hunger, without sap and salt, a world without family, without mothers, without children, almost without women? The instinctual life is tame ..."
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Isn’t it an artificial, sterilized, didactically pruned world, a mere sham world in which you cravenly vegetate, a world without vices, without passions without hunger, without sap and salt, a world without family, without mothers, without children, almost without women? The instinctual life is tamed by meditation. For generations you have left to others dangerous, daring, and responsible things like economics, law, and politics. Cowardly and wellprotected, fed by others, and having few burdensome duties, you lead your drones ’ lives, and so that they won’t be too boring you busy yourselves with all these erudite specialties, count syllables and letters, make music, and play the Glass Bead Game, while outside in the filth of the world poor harried people live real lives and do real work.
Abstract Minimal enclosing parallelepiped in 3D
"... We investigate the problem of finding a minimal volume parallelepiped enclosing a given set of n threedimensional points. We give two mathematical properties of these parallelepipeds, from which we derive two algorithms of theoretical complexity O(n 6). Experiments show that in practice our quickes ..."
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We investigate the problem of finding a minimal volume parallelepiped enclosing a given set of n threedimensional points. We give two mathematical properties of these parallelepipeds, from which we derive two algorithms of theoretical complexity O(n 6). Experiments show that in practice our quickest algorithm runs in O(n 2) (at least for n ≤ 10 5). We also present our application in structural biology.
Minimal enclosing parallelepiped in 3D
, 2002
"... We investigate the problem of finding a minimal volume parallelepiped enclosing a given set of n threedimensional points. We give two mathematical properties of these parallelepipeds, from which we derive two algorithms of theoretical complexity O(n 6). Experiments show that in practice our quickes ..."
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We investigate the problem of finding a minimal volume parallelepiped enclosing a given set of n threedimensional points. We give two mathematical properties of these parallelepipeds, from which we derive two algorithms of theoretical complexity O(n 6). Experiments show that in practice our quickest algorithm runs in O(n 2) (at least for n ≤ 10 5). We also present our application in structural biology.