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148
Efficiently Approximating the MinimumVolume Bounding Box of a Point Set in Three Dimensions
 In Proc. 10th ACMSIAM Sympos. Discrete Algorithms
, 2001
"... We present an efficient O(n + 1/ε^4.5)time algorithm for computing a (1 + 1/ε)approximation of the minimumvolume bounding box of n points in R³. We also present a simpler algorithm (for the same purpose) whose running time is O(n log n+n/ε³). ..."
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Cited by 93 (13 self)
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We present an efficient O(n + 1/&epsilon;^4.5)time algorithm for computing a (1 + 1/&epsilon;)approximation of the minimumvolume bounding box of n points in R&sup3;. We also present a simpler algorithm (for the same purpose) whose running time is O(n log n+n/&epsilon;&sup3;). We give some experimental results with implementations of various variants of the second algorithm. The implementation of the algorithm described in this paper is available online [Har00].
Computing the Width of a Set
 IEEE Trans. Pattern Anal. Mach. Intell
, 1988
"... Given a set of points P = {p 1 , p 2 ,..., p n } in three dimensions, the width of P, W(P), is defined as the minimum distance between parallel planes of support of P. It is shown that W(P) can be computed in O(n log n + I) time and O(n) space, where I is the number of antipodal pairs of edges ..."
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Cited by 65 (4 self)
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Given a set of points P = {p 1 , p 2 ,..., p n } in three dimensions, the width of P, W(P), is defined as the minimum distance between parallel planes of support of P. It is shown that W(P) can be computed in O(n log n + I) time and O(n) space, where I is the number of antipodal pairs of edges of the convex hull of P, and in the worst case I = W(n 2 ). For convex polyhedra, the time complexity becomes O(n + I). If P is a set of points in the plane, the complexity can be reduced to O(n log n). Finally, for simple polygons linear time suffices. Index Terms  Algorithms, antipodal pairs, artificial intelligence, computational geometry, convex hull, geometric complexity, geometric transforms, image processing, minimax approximating line, minimax approximating plane, pattern recognition, rotating calipers, width. 1. Introduction The width of a set of points P (or a simple polygon P) in two dimensions is the minimum distance between parallel lines of support of P (or P). In three d...
Practical shadow mapping
 Journal of Graphics Tools
, 2000
"... In this paper we propose several methods that can greatly improve image quality when using the shadow mapping algorithm. Shadow artifacts introduced by shadow mapping are mainly due to low resolution shadow maps and/or the limited numerical precision used when performing the shadow test. These probl ..."
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Cited by 64 (9 self)
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In this paper we propose several methods that can greatly improve image quality when using the shadow mapping algorithm. Shadow artifacts introduced by shadow mapping are mainly due to low resolution shadow maps and/or the limited numerical precision used when performing the shadow test. These problems especially arise when the light source’s viewing frustum, from which the shadow map is generated, is not adjusted to the actual camera view. We show how a tight fitting frustum can be computed such that the shadow mapping algorithm concentrates on the visible parts of the scene and takes advantage of nearly the full available precision. Furthermore, we recommend uniformly spaced depth values in contrast to perspectively spaced depths in order to equally sample the scene seen from the light source. 1.
Geometric SpeedUp Techniques for Finding Shortest Paths in Large Sparse Graphs
, 2003
"... In this paper, we consider Dijkstra's algorithm for the single source single target shortest paths problem in large sparse graphs. The goal is to reduce the response time for online queries by using precomputed information. For the result of the preprocessing, we admit at most linear space. ..."
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Cited by 59 (15 self)
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In this paper, we consider Dijkstra's algorithm for the single source single target shortest paths problem in large sparse graphs. The goal is to reduce the response time for online queries by using precomputed information. For the result of the preprocessing, we admit at most linear space. We assume that a layout of the graph is given. From this layout, in the preprocessing, we determine for each edge a geometric object containing all nodes that can be reached on a shortest path starting with that edge. Based on these geometric objects, the search space for online computation can be reduced significantly. We present an extensive experimental study comparing the impact of different types of objects. The test data we use are traffic networks, the typical field of application for this scenario.
Movable Separability of Sets
 Computational Geometry
, 1985
"... Spurred by developments in spatial planning in robotics, computer graphics, and VLSI layout, considerable attention has been devoted recently to the problem of moving sets of objects, such as line segments and polygons in the plane to polyhedra in three dimensions, without allowing collisions betwee ..."
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Cited by 40 (4 self)
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Spurred by developments in spatial planning in robotics, computer graphics, and VLSI layout, considerable attention has been devoted recently to the problem of moving sets of objects, such as line segments and polygons in the plane to polyhedra in three dimensions, without allowing collisions between the objects. One class of such problems considers the separability of sets of objects under different kinds of motions and various definitions of separation. This paper surveys this new area of research in a tutorial fashion, present new results, and provides a list of open problems and suggestions for further research. Key Words and Phrases: sofa problem, polygons, polyhedra, movable separability, visibility hulls, hidden lines, hidden surfaces, algorithms, complexity, computational geometry, spatial planning, collision avoidance, robotics, artificial intelligence. CR Categories: 3.36, 3.63, 5.25. 5.32. 5.5 * Research supported by NSERC Grant no. A9293 and FCAR Grant no.EQ1678.  2  ...
Interactive Rendering of Translucent Objects
, 2002
"... This paper presents a rendering method for translucent objects, in which view point and illumination can be modified at interactive rates. In a preprocessing step the impulse response to incoming light impinging at each surface point is computed and stored in two different ways: The local effect on ..."
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Cited by 39 (6 self)
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This paper presents a rendering method for translucent objects, in which view point and illumination can be modified at interactive rates. In a preprocessing step the impulse response to incoming light impinging at each surface point is computed and stored in two different ways: The local effect on closeby surface points is modeled as a pertexel filter kernel that is applied to a texture map representing the incident illumination. The global response (i.e. light shining through the object) is stored as vertextovertex throughput factors for the triangle mesh of the object. During rendering, the illumination map for the object is computed according to the current lighting situation and then filtered by the precomputed kernels. The illumination map is also used to derive the incident illumination on the vertices which is distributed via the vertextovertex throughput factors to the other vertices. The final image is obtained by combining the local and global response. We demonstrate the performance of our method for several models.
A Practical Approach for Computing the Diameter of a Point Set
 In Proc. 17th ACM Sympos. Comput. Geom
, 2001
"... We present an approximation algorithm for computing the diameter of a pointset in ddimensions. The new algorithm is sensitive to the \hardness" of computing the diameter of the given input, and for most inputs it is able to compute the exact diameter extremely fast. The new algorithm is si ..."
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Cited by 27 (1 self)
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We present an approximation algorithm for computing the diameter of a pointset in ddimensions. The new algorithm is sensitive to the \hardness" of computing the diameter of the given input, and for most inputs it is able to compute the exact diameter extremely fast. The new algorithm is simple, robust, has good empirical performance, and can be implemented quickly. As such, it seems to be the algorithm of choice in practice for computing/approximating the diameter.
A new convexity measure for polygons
 IEEE Transactions on Pattern Analysis and Machine Intelligence
"... Convexity estimators are commonly used in the analysis of shape. In this paper we define and evaluate a new easily computable measure of convexity for polygons. Let be an arbitrary polygon. If denotes the perimeter in the sense of metrics of the polygon obtained by the rotation of by angle ..."
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Cited by 25 (5 self)
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Convexity estimators are commonly used in the analysis of shape. In this paper we define and evaluate a new easily computable measure of convexity for polygons. Let be an arbitrary polygon. If denotes the perimeter in the sense of metrics of the polygon obtained by the rotation of by angle with the origin as the center of the applied rotation, and if is the Euclidean perimeter of the minimal rectangle having the edges parallel to coordinate axes which includes such a rotated polygon , then we show that defined as can be used as an estimate for the convexity of . Several desirable properties of are proved, as well.
D.: Compression of pointbased 3d models by shapeadaptive wavelet coding of multiheight fields
 In Eurographics Symposium on PointBased Graphics (2004
"... In order to efciently archive and transmit large 3D models, lossy and lossless compression methods are needed. We propose a compression scheme for coordinate data of pointbased 3D models of surfaces. A pointbased model is processed for compression in a pipeline of three subsequent operations, part ..."
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Cited by 22 (6 self)
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In order to efciently archive and transmit large 3D models, lossy and lossless compression methods are needed. We propose a compression scheme for coordinate data of pointbased 3D models of surfaces. A pointbased model is processed for compression in a pipeline of three subsequent operations, partitioning, parameterization, and coding. First the point set is partitioned yielding a suitable number of point clusters. Each cluster corresponds to a surface patch, that can be parameterized as a height eld and resampled on a regular grid. The domains of the height elds have irregular shapes that are encoded losslessly. The height elds themselves are encoded using a shapeadaptive wavelet coder, producing a progressive bitstream for each patch. A ratedistortion optimization provides for an optimal bit allocation for the individual patch codes. With this algorithm design compact codes are produced that are scalable with respect to rate, quality, and resolution. In our encodings of complex 3D models competitive ratedistortion performances were achieved with excellent reconstruction quality at under 3 bits per point (bpp).