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Simplification Envelopes
"... We propose the idea of simplification envelopes for generating a hierarchy of levelofdetail approximations for a given polygonal model. Our approach guarantees that all points of an approximation are within a userspecifiable distance # from the original model and that all points of the original m ..."
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Cited by 182 (16 self)
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We propose the idea of simplification envelopes for generating a hierarchy of levelofdetail approximations for a given polygonal model. Our approach guarantees that all points of an approximation are within a userspecifiable distance # from the original model and that all points of the original model are within a distance # from the approximation. Simplificationenvelopes provide a general framework within which a large collection of existing simplification algorithms can run. We demonstrate this technique in conjunction with two algorithms, one local, the other global. The local algorithm provides a fast method for generating approximations to large input meshes (at least hundreds of thousands of triangles). The global algorithm provides the opportunity to avoid local "minima" and possibly achieve better simplifications as a result. Each approximation attempts to minimize the total number of polygons required to satisfy the above # constraint. The key advantages of our approach are...
Discrete Geometric Shapes: Matching, Interpolation, and Approximation: A Survey
 Handbook of Computational Geometry
, 1996
"... In this survey we consider geometric techniques which have been used to measure the similarity or distance between shapes, as well as to approximate shapes, or interpolate between shapes. Shape is a modality which plays a key role in many disciplines, ranging from computer vision to molecular biolog ..."
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Cited by 129 (9 self)
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In this survey we consider geometric techniques which have been used to measure the similarity or distance between shapes, as well as to approximate shapes, or interpolate between shapes. Shape is a modality which plays a key role in many disciplines, ranging from computer vision to molecular biology. We focus on algorithmic techniques based on computational geometry that have been developed for shape matching, simplification, and morphing. 1 Introduction The matching and analysis of geometric patterns and shapes is of importance in various application areas, in particular in computer vision and pattern recognition, but also in other disciplines concerned with the form of objects such as cartography, molecular biology, and computer animation. The general situation is that we are given two objects A, B and want to know how much they resemble each other. Usually one of the objects may undergo certain transformations like translations, rotations or scalings in order to be matched with th...
Surface Approximation and Geometric Partitions
 IN PROC. 5TH ACMSIAM SYMPOS. DISCRETE ALGORITHMS
, 1994
"... Motivated by applications in computer graphics, visualization, and scientific computation, we study the computational complexity of the following problem: Given a set S of n points sampled from a bivariate function f(x; y) and an input parameter " ? 0, compute a piecewise linear function \Si ..."
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Cited by 94 (15 self)
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Motivated by applications in computer graphics, visualization, and scientific computation, we study the computational complexity of the following problem: Given a set S of n points sampled from a bivariate function f(x; y) and an input parameter " ? 0, compute a piecewise linear function \Sigma(x; y) of minimum complexity (that is, a xymonotone polyhedral surface, with a minimum number of vertices, edges, or faces) such that j\Sigma(x p ; y p ) \Gamma z p j "; for all (x p ; y p ; z p ) 2 S: We prove that the decision version of this problem is NPHard . The main result of our paper is a polynomialtime approximation algorithm that computes a piecewise linear surface of size O(K o log K o ), where K o is the complexity of an optimal surface satisfying the constraints of the problem. The technique
Controlled topology simplification
 IEEE Transactions on Visualization and Computer Graphics
, 1996
"... ..."
Efficient PiecewiseLinear Function Approximation Using the Uniform Metric
 Discrete & Computational Geometry
, 1994
"... We give an O(n log n)time method for finding a best klink piecewiselinear function approximating an npoint planar data set using the wellknown uniform metric to measure the error, ffl 0, of the approximation. Our method is based upon new characterizations of such functions, which we exploit to ..."
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Cited by 39 (0 self)
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We give an O(n log n)time method for finding a best klink piecewiselinear function approximating an npoint planar data set using the wellknown uniform metric to measure the error, ffl 0, of the approximation. Our method is based upon new characterizations of such functions, which we exploit to design an efficient algorithm using a plane sweep in "ffl space" followed by several applications of the parametric searching technique. The previous best running time for this problem was O(n 2 ). 1 Introduction Approximating a set S = f(x 1 ; y 1 ); (x 2 ; y 2 ); : : : ; (x n ; y n )g of points in the plane by a function is a classic problem in applied mathematics. The general goals in this area of research are to find a function F belonging to a class of functions F such that each F 2 F is simple to describe, represent, and compute and such that the chosen F approximates S well. For example, one may desire that F be the class of linear or piecewiselinear functions, and, for any parti...
Approximation schemes for NPhard geometric optimization problems: A survey
 Mathematical Programming
, 2003
"... NPhard geometric optimization problems arise in many disciplines. Perhaps the most famous one is the traveling salesman problem (TSP): given n nodes in ℜ 2 (more generally, in ℜ d), find the minimum length path that visits each node exactly once. If distance is computed using the Euclidean norm (di ..."
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Cited by 39 (2 self)
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NPhard geometric optimization problems arise in many disciplines. Perhaps the most famous one is the traveling salesman problem (TSP): given n nodes in ℜ 2 (more generally, in ℜ d), find the minimum length path that visits each node exactly once. If distance is computed using the Euclidean norm (distance between nodes (x1, y1) and (x2, y2) is ((x1−x2) 2 +(y1−y2) 2) 1/2) then the problem is called Euclidean TSP. More generally the distance could be defined using other norms, such as ℓp norms for any p> 1. All these are subcases of the more general notion of a geometric norm or Minkowski norm. We will refer to the version of the problem with a general geometric norm as geometric TSP. Some other NPhard geometric optimization problems are Minimum Steiner Tree (“Given n points, find the smallest network connecting them,”), kTSP(“Given n points and a number k, find the shortest salesman tour that visits k points”), kMST (“Given n points and a number k, find the shortest tree that contains k points”), vehicle routing, degree restricted minimum
Hierarchical Geometric Approximations
, 1994
"... This dissertation explores some techniques for automatic approximation of geometric objects. My thesis is that using and extending concepts from computational geometry can help us in devising efficient and parallelizable algorithms for automatically constructing useful detail hierarchies for geometr ..."
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Cited by 36 (4 self)
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This dissertation explores some techniques for automatic approximation of geometric objects. My thesis is that using and extending concepts from computational geometry can help us in devising efficient and parallelizable algorithms for automatically constructing useful detail hierarchies for geometric objects. We have demonstrated this by developing new algorithms for two kinds of geometric approximation problems that have been motivated by a single driving problem  the efficient computation and display of smooth solventaccessible molecular surfaces. The applications of these detail hierarchies are in biochemistry and computer graphics. The smooth solventaccessible surface of a molecule is useful in studying the structure and interactions of proteins, in particular for attacking the proteinsubstrate docking problem. We have developed a parallel lineartime algorithm for computing molecular surfaces. Molecular surfaces are equivalent to the weighted ffhulls. Thus our work is pot...
An Efficient Algorithm for Euclidean Shortest Paths Among Polygonal Obstacles in the Plane
, 1988
"... We give an algorithm to compute a (Euclidean) shortest path in a polygon with h holes and a total of n vertices. The algorithm uses O(n) space and requires O(n + h² log n) time. ..."
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Cited by 34 (1 self)
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We give an algorithm to compute a (Euclidean) shortest path in a polygon with h holes and a total of n vertices. The algorithm uses O(n) space and requires O(n + h² log n) time.