On Approximating Polygonal Curves in Two and Three Dimensions

Eu, David; Toussaint, Godfried T.

doi:10.1006/cgip.1994.1021

Cited by 51 publications

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“…The general problem of approximating a polygonal curve by a coarser one is of fundamental importance, and has been studied in disciplines such as geographic information systems [7], [11], [15], [21], [29], [31], digital image analysis [5], [24], [28], and computational geometry [8], [16], [18], [19], [25], [33], [35]. The problems considered fall in one of two categories, depending on whether the vertices of the approximating chain are required to be a subset of the vertices of the input chain.…”

Section: Motivation and Previous Resultsmentioning

confidence: 99%

Efficient Algorithms for Approximating Polygonal Chains

Agarwal¹,

Varadarajan

2000

Discrete Comput Geom

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We consider the problem of approximating a polygonal chain C by another polygonal chain C whose vertices are constrained to be a subset of the set of vertices of C. The goal is to minimize the number of vertices needed in the approximation C . Based on a framework introduced by Imai and Iri [25], we define an error criterion for measuring the quality of an approximation. We consider two problems. (1) Given a polygonal chain C and a parameter ε ≥ 0, compute an approximation of C, among all approximations whose error is at most ε, that has the smallest number of vertices. We present an O(n 4/3+δ )-time algorithm to solve this problem, for any δ > 0; the constant of proportionality in the running time depends on δ.(2) Given a polygonal chain C and an integer k, compute an approximation of C with at most k vertices whose error is the smallest among all approximations with at most k vertices. We present a simple randomized algorithm, with expected running time O(n 4/3+δ ), to solve this problem.

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Section: Motivation and Previous Resultsmentioning

confidence: 99%

Efficient Algorithms for Approximating Polygonal Chains

Agarwal¹,

Varadarajan

2000

Discrete Comput Geom

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“…The heuristic by Douglas and Peucker [11] is the standard technique for this problem, although it has no optimality guarantee and a worst-case running time of O(n 2 ). For the problem of simplification of a space curve, with the vertices of the output polygon being a subset of the input vertices, Eu and Toussaint [12] present an algorithm which finds an optimal solution within a running time of O(n 3 ). Agarwal et al [13] describe an efficient greedy approximation algorithm running in O(n log n) and requiring at most as many line segments as the optimal 2 approximating polygonal chain.…”

Section: Related Workmentioning

confidence: 99%

“…3 (d). -The polygonal curve is optionally simplified using Eu and Toussaint's algorithm [12] to reduce the problem complexity for further computations. -This polygonal curve is then approximated using an arc-line spline as described in Section 4, see Fig.…”

Section: Algorithm Overviewmentioning

confidence: 99%

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Generating parametric models of tubes from laser scans

Bauer

Polthier

2009

Computer-Aided Design

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We present an automatic method for computing an accurate parametric model of a piecewise defined pipe surface, consisting of cylinder and torus segments, from an unorganized point set. Our main contributions are reconstruction of the spine curve of a pipe surface from surface samples, and approximation of the spine curve by G 1 continuous circular arcs and line segments. Our algorithm accurately outputs the parametric data required for bending machines to create the reconstructed tube.

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“…Various error criteria have been used in solving polygonal path approximation problems (e.g., see [3]- [7], [9]- [15], [17], [20], [21], [22], [24]- [26], [27]- [33], [34], and [2]- [6]). In this paper we consider the error criterion used in [14], [22], [34], and [8], called the infinite beam or parallel-strip criterion, under the Euclidean L 2 measure of distance. With the infinite beam criterion, the ε-tolerance region of line segment p i p j is the set of points that are within distance ε from the line L( p i p j ) supporting the line segment p i p j (an infinite strip/cylinder of width 2ε centered at L( p i p j )).…”

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confidence: 99%

New Results on Path Approximation

Daescu

2003

Algorithmica

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In this paper we give bounds on the complexity of some algorithms for approximating 2-D and 3-D polygonal paths with the infinite beam measure of error. While the time/space complexities of the algorithms known for other error measures are well understood, path approximation with infinite beam measure seems to be harder due to the complexity of some geometric structures that arise in the known approaches. Our results answer some open problems left in previous work. We also present a more careful analysis of the time complexity of the general iterative algorithm for infinite beam measure and show that it could perform much better in practice than in the worst case. We then propose a new approach for path approximation that consists of a breadth first traversal of the path approximation graph, without constructing the graph. This approach can be applied to any error criterion in any constant dimension. The running time of the new algorithm depends on the size of the output: if the optimal approximating path has m vertices, the algorithm performs F(m) iterations rather than n − 1 in the previous approaches, where F(m) ≤ n − 1 is the number of vertices of the path approximation graph that can be reached with at most m − 2 edges. This is the first output sensitive path approximation algorithm.

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On Approximating Polygonal Curves in Two and Three Dimensions

Cited by 51 publications

References 0 publications

Efficient Algorithms for Approximating Polygonal Chains

Efficient Algorithms for Approximating Polygonal Chains

Generating parametric models of tubes from laser scans

New Results on Path Approximation

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