article

Near-Linear Time Approximation Algorithms for Curve Simplification

Authors:

Pankaj K. Agarwal,

Sariel Har-Peled,

Nabil H. Mustafa,

Yusu WangAuthors Info & Claims

Algorithmica, Volume 42, Issue 3-4

Pages 203 - 219

https://doi.org/10.1007/s00453-005-1165-y

Published: 01 July 2005 Publication History

Abstract

We consider the problem of approximating a polygonal curve P under a given error criterion by another polygonal curve P' whose vertices are a subset of the vertices of P. The goal is to minimize the number of vertices of P' while ensuring that the error between P' and P is below a certain threshold. We consider two different error measures: Hausdorff and Frechet. For both error criteria, we present near-linear time approximation algorithms that, given a parameter ź > 0, compute a simplified polygonal curve P' whose error is less than ź and size at most the size of an optimal simplified polygonal curve with error ź/2. We consider monotone curves in ź2 in the case of the Hausdorff error measure under the uniform distance metric and arbitrary curves in any dimension for the Frechet error measure under Lp metrics. We present experimental results demonstrating that our algorithms are simple and fast, and produce close to optimal simplifications in practice.

References

[1]

P. Agarwal and K. R. Varadarajan. Efficient algorithms for approximating polygonal chains. Discrete & Computational Geometry, 23:273-291, 2000.

Crossref

Google Scholar

[2]

H. Alt and M. Godau. Computing the Fréchet distance between two polygonal curves. International Journal of Computational Geometry, 5(1):75-91, 1995.

Crossref

Google Scholar

[3]

G. Barequet, D. Z. Chen, O. Daescu, M. T. Goodrich, and J. Snoeyink. Efficiently approximating polygonal paths in three and higher dimensions. Algorithmica, 33(2):150-167, 2002.

Digital Library

Google Scholar

[4]

W. S. Chan and F. Chin. Approximation of polygonal curves with minimum number of line segments. In Proc. 3rd Annual International Symposium on Algorithms and Computation, pages 378-387, 1992.

Crossref

Google Scholar

[5]

D. H. Douglas and T. K. Peucker. Algorithms for the reduction of the number of points required to represent a digitized line or its caricature. Canadian Cartographer, 10(2):112-122, 1973.

Crossref

Google Scholar

[6]

M. Godau. A natural metric for curves: computing the distance for polygonal chains and approximation algorithms. In Proc. 8th Annual Symposium on Theoretical Aspects of Computer Science, pages 127-136, 1991.

Crossref

Google Scholar

[7]

L. J. Guibas, J. E. Hershberger, J. B. Mitchell, and J. Snoeyink. Approximating polygons and subdivisions with minimum link paths. International Journal of Computational Geometry and Applications, 3(4):383-415, 1993.

Crossref

Google Scholar

[8]

P. Heckbert and M. Garland. Survey of polygonal surface simplification algorithms. In SIGGRAPH 97 Course Notes: Multiresolution Surface Modeling, 1997.

Google Scholar

[9]

J. Hershberger and J. Snoeyink. An O(n log n) implementation of the Douglas-Peucker algorithm for line simplification. In Proc. 10th Annual ACMSymposium on Computational Geometry, pages 383-384, 1994.

Crossref

Google Scholar

[10]

H. Imai and M. Iri. An optimal algorithm for approximating a piecewise linear function. Information Processing Letters, 9(3):159-162, 1986.

Google Scholar

[11]

H. Imai and M. Iri. Polygonal approximations of curve-formulations and algorithms. In G. T. Toussaint, editor, Computational Morphology, pages 71-86. North-Holland, Amsterdam, 1988.

Crossref

Google Scholar

[12]

A. Melkman and J. O'Rourke. On polygonal chain approximation. In G. T. Toussaint, editor, Computational Morphology, pages 87-95. North-Holland, Amsterdam, 1988.

Google Scholar

[13]

R. Weibel. Generalization of spatial data: principles and selected algorithms. In M. van Kreveld, J. Nievergelt, T. Roos, and P. Widmayer, editors, Algorithmic Foundations of Geographic Information System, pages 99-152. Springer-Verlag, Berlin, 1997.

Crossref

Google Scholar

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Published In

Algorithmica Volume 42, Issue 3-4

July 2005

133 pages

ISSN:0178-4617

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Publisher

Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 01 July 2005

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https://dl.acm.org/doi/10.1145/3539660
Gudmundsson JHorton MPfeifer JSeybold M(2021)A Practical Index Structure Supporting Fréchet Proximity Queries among TrajectoriesACM Transactions on Spatial Algorithms and Systems10.1145/34601217:3(1-33)Online publication date: 14-Jun-2021
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van de Kerkhof MKostitsyna ILöffler M(2021)Embedding Ray Intersection Graphs and Global Curve SimplificationGraph Drawing and Network Visualization10.1007/978-3-030-92931-2_26(358-371)Online publication date: 14-Sep-2021
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