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Discrete & Computational Geometry

, Volume 39, Issue 1–3, pp 38–58 | Cite as

Robust Shape Fitting via Peeling and Grating Coresets

  • Pankaj K. AgarwalEmail author
  • Sariel Har-Peled
  • Hai Yu
Article

Abstract

Let P be a set of n points in ℝ d . A subset \(\mathcal {S}\) of P is called a (k,ε)-kernel if for every direction, the directional width of \(\mathcal {S}\) ε-approximates that of P, when k “outliers” can be ignored in that direction. We show that a (k,ε)-kernel of P of size O(k/ε (d−1)/2) can be computed in time O(n+k 2/ε d−1). The new algorithm works by repeatedly “peeling” away (0,ε)-kernels from the point set.

We also present a simple ε-approximation algorithm for fitting various shapes through a set of points with at most k outliers. The algorithm is incremental and works by repeatedly “grating” critical points into a working set, till the working set provides the required approximation. We prove that the size of the working set is independent of n, and thus results in a simple and practical, near-linear ε-approximation algorithm for shape fitting with outliers in low dimensions.

We demonstrate the practicality of our algorithms by showing their empirical performance on various inputs and problems.

Keywords

Shape fitting Coresets Geometric approximation algorithms 

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Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of Computer ScienceDuke UniversityDurhamUSA
  2. 2.Department of Computer ScienceUniversity of IllinoisUrbanaUSA

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