Discrete & Computational Geometry

, Volume 39, Issue 1–3, pp 38–58

# Robust Shape Fitting via Peeling and Grating Coresets

• Pankaj K. Agarwal
• 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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