# Robust shape fitting via peeling and grating coresets

Published

Journal Article

Let P be a set of n points in ℝd. We show that a (k, ε)-kernel of P of size O(k/ε(d-1)/2) can be computed in time O(n + k2/εd-1), where a (k, ε)-kernel is a subset of P that ε-approximates the directional width of P, for any direction, when k outliers can be ignored in that direction. A (k, ε)-kernel is instrumental in solving shape fitting problems with k outliers, like computing the minimum-width annulus covering all but k of the input points. The size of the new kernel improves over the previous known upper bound O(k/εd-1) [17], and is tight in the worst case. The new algorithm works by repeatedly "peeling" away (0, ε)-kernels. We demonstrate the practicality of our algorithm by showing its empirical performance on various inputs. We also present a simple incremental algorithm for (1 + ε)-fitting various shapes through a set of points with at most k outliers. The algorithm 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-time algorithm for shape fitting with outliers. We illustrate the versatility and practicality of this technique by implementing approximation algorithms for minimum enclosing circle and minimum-width annulus.

### Full Text

### Duke Authors

### Cited Authors

- Agarwal, PK; Har-Peled, S; Yu, H

### Published Date

- February 28, 2006

### Published In

- Proceedings of the Annual Acm Siam Symposium on Discrete Algorithms

### Start / End Page

- 182 - 191

### Digital Object Identifier (DOI)

- 10.1145/1109557.1109579

### Citation Source

- Scopus