Efficient randomized algorithms for robust estimation of circular arcs and aligned ellipses

David M. Mount, Nathan S. Netanyahu

Research output: Contribution to journalArticlepeer-review

20 Scopus citations


Fitting two-dimensional conic sections (e.g., circular and elliptical arcs) to a finite collection of points in the plane is an important problem in statistical estimation and has significant industrial applications. Recently there has been a great deal of interest in robust estimators, because of their lack of sensitivity to outlying data points. The basic measure of the robustness of an estimator is its breakdown point, that is, the fraction (up to 50%) of outlying data points that can corrupt the estimator. In this paper we introduce nonlinear Theil-Sen and repeated median (RM) variants for estimating the center and radius of a circular arc, and for estimating the center and horizontal and vertical radii of an axis-aligned ellipse. The circular arc estimators have breakdown points of ≈ 21% and 50%, respectively, and the ellipse estimators have breakdown points of ≈ 16% and 50%, respectively. We present randomized algorithms for these estimators, whose expected running times are O(n2logn) for the circular case and O(n3logn) for the elliptical case. All algorithms use O(n) space in the worst case.

Original languageEnglish
Pages (from-to)1-33
Number of pages33
JournalComputational Geometry: Theory and Applications
Issue number1
StatePublished - Jun 2001

Bibliographical note

Funding Information:
✩A preliminary version of this report was presented at the Fifth Canadian Conference on Computational Geometry [32]. *Corresponding author. E-mail addresses: mount@cs.umd.edu (D.M. Mount), nathan@macs.biu.ac.il (N.S. Netanyahu). 1 Part of this research was done while the author was visiting the Max-Planck Institut für Informatik, Saarbrücken, Germany. The support of the National Science Foundation under grant CCR–9310705 is gratefully acknowledged. 2 This research was carried out, in part, while the author was also affiliated with the Center of Excellence in Space Data and Information Sciences, Code 930.5, NASA Goddard Space Flight Center, Greenbelt, MD 20771.


  • Aligned ellipse fitting
  • Arrangements
  • Circular arc fitting
  • Computational geometry
  • RM estimator
  • Randomized algorithms
  • Range searching
  • Robust estimation
  • Theil-Sen estimator


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