Journal of Mathematical Imaging and Vision

, Volume 52, Issue 2, pp 173–199 | Cite as

Guaranteed Ellipse Fitting with a Confidence Region and an Uncertainty Measure for Centre, Axes, and Orientation

  • Zygmunt L. SzpakEmail author
  • Wojciech Chojnacki
  • Anton van den Hengel


A simple and fast ellipse estimation method is presented based on optimisation of the Sampson distance serving as a measure of the quality of fit between a candidate ellipse and data points. Generation of ellipses, not just conics, as estimates is ensured through the use of a parametrisation of the set of all ellipses. Optimisation of the Sampson distance is performed with the aid of a custom variant of the Levenberg–Marquardt algorithm. The method is supplemented with a measure of uncertainty of an ellipse fit in two closely related forms. One of these concerns the uncertainty in the algebraic parameters of the fit and the other pertains to the uncertainty in the geometrically meaningful parameters of the fit such as the centre, axes, and major axis orientation. In addition, a means is provided for visualising the uncertainty of an ellipse fit in the form of planar confidence regions. For moderate noise levels, the proposed estimator produces results that are fully comparable in accuracy to those produced by the much slower maximum likelihood estimator. Due to its speed and simplicity, the method may prove useful in numerous industrial applications where a measure of reliability for geometric ellipse parameters is required.


Ellipse fitting Maximum likelihood Uncertainty measure Simultaneous confidence region Centre Semi-major and semi-minor axes Orientation 



This work was partially supported by the Australian Research Council.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Zygmunt L. Szpak
    • 1
    Email author
  • Wojciech Chojnacki
    • 1
  • Anton van den Hengel
    • 1
  1. 1.School of Computer ScienceThe University of AdelaideAdelaideAustralia

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