Circular Arcs Fitted on a Riemann Sphere

doi:10.1006/cviu.1997.0529

Computer Vision and Image Understanding

Volume 67, Issue 3, September 1997, Pages 311-317

https://doi.org/10.1006/cviu.1997.0529 Get rights and content

Abstract

A method for fitting a circle to a set of points in two dimensions is presented. The method applies a stereographic projection to map the 2D points onto a cricle on a 3D Riemann sphere. Then, a plane is fitted to the transformed points, and the parameters of the sought circle are determined from the identified plane parameters. The method is direct, and it is shown that the estimator is unbiased to first-order accuracy. The best performance is obtained when the circle has unit radius and is centered at the origin. The method is unbiased to second order in this case, and experiments with synthetic data show that the method is practically unbiased. In addition, the simulations show that origin centering and unit radius requirements are not critical. Circular arcs down to 45° can be estimated at realistic noise levels. Comparisons with three alternative methods are done, and the accuracy and computational simplicity of this method compares favorably with the alternatives.

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We present a novel method of fitting trajectories of charged particles in high-energy physics particle detectors. The method fits a circular arc to two-dimensional measurements by mapping the measurements onto the Riemann sphere and fitting a plane to the transformed coordinates of the measurements. In this way, the non-linear task of circle fitting, which in general requires the application of some iterative procedure, is turned into a linear problem which can be solved in a fast, direct and non-iterative manner. We illustrate the usefulness of our approach by stating results from two simulation experiments of tracks from the ATLAS Inner Detector Transition Radiation Tracker (TRT). The first experiment shows that with a significantly lower execution time, the accuracy of the estimated track parameters is virtually as good as the accuracy obtained by applying an optimal, non-linear least-squares procedure. The second experiment focuses on track parameter estimation in the presence of ambiguous measurements. For this purpose, we have developed a new version of the Elastic Arms algorithm called the Elastic Planes algorithm. The algorithm produces results which are almost identical to the results from an optimal version of the Elastic Arms algorithm. The computational cost of our algorithm, however, is much lower.
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NoteCircular Arcs Fitted on a Riemann Sphere

Abstract

CVGIP: Graph. Models Image Process.

Nucl. Instrum. Methods Phys. Res. A

Pattern Recognit. Lett.

Pattern Recognit.

Pattern Recognit. Lett.

CVGIP: Image Understanding

CVGIP: Image Understanding

Computer Vision, Graphics and Image Processing

Computer Vision, Graphics and Image Processing

Computer Vision, Graphics, and Image Processing

Pattern Recognit. Lett.

Computer Graphics and Image Processing

Comput. Vision Image Understanding

Image and Vision Computing

Note
Circular Arcs Fitted on a Riemann Sphere