Abstract
In computer vision and computer aided manufacturing, it is often necessary to fit a circular arc through a number of noisy points. Determining the arc center and radius from a set of points is inherently a nonlinear problem and all estimators will exhibit the so-called threshold phenomenon. A combination of a short arc, small number of points, and large noise magnitude will create a threshold region (THR) whereby the estimation errors are several times larger than those above the THR. The transition into the THR is sudden. It is also difficult to determine the THR for an estimator. This paper presents an estimation scheme for the circle parameters by first computing different centers from all combinations of N data points, taken three at a time. A weighted average of those centers gives the final estimate. The procedure is simple, noniterative and simulation results show that it has a smaller THR than an estimator which is near-optimal when not operating inside the THR.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
KARIMAKI, V., Effective Circle Fitting for Particle Trajectories, Nuclear Instrumentation Methods in Physics Research, Vol. 305A, pp. 187-191, 1991.
JOSEPH, S. H., Unbiased Least Squares Fitting of Circular Arcs, CVGIP: Computer Vision Graphics and Image Processing, Vol. 56, pp. 424-432, 1994.
SHAPIRO, S. D., Properties of Transforms for the Detection of Curves in Noisy Pictures, Computer Vision Graphics and Image Processing, Vol. 8, pp. 129-143, 1978.
KIM, C. E., Digital Disks, IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 6, pp. 372-374, 1984.
LANDAU, V. M., Estimation of Circular Arc Center and Its Radius, Computer Vision Graphics and Image Processing, Vol. 38, pp. 317-332, 1987.
COOPE, I. D., Circle Fitting by Linear and Nonlinear Least Squares, Journal of Optimization Theory and Applications, Vol. 76, pp. 381-388, 1993.
WU, Z. Q., WU, L. D., and WU, A. C., The Robot Algorithms for Finding the Center of an Arc, Computer Vision and Image Understanding, Vol. 62, pp. 269-278, 1995.
THOMAS, S. M., and CHAN, Y. T., A Simple Approach for the Estimation of Circular Arc and Its Radius, Computer Vision Graphics and Image Processing, Vol. 45, pp. 362-370, 1989.
GANDER, W., GOLUB, G. H., and TREBEL, R. S., Least-Squares Fitting of Circles and Ellipses, Bit, Vol. 34, pp. 558-578, 1994.
CHAN, Y. T., and THOMAS S. M., An Approximate Maximum Likelihood Linear Estimator of Circle Parameters, CVGIP: Graphical Models and Image Processing, Vol. 59, pp. 173-178, 1997.
CHAN, Y. T., LEE, B. H., and THOMAS, S. M., Unbiased Estimation of Circle Parameters, Journal of Optimization Theory and Application, Vol. 106, pp. 49-60, 2000.
LATHI, B. P., Modern Digital and Analog Communication Systems, 3rd Edition, Oxford University Press, Oxford, England, 1998.
VAN TREES, H. I., Detection, Estimation, and Modulation Theory, Part 1, Wiley, New York, NY, 1968.
CHAN, Y. T., and THOMAS, S. M., Cramer-Rao Lower Bounds for Estimation of a Circular Arc Center and Its Radius, CVGIP: Graphical Models and Image Processing, Vol. 57, pp. 527-532, 1995.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Chan, Y., Elhalwagy, Y. & Thomas, S. Estimation of Circle Parameters by Centroiding. Journal of Optimization Theory and Applications 114, 363–371 (2002). https://doi.org/10.1023/A:1016087702231
Issue Date:
DOI: https://doi.org/10.1023/A:1016087702231