### CONSIDERATIONS ABOUT ALGEBRAIC FITTING OF AN ELLIPSE TO SCATTERED 2D DATA

#### Abstract

The our aim of this work is to present some theoretical aspects and numerical results about the algebraic ellipse fitting to 2D data, showing the advantages and also the drawbacks of this procedure. All the necessary theoretical aspects are rigorously presented, as well as many examples that justify the usefulness of the method of algebraic fitting. Consulting the graphical obtained results, one may notice that the accuracy corresponds for the practical use of this method. Consulting the literature, we observe that the main part of scientific works deal with the geometric fitting and the algebraic fitting is considered only to show its deficiencies with respect to the geometric fitting. Our hope is that the algebraic fitting will be reconsidered and used in the cases when it is applicable.

Key words: ellipse, algebraic fitting, least-squares method, parametric equations, accuracy

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Ahn, S. J., Rauh, W., Recknagel, M., Geometric fitting of line, plane, circle, sphere and ellipse, ABW-Workshop 3D-Bildverarbeitung an der Technischen Akademie Esslingen, 25-26 I 1999, 8 pp.

Ahn, S. J., Rauh, W., Warnecke, H.-J., Best-fit of implicit surfaces and plane curves, in Tom Lyche and Larry L. Schumaker (eds.), Mathematical Methods for Curves and Surfaces, pp. 1-14, Oslo, 2000, ISBN 0-8265-1378-6

Ahn, S. J., Rauh, W,., Warneke, H.-J., Least-squares orthogonal distances fitting of circle, ellipse, hyperbola and parabola, Pattern Recognition, 2001, Vol. 34, No. 12, pp. 2283-2303

Bookstein, F. L., Fitting conic sections to scattered data, Computer Graphics and Image Processing, 1979, vol. 9, pp. 56-71

Fitzgibbon, A., Pilu, M., Fisher, R. B., Direct least square fitting of ellipses, IEEE Transactions on Pattern Analysis and Machine Intelligence, 1999, Vol. 21, No. 5, pp. 476-480

Gander, W., Golub, G. H., Strebel, R., Least-squares fitting of circles and ellipses, BIT 1994, Vol. 34, pp. 558-578

Kanatani, K., Ellipse fitting with hyperaccuracy, IEICE Trans. Inform. Syst., 2006, E89-D, pp. 2653–2660.

Kanatani, K., Sugaya, Y., Compact algorithm for strictly ML ellipse fitting, Proceedings of 19th International Conference in Pattern Recognition, Tampa, FL., U.S., 2008

Kanatani, K., Rangarajan, P., Hyperaccurate ellipse fitting without iterations, Proc. Int. Conf. Computer Vision Theory and Applications, (VISAPP 2010), Angers, France, May 2010, Vol. 2, pp. 5-12

Kåsa, I., A circle fitting procedure and its error analysis, IEEE Transactions on Instrumentation and Measurement, 1976, Vol. 25, pp. 8-14

Murgulescu, Elena a. o., Analytical and Differential Geometry, sec. ed. (in Romanian), Editura Didactică şi Pedagogică, Bucureşti, 1965, 772 pp.

Pearson, K., On lines and planes of closest fit to systems of points in space, The Philosophical Magazine, 1901, Ser. 6, Vol. 2, No. 11, pp. 559-572

Pratt, V., Direct least-squares fitting of algebraic surfaces, Computer Graphics, 1987, Vol. 21, pp. 145–152

Press, W. H. a. o., Numerical Recipes in C++. The Art of Scientific Computing, Cambridge University Press, 2003, 1002 pp., ISBN 0-521-75033-4

Rosin, P. I., A note on the least square fitting of ellipses, Pattern Recognition Letters, 1993, Vol. 14, pp. 799-808

Rosin, P. I., Analyzing error of fit functions for ellipses, British Machine Vision Conference, Pattern Recognition Letters, 1996, Vol. 17, No. 14, pp. 1461-1470

Taubin, G., Estimation of planar curves, surfaces and nonplanar space curves defined by implicit equations, with applications to edge and range image segmentation, IEEE Transactions on Pattern Analysis and Machine Intelligence, 1991, Vol. 13, No. 11, pp. 1115–1138

Ursu-Fischer, N., Ursu, M., Numerical Methods in Engineering and Programs in C/C++ (in Romanian), vol. I, Cluj-Napoca, Casa Cărţii de Ştiinţă, 2000, 282 pp., ISBN 973-686-039-6

Ursu-Fischer, N., Ursu, M., Numerical Methods in Engineering and Programs in C/C++ (in Romanian), vol. II, Cluj-Napoca, Casa Cărţii de Ştiinţă, 2003, 288 pp., ISBN 973-686-463-4

Ursu-Fischer, N., Ursu, M., A new and efficient method to perform the circle fitting, Acta Technica Napocensis, Series: Applied Mathematics and Mechanics, 2004, No. 47, Vol. III, pp. 21-30, ISSN 1221-5872

Wang, Hao a. o., Robust real-time ellipse fitting based on Lagrange programming neural network and locally competitive algorithm, IEEE Transactions, 2018, 30 pp.

Watson, G. A., Least squares fitting of circles and ellipses to measured data, BIT, 1999, Vol. 39, No. 1, pp. 176-191

Watson, G. A., Least squares fitting of parametric surfaces to measured data, ANZIAM J., 42(E), pp. C68-C96

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