# HYPERACCURATE ELLIPSE FITTING WITHOUT ITERATIONS

### Kenichi Kanatani, Prasanna Rangarajan

#### Abstract

This paper presents a new method for fitting an ellipse to a point sequence extracted from images. It is widely known that the best fit is obtained by maximum likelihood. However, it requires iterations, which may not converge in the presence of large noise. Our approach is algebraic distance minimization; no iterations are required. Exploiting the fact that the solution depends on the way the scale is normalized, we analyze the accuracy to high order error terms with the scale normalization weight unspecified and determine it so that the bias is zero up to the second order. We demonstrate by experiments that our method is superior to the Taubin method, also algebraic and known to be highly accurate.

#### References

- Albano, R. (1974). Representation of digitized contours in terms of conics and straight-line segments, Comput. Graphics Image Process., 3, 23-33.
- Al-Sharadqah, A. and Chernov, N. (2009). Error analysis for circle fitting algorithms, Elec. J. Stat., 3, 886-911.
- Bookstein, F. J. (1979). Fitting conic sections to scattered data, Comput. Graphics Image Process., 9, 56-71.
- Chernov, N. and Lesort, C. (2004). Statistical efficiency of curve fitting algorithms, Comput. Stat. Data Anal., 47, 713-728.
- Chojnacki, W., Brooks, M. J., van den Hengel, A. and D. Gawley, D. (2000). On the fitting of surfaces to data with covariances, IEEE Trans. Patt. Anal. Mach. Intell., 22, 1294-1303.
- Cooper, D. B. and Yalabik, N. (1976). On the computational cost of approximating and recognizing noiseperturbed straight lines and quadratic arcs in the plane, IEEE Trans. Computers, 25, 1020-1032.
- Fitzgibbon, A., Pilu, M. and Fisher, R. B. (1999). Direct least square fitting of ellipses, IEEE Trans. Patt. Anal. Mach. Intell., 21, 476-480.
- Gander, W., Golub, H. and Strebel, R. (1995). Least-squares fitting of circles and ellipses, BIT , 34, 558-578.
- Gnanadesikan, R. (1977). Methods for Statistical Data Analysis of Multivariable Observations (2nd ed.), Hoboken, NJ: Wiley.
- Hartley, R. and Zisserman, A. (2004). Multiple View Geometry in Computer Vision (2nd ed.), Cambridge: Cambridge University Press.
- Kanatani, K. (2006). Ellipse fitting with hyperaccuracy, IEICE Trans. Inf. & Syst., E89-D, 2653-2660.
- Kanatani, K. (1993). Geometric Computation for Machine Vision, Oxford: Oxford University Press.
- Kanatani, K. (1996). Statistical Optimization for Geometric Computation: Theory and Practice, Amsterdam: Elsevier. Reprinted (2005) New York: Dover.
- Kanatani, K. (2008). Statistical optimization for geometric fitting: Theoretical accuracy analysis and high order error analysis, Int. J. Comp. Vis. 80, 167-188.
- Kanatani, K. and Ohta, N. (2004). Automatic detection of circular objects by ellipse growing, Int. J. Image Graphics, 4, 35-50.
- Kanatani, K. and Sugaya, Y. (2008). Compact algorithm for strictly ML ellipse fitting, Proc. 19th Int. Conf. Pattern Recognition, Tampa, FL.
- Kanatani, K. and Sugaya, Y. (2007). Performance evaluation of iterative geometric fitting algorithms, Comp. Stat. Data Anal., 52, 1208-1222.
- Leedan, Y. and Meer, P. (2000). Heteroscedastic regression in computer vision: Problems with bilinear constraint, Int. J. Comput. Vision., 37, 127-150.
- Matei, B. C. and Meer, P. (2006). Estimation of nonlinear errors-in-variables models for computer vision applications, IEEE Trans. Patt. Anal. Mach. Intell., 28, 1537-1552.
- Paton, K. A. (1970). Conic sections in chromosome analysis, Patt. Recog., 2, 39-40.
- Rangarajan, P. and Kanatani, K. (2009). Improved algebraic methods for circle fitting, Elec. J. Stat., 3, 1075-1082.
- Rosin, P. L. (1993). A note on the least squares fitting of ellipses, Patt. Recog. Lett., 14, 799-808.
- Rosin, P. L. and West, G. A. W. (1995). Nonparametric segmentation of curves into various representations, IEEE. Trans. Patt. Anal. Mach. Intell., 17, 1140-1153.
- Taubin, G. (1991). Estimation of planar curves, surfaces, and non-planar space curves defined by implicit equations with applications to edge and range image segmentation, IEEE Trans. Patt. Anal. Mach. Intell., 13, 1115-1138.

#### Paper Citation

#### in Harvard Style

Kanatani K. and Rangarajan P. (2010). **HYPERACCURATE ELLIPSE FITTING WITHOUT ITERATIONS** . In *Proceedings of the International Conference on Computer Vision Theory and Applications - Volume 2: VISAPP, (VISIGRAPP 2010)* ISBN 978-989-674-029-0, pages 5-12. DOI: 10.5220/0002814500050012

#### in Bibtex Style

@conference{visapp10,

author={Kenichi Kanatani and Prasanna Rangarajan},

title={HYPERACCURATE ELLIPSE FITTING WITHOUT ITERATIONS},

booktitle={Proceedings of the International Conference on Computer Vision Theory and Applications - Volume 2: VISAPP, (VISIGRAPP 2010)},

year={2010},

pages={5-12},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0002814500050012},

isbn={978-989-674-029-0},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the International Conference on Computer Vision Theory and Applications - Volume 2: VISAPP, (VISIGRAPP 2010)

TI - HYPERACCURATE ELLIPSE FITTING WITHOUT ITERATIONS

SN - 978-989-674-029-0

AU - Kanatani K.

AU - Rangarajan P.

PY - 2010

SP - 5

EP - 12

DO - 10.5220/0002814500050012