Electronic Journal of Statistics

Error analysis for circle fitting algorithms

Ali Al-Sharadqah and Nikolai Chernov

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We study the problem of fitting circles (or circular arcs) to data points observed with errors in both variables. A detailed error analysis for all popular circle fitting methods – geometric fit, Kåsa fit, Pratt fit, and Taubin fit – is presented. Our error analysis goes deeper than the traditional expansion to the leading order. We obtain higher order terms, which show exactly why and by how much circle fits differ from each other. Our analysis allows us to construct a new algebraic (non-iterative) circle fitting algorithm that outperforms all the existing methods, including the (previously regarded as unbeatable) geometric fit.

Article information

Electron. J. Statist., Volume 3 (2009), 886-911.

First available in Project Euclid: 24 August 2009

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Least squares fit curve fitting circle fitting algebraic fit error analysis variance bias functional model


Al-Sharadqah, Ali; Chernov, Nikolai. Error analysis for circle fitting algorithms. Electron. J. Statist. 3 (2009), 886--911. doi:10.1214/09-EJS419. https://projecteuclid.org/euclid.ejs/1251119958

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  • [1] Amemiya, Y. and Fuller, W.A. (1988). Estimation for the nonlinear functional relationship., Annals Statist. 16 147–160.
  • [2] Anderson, T.W. (1976). Estimation of linear functional relationships: Approximate distributions and connections with simultaneous equations in econometrics., J. R. Statist. Soc. B 38 1–36.
  • [3] Anderson, T.W. and Sawa, T. (1982). Exact and approximate distributions of the maximum likelihood estimator of a slope coefficient., J. R. Statist. Soc. B 44 52–62.
  • [4] Atieg, A. and Watson, G.A. (2004). Fitting circular arcs by orthogonal distance regression., Appl. Numer. Anal. Comput. Math. 1 66–76.
  • [5] Berman, M. (1989). Large sample bias in least squares estimators of a circular arc center and its radius., CVGIP: Image Understanding 45 126–128.
  • [6] Chan, N.N. (1965). On circular functional relationships., J. R. Statist. Soc. B 27 45–56.
  • [7] Chernov, N. Fitting circles to scattered data: parameter estimates have no moments., Manuscript, see http://www.math.uab.edu/~chernov/cl.
  • [8] Chernov, N. and Lesort, C. (2004). Statistical efficiency of curve fitting algorithms., Comp. Stat. Data Anal 47 713–728.
  • [9] Chernov, N. and Lesort, C. (2005). Least squares fitting of circles., J. Math. Imag. Vision 23 239–251.
  • [10] Chernov, N. and Sapirstein, P. (2008). Fitting circles to data with correlated noise., Comput. Statist. Data Anal 52 5328–5337.
  • [11] Delogne, P. (1972). Computer optimization of Deschamps’ method and error cancellation in reflectometry., In Proc. IMEKO-Symp. Microwave Measurement (Budapest) 117–123.
  • [12] Fuller, W.A. (1987)., Measurement Error Models. L. Wiley & Son, New York.
  • [13] Gander, W., Golub, G.H., and Strebel, R. (1994). Least squares fitting of circles and ellipses., BIT 34 558–578.
  • [14] http://www.math.uab.edu/, chernov/cl.
  • [15] Joseph, S.H. (1994). Unbiased least-squares fitting of circular arcs., Graph. Mod. Image Process. 56 424–432.
  • [16] Kadane, J.B. (1970). Testing overidentifying restrictions when the disturbances are small., J. Amer. Statist. Assoc. 65 182–185.
  • [17] Kanatani, K. (1996)., Statistical Optimization for Geometric Computation: Theory and Practice. Elsevier Science, Amsterdam, Netherlands.
  • [18] Kanatani, K. (1998). Cramer-Rao lower bounds for curve fitting., Graph. Mod. Image Process 60 93–99.
  • [19] Kanatani, K. (2004). For geometric inference from images, what kind of statistical model is necessary?, Syst. Comp. Japan 35 1–9.
  • [20] Kanatani, K. (2005). Optimality of maximum likelihood estimation for geometric fitting and the KCR lower bound., Memoirs Fac. Engin. Okayama Univ 39 63–70.
  • [21] Kanatani, K. (2006). Ellipse fitting with hyperaccuracy., IEICE Trans. Inform. Syst. E89-D, 2653–2660.
  • [22] Kanatani, K. (2008). Statistical optimization for geometric fitting: Theoretical accuracy bound and high order error analysis., Int. J. Computer Vision 80 167–188.
  • [23] Kåsa, I. (1976). A curve fitting procedure and its error analysis., IEEE Trans. Inst. Meas. 25 8–14.
  • [24] Landau, U.M. (1987). Estimation of a circular arc center and its radius., CVGIP: Image Understanding 38 317–326.
  • [25] Meer, P. (2004)., Robust techniques for computer vision. In G. Medioni and S. B. Kang, editors. Emerging Topics in Computer Vision, Prentice Hall, pages 107–190.
  • [26] Nievergelt, Y. (2002). A finite algorithm to fit geometrically all midrange lines, circles, planes, spheres, hyperplanes, and hyperspheres., J. Numerische Math. 91 257–303.
  • [27] Pratt, V. (1987). Direct least-squares fitting of algebraic surfaces., Computer Graphics 21 145–152.
  • [28] Rusu, C., Tico, M., Kuosmanen, P., and Delp, E.J. (2003). Classical geometrical approach to circle fitting – review and new developments., J. Electron. Imaging 12 179–193.
  • [29] Sarkar, B., SinghL, K., and Sarkar, D. (2003). Approximation of digital curves with line segments and circular arcs using genetic algorithms., Pattern Recogn. Letters 24 2585–2595.
  • [30] Späth, H. (1996). Least-squares fitting by circles., Computing 57 179–185.
  • [31] Strandlie, A., Wroldsen, J., Frühwirth, R. and Lillekjendlie, B. (2000). Particle tracks fitted on the Riemann sphere., Computer Physics Commun. 131 95–108.
  • [32] Taubin, G. (1991). Estimation of planar curves, surfaces and nonplanar space curves defined by implicit equations, with applications to edge and range image segmentation., IEEE Trans. Pattern Analysis Machine Intelligence 13 1115–1138.
  • [33] Umbach, D. and Jones, K.N. (2003). A few methods for fitting circles to data., IEEE Trans. Instrument. Measur. 52 1181–1885.
  • [34] Wolter, K.M. and Fuller, W.A. (1982). Estimation of nonlinear errors-in-variables models., Annals Statist. 10 539–548.
  • [35] Yin, S.J. and Wang, S.G. (2004). Estimating the parameters of a circle by heteroscedastic regression models., J. Statist. Planning Infer. 124 439–451.
  • [36] Zelniker, E. and Clarkson, V. (2006). Maximum-likelihood estimation of circle parameters via convolution., IEEE Trans. Image Proc. 15 865–876.
  • [37] Zelniker, E. and Clarkson, V. (2006). A statistical analysis of the Delogne-Kåsa method for fitting circles., Digital Signal Proc. 16 498–522.
  • [38] Zhang, S. and Xie, L. and Adams, M.D. (2006) Feature extraction for outdoor mobile robot navigation based on a modified Gauss-Newton optimization approach., Robotics Autonom. Syst. 54 277–287.