Journal of Mathematical Imaging and Vision

, Volume 23, Issue 3, pp 239–252 | Cite as

Least Squares Fitting of Circles

  • N. Chernov
  • C. Lesort


Fitting standard shapes or curves to incomplete data (which represent only a small part of the curve) is a notoriously difficult problem. Even if the curve is quite simple, such as an ellipse or a circle, it is hard to reconstruct it from noisy data sampled along a short arc. Here we study the least squares fit (LSF) of circular arcs to incomplete scattered data. We analyze theoretical aspects of the problem and reveal the cause of unstable behavior of conventional algorithms. We also find a remedy that allows us to build another algorithm that accurately fits circles to data sampled along arbitrarily short arcs.


least squares fit circle fitting Levenberg-Marquardt algorithm 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    G.J. Agin, “Fitting Ellipses and General Second-Order Curves,” Carnegi Mellon University, Robotics Institute, Technical Report 81–5, 1981.Google Scholar
  2. 2.
    S.J. Ahn, W. Rauh, and H.J. Warnecke, “Least-squares orthogonal distances fitting of circle, sphere, ellipse, hyperbola, and parabola,” Pattern Recog., Vol. 34, pp. 2283–2303, 2001.CrossRefGoogle Scholar
  3. 3.
    M. Berman and D. Culpin, “The statistical behaviour of some least squares estimators of the centre and radius of a circle,” J. R. Statist. Soc. B, Vol. 48, pp. 183–196, 1986.Google Scholar
  4. 4.
    R.H. Biggerstaff, “Three variations in dental arch form estimated by a quadratic equation,” J. Dental Res., Vol. 51, pp. 1509, 1972.Google Scholar
  5. 5.
    F.L. Bookstein, “Fitting conic sections to scattered data,” Comp. Graph. Image Proc., Vol. 9, pp. 56–71, 1979.Google Scholar
  6. 6.
    Y.T. Chan and S.M. Thomas, “CramerRao Lower Bounds for Estimation of a Circular Arc Center and Its Radius,” Graph. Models Image Proc., Vol. 57, pp. 527–532, 1995.CrossRefGoogle Scholar
  7. 7.
    B.B. Chaudhuri and P. Kundu, “Optimum Circular Fit to Weighted Data in Multi-Dimensional Space,” Patt. Recog. Lett., Vol. 14, pp. 1–6, 1993.CrossRefGoogle Scholar
  8. 8.
    N. Chernov and C. Lesort, “Fitting circles and lines by least squares: theory and experiment, preprint,” available at
  9. 9.
    N. Chernov and C. Lesort, “Statistical efficiency of curve fitting algorithms,” Comput. Statist.&Data Analysis, Vol. 47, pp. 713–728, 2004.Google Scholar
  10. 10.
    N.I. Chernov and G.A. Ososkov, “Effective algorithms for circle fitting,” Comp. Phys. Comm., Vol. 33, pp. 329–333, 1984.CrossRefGoogle Scholar
  11. 11.
    W. Chojnacki, M.J. Brooks, and A. van den Hengel, “Rationalising the renormalisation method of Kanatani,” J. Math. Imaging&Vision, Vol. 14, pp. 21–38, 2001.Google Scholar
  12. 12.
    J.F. Crawford, “A non-iterative method for fitting circular arcs to measured points,” Nucl. Instr. Meth., Vol. 211, pp. 223–225, 1983.CrossRefGoogle Scholar
  13. 13.
    P. Delonge, “Computer optimization of Deschamps’ method and error cancellation in reflectometry,” in Proceedings IMEKO-Symp. Microwave Measurement, Budapest, 1972, pp. 117–123.Google Scholar
  14. 14.
    P.R. Freeman, “Note: Thom’s survey of the Avebury ring,” J. Hist. Astronom., Vol. 8, pp. 134–136, 1977.Google Scholar
  15. 15.
    W. Gander, G.H. Golub, and R. Strebel, “Least squares fitting of circles and ellipses,” BIT, Vol. 34, pp. 558–578, 1994.CrossRefGoogle Scholar
  16. 16.
    S.H. Joseph, “Unbiased Least-Squares Fitting Of Circular Arcs,” Graph. Models Image Proc., Vol. 56, pp. 424–432, 1994.Google Scholar
  17. 17.
    K. Kanatani, “Cramer-Rao lower bounds for curve fitting,” Graph. Models Image Proc., Vol. 60, pp. 93–99, 1998.CrossRefGoogle Scholar
  18. 18.
    V. Karimäki, “Effective circle fitting for particle trajectories,” Nucl. Instr. Meth. Phys. Res. A, Vol. 305, pp. 187–191, 1991.Google Scholar
  19. 19.
    I. Kasa, “A curve fitting procedure and its error analysis,” IEEE Trans. Inst. Meas., Vol. 25, pp. 8–14, 1976.Google Scholar
  20. 20.
    U.M. Landau, “Estimation of a circular arc center and its radius,” Computer Vision, Graphics and Image Processing, Vol. 38, pp. 317–326, 1987.Google Scholar
  21. 21.
    Y. Leedan and P. Meer, “Heteroscedastic regression in computer vision: Problems with bilinear constraint,” Intern. J. Comp. Vision, Vol. 37, pp. 127–150, 2000.CrossRefGoogle Scholar
  22. 22.
    K. Levenberg, “A Method for the Solution of Certain Non-linear Problems in Least Squares,” Quart. Appl. Math., Vol. 2, pp. 164–168, 1944.Google Scholar
  23. 23.
    D. Marquardt, “An Algorithm for Least Squares Estimation of Nonlinear Parameters,” SIAM J. Appl. Math., Vol. 11, pp. 431–441, 1963.CrossRefGoogle Scholar
  24. 24.
    L. Moura and R.I. Kitney, “A direct method for least-squares circle fitting,” Comp. Phys. Comm., Vol. 64, pp. 57–63, 1991.CrossRefGoogle Scholar
  25. 25.
    G.A. Ososkov, “JINR technical report P10-83-187,” Dubna, 1983, pp. 40 (in Russian).Google Scholar
  26. 26.
    K. Paton, “Conic sections in chromosome analysis,” Pattern Recogn., Vol. 2, pp. 39–51, 1970.CrossRefGoogle Scholar
  27. 27.
    V. Pratt, “Direct least-squares fitting of algebraic surfaces,” Computer Graphics, Vol. 21, pp. 145–152, 1987.Google Scholar
  28. 28.
    S.M. Robinson, “Fitting spheres by the method of least squares,” Commun. Assoc. Comput. Mach., Vol. 4, p. 491, 1961.Google Scholar
  29. 29.
    P.D. Sampson, “Fitting conic sections to very scattered data: An iterative refinement of the Bookstein algorithm,” Comp. Graphics Image Proc., Vol. 18, pp. 97–108, 1982.CrossRefGoogle Scholar
  30. 30.
    C. Shakarji, “Least-squares fitting algorithms of the NIST algorithm testing system,” J. Res. Nat. Inst. Stand. Techn., Vol. 103, pp. 633–641, 1998.Google Scholar
  31. 31.
    H. Spath, “LeastSquares Fitting By Circles,” Computing, Vol. 57, pp. 179–185, 1996.Google Scholar
  32. 32.
    H. Spath, “Orthogonal least squares fitting by conic sections,” in Recent Advances in Total Least Squares techniques and Errors-in-Variables Modeling, SIAM, 1997, pp. 259–264.Google Scholar
  33. 33.
    G. Taubin, “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 Vol. 13, pp. 1115–1138, 1991.CrossRefGoogle Scholar
  34. 34.
    S.M. Thomas and Y.T. Chan, “A simple approach for the estimation of circular arc center and its radius,” Computer Vision, Graphics and Image Processing, Vol. 45, pp. 362–370, 1989.Google Scholar
  35. 35.
    K. Turner, “Computer perception of curved objects using a television camera,” Ph.D. Thesis, Dept. of Machine Intelligence, University of Edinburgh, 1974.Google Scholar
  36. 36.
    Z. Wu, L. Wu, and A. Wu, “The Robust Algorithms for Finding the Center of an Arc,” Comp. Vision Image Under., Vol. 62, pp. 269–278, 1995.CrossRefGoogle Scholar
  37. 37.
    Z. Zhang, “Parameter Estimation Techniques: A Tutorial with Application to Conic Fitting,” International Journal of Image and Vision Computing, Vol. 15, pp. 59–76, 1997.CrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  • N. Chernov
    • 1
  • C. Lesort
    • 1
  1. 1.Department of MathematicsUniversity of Alabama at BirminghamBirminghamUSA

Personalised recommendations