Journal of Mathematical Imaging and Vision

, Volume 23, Issue 3, pp 239–252

# Least Squares Fitting of Circles

• N. Chernov
• C. Lesort
Article

## Abstract

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.

## Keywords

least squares fit circle fitting Levenberg-Marquardt algorithm

## References

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.
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.
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.
8. 8.
N. Chernov and C. Lesort, “Fitting circles and lines by least squares: theory and experiment, preprint,” available at http://www.math.uab.edu/cl/cl1
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.