Least-squares fitting of circles and ellipses
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Fitting circles and ellipses to given points in the plane is a problem that arises in many application areas, e.g., computer graphics, coordinate meteorology, petroleum engineering, statistics. In the past, algorithms have been given which fit circles and ellipses insome least-squares sense without minimizing the geometric distance to the given points.
In this paper we present several algorithms which compute the ellipse for which thesum of the squares of the distances to the given points is minimal. These algorithms are compared with classical simple and iterative methods.
Circles and ellipses may be represented algebraically, i.e., by an equation of the formF(x)=0. If a point is on the curve, then its coordinates x are a zero of the functionF. Alternatively, curves may be represented in parametric form, which is well suited for minimizing the sum of the squares of the distances.
AMS subject classifications65H10 65F20
Key wordsLeast squares curve fitting singular value decomposition
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- 1.V. Pratt,Direct Least Squares Fitting of Algebraic Surfaces, ACM J. Computer Graphics, Volume 21, Number 4, July 1987.Google Scholar
- 2.M. G. Cox, A. B. Forbes,Strategies for Testing Assessment Software, Technical Report NPL DITC 211/92, National Physical Laboratory, Teddington, UK., 1992.Google Scholar
- 3.G. H. Golub, Ch. Van Loan,Matrix Computations, 2nd ed., The Johns Hopkins University Press, Baltimore, MD., 1989.Google Scholar
- 4.D. Sourlier, A. Bucher,Normgerechter Best-fit-Algorithms für Freiform-Flächen oder andere nicht-reguläre Ausgleichs-Flächen in Parameter-Form, Technisches Messen 59 (1992), pp. 293–302.Google Scholar
- 5.H. Späth,Orthogonal least squares fitting with linear manifolds, Numer. Math., 48 (1986), pp. 441–445.Google Scholar
- 6.L. B. Smith,The use of man-machine interaction in data fitting problems, TR CS 131, Computer Science Department, Stanford University, Stanford, CA, Ph.D. thesis, March 1969.Google Scholar
- 7.Y. V. Linnik,Method of least squares and principles of the theory of observations, Pergamon Press, New York, 1961.Google Scholar
- 8.C. L. Lawson,Contributions to the theory of linear least maximum approximation, UCLA, Los Angeles, Ph.D. thesis, 1961.Google Scholar
- 9.Fred L. Bookstein,Fitting conic sections to scattered data, Computer Graphics and Image Processing 9, (1979), pp. 56–71.Google Scholar
- 10.G. H. golub, V. Pereyra,The differentiation of pseudo-inverses and nonlinear least squares problems whose variables separate, SIAM J. Numer. Anal. 10, No. 2, (1973), pp. 413–432.Google Scholar
- 11.M. Heidari, P. C. Heigold,Determination of Hydraulic Conductivity Tensor Using a Nonlinear Least Squares Estimator, Water Resources Bulletin, June 1993, pp. 415–424.Google Scholar
- 12.W. Gander, U. von Matt,Some Least Squares Problems, in Solving Problems in Scientific Computing Using Maple and Matlab, W. Gander and J. Hřebíček, eds., Springer-Verlag, 1993, pp. 251–266.Google Scholar
- 13.Paul T. Boggs, Richard H. Byrd and Robert B. Schnabel,A stable and efficient algorithm for nonlinear orthogonal distance regression, SIAM J. Sci. and Statist. Comput. 8 (1987), pp. 1052–1078.Google Scholar
- 14.Paul T. Boggs, Richard H. Byrd, Janet E. Rogers and Robert B. Schnabel,User's Reference Guide for ODRPACK Verion 2.01—Software for Weighted Orthogonal Distance Regression, National Institute of Standards and Technology, Gaithersburg, June 1992.Google Scholar
- 15.P. E. Gill, W. Murray and M. H. Wright,Practical Optimization, Academic Press, New York, 1981.Google Scholar
- 16.Gene H. Golub, Alan Hoffman, G. W. Stewart,A generalization of the Eckart-Young-Mirsky Matrix Approximation Theorem, Linear Algebra Appl., 88/89 (1987), pp. 317–327.Google Scholar
- 17.Walter Gander, Gene H. Golub, and Rolf Strebel,Fitting of circles and ellipses—least squares solution, Technical Report 217, Insitiut für Wisenschaftliches Rechnen, ETH Zürich, June 1994. Available via anonymous ftp from fip.inf.ethz.ch as doc/tech-reports/1994/217.ps.Google Scholar