Least-squares fitting of circles and ellipses
Tóm tắt
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.
Tài liệu tham khảo
V. Pratt,Direct Least Squares Fitting of Algebraic Surfaces, ACM J. Computer Graphics, Volume 21, Number 4, July 1987.
M. G. Cox, A. B. Forbes,Strategies for Testing Assessment Software, Technical Report NPL DITC 211/92, National Physical Laboratory, Teddington, UK., 1992.
G. H. Golub, Ch. Van Loan,Matrix Computations, 2nd ed., The Johns Hopkins University Press, Baltimore, MD., 1989.
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.
H. Späth,Orthogonal least squares fitting with linear manifolds, Numer. Math., 48 (1986), pp. 441–445.
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.
Y. V. Linnik,Method of least squares and principles of the theory of observations, Pergamon Press, New York, 1961.
C. L. Lawson,Contributions to the theory of linear least maximum approximation, UCLA, Los Angeles, Ph.D. thesis, 1961.
Fred L. Bookstein,Fitting conic sections to scattered data, Computer Graphics and Image Processing 9, (1979), pp. 56–71.
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.
M. Heidari, P. C. Heigold,Determination of Hydraulic Conductivity Tensor Using a Nonlinear Least Squares Estimator, Water Resources Bulletin, June 1993, pp. 415–424.
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.
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.
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.
P. E. Gill, W. Murray and M. H. Wright,Practical Optimization, Academic Press, New York, 1981.
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.
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.