Perturbation analysis for circles, spheres, and generalized hyperspheres fitted to data by geometric total least-squares

Nievergelt, Yves

doi:10.1090/S0025-5718-03-01613-2

Perturbation analysis for circles, spheres, and generalized hyperspheres fitted to data by geometric total least-squares
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Math. Comp. 73 (2004), 169-180 Request permission

Abstract:

A continuous extension of the objective function to a projective space guarantees that for each data set there exists at least one hyperplane or hypersphere minimizing the average squared distance to the data. For data sufficiently close to a hypersphere, as the collinearity of the data increases, so does the sensitivity of the fitted hypersphere to perturbations of the data.

References

Åke Björck, Numerical methods for least squares problems, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. MR 1386889, DOI 10.1137/1.9781611971484
T. Altherr and J. Seixas, Cherenkov ring recognition using a nonadaptable network, Nuclear Instruments & Methods in Physics Research, Section A A317 (1992), no. 1, 2, 335–338.
Fred L. Bookstein, Fitting conic sections to scattered data, Computer Graphics and Image Processing 9 (1979), 56–71.
I. D. Coope, Circle fitting by linear and nonlinear least squares, J. Optim. Theory Appl. 76 (1993), no. 2, 381–388. MR 1203908, DOI 10.1007/BF00939613
J. F. Crawford, A noniterative method for fitting circular arcs to measured points, Nuclear Instruments and Methods 211 (1983), no. 2, 223–225.
I. Duerdoth, Track fitting and resolution with digital detectors, Nuclear Instruments and Methods 203 (1982), no. 1–3, 291–297.
Wendell Helms Fleming, Functions of several variables, second ed., Springer-Verlag, New York, NY, 1987, Corrected third printing.
Walter Gander, Gene H. Golub, and Rolf Strebel, Least-squares fitting of circles and ellipses, BIT 34 (1994), no. 4, 558–578. MR 1430909, DOI 10.1007/BF01934268
M. Hansroul, H. Jeremie, and D. Savard, Fast circle fit with the conformal mapping method, Nuclear Instruments & Methods in Physics Research, Section A A270 (1988), no. 2, 3, 498–501.
Peter Henrici, Essentials of numerical analysis with pocket calculator demonstrations, John Wiley & Sons, Inc., New York, 1982. MR 655251
—, Solutions manual essentials of numerical analysis with pocket calculator demonstrations, Wiley, New York, NY, 1982.
Nicholas J. Higham, Accuracy and stability of numerical algorithms, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. MR 1368629
V. Karimäki, Effective circle fitting for particle trajectories, Nuclear Instruments & Methods in Physics Research, Section A A305 (1991), no. 1, 187–191.
I. Kȧsa, A circle fitting procedure and its error analysis, IEEE Transactions on Instrumentation and Measurement 25 (1976), no. 1, 8–14.
Charles L. Lawson and Richard J. Hanson, Solving least squares problems, Classics in Applied Mathematics, vol. 15, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. Revised reprint of the 1974 original. MR 1349828, DOI 10.1137/1.9781611971217
L. Moura and R. Kitney, A direct method for least-squares circle fitting, Comput. Phys. Comm. 64 (1991), no. 1, 57–63. MR 1099915, DOI 10.1016/0010-4655(91)90049-Q
Yves Nievergelt, A tutorial history of least squares with applications to astronomy and geodesy, J. Comput. Appl. Math. 121 (2000), no. 1-2, 37–72. Numerical analysis in the 20th century, Vol. I, Approximation theory. MR 1780042, DOI 10.1016/S0377-0427(00)00343-5
Vaughan Pratt, Direct least-squares fitting of algebraic surfaces, Comput. Graphics 21 (1987), no. 4, 145–152. SIGGRAPH ’87 (Anaheim, CA, 1987). MR 987653, DOI 10.1145/37402.37420
Chris Rorres and David Gilman Romano, Finding the center of a circular starting line in an ancient Greek stadium, SIAM Rev. 39 (1997), no. 4, 745–754. MR 1491056, DOI 10.1137/S0036144596305727
H. Späth, Least-square fitting with spheres, J. Optim. Theory Appl. 96 (1998), no. 1, 191–199. MR 1608054, DOI 10.1023/A:1022675403441
G. W. Stewart, On the early history of the singular value decomposition, SIAM Rev. 35 (1993), no. 4, 551–566. MR 1247916, DOI 10.1137/1035134
Sabine Van Huffel and Joos Vandewalle, The total least squares problem, Frontiers in Applied Mathematics, vol. 9, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1991. Computational aspects and analysis; With a foreword by Gene H. Golub. MR 1118607, DOI 10.1137/1.9781611971002
Per-Ȧke Wedin, Perturbation bounds in connection with singular value decomposition, Nordisk Tidskr. Informationsbehandling (BIT) 12 (1972), 99–111. MR 309968, DOI 10.1007/bf01932678