Total Least-Squares Spiral Curve Fitting
Publication: Journal of Surveying Engineering
Volume 125, Issue 4
Abstract
A rapidly convergent algorithm for fitting clothoids to measured points is developed and tested. The second-order, reduced Hessian method, broadly applicable to the class of scalable, C2 parametrizations, is orthogonal distance regression with four-parameter similarity transformations. The local parameters, or state variables, are implicitly eliminated, and second-order solutions are rigorously computed in the model parameter space (rank ≤4). The algorithm is further distinguished from earlier works by the inclusion of approximation procedures that yield very good starting values. Additionally, a strong connection between the Helmert transformation and the total least-squares problem is established, and a fixed point method is suggested.
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Adcock, R. J. (1978). “A problem in least squares.” Analyst, 5, 53–54.
2.
Davis, T. G. ( 1998). “Total least squares curve fitting,” PhD dissertation, University of South Florida, Tampa, Fla.
3.
Davis, T. G., and Lin, S. (1996). “Discussion of `Intersection of spiral curve with circle,' by Olcay Öztan, Orhan Baykal, Oguz Müftüoglu, and Muhammed Sahin.”J. Surv. Engrg., ASCE, 122(4), 181–183.
4.
Gander, W., Golub, G. H., and Strebel, R. (1994). “Least-squares fitting of circles and ellipses.” BIT, 34, 558–578.
5.
Golub, G. H., and Van Loan, C. F. (1980). “An analysis of the total least squares problem.” SIAM J. Numer. Anal., 17(6), 883–393.
6.
Gulliksson, M., and Söderkvist, I. (1995). “Surface fitting and parameter estimation with nonlinear least squares.” Opt. Meth. and Soft., 5, 247–269.
7.
Hager, W. W. (1988). Applied numerical linear algebra. Prentice-Hall, Englewood Cliffs, N.J.
8.
Kahmen, H., and Faig, W. (1988). Surveying. Walter de Gruyter, Hawthorne, N.Y.
9.
Karimäki, V. (1992). “Fast code to fit circular arcs.” Comput. Phys. Comm., 69, 133–141.
10.
Marquardt, D. W. (1963). “An algorithm for least-squares estimation of nonlinear parameters.” J. Soc. Indust. Appl. Math., 11(2), 431–441.
11.
Marsden, J. E., and Tromba, A. J. (1981). Vector calculus. Freeman, San Francisco.
12.
Murray, W., ed. (1972). Numerical methods for unconstrained optimization. Academic, New York.
13.
Pearson, K. (1901). “On lines and planes of closest fit to systems of points in space.” Philosophical Mag., 2, 559–572.
14.
Reklaitis, G. V., Ravindran, A., and Ragsdell, K. M. (1983). Engineering optimization. Wiley, New York.
15.
Späth, H. ( 1997). “Orthogonal least squares fitting by conic sections.” Recent advances in total least squares techniques and errors-in-variables modeling, S. Van Huffel, ed., SIAM, Philadelphia, 259–264.
16.
Stoer, J. (1982). “Curve fitting with clothoidal splines.” J. Res. Natl. Bur. Stand., 87(4), 317–346.
17.
Van Huffel, S., ed. ( 1997). Recent advances in total least squares techniques and errors-in-variables modeling. SIAM, Philadelphia.
18.
Van Huffel, S., and Vandewalle, J. (1991). The total least squares problem: Computational aspects and analysis. SIAM, Philadelphia.
19.
Walton, D. J., and Meek, D. S. (1990). “Clothoidal splines.” Comp. and Graphics, 14(1), 95–100.
20.
Wolf, H. (1968). Ausgleichungsrechung nach der Methode der kleinstein Quadrate. Dümmlers Verlag, Bonn, Germany (in German).
Information & Authors
Information
Published In
Journal of Surveying Engineering
Volume 125 • Issue 4 • November 1999
Pages: 159 - 176
History
Received: Nov 23, 1998
Published online: Nov 1, 1999
Published in print: Nov 1999
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.