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Approximating surfaces by moving total least squares method. (English) Zbl 0943.65026
Summary: We suggest a method for generating a surface approximating the given data \((x_i,y_i,z_i)\) \(\in \mathbb{R}^3\), \(i= 1,\dots, m\), assuming that the errors can occur both in the independent variables \(x\) and \(y\), as well as in the dependent variable \(z\). Our approach is based on the moving total least squares method, where the local approximants (local planes) are determined in the sense of total least squares. The parameters of the local approximants arere obtained by finding the eigenvector, corresponding to the smallest eigenvalue of a certain symmetric matrix. To this end, we develop a procedure based on the inverse power method. The method is tested on several examples.

65D17 Computer-aided design (modeling of curves and surfaces)
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
65F20 Numerical solutions to overdetermined systems, pseudoinverses
Full Text: DOI
[1] Cleveland, W.S.; Devlin, S.J.; Grose, E., Regression by local Fitting: methods, properties, and computational algorithms, J. econometrics, 37, 87-114, (1988)
[2] Farwig, R., Multivariate interpolation of scattered data by moving least squares methods, (), 193-211 · Zbl 0615.41001
[3] Franke, R.; Salkauskas, K., Localization of multivariate interpolation and smoothing methods, J. comput. appl. math., 73, 79-94, (1996) · Zbl 0876.65003
[4] Franke, R., Approximation of scattered data for meteorological applications, Int. series. num. meth., 94, 107-120, (1990) · Zbl 0708.41013
[5] Franke, R.; Hagen, H., Least squares surface approximation to scattered data using multiquadratic functions, Adv. comput. math., 2, 81-99, (1994) · Zbl 0831.65015
[6] Hagen, H., Topics in surface modelling, (1992), SIAM Philadelphia, PA
[7] Lancaster, P.; Salkauskas, K., Curve and surface Fitting: an introduction, (1986), Academic Press London · Zbl 0649.65012
[8] Schumaker, L., Fitting surfaces to scattered data, () · Zbl 0343.41003
[9] Scitovski, R.; Jukić, D.; Basić, I., Application of the moving least squares method in surface generating, (), 461-467
[10] Franke, R., Scattered data interpolation: test of some methods, Math. comp., 38, 99-106, (1982)
[11] Golub, G.H.; van Loan, C.F., Matrix computations, (1989), The Johns Hopkins Univ. Press Baltimore, MD · Zbl 0733.65016
[12] van Huffel, S.; Vandewalle, J., The total least squares problem: computation aspects and analysis, () · Zbl 0645.93060
[13] Neivergelt, Y., Total least squares: state-of-the-art regression in numerical analysis, SIAM rev., 36, 258-264, (1994)
[14] Nyquist, H., Last orthogonal absolute deviations, Comput. stat. data anal., 6, 361-367, (1988) · Zbl 0726.62111
[15] Scitovski, R.; Ungar, S.; Jukić, D.; Crnjac, M., Moving total least squares for parameter identification in mathematical model, (), 196-201
[16] Scitovski, R.; Jukić, D., A method for solving the parameter identification problem for ordinary differential equations of the second order, Appl. math. comput., 74, 273-291, (1996) · Zbl 0854.65070
[17] Björck, A., Numerical methods for least squares problems, (1996), SIAM PA, Philadelphia · Zbl 0847.65023
[18] Mc Lain, D.H., Drawing contours from arbitrary data point, Comput. J., 17, 318-324, (1974)
[19] Friedman, J.H.; Bentley, J.L.; Finkel, R.A., An algorithm for finding best matches in logarithmic expected time, ACM trans. math. software, 3, 209-226, (1977) · Zbl 0364.68037
[20] Bowyer, A., Computing Dirichlet tessellation, Computer J., 24, 162-166, (1981)
[21] Farin, G., Surfaces over Dirichlet tessellation, Computer aided geometric design, 7, 281-292, (1990) · Zbl 0728.65013
[22] Green, P.J.; Sibson, R., Computing Dirichlet tessellation in the plane, Computer J., 21, 168-173, (1978) · Zbl 0377.52001
[23] Watson, D.F., Computing the n-dimensional Delaunay tessellation with application to Voronoi polytopes, Computer J., 24, 167-172, (1981)
[24] Slapnicar, I., Accurate computation of singular values and eigenvalues of symmetric matrices, Math. commun., 1, 153-167, (1996) · Zbl 0878.65026
[25] Parlet, B.N., The symmetric eigenvalue problem, (1980), Prentice-Hall Englewood Cliffs, NJ
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