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Interpolation of track data with radial basis methods. (English) Zbl 0766.65005

The authors study some interpolation problems with data sets \(S\), where \(S=\{(x_ i,y_ i),1\leq i\leq n\}\), of \(n\) distinct points in \(\mathbb{R}^ 2\), in which the points lie along tracks or paths in \(\mathbb{R}^ 2\). Four scattered data interpolation methods that perform well in a critical comparison of R. Franke [Math. Comput. 38, 181-200 (1982; Zbl 0476.65005)] are the multiquadric method of R. L. Hardy [Comput. Math. Appl. 19, No. 8/9, 163-208 (1990; Zbl 0692.65003)], the thin plate spline method of J. Duchon [Lect. Notes Math. 571, 85-100 (1977; Zbl 0342.41012)], the minimum norm network method of G. M. Nielson [Math. Comput. 40, 253-271 (1983; Zbl 0549.65005)], and the modified quadratic Shepard method of R. Franke and G. M. Nielson [Int. J. Numer. Methods Eng. 15, 1691-1704 (1980; Zbl 0444.65011)].
These methods are discussed and applied to several track data samples. Also, the multiquadric and thin plate spline radial basis methods are applied to various sets of data that are sampled densely along tracks in the plane.

MSC:

65D05 Numerical interpolation
65D07 Numerical computation using splines

Software:

QSHEP2D
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Franke, R., A critical comparison of some methods for interpolation of scattered data, (Technical Report NPS-53-79-003 (1979), Naval Postgraduate School)
[2] Franke, R., Scattered data interpolation: Tests of some methods, Math. Comp., 38, 181-200 (1982) · Zbl 0476.65005
[3] Hardy, R. L., Multiquadric equations of topography and other irregular surfaces, J. Geophys. Res., 76, 1905-1915 (1971)
[4] Hardy, R. L., Theory and applications of the multiquadric-biharmonic method, Comput. Math. Appl., 19, 8/9, 163-208 (1990) · Zbl 0692.65003
[5] Duchon, J., Splines minimizing rotation invariant semi-norms in Sobelev spaces, (Schempp, W.; Zeller, K., Multivariate Approximation Theory (1975), Birkhauser: Birkhauser Basel), 85-100
[6] Nielson, G. M., A method for interpolation of scattered data based upon a minimum norm network, Math. Comp., 40, 253-271 (1983) · Zbl 0549.65005
[7] Franke, R.; Nielson, G. M., Smooth interpolation to large sets of scattered data, Intern. J. Numer. Methods Engr., 15, 1691-1704 (1980) · Zbl 0444.65011
[8] Carlson, R. E.; Foley, T. A., The parameter \(R^2\) in multiquadric interpolation, Comput. Math. Appl., 21, 9, 29-42 (1991) · Zbl 0725.65009
[9] Buhmann, M. D., Convergence of univariate quasi-interpolation using multiquadrics, IMA J. Numer. Anal., 8, 365-383 (1988) · Zbl 0659.41003
[10] Micchelli, C. A., Interpolation of scattered data: Distance matrices and conditionally positive definite functions, Constr. Approx., 2, 11-22 (1986) · Zbl 0625.41005
[11] Powell, M. J.D., Radial basis functions for multivariate interpolation: A review, (Mason, J. C.; Cox, M. G., Algorithms for Approximation (1987), Oxford University Press: Oxford University Press Oxford), 143-167 · Zbl 0638.41001
[12] Tarwater, A. E., A parameter study of Hardy’s multiquadric method for scattered data interpolation, (Technical Report UCRL-563670 (1985), Lawrence Livermore National Laboratory)
[13] Lawson, C. L., Software for \(C^1\) surface interpolation, (Rice, J. R., Mathematical Software (1977), Academic Press: Academic Press New York), 161-194 · Zbl 0407.68033
[14] Schumaker, L. L., Triangulation methods, (Chui, C.; Schumaker, L. L.; Utreras, F., Topics in Multivariate Approximation (1987), Academic Press: Academic Press New York), 219-232 · Zbl 0632.65120
[15] Renka, R. L., Algorithm 660: QSHEP2D: Quadratic Shepard method for bivariate interpolation to scattered data, ACM Trans. Math. Soft., 14, 149-150 (1988) · Zbl 0709.65504
[16] Foley, T., Interpolation and approximation of 3-D and 4-D scattered data, Comput. Math. Appl., 13, 8, 711-740 (1987) · Zbl 0635.65007
[17] Nielson, G. M.; Franke, R., A method for construction of surfaces under tension, Rocky Mountain J. Math., 14, 203-221 (1984) · Zbl 0551.65002
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