Journal article

Publication year: 2004

Pages: 223-271

Journal: Advances in Computational Mathematics

Volume number: 21

Issue number: 3-4

ISSN: 1019-7168

eISSN: 1572-9044

Open access status: Green

DOI Link: https://doi.org/10.1023/B:ACOM.0000032041.68678.fa

Publisher: Springer

Abstract: 
We present a new scattered data fitting method, where local approximating polynomials are directly extended to smooth (C-1 or C-2) splines on a uniform triangulation Delta ( the four-directional mesh). The method is based on designing appropriate minimal determining sets consisting of whole triangles of domain points for a uniformly distributed subset of Delta. This construction allows to use discrete polynomial least squares approximations to the local portions of the data directly as parts of the approximating spline. The remaining Bernstein-Bezier coefficients are efficiently computed by extension, i.e., using the smoothness conditions. To obtain high quality local polynomial approximations even for difficult point constellations (e.g., with voids, clusters, tracks), we adaptively choose the polynomial degrees by controlling the smallest singular value of the local collocation matrices. The computational complexity of the method grows linearly with the number of data points, which facilitates its application to large data sets. Numerical tests involving standard benchmarks as well as real world scattered data sets illustrate the approximation power of the method, its efficiency and ability to produce surfaces of high visual quality, to deal with noisy data, and to be used for surface compression.

Harvard Citation style: Davydov, O. and Zeilfelder, F. (2004) Scattered data fitting by direct extension of local polynomials to bivariate splines, Advances in Computational Mathematics, 21(3-4), pp. 223-271. https://doi.org/10.1023/B:ACOM.0000032041.68678.fa

APA Citation style: Davydov, O., & Zeilfelder, F. (2004). Scattered data fitting by direct extension of local polynomials to bivariate splines. Advances in Computational Mathematics. 21(3-4), 223-271. https://doi.org/10.1023/B:ACOM.0000032041.68678.fa