Davydov, O. and Zeilfelder, F. (2004) Scattered data fitting by direct extension of local polynomials to bivariate splines. Advances in Computational Mathematics, 21 (34). pp. 223271. ISSN 10197168

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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 (the fourdirectional mesh). The method is based on designing appropriate minimal determining sets consisting of whole triangles of domain points for a uniformly distributed subset of . 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 BernsteinBæ#169;zier 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.
Item type:  Article 

ID code:  2172 
Keywords:  scattered data fitting, bivariate splines, fourdirectional mesh, local polynomial least squares approximation, BernsteinBézier techniques, minimal determining set, Mathematics, Computational Mathematics, Applied Mathematics 
Subjects:  Science > Mathematics 
Department:  Faculty of Science > Mathematics and Statistics 
Depositing user:  Strathprints Administrator 
Date Deposited:  04 Jan 2007 
Last modified:  18 Jun 2015 09:11 
URI:  http://strathprints.strath.ac.uk/id/eprint/2172 
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