Computational geometry: an introduction
Computational geometry: an introduction
Algorithms
Multiresolution surface modeling based on hierarchical triangulation
Computer Vision and Image Understanding
Multistep scattered data interpolation using compactly supported radial basis functions
Journal of Computational and Applied Mathematics - Special issue on scattered data
Mathematical Methods for Curves and Surfaces
Hierarchical scattered data filtering for multilevel interpolation schemes
Mathematical Methods for Curves and Surfaces
Approximating and intersecting surfaces from points
Proceedings of the 2003 Eurographics/ACM SIGGRAPH symposium on Geometry processing
Progressive scattered data filtering
Journal of Computational and Applied Mathematics
Proceedings of the 2004 Eurographics/ACM SIGGRAPH symposium on Geometry processing
Image compression by linear splines over adaptive triangulations
Signal Processing
Enhancement of spatially adaptive smoothing splines via parameterization of smoothing parameters
Computational Statistics & Data Analysis
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This paper studies adaptive thinning strategies for approximating a large set of scattered data by piecewise linear functions over triangulated subsets. Our strategies depend on both the locations of the data points in the plane, and the values of the sampled function at these points--adaptive thinning. All our thinning strategies remove data points one by one, so as to minimize an estimate of the error that results by the removal of a point from the current set of points (this estimate is termed "anticipated error"). The thinning process generates subsets of "most significant" points, such that the piecewise linear interpolants over the Delaunay triangulations of these subsets approximate progressively the function values sampled at the original scattered points, and such that the approximation errors are small relative to the number of points in the subsets. We design various methods for computing the anticipated error at reasonable cost, and compare and test the performance of the methods. It is proved that for data sampled from a convex function, with the strategy of convex triangulation, the actual error is minimized by minimizing the best performing measure of anticipated error. It is also shown that for data sampled from certain quadratic polynomials, adaptive thinning is equivalent to thinning which depends only on the locations of the data points--nonadaptive thinning. Based on our numerical tests and comparisons, two practical adaptive thinning algorithms are proposed for thinning large data sets, one which is more accurate and another which is faster.