Adaptive thinning for bivariate scattered data

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.