Univariate Adaptive Thinning

In this paper we approximate large sets of univariate data by piecewise linear functions which interpolate subsets of the data using adaptive thinning strategies Rather than minimize the global error at each removal AT we propose a much cheaper thinning strategy AT which only minimizes errors locally Interestingly the two strategies are equivalent in all our numerical tests and we prove this to be true for convex data We also compare with non adaptive thinning strategies x Introduction In applications such as visualization it is often desirable to generate a hierar chy of coarser and coarser representations of a given discrete data set Though we are primarily interested in hierarchies of scattered data sets and in par ticular piecewise linear approximations over triangulations in the plane we focus in this paper on univariate data sets and propose several adaptive thinning strategies Thinning algorithms generate hierarchies of subsets by removing points from the given data set one by one in such a way that the least signi cant point is removed at each step according to some desirable criterion Our criterion here will primarily be the minimization of approxi mation error so our thinning algorithms are adaptive This is in contrast for example to the thinning strategies of where the criterion was to generate subsets of well distributed points independent of the height values Thinning algorithms for piecewise linear approximation to univariate data have appeared before in the literature as decimation algorithms as in Heckbert and Garland and as knot removal for linear splines as in Lyche In this paper we design test and compare four methods for anticipating the error incurred by the removal of a point from the current subset Our algorithms choose the point to be removed as the one of minimal anticipated error Our main conclusion is that the algorithm AT which is based on making a local error estimate but taking account of all previously removed points is the best algorithm from the point of view of our numerical results Mathematical Methods for Curves and Surfaces Oslo Tom Lyche and Larry L Schumaker eds pp Copyright oc by Vanderbilt University Press Nashville TN ISBN All rights of reproduction in any form reserved N Dyn M S Floater and A Iske and theoretical analysis In fact our theoretical analysis shows that its com putational complexity is O N logN with N the number of points in the data set provided one uses a heap to store the anticipated errors Moreover we prove that for data sampled from a convex function AT minimizes the global approximation error at every step These latter two results extend to piecewise linear functions over triangulations for scattered data in the plane see x Adaptive Thinning Suppose a b is a real interval and that X x xN is a given sequence of points in a b such that a x x xN b Suppose further that some unknown function f a b IR is sampled at these points giving the values f x f xN For each n n N we are interested in nding a subset Y y yn of X such that a x y y yn xN b and such that the piecewise linear interpolant L f Y to the data f y f y y Y g is close to the given data f x f x x Xg in the sense that the error E Y X f max x X jL f Y x f x j is small relative to the errors corresponding to other subsets ofX of cardinality n To guarantee that E Y X f is well de ned we refrain from removing the points x xN so that L f Y is de ned on a b Ideally for any given n n N we would like to nd a subset Y of X of cardinality n for which the error in is minimal However it is clearly impractical to search amongst all possible subsets and this motivates the more pragmatic approach of thinning The idea of thinning is to remove points from X one by one in order to reach a subset Y of a certain size In general we want to remove a point of least signi cance Our criterion for removing a point from the current subset is to minimize its anticipated error which is an estimate of the error incurred by the removal of the point with respect to some error measure Thus the thinning algorithm is a greedy algorithm choosing the current step to do the optimal step in the current situation We de ne our thinning algorithm by saying that a point yi in Y i n is removable if e yi min j n e yj