Spline Subdivision Schemes for Compact Sets with Metric Averages

To de ne spline subdivision schemes for general compact sets we use the representation of spline subdivision schemes in terms of repeated averages and replace the usual average convex combination by a binary averaging operation between two compact sets introduced in and termed here the metric average These schemes are shown to converge in the Hausdor metric and to provide O h approximation x Introduction In this paper we introduce spline subdivision schemes for general compact sets Motivated by the problem of the reconstruction of D objects from their D cross sections we consider spline subdivision schemes operating on data consisting of compact sets A spline subdivision scheme generates from such initial data a sequence of set valued functions with compact sets as images which converges in the Hausdor metric to a limit set valued function In the case of D sets the limit set valued function with D sets as images describes a D object For the case of initial data consisting of convex compact sets we intro duced in spline subdivision schemes where the usual addition of numbers is replaced by Minkowski sums of sets Then the spline subdivision schemes generate limit set valued functions with convex compact images which can be expressed as linear combinations of integer shifts of a B spline with the initial sets as coe cients The subdivision techniques are used to conclude that these limit set valued spline functions have shape preserving properties similar to those of scalar spline functions for shape properties de ned on sequences of sets and on set valued functions For the case of non convex initial sets it is shown in that the limit set valued function generated by a spline subdivision scheme using the Minkow ski sums coincides with the limit set valued function generated by the same Trends in Approximation Theory Kirill Kopotun Tom Lyche and Marian Neamtu eds pp Copyright oc by Vanderbilt University Press Nashville TN ISBN All rights of reproduction in any form reserved