The metric average of 1D compact sets. (English) Zbl 1043.65034

Baier, Robert; Dyn, Nira; Farkhi, Elza

Chui, Charles K. (ed.) et al., Approximation theory X. Abstract and classical analysis. Papers from the 10th international symposium, St. Louis, MO, USA, March 26–29, 2001. Nashville, TN: Vanderbilt University Press (ISBN 0-8265-1415-4/hbk). Innovations in Applied Mathematics, 9-22 (2002).

Summary: We study properties of a binary operation between two compact sets depending on a weight in \([0,1]\), termed metric average. The metric average is used in spline subdivision schemes for compact sets in \(\mathbb{R}^n\), instead of the Minkowski convex combination of sets, to retain non-convexity. Some properties of the metric average of sets in \(\mathbb{R}\), like the cancellation property, and the linear behavior of the Lebesgue measure of the metric average with respect to the weight, are proven. We present an algorithm for computing the metric average of two compact sets in \(\mathbb{R}\) which are finite unions of intervals, as well as an algorithm for reconstructing one of the metric average’s operands, given the second operand, the metric average and the weight.
For the entire collection see [Zbl 1028.00020].

MSC:

65D18

Numerical aspects of computer graphics, image analysis, and computational geometry

Keywords:

metric average; spline subdivision schemes; compact sets; algorithm