Non - Stationary Subdivision for Inhomogenous

This paper provides a methodology for the systematic derivation of subdivision schemes that model solutions to inhomogeneous linear diierential equations. In previous work, we showed that subdivision can be used to capture very eeciently the solutions of homogeneous, linear diierential equations. The resulting subdivision masks are stationary and can be precomputed, allowing for very simple and fast application of these schemes. In this paper, we show that this method can be extended to express solutions of systems of inhomogeneous, linear diierential equations. Even though the resulting subdivision masks may be non-stationary, the masks can again be precomputed. Thus, the resulting subdivision schemes capture very eeciently solutions of inhomogeneous, linear partial diieren-tial equations. Subdivision is a popular and eecient method for modeling shapes. In particular , subdivision describes a continuous shape p p p p p p p p p as the limit of a sequence p k ; k 0 of discrete shapes, lim k!1 p k = p p p p p p p p p: The beauty of subdivision lies in the fact that these discrete shapes p k are linked by a simple linear transformation S which is based on splitting and averaging, p k = S k?1 p k?1 : Figure 1 shows an example of a subdivision scheme. Starting from the coarse shape p 0 on the left, application of the subdivision matrix S 0 yields the denser shape p 1. As we continue the process, the sequence of discrete shapes ISBN 1-xxxxx-xxx-x. All rights of reproduction in any form reserved.