Smoothness Analysis of Subdivision Schemes on Regular Grids by Proximity

Abstract

Subdivision is a powerful way of approximating a continuous object $f(x,y)$ by a sequence $( (S^lp_{i,j})_{i,j\in \mathbb{Z}})_{l\in \mathbb{N}}$ of discrete data on finer and finer grids. The rule $S$ that maps an approximation on a coarse grid $S^lp$ to the approximation on the next finer grid $S^{l+1}p$ is called a subdivision scheme. If for a given scheme $S$ every continuous object $f(x,y)$ constructed by $S$ is of $C^k$ smoothness, then $S$ is said to have smoothness order $k$. Subdivision schemes are well understood if they are linear. However, for various applications the data have values in a manifold which is not a vector space (for example, when our data are positions of a moving rigid body). Under these circumstances, subdivision schemes become nonlinear and much harder to analyze. One way of analyzing such schemes is to relate them to a given linear scheme and to establish a so-called proximity condition between the two schemes, which helps in proving that the two schemes share the same smoothness. The present paper uses this method to show the $C^1$-smoothness of a wide class of nonlinear multivariate schemes.