SMOOTHNESS ANALYSIS OF SUBDIVISION SCHEMES ONREGULAR GRIDS BY PROXIMITY

摘要

Subdivision is a powerful way of approximating a continuous object f(x,y) by a sequence ((S~lp_(i,j))i,j∈Z)l∈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.