Analysis of quasi-uniform subdivision

We study the smoothness of quasi-uniform bivariate subdivision. A quasi-uniform bivariate scheme consists of different uniform rules on each side of the y-axis, far enough from the axis, some different rules near the y-axis, and is uniform in the y-direction. For schemes that generate polynomials up to degree m, we derive a sufficient condition for C^m continuity of the limit function, which is simple enough to be used in practice. It amounts to showing that the joint spectral radius of a certain pair of matrices has to be less than 2^-m. We also relate the Hölder exponent of the mth order derivatives to that joint spectral radius. The main tool is an extension of existing analysis techniques for uniform subdivision schemes, although a different proof is required for the quasi-uniform case. The same idea is also applicable to the analysis of quasi-uniform subdivision processes in higher dimension. Along with the analysis we present a 'tri-quad' scheme, which is combined of a scheme on a triangular grid on the half plane x < 0 and a scheme on a square grid on the other half plane x > 0 and special rules near the y-axis. Using the new analysis tools it is shown that the tri-quad scheme is globally C².