Convergence of Cascade Algorithms and Subdivision Schemes
by
S. L. Lee
National University of Singapore

A cascade algorithm is a Picard-type iteration for the solution of equations of the form

\phi(x) = å j in Zs  2s h(j) \phi(2x-j),     x in Rs,

called a refinement equation. Here, h is assumed to be a finitely suppor ted sequence that sums to 1. The cascade algorithm starts with a chosen functi on \phi₀, and defines a sequence \phi_n iteratively by

\phin(x) = å j in Zs  2s h(j) \phin-1(2x-j),     x in Rs.

The limit of the cascade algorithm if it exists is the solution of the corresponding refinement equation. Solution of refinement equations has been studied extensively in conjunction with the construction of wavelets. However, interest in cascade algorithms is not confined only to wavelet analysis, but also extends to geometric modelling and computer graphics because of its close connection with subdivision schemes. In this talk we consider the convergence of cascade algorithms and the corresponding subdivision schemes, and their extensions to nonstationary and nonuniform cases.

Date received: February 9, 1999