Multiresolutions Numerically from Subdivisions

Abstract

In previous work we introduced a construction to produce multiresolutions from given subdivisions. A portion of that construction required solving bilinear equations using a symbolic algebra system. Here we replace the bilinear equations with a pair of linear equation systems, resulting in a completely numerical construction. Diagrammatic tools provide assistance in carrying this out. The construction is shown for an example of univariate subdivision. The results for a bivariate subdivision are given to illustrate the construction's ability to handle multivariate meshes, as well as special points, without requiring any modi cation of approach. The construction usually results in analysis and reconstruction lters that are nite, since it seeks each lter locally for the neighborhood of the mesh to which it applies. The use of a set of lters constructed in this way is compared with lters based on spline wavelets for image compression to show that the construction can yield satisfactory results.