Multiresolutions numerically from subdivisions

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 modification of approach. The construction usually results in analysis and reconstruction filters that are finite, since it seeks each filter locally for the neighborhood of the mesh to which it applies. The use of a set of filters constructed in this way is compared with filters based on spline wavelets for image compression to show that the construction can yield satisfactory results.