Diagrammatic Tools for Generating Biothogonal Multiresolutions

Abstract

In a previous work we introduced a construction designed to produce biorthogonal multiresolutions from given subdivisions. This construction was formulated in matrix terms, which is appropriate for curves and tensor-product surfaces. For mesh surfaces of non-tensor connectivity, however, matrix notation is inconvenient. This work introduces diagrams and diagram interactions to replace matricies and matrix multiplication. The diagrams we use are patterns of value-labeled nodes, one type of diagram corresponding to each row or column of one of the matricies of a biorthogonal system. All types of diagrams used in the construction are defined on a common mesh of the multiresolution.