√ 3 Multiresolution by Local Least Squares: The Diagrammatic Approach

In [2, 3, 20, 21] the authors explored a construction to produce multiresolutions from given subdivisions. Certain assumptions carried through that work, two of which we wish to challenge: (1) that multiresolutions for irregular meshes have to be constructed on the fly rather than being prepared beforehand and (2) that the connectivity graph of the coarse mesh would have to be a subgraph of the connectivity graph of the fine mesh. Kobbelt's √3 subdivision [11] lets us engage both of these assumptions. With respect to (2), the √3, post-subdivision connectivity graph shares no interior edges with the pre-subdivision connectivity graph. With respect to (1), we observe that subdivision does not produce an arbitrary connectivity graph. Rather, there are local regularities that subdivision imposes on the fine mesh that are exploitable to establish, in advance, the decomposition and reconstruction filters of a multiresolution for an irregular coarse mesh.