Lossless Topological Subdivision of Triangle Meshes

In this paper, we investigate subdivision tree representations of arbitrary triangle meshes. By subdivision, we mean the recursive topological partitioning of a triangle into subtriangles. Such a process can be represented by a subdivision tree. We identify the class of regular triangle meshes, those that can be losslessly represented by a nontrivial subdivision tree, i.e. by a subdivision tree which uses subdivisions other than the single step partitioning of a triangle into the entire mesh. We prove that, surprisingly, any regular mesh can be represented by a BTQM subdivision tree, in which each step is one of four simple elementary subdivision steps. We show that our four elementary subdivision schemes form the minimal set of subdivision schemes that can generate regular meshes. We also show that the BTQM subdivision tree representation provides an eecient way to encode the connectivity of triangle meshes.