Grow & fold: compressing the connectivity of tetrahedral meshes

Standard representations of irregular nite element meshes combine vertex data (sample coordinates and node values) and connectivity (tetrahedron-vertex incidence). Connectivity speciies how the samples should be interpolated. It may be encoded as four vertex-references for each tetrahedron, which requires 128m bits where m is the number of tetrahedra in the mesh. Our`Grow&Fold' format reduces the connectivity storage down to 7 bits per tetrahedron: 3 of these are used to encode the presence of children in a tetrahedron spanning tree; the other 4 constrain sequences of`folding' operations, so that they produce the connectivity graph of the original mesh. Additional bits must be used for each handle in the mesh and for each topologicaìlock' in the tree. However, as our experiments with a prototype implementation show, the increase of the storage cost due to this extra information is typically no more than 1-2%. By storing vertex data in an order deened by the tree, we avoid the need to store tetrahedron-vertex references and facilitate variable length coding techniques for the vertex data. We provide the details of simple, loss-less compression and decompression algorithms and discuss a way of decreasing the storage cost to about 6 bits per tetrahedron. This paper addresses the problem of a bit-eecient loss-less encoding of the incidence of a tetrahedral mesh whose boundary is a manifold surface. A simple representation of such a mesh consists of two tables: the vertex table keeping vertex coordinates and vertex data, such as temperature or pressure, and the tetrahedron table storing quadruples of vertex indices, representing vertex sets for each one of the m tetrahedra in the mesh. The tetrahedron table describes explicitly only the vertex incidence for each tetrahedron. However, all other connectivity information, like tetrahedron-face or triangle-vertex incidence can be derived from it algorithmically. For a mesh with one Million vertices and six