Multiresolution of arbitrary triangular meshes

Current multiresolution and topology-preserving representations of the topology of triangular meshes can be classified into two major categories: lossless and lossy. Lossless methods usually refine meshes progressively with vertex-adding techniques. 2,3 Such approaches can reconstruct the original mesh perfectly from a simple initial mesh, but they do not provide good parametrizations relating surface levels. Lossy methods apply regular (quaternary) surface subdivision schemes to base meshes to approximate irregular meshes. 1,4 Although they can not recover the original topology (connec-tivity) exactly, the regular subdivision schemes enable us to construct 2D wavelets on triangular meshes so that geometry and color information of meshes can be expressed hierarchically and efficiently. To combine the advantages of these two different approaches , we are developing a new topology representation for arbitrary triangular meshes. Our method expresses the topology of meshes hierarchically and losslessly, and enables us to construct 2D wavelets on the arbitrary triangular meshes. 2 The Approach The key theory (general subdivision theorem) supporting our approach is follows (see 6 for detail). A very general class of triangular meshes, called normal triangular meshes, can be represented as subdivision trees. Each interior node of the tree, representing a triangle, is subdivided using one of four elementary subdivision operations (shown in Fig.1). According to the theorem, a normal triangular mesh can be reconstructed by applying a sequence of elementary subdivisions to an initial single triangle. The necessary and sufficient conditions for a mesh to be normal are simple and easy to verify. 6 We also present some methods to convert abnormal meshes into normal meshes at a small additional cost. These methods still allow us to reconstruct the initial input mesh losslessly. A subdivision tree representation creates a sequence of nested spaces for the topology. These spaces give us a natural way to parameterize surfaces between different levels, thus making it possible to construct surface wavelets on the given meshes. Using the lifting scheme of 5, we construct wavelet-based representations of arbitrary triangular meshes with piecewise triangular Bézier patches as the basis functions. For piecewise linear functions, our wavelets are a natural extension of wavelets generated by hat functions. Unlike existing methods, our wavelets have closed forms for high-order cases, although this restricts us to using Bézier rather than B-spline surfaces for our wavelets, and thus orders of continuity higher than C 0 between patches can only be achieved with inter-patch geometric constraints as is usual …