Slow Growing Subdivision (SGS) in Any Dimension: Towards Removing the Curse of Dimensionality

In recent years subdivision methods have been one of the most successful techniques applied to the multi-resolution representation and visualization of surface meshes. Extension these techniques to the volumetric case would enable their use in a broad class of applications including solid modeling, scientific visualization and mesh generation. Unfortunately, major challenges remain unsolved both in the generalization of the combinatorial structure of the refinement procedure and in the analysis of the smoothness of the limit mesh. In this paper we mainly tackle the first part of the problem introducing a subdivision scheme that generalizes to 3D and higher dimensional meshes without the excessive vertex proliferation typical of tensor-product refinements. The main four qualities of our subdivision procedure are: (i) the rate of refinement does not grow with the dimension of the mesh, (ii) adaptive refinement of the mesh is possible without introducing special temporary cell decompositions, (iii) the cells of the base meshes can have virtually unrestricted topology, and (iv) “sharp” features of different dimensions can be incorporated naturally. We use a narrow averaging mask that is applied to the vertices of the mesh and/or to eventual functions defined on the mesh. The general study of the limit smoothness of the approach requires new analysis techniques that are beyond the scope of this paper.