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

Abstract

The efficient representation of volumetric meshes is a central problem in scientific visualization. The difference in performance between most visualization algorithm for rectilinear grids and for unstructured mesh is mostly due to fundamental difference in efficiency of their representations. In Computer Graphics the gap in performance between 2D rectilinear grids and unstructured mesh has been overcome with the development of representation schemes based on the concept of subdivision surfaces. This gap has not been bridged in the volumetric cases which is fundamental interest for Scientific Visualization. In this paper we introduce a slow-growing volumetric subdivision scheme for meshes of any topology, any intrinsic dimension d and composed of a general type of polyhedral cells (topological balls). The main feature of this approach is the ability to split in different stages cells of different dimensions. This allows to increase the resolution of the mesh slowly using small stencils for the smoothing rules. ''Sharp features'' of dimension lower than d are embedded naturally in the subdivision procedure. Automatic adaptation is provided for variable resolution. In the uniform case the slow subdivision doubles the number of vertices in the mesh at each refinement independent of its dimension d. The bisection of all themore » edges in a d-dimensional simplicial mesh requires d subdivision steps. Hence the slow subdivision is a d{radical}2 subdivision scheme. This algorithm generalizes a recently developed {radical}2 subdivision scheme to 3D and higher dimensional meshes where the vertex proliferation becomes increasingly problematic as d grows (the curse of dimensionality). We introduce a smoothing rule for both the domain mesh and for functions defined on it. Empirical evidence demonstrates the smoothness of the scheme directly on the mesh and indirectly on the isosurfaces of the functions.« less