Semi-regular 4-8 refinement and box spline surfaces

We introduce a new mesh refinement method for subdivision surfaces. It generates a semiregular 3-direction hierarchical structure from control meshes representing 2D manifolds of arbitrary topology. The main advantage of this structure is that it allows the extraction of conforming variable-resolution meshes based on spatially varying adaptation functions. We also present a smoothing method designed to work in conjunction with our semi-regular refinement. It produces generalized four direction box spline surfaces of class C¹. Together the refinement and smoothing operators result in a subdivision scheme that is very effective in multiresolution applications