Construction of multiresolution triangular B-spline surfaces using hexagonal filters

We present multiresolution B-spline surfaces of arbitrary order defined over triangular domains. Unlike existing methods, the basic idea of our approach is to construct the triangular basis functions from their tensor-product relatives in the spirit of box splines by projecting: them onto the barycentric plane. The scheme works for splines of any order where the fundamental building blocks of the surface are hierarchies of triangular B-spline scaling functions and wavelets: spanning the complement spaces between levels of different resolution. Although our basis functions have been deduced from the corresponding 3D bases, our decomposition and reconstruction scheme operates directly on the triangular mesh using hexagonal filters. The resulting basis functions are used to approximate triangular surfaces and possess many useful properties, such as multiresolution editing, local level of detail, continuity control, surface compression, and many more. The performance of our approach is illustrated by various examples, including parametric and nonparametric surface editing and compression.