Construction of multiresolution triangular B-spline surfaces using hexagonal filters

We present multiresolution B-spline surfaces of arbitrary order defined ove r triangular domains. Unlike existing methods, the basic idea of our approa ch is to construct the triangular basis functions from their tensor-product relatives in the spirit of box splines by projecting: them onto the baryce ntric plane. The scheme works for splines of any order where the fundamenta l building blocks of the surface are hierarchies of triangular B-spline sca ling functions and wavelets: spanning the complement spaces between levels of different resolution. Although our basis functions have been deduced fro m the corresponding 3D bases, our decomposition and reconstruction scheme o perates directly on the triangular mesh using hexagonal filters. The result ing basis functions are used to approximate triangular surfaces and possess many useful properties, such as multiresolution editing, local level of de tail, continuity control, surface compression, and many more. The performan ce of our approach is illustrated by various examples, including parametric and nonparametric surface editing and compression.