Multiresolution triangular B-spline surfaces

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 into 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 decomposition and reconstruction schemes operate in principle on a tensor product grid in 3D, the sparsity of the arrangement enables us to design efficient linear time algorithms. The resulting basis functions are used to approximate triangular surfaces and provide many useful properties, such as multiresolution editing, local level of detail, continuity control, surface compression and much more. The performance of our approach is illustrated by various examples including parametric and nonparametric surface editing and compression.