Beta-function B-spline smoothing on triangulations

Abstract

In this work we investigate a novel family of C k -smooth rational basis functions on triangulations for fitting, smoothing, and denoising geometric data. The introduced basis function is closely related to a recently introduced general method introduced in utilizing generalized expo-rational B-splines, which provides C k -smooth convex resolutions of unity on very general disjoint partitions and overlapping covers of multidimensional domains with complex geometry. One of the major advantages of this new triangular construction is its locality with respect to the star-1 neighborhood of the vertex on which the said base is providing Hermite interpolation. This locality of the basis functions can be in turn utilized in adaptive methods, where, for instance a local refinement of the underlying triangular mesh affects only the refined domain, whereas, in other method one needs to investigate what changes are occurring outside of the refined domain. Both the triangular and the general smooth constructions have the potential to become a new versatile tool of Computer Aided Geometric Design (CAGD), Finite and Boundary Element Analysis (FEA/BEA) and Iso-geometric Analysis (IGA).