Minimal energy Cr-surfaces on uniform Powell–Sabin-type meshes for noisy data

Abstract

In this paper we present a method to obtain for noisy data, a C r -surface, for any r 1, on a polygonal domain which approximates a Lagrangian data set and minimizes a quadratic functional that contains some terms associated with Sobolev semi-norms weighted by some smoothing parameters. The minimization space is composed of bivariate spline functions constructed on a uniform Powell-Sabin-type triangulation of the domain. We prove a result of almost sure convergence and we choose optimal values of the smoothing parameters by adapting the generalized cross-validation method. We finish with some numerical and graphical examples.