Filling polygonal holes with minimal energy surfaces on Powell–Sabin type triangulations

Abstract

In this paper we present two different methods for filling in a hole in an explicit 3D surface, defined by a smooth function $f$ in a part of a polygonal domain $D\subset {\mathbb{R}}^2$. We obtain the final reconstructed surface over the whole domain $D$. We do the filling in two different ways: discontinuous and continuous. In the discontinuous case, we fill the hole with a function in a Powell–Sabin spline space that minimizes a linear combination of the usual seminorms in an adequate Sobolev space, and approximates (in the least squares sense) the values of $f$ and those of its normal derivatives at an adequate set of points. In the continuous case, we will first replace $f$ outside the hole by a smoothing bivariate spline $s_f$, and then we fill the hole also with a Powell–Sabin spline minimizing a linear combination of given seminorms. In both cases, we obtain existence and uniqueness of solutions and we present some graphical examples, and, in the continuous case, we also give a local convergence result.