Pedro Pablo Fuentes González

Multiresolution analysis for minimal energy C r -surfaces on Powell-Sabin type meshes

This paper is intended to provide a multiresolution analysis (MRA) scheme to obtain a sequence of Cr-spline surfaces over a Powell-Sabin triangulation of a polygonal domain approximating a Lagrangian data set and minimizing a certain “energy functional”. We define certain non separable scaling and wavelet functions in bidimensional domains, and we give the decomposition and reconstruction formulas in the framework of lifting schemes. Two important applications of the theory are given: In the first one we develop an algorithm for noise reduction of signals. The second one is related to the localization of the regions where the energy of a given function is mostly concentrated. Some numerical and graphical examples for different test functions and resolution levels are given.

Original article: A hole filling method for explicit and parametric surfaces by using C1-Powell Sabin splines

In this work we develop a method to fill a hole in a surface, either explicit or in parametric form, or just in a set of three dimensional scattered data. We will construct a new surface which is very close to the original one where it is known and that fills the hole in a homogeneous way, in such a way that the final reconstruction is of class C^1. We give results which prove the existence and uniqueness of solution of the proposed method, and we present several graphical examples which show the efficiency of the theory developed.