Hole filling on surfaces by discrete variational splines

This paper concerns about a method to fill polygonal holes in a given surface by using smoothing variational splines. We develop two different approaches of this technique: discontinuous filling and regular filling. In the discontinuous case, we fill the holes with spline functions in a finite element space that minimizes an energy functional. Such fillings are chosen to be smooth and as close as possible to the original surface in the neighborhoods of the holes. In this approach the global reconstructed surface will not be continuous. On the contrary, in the regular case, we do not only fill the holes, but we also replace the known surface with another very similar in such a way that the global reconstructed surface will have the desired regularity. We give convergence results and we illustrate the developed theory with several graphical and numerical examples.