Filling holes with geometric and volumetric constraints

We propose and analyze different methods to reconstruct a function that is defined outside a sub-domain (hole) of a given domain. The reconstructed function is a smooth Powell-Sabin spline that is defined also inside this hole, filling then this lack of information, and, at the same time, fulfills certain global geometric considerations and other local volume constraints on the hole. We give several examples and we include a technique to estimate the volume of the function inside the hole by using just the data of the function where it is known, that is, outside the hole.