Hole Filling Splines with Volumen Constrains by Radial Basis Functions

In many situations we have to fill one or several holes of certain function defined in a domain where there is a lack of information inside some sub-domains. For this we have developped some methods (see [1, 2, 3, 4]). But in some practical cases we just know some specific geometrical constraints, of industrial or design type, as the special case of a specified volume inside each one of these sub-domains. In this work we study this particular issue, giving both some theoretical and computational results that assures the feasibility of the corresponding procedures. The studied method in this work manage to find a function of a vector space generated by a radial function basis that minimizes certain quadratic functional that includes some terms associated with the volume constrain and the usual semi-norms in a Sobolev space. In this way, some approximation methods have been developed (see [5, 6]). In next Section 2 we establish some general and specific notation as the functional spaces where we obtain the reconstructed functions. In section 3 we pose the problem of finding a function that fill a given hole and fulfils a volume restriction . In Section 4 we establish the computation algorithm and a convergence result.