Scattered data interpolation subject to piecewise quadratic range restrictions

The construction of range restricted univariate and bivariate C-1 interpolants to scattered data is considered. In particular, we deal with quadratic spline interpolation on a refined univariate grid (respectively on a Powell-Sabin refinement of a triangulation of the data sites) subject to piecewise quadratic lower and upper obstacles to the values of the interpolant. The derived sufficient conditions for the fulfillment of the range restrictions result in a system of linear inequalities for the slopes (respectively gradients) as parameters, which is separated with respect to the data sites. This system is shown to be always solvable for important special forms of the obstacles. If at all, in general there exist an infinite number of spline interpolants meeting the constraints. The selection of a visually pleasant one is based on the minimization of a suitable choice functional.