POWELL-SABIN SPLINES IN RANGE RESTRICTED INTERPOLATION OF SCATTERED DATA

The construction of range restricted bivariate C-1 interpolants to sca ttered data is considered. In particular, we deal with quadratic splin e interpolation on a Powell-Sabin refinement of a triangulation of the data sites subject to piecewise constant lower and upper bounds on th e values of the interpolant. The derived sufficient conditions for the fulfillment of the range restrictions result in a solvable system of linear inequalities for the gradients as parameters, which is separate d with respect to the data sites. Since there exists an infinite numbe r of spline interpolants meeting the constraints, the selection of a v isually pleasant solution is based on the minimum norm modification of a suitable initial interpolant or on the minimization of the thin pla te functional. While the first proposal reduces to the solution of ind ependent local quadratic programs, the second proposal results in a gl obal quadratic optimization problem.