Scattered data interpolation with nonnegative preservation using bivariate splines and its application

We study how to use bivariate splines for scattered data interpolation with nonnegativity preservation of the given data values. That is, we propose a constrained minimal energy method to find a C1C1 smooth interpolation of nonnegative data values over scattered locations using bivariate splines. We establish the existence and uniqueness of the minimizer under mild assumptions on the data locations and triangulations. Next we show some approximation properties of the minimizer. We then use the classic projected gradient algorithm to find the minimizer using a simplified nonnegative constraint. Some synthetic as well as a real life example demonstrate the performance, producing nonnegative interpolatory spline surfaces from nonnegative data values. Experimental results also show that the approximation of the nonnegative interpolatory spline surfaces is slightly better than the approximation of the classic minimal energy interpolatory splines.