Scattered Data Fitting with Nonnegative Preservation using Bivariate Splines and Its Application

We study how to use bivariate splines for scattered data interpolation and fitting with preservation of non-negativity of the data values. We propose a minimal energy method to find a C1 smooth interpolation/fitting of non-negative data values from scattered locations based on bivariate splines. We establish the existence and uniqueness of the minimizer under mild assumptions on the data locations and triangulations. We then use Uzawa’s algorithm to compute the nonnegative preserving interpolatory or fitting spline surface by linearizing the nonnegativity conditions. The convergence of the algorithm will be shown under a sufficient condition that there is a nonnegative preserving spline interpolant satisfying the linearized nonnegativity conditions. Some artificial examples and one real life example demonstrate that our algorithm works well and improves the nonnegativity of the spline surfaces even without having the sufficient condition.