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Surface least squares approximation: a shape preserving approach

A new approach for fitting surfaces to scattered data is presented. In particular, the desired values of an approximating surface evaluated at points on a rectangular grid are given. The aim of the work is to take into account more information than just positional values, namely the intrinsic geometric properties of the surfaces. A local polynomial least squares surface approximation is defined for each data point, in order to minimise an objective function; for this aim only the data points whose gaussian and mean curvatures are close (in a specified sense) to those at the point in question are taken into account, in order to obtain a shape preserving approximation of the surface. Finally a weighted mean of the values assumed at each grid point Gj, by the polynomials corresponding at suitable data points surrounding Gj, is assumed to be the value of the approximating surface at Gj.