3D Interpolation and Approximation
In this chapter we extend to n dimensions the spline theory developed in Chap. 4 on cubic splines for interpolation and approximation of functions; in particular we deal with 3D data. As a consequence of interpolation and smoothness conditions we obtain by a constructive proof, the Thin Plate Spline (TPS), whose explicit expression is given in terms of a convolution with the fundamental solutions of the biharmonic differential operator. In this construction no geometry on data is assumed. Thus, the data do not have to conform a regular grid, so we speak about reconstruction from scattered data.