Discrete smooth interpolation

Interpolation of a function ƒ (.) known at some data points of R P is a common problem. Many computer applications (e.g., automatic contouring) need to perform interpolation only at the nodes of a given grid. Whereas most classical methods solve the problem by finding a function defined everywhere, the proposed method avoids explicitly computing such a function and instead produces values only at the grid points. For two-dimensional regular grids, a special case of this method is identical to the Briggs method (see “Machine Contouring Using Minimum Curvature,” Geophysics 17 , 1 (1974)), while another special case is equivalent to a discrete version of thin plate splines (see J. Duchon, Fonctions Splines du type Plaque Mince en Dimention 2 , Séminaire d'analyse numérique, n 231, U.S.M.G., Grenoble, 1975; and J. Enriquez, J. Thomann, and M. Goupillot, Application of bidimensional spline functions to geophysics, Geophysics 48 , 9 (1983)).