Scattered data near-interpolation with application to discontinuous surfaces

Abstract : This paper discusses a particular type of function approximation on scattered data in a general number of variables, and its application to surface representation with imposed conditions. If the given function values are subject to errors, it is not appropriate to interpolate the function at the data in the sense of exact matching. As a consequence, we formulate a weakened version of the classical scattered data interpolation problem, and give a simple and efficient procedure to obtain near-interpolation formulas. Near-interpolants enjoy many remarkable properties, which are very useful from both theoretical and practical points of view (shape preserving properties, operator positivity, subdivision techniques, parallel and multistage computation). Applications of near-interpolants to the representation of surfaces, in particular with faults, are discussed in detail (parameter values, localizing weights, etc.).