Progressive RBF interpolation

Abstract

Interpolation based on Radial Basis Functions (RBF) is very often used for scattered scalar data interpolation in n-dimensional space in general. RBFs are used for surface reconstruction of 3D objects, reconstruction of corrupted images etc. As there is no explicit order in data sets, computations are quite time consuming that leads to limitation of usability even for static data sets. Generally the complexity of computation of RBF interpolation for N points is of O(N3) or O(k N2), k is a number of iterations if iterative methods are used, which is prohibitive for real applications. The inverse matrix can also be computed by the Strassen algorithm based on matrix block notation with O(N2.807) complexity. Even worst situation occurs when interpolation has to be made over non-constant data sets, as the whole set of equations for determining RBFs has to be recomputed. This situation is typical for applications in which some points are becoming invalid and new points are acquired. In this paper a new technique for incremental RBFs computation with complexity of O(N2) is presented. This technique enables efficient insertion of new points and removal of selected or invalid points. Due to the formulation it is possible to determine an error if one point is removed that leads to a possibility to determine the most important points from the precision of interpolation point of view and insert gradually new points, which will progressively decrease the error of interpolation using RBFs. The Progressive RBF Interpolation enables also fast interpolation on "sliding window" data due to insert/remove operations which will also lead to a faster rendering.