Reconstruction and representation of 3D objects with radial basis functions

Reviewer: Martin L. Brady

An application of polyharmonic radial basis functions (RBFs) to the modeling of 3D surfaces is described. The zero set of the RBF implicitly defines a surface that passes through a set of data points, and the RBF smoothly interpolates between these centers. RBFs have previously been applied to various problems in surface modeling, but problem sizes have been limited because the direct methods used to fit the surface data are storage and computation intensive ( O(N 2) and O(N 3) , respectively). The authors employ the fast multipole method (FMM) of Greengard and Rokhlin [1], which reduces the storage to O(N) and the fitting time to O(N log N) . Evaluation time (after O(N log N) setup) is reduced to O(1) from O(N) . The FMM is an approximation technique that introduces accuracy parameters for both the fitting and the evaluation. The authors further improve the efficiency using a greedy algorithm to reduce the number of RBF centers without exceeding the specified accuracy. The paper explores several applications of this technique. First, the method is applied to the approximation of noisy range data by relaxing the fitting constraints. It is also used to polygonalize large point-cloud data sets, by first fitting an RBF, and then iso-surfacing using a surface-following algorithm. The results show that complicated surfaces represented by hundreds of thousands of data points can be fit and evaluated accurately on a standard desktop computer. Online Computing Reviews Service