J. C. Carr
ABSTRACT
We use polyharmonic Radial Basis Functions (RBFs) to reconstruct smooth, manifold surfaces from point-cloud data and to repair incomplete meshes. An object's surface is defined implicitly as the zero set of an RBF fitted to the given surface data. Fast methods for fitting and evaluating RBFs allow us to model large data sets, consisting of millions of surface points, by a single RBF — previously an impossible task. A greedy algorithm in the fitting process reduces the number of RBF centers required to represent a surface and results in significant compression and further computational advantages. The energy-minimisation characterisation of polyharmonic splines result in a “smoothest” interpolant. This scale-independent characterisation is well-suited to reconstructing surfaces from non-uniformly sampled data. Holes are smoothly filled and surfaces smoothly extrapolated. We use a non-interpolating approximation when the data is noisy. The functional representation is in effect a solid model, which means that gradients and surface normals can be determined analytically. This helps generate uniform meshes and we show that the RBF representation has advantages for mesh simplification and remeshing applications. Results are presented for real-world rangefinder data.
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Index Terms
Reconstruction and representation of 3D objects with radial basis functions
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Reviews
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
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