R. K. Beatson

Reconstruction and representation of 3D objects with radial basis functions

We use polyharmonic Radial Basis Functions (RBFs) to reconstruct smooth, manifold surfaces from point-cloud data and to repair incomplete meshes. An objects 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.

Surface interpolation with radial basis functions for medical imaging

Radial basis functions are presented as a practical solution to the problem of interpolating incomplete surfaces derived from three-dimensional (3-D) medical graphics. The specific application considered is the design of cranial implants for the repair of defects, usually holes, in the skull. Radial basis functions impose few restrictions on the geometry of the interpolation centers and are suited to problems where the Interpolation centers do not form a regular grid. However, their high computational requirements have previously limited their use to problems where the number of interpolation centers is small (<300). Recently developed fast evaluation techniques have overcome these limitations and made radial basis interpolation a practical approach for larger data sets. In this paper radial basis functions are fitted to depth-maps of the skulls surface, obtained from X-ray computed tomography (CT) data using ray-tracing techniques. They are used to smoothly interpolate the surface of the skull across defect regions. The resulting mathematical description of the skulls surface can be evaluated at any desired resolution to be rendered on a graphics workstation or to generate instructions for operating a computer numerically controlled (CNC) mill.

Fast Solution of the Radial Basis Function Interpolation Equations: Domain Decomposition Methods

In this paper we consider domain decomposition methods for solving the radial basis function interpolation equations. There are three interwoven threads to the paper. The first thread provides good ways of setting up and solving small- to medium-sized radial basis function interpolation problems. These may occur as subproblems in a domain decomposition solution of a larger interpolation problem. The usual formulation of such a problem can suffer from an unfortunate scale dependence not intrinsic in the problem itself. This scale dependence occurs, for instance, when fitting polyharmonic splines in even dimensions. We present and analyze an alternative formulation, available for all strictly conditionally positive definite basic functions, which does not suffer from this drawback, at least for the very important example previously mentioned. This formulation changes the problem into one involving a strictly positive definite symmetric system, which can be easily and efficiently solved by Cholesky factorization. The second section considers a natural domain decomposition method for the interpolation equations and views it as an instance of von Neumanns alternating projection algorithm. Here the underlying Hilbert space is the reproducing kernel Hilbert space induced by the strictly conditionally positive definite basic function. We show that the domain decomposition method presented converges linearly under very weak nondegeneracy conditions on the possibly overlapping subdomains. The last section presents some algorithmic details and numerical results of a domain decomposition interpolatory code for polyharmonic splines in 2 and 3 dimensions. This code has solved problems with 5 million centers and can fit splines with 10,000 centers in approximately 7 seconds on very modest hardware.

Smooth surface reconstruction from noisy range data

This paper shows that scattered range data can be smoothed at low cost by fitting a Radial Basis Function (RBF) to the data and convolving with a smoothing kernel (low pass filtering). The RBF exactly describes the range data and interpolates across holes and gaps. The data is smoothed during evaluation of the RBF by simply changing the basic function. The amount of smoothing can be varied as required without having to fit a new RBF to the data. The key feature of our approach is that it avoids resampling the RBF on a fine grid or performing a numerical convolution. Furthermore, the computation required is independent of the extent of the smoothing kernel, i.e., the amount of smoothing. We show that particular smoothing kernels result in the applicability of fast numerical methods. We also discuss an alternative approach in which a discrete approximation to the smoothing kernel achieves similar results by adding new centres to the original RBF during evaluation. This approach allows arbitrary filter kernels, including anisotropic and spatially varying filters, to be applied while also using established fast evaluation methods. We illustrate both techniques with LIDAR laser scan data and noisy synthetic data.

Fast fitting of radial basis functions: Methods based on preconditioned GMRES iteration

Solving large radial basis function (RBF) interpolation problems with non‐customised methods is computationally expensive and the matrices that occur are typically badly conditioned. For example, using the usual direct methods to fit an RBF with N centres requires O(N2) storage and O(N3) flops. Thus such an approach is not viable for large problems with N≥10,000.In this paper we present preconditioning strategies which, in combination with fast matrix–vector multiplication and GMRES iteration, make the solution of large RBF interpolation problems orders of magnitude less expensive in storage and operations. In numerical experiments with thin‐plate spline and multiquadric RBFs the preconditioning typically results in dramatic clustering of eigenvalues and improves the condition numbers of the interpolation problem by several orders of magnitude. As a result of the eigenvalue clustering the number of GMRES iterations required to solve the preconditioned problem is of the order of 10-20. Taken together, the combination of a suitable approximate cardinal function preconditioner, the GMRES iterative method, and existing fast matrix–vector algorithms for RBFs [4,5] reduce the computational cost of solving an RBF interpolation problem to O(N) storage, and O(N \log N) operations.

Fast evaluation of radial basis functions: I

This paper describes some new techniques for the rapid evaluation and fitting of radial basic functions. The techniques are based on the hierarchical and multipole expansions recently introduced by several authors for the calculation of many-body potentials. Consider in particular the N term thin-plate spline, s(x) = Σj=1Ndjφ(x−xj), where φ(u) = |u|2log|u|, in 2-dimensions. The direct evaluation of s at a single extra point requires an extra O(N) operations. This paper shows that, with judicious use of series expansions, the incremental cost of evaluating s(x) to within precision ϵ, can be cut to O(1+|log ϵ|) operations. In particular, if A is the interpolation matrix, ai,j = φ(xi−xj, the technique allows computation of the matrix-vector product Ad in O(N), rather than the previously required O(N2) operations, and using only O(N) storage. Fast, storage-efficient, computation of this matrix-vector product makes pre-conditioned conjugate-gradient methods very attractive as solvers of the interpolation equations, Ad = y, when N is large.

Univariate multiquadric approximation: Quasi-interpolation to scattered data

The univariate multiquadric function with centerxj∈R has the form {ϕj(x)=[(x−xj)2+c2]1/2, x∈R} wherec is a positive constant. We consider three approximations, namely, ℒAf, ℒℬf, and ℒCf, to a function {f(x),x0≤x≤xN} from the space that is spanned by the multiquadrics {ϕj:j=0, 1, ...,N} and by linear polynomials, the centers {xj:j=0, 1,...,N} being given distinct points of the interval [x0,xN]. The coefficients of ℒAf and ℒℬf depend just on the function values {f(xj):j=0, 1,...,N}. while ℒAf, ℒCf also depends on the extreme derivativesf′(x0) andf′(xN). These approximations are defined by quasi-interpolation formulas that are shown to give good accuracy even if the distribution of the centers in [x0,xN] is very irregular. Whenf is smooth andc=O(h), whereh is the maximum distance between adjacent centers, we find that the error of each quasi-interpolant isO(h2|logh|) away from the ends of the rangex0≤x≤xN. Near the ends of the range, however, the accuracy of ℒAf and ℒℬf is onlyO(h), because the polynomial terms of these approximations are zero and a constant, respectively. Thus, some of the known accuracy properties of quasiinterpolation when there is an infinite regular grid of centers {xj=jh:j ∈F} given by Buhmann (1988), are preserved in the case of a finite rangex0≤x≤xN, and there is no need for the centers {xj:j=0, 1, ...,N} to be equally spaced.

Fast Evaluation of Radial Basis Functions: Methods for Generalized Multiquadrics in

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monotone quadratic splines are analyzed. This motivates an algorithm for interpolating monotone data, given on a rectangular grid, with a