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Reconstruction and Representation of 3D Objects with Radial Basis Functions
 Computer Graphics (SIGGRAPH ’01 Conf. Proc.), pages 67–76. ACM SIGGRAPH
, 2001
"... We use polyharmonic Radial Basis Functions (RBFs) to reconstruct smooth, manifold surfaces from pointcloud 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 al ..."
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Cited by 505 (1 self)
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We use polyharmonic Radial Basis Functions (RBFs) to reconstruct smooth, manifold surfaces from pointcloud 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 RBFpreviously 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 energyminimisation characterisation of polyharmonic splines result in a "smoothest" interpolant. This scaleindependent characterisation is wellsuited to reconstructing surfaces from nonuniformly sampled data. Holes are smoothly filled and surfaces smoothly extrapolated. We use a noninterpolating 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 realworld rangefinder data.
Fast evaluation of radial basis functions: Methods for twodimensional polyharmonic splines
 IMA Journal of Numerical Analysis
, 1997
"... Abstract. A generalised multiquadric radial basis function is a function of the form s(x) =∑N i=1 diφ(x − ti), where φ(r) = r2 + τ2 ..."
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Cited by 90 (5 self)
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Abstract. A generalised multiquadric radial basis function is a function of the form s(x) =∑N i=1 diφ(x − ti), where φ(r) = r2 + τ2
Fast solution of the radial basis function interpolation equations: Domain decomposition methods
 SIAM Journal of Scientific Computing
, 2000
"... Abstract. 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 mediumsized radial basis function interpolation pr ..."
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Cited by 76 (3 self)
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Abstract. 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 mediumsized 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 Neumann’s 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.
Transport schemes on a sphere using radial basis functions
 J. COMP. PHYS
, 2007
"... The aim of this work is to introduce the physics community to the high performance of radial basis functions (RBFs) compared to other spectral methods for modeling transport (pure advection) and to provide the first known application of the RBF methodology to hyperbolic partial differential equation ..."
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Cited by 31 (8 self)
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The aim of this work is to introduce the physics community to the high performance of radial basis functions (RBFs) compared to other spectral methods for modeling transport (pure advection) and to provide the first known application of the RBF methodology to hyperbolic partial differential equations on a sphere. First, it is shown that even when the advective operator is posed in spherical coordinates (thus having singularities at the poles), the RBF formulation of it is completely singularityfree. Then, two classical test cases are conducted: 1) linear advection, where the initial condition is simply transported around the sphere and 2) deformational flow (idealized cyclogenesis), where an angular velocity is applied to the initial condition, spinning it up around an axis of rotation. The results show that RBFs allow for a much lower spatial resolution (i.e. lower number of nodes) while being able to take unusually large timesteps to achieve the same accuracy as compared to other commonly used spectral methods on a sphere such as spherical harmonics, double Fourier series, and spectral element methods. Furthermore, RBFs are algorithmically much simpler to program.
Adaptive residual subsampling methods for radial basis function interpolation and collocation problems
, 2006
"... We construct a new adaptive algorithm for radial basis functions (RBFs) method applied to interpolation, boundaryvalue, and initialboundaryvalue problems with localized features. Nodes can be added and removed based on residuals evaluated at a finer point set. We also adapt the shape parameters ..."
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Cited by 23 (1 self)
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We construct a new adaptive algorithm for radial basis functions (RBFs) method applied to interpolation, boundaryvalue, and initialboundaryvalue problems with localized features. Nodes can be added and removed based on residuals evaluated at a finer point set. We also adapt the shape parameters of RBFs based on the node spacings to prevent the growth of the conditioning of the interpolation matrix. The performance of the method is shown in numerical examples in one and two space dimensions with nontrivial domains.
Approximate Moving LeastSquares Approximation with Compactly Supported Radial Weights
 Vienna University of Technology
, 2002
"... We use Maz'ya and Schmidt's theory of approximate approximation to devise a fast and accurate approximate moving leastsquares approximation method which does not require the solution of any linear systems. Since we use compactly supported weight functions, the remaining summation is also ..."
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Cited by 18 (7 self)
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We use Maz'ya and Schmidt's theory of approximate approximation to devise a fast and accurate approximate moving leastsquares approximation method which does not require the solution of any linear systems. Since we use compactly supported weight functions, the remaining summation is also ecient. We compare our new algorithm with three other approximation methods based on compactly supported radial functions: multilevel interpolation, the standard moving leastsquares approximation method, and a multilevel moving leastsquares algorithm. A multilevel approximate moving leastsquares approximation algorithm is also included.
A Volumetric Integral Radial Basis Function Method for TimeDependent Partial Differential Equations: I. Formulation
"... A. Local rotational and Galilean translational transformations can be obtainedto reduce the conservation equations into steady state forms for the inviscid Euler equations or NavierStokes equations. B. The entire set of PDEs are transformed into the method of lines approachyielding a set of coupled ..."
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Cited by 14 (1 self)
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A. Local rotational and Galilean translational transformations can be obtainedto reduce the conservation equations into steady state forms for the inviscid Euler equations or NavierStokes equations. B. The entire set of PDEs are transformed into the method of lines approachyielding a set of coupled ordinary differential equations whose homogeneous solution is exact in time. C. The spatial components are approximated by expansions of meshless RBFs;each individual RBF is volumetrically integrated at one of the sampling knots xi, yielding a collocation formulation of the method of lines structure of theODEs. D. Because the volume integrated RBFs increase more rapidly away from thedata center than the commonly used RBFs, we use a higher order preconditioner to counteract the illconditioning problem. Domain decomposition isused over each piecewise continuous subdomain.
Fast radial basis function interpolation via preconditioned Krylov iteration
 SIAM Journal on Scientific Computing
"... Abstract. We consider a preconditioned Krylov subspace iterative algorithm presented by Faul et al. (IMA Journal of Numerical Analysis (2005) 25, 1—24) for computing the coefficients of a radial basis function interpolant over N data points. This preconditioned Krylov iteration has been demonstrated ..."
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Cited by 12 (1 self)
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Abstract. We consider a preconditioned Krylov subspace iterative algorithm presented by Faul et al. (IMA Journal of Numerical Analysis (2005) 25, 1—24) for computing the coefficients of a radial basis function interpolant over N data points. This preconditioned Krylov iteration has been demonstrated to be extremely robust to the distribution of the points and the iteration rapidly convergent. However, the iterative method has several steps whose computational and memory costs scale as O(N 2), both in preliminary computations that compute the preconditioner and in the matrixvector product involved in each step of the iteration. We effectively accelerate the iterative method to achieve an overall cost of O(N log N). The matrix vector product is accelerated via the use of the fast multipole method. The preconditioner requires the computation of a set of closest points to each point. We develop an O (N log N) algorithm for this step as well. Results are presented for multiquadric interpolation in R2 and biharmonic interpolation in R3. A novel FMM algorithm for the evaluation of sums involving multiquadric functions in R2 is presented as well.
On approximate cardinal preconditioning methods for solving PDEs with radial basis functions. Engineering Analysis with Boundary Elements
, 2005
"... Abstract The approximate cardinal basis function (ACBF) preconditioning technique has been used to solve partial differential equations (PDEs) with radial basis functions (RBFs). In In this paper, we review the ACBF preconditioning techniques previously used for interpolation problems and investig ..."
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Cited by 11 (2 self)
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Abstract The approximate cardinal basis function (ACBF) preconditioning technique has been used to solve partial differential equations (PDEs) with radial basis functions (RBFs). In In this paper, we review the ACBF preconditioning techniques previously used for interpolation problems and investigate a class of preconditioners based on the one proposed in [31] when a cardinality condition is enforced on different subsets. We numerically compare the ACBF preconditioners on several numerical examples of Poisson's, modified Helmholtz and Helmholtz equations, as well as a diffusion equation and discuss their performance. Hardy's multiquadric φ(r) = √ r 2 + c 2 , Inverse multiquadric φ(r) = 1/ √ r 2 + c 2 , Gaussian spline φ(r) = e −cr 2 ,
Using Meshfree Approximation for MultiAsset American Option Problems
, 2003
"... We study the applicability of meshfree approximation schemes for the solution of multiasset American option problems. In particular, we consider a penalty method which allows us to remove the free and moving boundary by adding a small and continuous penalty term to the BlackScholes equation. A com ..."
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Cited by 10 (2 self)
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We study the applicability of meshfree approximation schemes for the solution of multiasset American option problems. In particular, we consider a penalty method which allows us to remove the free and moving boundary by adding a small and continuous penalty term to the BlackScholes equation. A comparison with results obtained recently by two of the authors using a linearly implicit finite difference method is included.