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Fast Evaluation of Radial Basis Functions: Methods Based on Partition of Unity
 Approximation Theory X: Wavelets, Splines, and Applications
, 2002
"... We combine the theory of radial basis function interpolation with a partition of unity method to solve largescale, scattered data problems. We analyze the computational complexity and pay special attention to the underlying data structure. Finally, we give a numerical example. ..."
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We combine the theory of radial basis function interpolation with a partition of unity method to solve largescale, scattered data problems. We analyze the computational complexity and pay special attention to the underlying data structure. Finally, we give a numerical example.
Efficient Reconstruction of Large Scattered Geometric Datasets Using the Partition of Unity and Radial Basis Functions
, 2004
"... We present a new scheme for the reconstruction of large geometric data. It is based on the wellknown radial basis function model combined with an adaptive spatial and functional subdivision associated with a family of functions forming a partition of unity. This combination offers robust and effi ..."
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Cited by 22 (1 self)
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We present a new scheme for the reconstruction of large geometric data. It is based on the wellknown radial basis function model combined with an adaptive spatial and functional subdivision associated with a family of functions forming a partition of unity. This combination offers robust and efficient solution to a great variety of 2D and 3D reconstruction problems, such as the reconstruction of implicit curves or surfaces with attributes starting from unorganized point sets, image or mesh repairing, shape morphing or shape deformation, etc. After having presented the theoretical background, the paper mainly focuses on implementation details and issues, as well as on applications and experimental results.
Recent research at Cambridge on Radial Basis Functions
 IN 1990, IN ADVANCES IN NUMERICAL ANALYSIS II: WAVELETS, SUBDIVISION, AND RADIAL BASIS FUNCTIONS, W. LIGHT (ED
, 1998
"... Much of the research at Cambridge on radial basis functions during the last four years has addressed the solution of the thin plate spline interpolation equations in two dimensions when the number of interpolation points, n say, is very large. It has provided some techniques that will be surveyed be ..."
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Cited by 16 (0 self)
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Much of the research at Cambridge on radial basis functions during the last four years has addressed the solution of the thin plate spline interpolation equations in two dimensions when the number of interpolation points, n say, is very large. It has provided some techniques that will be surveyed because they allow values of n up to 10^5, even when the positions of the points are general. A close relation between these techniques and Newton's interpolation method is explained. Another subject of current research is a new way of calculating the global minimum of a function of several variables. It is described briefly, because it employs a seminorm of a large space of radial basis functions. Further, it is shown that radial basis function interpolation minimizes this seminorm in a way that is a generalisation of the wellknown variational property of thin plate spline interpolation in two dimensions. The final subject is the deterioration in accuracy of thin plate spline interpolation near the edges of finite grids. Several numerical experiments in the onedimensional case are reported that suggest some interesting conjectures that are still under investigation.
Numerical techniques based on radial basis functions
 Curve and Surface Fitting: SaintMalo 1999
, 2000
"... Radial basis functions are tools for reconstruction of multivariate functions from scattered data. This includes, for instance, reconstruction of surfaces from large sets of measurements, and solving partial differential equations by collocation. The resulting very large linear N x N systems require ..."
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Cited by 16 (4 self)
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Radial basis functions are tools for reconstruction of multivariate functions from scattered data. This includes, for instance, reconstruction of surfaces from large sets of measurements, and solving partial differential equations by collocation. The resulting very large linear N x N systems require efficient techniques for their solution, preferably of O(N) or O(N log N) computational complexity. This contribution describes some special lines of research towards this future goal. Theoretical results are accompanied by numerical examples, and various open problems are pointed out.
Finite Element Thin Plate Splines for Data Mining Applications
, 1998
"... Thin plate splines have been used successfully to model curves and surfaces. A new application is in data mining where they are used to model interaction terms. These interaction splines break the "curse of dimensionality" by reducing the highdimensional nonparametric regression problem t ..."
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Cited by 13 (8 self)
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Thin plate splines have been used successfully to model curves and surfaces. A new application is in data mining where they are used to model interaction terms. These interaction splines break the "curse of dimensionality" by reducing the highdimensional nonparametric regression problem to the determination of a set of interdependent surfaces. However, the determination of the corresponding thin plate splines requires the solution of a dense linear system of equations of order n where n is the number of observations. For data mining applications n can be in the millions, and so standard thin plate splines, even using fast algorithms may not be practical. A finite element approximation of the thin plate splines will be described. The method uses H¹ elements in a formulation which only needs first order derivatives. The resolution of the method is chosen independently of the number of observations which only need to be read from secondary storage once and do not need to be stored ...
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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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 NFFT based summation of radial functions
 Sampl. Theory Signal Image Process., 3:1
"... on the occasion of his 75 th birthday. This paper is concerned with the fast summation of radial functions by the fast Fourier transform for nonequispaced data. We enhance the fast summation algorithm proposed in [20] by introducing a new regularization procedure based on the twopoint Taylor int ..."
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on the occasion of his 75 th birthday. This paper is concerned with the fast summation of radial functions by the fast Fourier transform for nonequispaced data. We enhance the fast summation algorithm proposed in [20] by introducing a new regularization procedure based on the twopoint Taylor interpolation by algebraic polynomials and estimate the corresponding approximation error. Our error estimates are more sophisticated than those in [20]. Beyond the kernels K (x) = 1=jxj ( 2 N) we are also interested in the generalized multiquadrics which play an important role in the approximation of functions by radial basis functions. Key words and phrases: fast discrete summation, fast Fourier transform at nonequispaced knots, generalized multiquadric
PetRBF—A parallel O(N) algorithm for radial basis function interpolation
, 909
"... We have developed a parallel algorithm for radial basis function (rbf) interpolation that exhibits O(N) complexity, requires O(N) storage, and scales excellently up to a thousand processes. The algorithm uses a gmres iterative solver with a restricted additive Schwarz method (rasm) as a precondition ..."
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We have developed a parallel algorithm for radial basis function (rbf) interpolation that exhibits O(N) complexity, requires O(N) storage, and scales excellently up to a thousand processes. The algorithm uses a gmres iterative solver with a restricted additive Schwarz method (rasm) as a preconditioner and a fast matrixvector algorithm. Previous fast rbf methods — achieving at most O(N log N) complexity—were developed using multiquadric and polyharmonic basis functions. In contrast, the present method uses Gaussians with a small variance (a common choice in particle methods for fluid simulation, our main target application). The fast decay of the Gaussian basis function allows rapid convergence of the iterative solver even when the subdomains in the rasm are very small. The present method was implemented in parallel using the petsc library (developer version). Numerical experiments demonstrate its capability in problems of rbf interpolation with more than 50 million data points, timing at 106 seconds (19 iterations for an error tolerance of 10 −15) on 1024 processors of a Blue Gene/L (700 MHz PowerPC processors). The parallel code is freely available in the opensource model. Key words: radial basis function interpolation, domain decomposition methods, gmres, orderN algorithms, particle methods, parallel computing
Multiscale reconstruction of implicit surfaces with attributes from large unorganized point sets
 In Shape Modeling International 2004
, 2004
"... We present a new method for the multiscale reconstruction of implicit surfaces with attributes from large unorganized point sets. The implicit surface is reconstructed by subdividing the global domain into overlapping local subdomains using a perfectly balanced binary tree, reconstructing the surfa ..."
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We present a new method for the multiscale reconstruction of implicit surfaces with attributes from large unorganized point sets. The implicit surface is reconstructed by subdividing the global domain into overlapping local subdomains using a perfectly balanced binary tree, reconstructing the surface parts in the local subdomains from nondisjunct subsets of the points by variational techniques using radial basis functions, and hierarchically blending together the surface parts of the local subdomains by using a family of functions called partition of unity. The subsets of the points in the inner nodes of the tree for intermediate resolutions are obtained by thinning algorithms. The reconstruction is particularly robust since the number of data points in the partition of unity blending zones can be specified explicitly. Furthermore, the new reconstruction method is valid for discrete datasets in any dimension, so we can use it also to reconstruct continuous functions for the surface’s attributes. In a short discussion, we evaluate the advantages and drawbacks of our reconstruction method compared to existing reconstruction methods for implicit surfaces. 1