Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree

Wendland, Holger

doi:10.1007/bf02123482

Cited by 2,293 publications

(1,179 citation statements)

References 6 publications

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“…In the following we define a positive definite kernel Φ from a univariate function φ satisfying (15) by using Wendland's compactly supported radial basis functions [27]. For any nonnegative integer j, let…”

Section: Spherical Basis Functionsmentioning

confidence: 99%

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Overlapping additive Schwarz preconditioners for elliptic PDEs on the unit sphere

2009

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Abstract. We present an overlapping domain decomposition technique for solving elliptic partial differential equations on the sphere. The approximate solution is constructed using shifts of a strictly positive definite kernel on the sphere. The condition number of the Schwarz operator depends on the way we decompose the scattered set into smaller subsets. The method is illustrated by numerical experiments on relatively large scattered point sets taken from MAGSAT satellite data.

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Section: Spherical Basis Functionsmentioning

confidence: 99%

“…where the right-hand side is given by g = J j=0 g j , with g j ∈ V j being solutions of (27) a(g j , w) = f, w , for any w ∈ V j .…”

Section: Lemma 32 For Any Two Zonal Functions ψ(T) and φ(T) Of The mentioning

confidence: 99%

Overlapping additive Schwarz preconditioners for elliptic PDEs on the unit sphere

2009

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“…Moving least square (MLS) [3][4][5][6] and radial basis function (RBF) [7][8][9][10][11][12][13][14][15] are two popular 3D implicit surface reconstruction methods.…”

Section: Introductionmentioning

confidence: 99%

“…Using this technique, an implicit surface is constructed by calculating the weights of a set of radial basis functions such they interpolate the given data points. From the pioneering work [7,8] to recent researches, such as compactly-supported RBF [9,10], fast RBF [11][12][13] and multi-scale RBF [14,15], the established algorithms can generate more and more faithful models of real objects in last twenty years, unfortunately, most of them are not feasible for the approximations of unorganized point clouds containing noise and outliers.…”

Section: Introductionmentioning

confidence: 99%

Implicit Surface Reconstruction from Scattered Point Data with Noise

Yang

Wang

Zhu

et al. 2007

Computational Science – ICCS 2007

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Abstract. This paper addresses the problem of reconstructing implicit function from point clouds with noise and outliers acquired with 3D scanners. We introduce a filtering operator based on mean shift scheme, which shift each point to local maximum of kernel density function, resulting in suppression of noise with different amplitudes and removal of outliers. The "clean" data points are then divided into subdomains using an adaptive octree subdivision method, and a local radial basis function is constructed at each octree leaf cell. Finally, we blend these local shape functions together with partition of unity to approximate the entire global domain. Numerical experiments demonstrate robust and high quality performance of the proposed method in processing a great variety of 3D reconstruction from point clouds containing noise and outliers.

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“…In [14] radially symmetric functions with positive Fourier transform are obtained by convolving characteristic functions of balls with themselves or by using tensor product splines and radializing the result, mainly in two and three dimensions (the former are called "Euclid's hats"). Finally, Wendland [17] finds functions that are closely related to Wu's radial basis functions but proves that these have minimal polynomial degree, given the support size and the smoothness.…”

Section: Introductionmentioning

confidence: 99%

Radial functions on compact support

Buhmann

1998

Proceedings of the Edinburgh Mathematical Society

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In this paper, radial basis functions that are compactly supported and give rise to positive definite interpolation matrices for scattered data are discussed. They are related to the well-known thin plate spline radial functions which are highly useful in applications for gridfree approximation methods. Also, encouraging approximation results for the compactly supported radial functions are shown.1991 Mathematics subject classification: 41A05, 41A25, 41A63, 42A82, 65D05, 65D07.

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Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree

Cited by 2,293 publications

References 6 publications

Overlapping additive Schwarz preconditioners for elliptic PDEs on the unit sphere

Overlapping additive Schwarz preconditioners for elliptic PDEs on the unit sphere

Implicit Surface Reconstruction from Scattered Point Data with Noise

Radial functions on compact support

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