Kernel Approximation on Manifolds II: The $L_{\infty}$ Norm of the $L_2$ Projector

Hangelbroek, Thomas; Narcowich, Francis J.; Sun, Xingping; Ward, Joseph D.

doi:10.1137/100795334

Cited by 29 publications

(27 citation statements)

References 20 publications

(55 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…At this point, we are able to state three important corollaries to Theorem 5.5 that satisfactorily answer the questions concerning bases and approximation properties of V X discussed in Section 1. These results were previously obtained in [16,15] for a class of Sobolev kernels. Here, we get them for a much broader, computationally implementable class of kernels.…”

Section: Implications For Interpolation and Approximationsupporting

confidence: 80%

Polyharmonic and Related Kernels on Manifolds: Interpolation and Approximation

2012

Self Cite

View full text Add to dashboard Cite

This article is devoted to developing a theory for effective kernel interpolation and approximation in a general setting. For a wide class of compact, connected C ∞ Riemannian manifolds, including the important cases of spheres and SO(3), we establish, using techniques involving differential geometry and Lie groups, that the kernels obtained as fundamental solutions of certain partial differential operators generate Lagrange functions that are uniformly bounded and decay away from their center at an algebraic rate, and in certain cases, an exponential rate. An immediate corollary is that the corresponding Lebesgue constants for interpolation as well as for L 2 minimization are uniformly bounded with a constant whose only dependence on the set of data sites is reflected in the mesh ratio, which measures the uniformity of the data. The kernels considered here include the restricted surface splines on spheres, as well as surface splines for SO(3), both of which have elementary closed-form representations that are computationally implementable. In addition to obtaining bounded Lebesgue constants in this setting, we also establish a "zeros lemma" for domains on compact Riemannian manifolds -one that holds in as much generality as the corresponding Euclidean zeros lemma (on Lipschitz domains satisfying interior cone conditions) with constants that clearly demonstrate the influence of the geometry of the boundary (via cone parameters) as well as that of the Riemannian metric.

show abstract

Section: Implications For Interpolation and Approximationsupporting

confidence: 80%

Polyharmonic and Related Kernels on Manifolds: Interpolation and Approximation

2012

Self Cite

View full text Add to dashboard Cite

show abstract

“…the span of all Wigner functions T k ij with k(k + 1) ≤ ω. As the formulas (17) and (20) show the Radon transform of such function is ω-bandlimited on S 2 × S 2 in the sense its Fourier expansion involves only functions Y k i Y k j which are eigenfunctions of ∆ S 2 ×S 2 with eigenvalue −k(k+1). Under our assumption about k the following Bernstein-type inequality holds for any function in the span of…”

Section: Approximation By Splines and A Sampling Theorem For Radon Trmentioning

confidence: 99%

Generalized Splines for Radon Transform on Compact Lie Groups with Applications to Crystallography

Bernstein

Ebert

Pesenson

2012

J Fourier Anal Appl

View full text Add to dashboard Cite

Abstract. The Radon transform Rf of functions f on SO(3) has recently been applied extensively in texture analysis, i.e. the analysis of preferred crystallographic orientation. In practice one has to determine the orientation probability density function f ∈ L 2 (SO(3)) from Rf ∈ L 2 (S 2 × S 2 ) which is known only on a discrete set of points. Since one has only partial information about Rf the inversion of the Radon transform becomes an ill-posed inverse problem. Motivated by this problem we define a new notion of the Radon transform Rf of functions f on general compact Lie groups and introduce two approximate inversion algorithms which utilize our previously developed generalized variational splines on manifolds. Our new algorithms fit very well to the application of Radon transform on SO(3) to texture analysis.

show abstract

“…[19, Theorem 4.6] [18, Proposition 3.6]). Let M be a compact Riemannian manifold of dimension n, and assume m > n/2.…”

mentioning

confidence: 99%

Kernel based quadrature on spheres and other homogeneous spaces

et al. 2013

Self Cite

View full text Add to dashboard Cite

Quadrature formulas for spheres, the rotation group, and other compact, homogeneous manifolds are important in a number of applications and have been the subject of recent research. The main purpose of this paper is to study coordinate independent quadrature (or cubature) formulas associated with certain classes of positive definite and conditionally positive definite kernels that are invariant under the group action of the homogeneous manifold. In particular, we show that these formulas are accurate -optimally so in many cases -, and stable under an increasing number of nodes and in the presence of noise, provided the set X of quadrature nodes is quasi-uniform. The stability results are new in all cases. In addition, we may use these quadrature formulas to obtain similar formulas for manifolds diffeomorphic to S n , oblate spheroids for instance. The weights are obtained by solving a single linear system. For S 2 , and the restricted thin plate spline kernel r 2 log r, these weights can be computed for two-thirds of a million nodes, using a preconditioned iterative technique introduced by us.

show abstract

Kernel Approximation on Manifolds II: The $L_{\infty}$ Norm of the $L_2$ Projector

Cited by 29 publications

References 20 publications

Polyharmonic and Related Kernels on Manifolds: Interpolation and Approximation

Polyharmonic and Related Kernels on Manifolds: Interpolation and Approximation

Generalized Splines for Radon Transform on Compact Lie Groups with Applications to Crystallography

Kernel based quadrature on spheres and other homogeneous spaces

Contact Info

Product

Resources

About