6,744 research outputs found
Smooth approximation of data on the sphere with splines
A computable function, defined over the sphere, is constructed, which is of classC1 at least and which approximates a given set of data. The construction is based upon tensor product spline basisfunctions, while at the poles of the spherical system of coordinates modified basisfunctions, suggested by the spherical harmonics expansion, are introduced to recover the continuity order at these points. Convergence experiments, refining the grid, are performed and results are compared with similar results available in literature.\ud
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The approximation accuracy is compared with that of the expansion in terms of spherical harmonics. The use of piecewise approximation, with locally supported basisfunctions, versus approximation with spherical harmonics is discussed
Splines and Wavelets on Geophysically Relevant Manifolds
Analysis on the unit sphere found many applications in
seismology, weather prediction, astrophysics, signal analysis, crystallography,
computer vision, computerized tomography, neuroscience, and statistics.
In the last two decades, the importance of these and other applications
triggered the development of various tools such as splines and wavelet bases
suitable for the unit spheres , and the
rotation group . Present paper is a summary of some of results of the
author and his collaborators on generalized (average) variational splines and
localized frames (wavelets) on compact Riemannian manifolds. The results are
illustrated by applications to Radon-type transforms on and
.Comment: The final publication is available at http://www.springerlink.co
Scattered data fitting on surfaces using projected Powell-Sabin splines
We present C1 methods for either interpolating data or for fitting scattered data associated with a smooth function on a two-dimensional smooth manifold Ī© embedded into R3. The methods are based on a local bivariate Powell-Sabin interpolation scheme, and make use of local projections on the tangent planes. The data fitting method is a two-stage method. We illustrate the performance of the algorithms with some numerical examples, which, in particular, confirm the O(h3) order of convergence as the data becomes dens
Polyharmonic approximation on the sphere
The purpose of this article is to provide new error estimates for a popular
type of SBF approximation on the sphere: approximating by linear combinations
of Green's functions of polyharmonic differential operators. We show that the
approximation order for this kind of approximation is for
functions having smoothness (for up to the order of the
underlying differential operator, just as in univariate spline theory). This is
an improvement over previous error estimates, which penalized the approximation
order when measuring error in , p>2 and held only in a restrictive setting
when measuring error in , p<2.Comment: 16 pages; revised version; to appear in Constr. Appro
Interpolation and scattered data fitting on manifolds using projected PowellāSabin splines
We present methods for either interpolating data or for fitting scattered data on a two-dimensional smooth manifold. The methods are based on a local bivariate Powell-Sabin interpolation scheme, and make use of a family of charts {(UĪ¾ , Ī¾)}Ī¾ā satisfying certain conditions of smooth dependence on Ī¾. If is a C2-manifold embedded into R3, then projections into tangent planes can be employed. The data fitting method is a two-stage method. We prove that the resulting function on the manifold is continuously differentiable, and establish error bounds for both methods for the case when the data are generated by a smooth function
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