A multiresolution method for fitting scattered data on the sphere
type: Item
instance of: scholarly article
title: A multiresolution method for fitting scattered data on the sphere
zbMATH Open document ID: 1187.65012
author: A. Lamnii
author: H. Mraoui
author: D. Sbibih
author: A. Zidna
published in: BIT
publication date: 2009-11-04T00:00:00Z
review text: A sphere-like surface (closed and bounded surface in \(\mathbb{R}^{3}\), topologically equivalent to a sphere) and a set of scattered points located on this surface along with data values corresponding to these points are considered. The goal is to find a smooth function which approximates the given data set such that its associated closed surface be a class of \(C^{1}\) functions. This type of approximation problem is addressed to the problem of surface reconstruction or visualization used in geophysics, meteorology, medical modeling. In order to obtain fast computational algorithms for a good fit of scattered data on a sphere (the unit sphere centered at the origin is taken here), the authors construct a local quasiinterpolant based on the tensor product of \(2\pi\) periodic uniform algebraic trigonometric B-splines generated over the space spanned by \(\left\{ 1, \cos \phi, \sin \phi \right\}\) and quadratic polynomial B-splines. Since many coefficients are implied in this fitting multiresolution representations are needed. More precisely, a smooth \(2\pi\)-periodic algebraic trigonometric wavelet decomposition of functions defined on \(\left[0, 2\pi \right]\) is constructed and then combined with quadratic polynomial spline wavelets on \(\left[-\frac{\pi}{2}, \frac{\pi} {2}\right]\) in order to construct a new multiresolution decomposition of \(f\) leading to \(C^{1}\) functions on the sphere. It is shown that some coefficients can be omitted by thresholding out insignificant wavelet coefficients without lose the quality of approximation. Finally, an algorithm of decomposition and reconstruction which allows to compress scattered data on sphere-like surfaces is applied to two different sets of 3D real medical data in order to illustrate the performance of the proposed method.
reviewed by: Nicoleta Breaz
Mathematics Subject Classification ID: 65D10
Mathematics Subject Classification ID: 65D05
Mathematics Subject Classification ID: 65D07
Mathematics Subject Classification ID: 65T60
Mathematics Subject Classification ID: 65D18
zbMATH DE Number: 5624999
zbMATH Keywords: multiresolution analysis
zbMATH Keywords: B-spline wavelets
zbMATH Keywords: compression of a closed surface
zbMATH Keywords: scattered data fitting
zbMATH Keywords: quasi-interpolation
zbMATH Keywords: sphere-like surface
MaRDI profile type: Publication
full work available at URL: s10543-009-0230-3
OpenAlex ID: W2068771573
cites work: Q4273944
cites work: Q4328447
cites work: Q4880359
cites work: A Multiresolution Tensor Spline Method for Fitting Functions on the Sphere
cites work: Fitting scattered data on spherelike surfaces using tensor products of trigonometric and polynomial splines
cites work: Optimal properties of the uniform algebraic trigonometric B-splines
cites work: Nuat B-spline curves
DOI: 10.1007/S10543-009-0230-3
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description: scientific article; zbMATH DE number 5624999