A multiresolution method for fitting scattered data on the sphere
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.
scientific article; zbMATH DE number 5624999