Sampling, splines and frames on compact manifolds

TL;DR

This paper reviews recent advances in sampling, splines, and frames on compact Riemannian manifolds, with applications to Radon transforms on spheres and rotation groups, enhancing analysis tools for various scientific fields.

Contribution

It summarizes new results on Shannon-type sampling, variational splines, and localized frames on compact manifolds, extending classical analysis to complex geometries.

Findings

Development of Shannon-type sampling on manifolds

Construction of generalized variational splines

Application to Radon-type transforms on spheres and SO(3)

Abstract

Analysis on the unit sphere $\mathbb{S}^{2}$ 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 $\mathbb{S}^{2}$, $\>\>\mathbb{S}^{3}$ and the rotation group $SO(3)$. Present paper is a summary of some of results of the author and his collaborators on the Shannon-type sampling, generalized (average) variational splines and localized frames (wavelets) on compact Riemannian manifolds. The results are illustrated by applications to Radon-type transforms on $\mathbb{S}^{d}$ and $SO(3)$.