Tight framelets and fast framelet filter bank transforms on manifolds

TL;DR

This paper develops a theory for tight framelets on manifolds, including their construction, characterization, and fast computational algorithms, with applications to data analysis on curved surfaces like the sphere.

Contribution

It introduces a new framework for constructing and analyzing tight framelets on manifolds, enabling efficient fast transforms with nearly linear complexity.

Findings

Explicit construction of tight framelets on the sphere $ ext{S}^2$

Development of nearly linear complexity algorithms for framelet transforms

Numerical demonstrations validating the methods

Abstract

Tight framelets on a smooth and compact Riemannian manifold $\mathcal{M}$ provide a tool of multiresolution analysis for data from geosciences, astrophysics, medical sciences, etc. This work investigates the construction, characterizations, and applications of tight framelets on such a manifold $\mathcal{M}$. Characterizations of the tightness of a sequence of framelet systems for $L_{2}(\mathcal{M})$ in both the continuous and semi-discrete settings are provided. Tight framelets associated with framelet filter banks on $\mathcal{M}$ can then be easily designed and fast framelet filter bank transforms on $\mathcal{M}$ are shown to be realizable with nearly linear computational complexity. Explicit construction of tight framelets on the sphere $\mathbb{S}^{2}$ as well as numerical examples are given.