Fast Tensor Needlet Transforms for Tangent Vector Fields on the Sphere

TL;DR

This paper introduces fast algorithms for tensor needlet transforms on the sphere, enabling efficient multiscale analysis of tangent vector fields with near linear computational cost.

Contribution

It constructs a semi-discrete tight frame of tensor needlets for tangent vector fields on the sphere and develops fast transform algorithms based on multiresolution analysis.

Findings

Near linear computational cost for transforms

Efficient decomposition and reconstruction algorithms

Numerical validation with simulated and real data

Abstract

This paper constructs a semi-discrete tight frame of tensor needlets associated with a quadrature rule for tangent vector fields on the unit sphere $\mathbb{S}^2$ of $\mathbb{R}^3$ --- tensor needlets. The proposed tight tensor needlets provide a multiscale representation of any square integrable tangent vector field on $\mathbb{S}^2$, which leads to a multiresolution analysis (MRA) for the field. From the MRA, we develop fast algorithms for tensor needlet transforms, including the decomposition and reconstruction of the needlet coefficients between levels, via a set of filter banks and scalar FFTs. The fast tensor needlet transforms have near linear computational cost proportional to $N\log \sqrt{N}$ for $N$ evaluation points or coefficients. Numerical examples for the simulated and real data demonstrate the efficiency of the proposed algorithm.