Practical wavelet design on the sphere

TL;DR

This paper presents a method for designing isotropic wavelets on the sphere that are spectrally limited and spatially localized, with applications in analyzing spherical data like the Cosmic Microwave Background.

Contribution

It introduces a practical construction of localized spherical wavelets derived from existing frames, optimized for specific applications and evaluated numerically.

Findings

Wavelets are perfectly limited in the spectral domain.

Wavelets are optimally localized in the spatial domain.

Numerical comparisons demonstrate effectiveness for spherical data analysis.

Abstract

We address the question of designing isotropic analysis functions on the sphere which are perfectly limited in the spectral domain and optimally localized in the spatial domain. This work is motivated by the need of localized analysis tools in domains where the data is lying on the sphere, e.g.{} the science of the Cosmic Microwave Background. Our construction is derived from the localized frames introduced by Narcowich, Petrushev, Ward, 2006. The analysis frames are optimized for given applications and compared numerically using various criteria.