A Simple Proposal for Radial 3D Needlets

TL;DR

This paper introduces Radial 3D Needlets, a simple, computationally efficient wavelet system for the three-dimensional ball with excellent localization and exact reconstruction properties, suitable for data analysis in spherical environments.

Contribution

It proposes a new wavelet system for 3D data on the ball, emphasizing simplicity, computational efficiency, and good localization, with theoretical and numerical validation.

Findings

Wavelets have excellent localization in real and harmonic domains.

The system allows for exact reconstruction as a tight frame.

Numerical analysis confirms theoretical properties.

Abstract

We present here a simple construction of a wavelet system for the three-dimensional ball, which we label \emph{Radial 3D Needlets}. The construction envisages a data collection environment where an observer located at the centre of the ball is surrounded by concentric spheres with the same pixelization at different radial distances, for any given resolution. The system is then obtained by weighting the projector operator built on the corresponding set of eigenfunctions, and performing a discretization step which turns out to be computationally very convenient. The resulting wavelets can be shown to have very good localization properties in the real and harmonic domain; their implementation is computationally very convenient, and they allow for exact reconstruction as they form a tight frame systems. Our theoretical results are supported by an extensive numerical analysis.