Orhonormal wavelet bases on the 3D ball via volume preserving map from the regular octahedron

TL;DR

This paper introduces a volume-preserving map from the 3D ball to an octahedron, enabling the construction of orthonormal wavelet bases on the ball with refinable grids and localized support.

Contribution

It presents a novel volume-preserving mapping and constructs orthonormal wavelet bases on the 3D ball, facilitating multiresolution analysis with localized functions.

Findings

Constructed a volume-preserving map between the 3D ball and octahedron.

Developed a multiresolution analysis on the 3D ball.

Built orthonormal wavelet bases with small local support.

Abstract

We construct a new volume preserving map from the unit ball $\mathbb B^3$ to the regular 3D octahedron, both centered at the origin, and its inverse. This map will help us to construct refinable grids of the 3D ball, consisting in diameter bounded cells having the same volume. On this 3D uniform grid, we construct a multiresolution analysis and orthonormal wavelet bases of $L^2(\mathbb B^3)$, consisting in piecewise constant functions with small local support.