Fast expansion into harmonics on the ball

TL;DR

This paper introduces fast, accurate algorithms for transforming 3D voxel data into ball harmonic expansions, significantly reducing computational complexity from O(N^6) to nearly O(N^3) with logarithmic factors.

Contribution

The authors develop provably accurate algorithms for efficient transformation between voxel representations and ball harmonic expansions in three dimensions.

Findings

Achieve relative accuracy ε with complexity O(N^3 (log N)^2 + N^3 |log ε|^2).

Reduce computational complexity from O(N^6) to nearly O(N^3) for the transformation.

Demonstrate effectiveness through numerical examples.

Abstract

We devise fast and provably accurate algorithms to transform between an $N\times N \times N$ Cartesian voxel representation of a three-dimensional function and its expansion into the {ball harmonics}, that is, the eigenbasis of the Dirichlet Laplacian on the unit ball in $\mathbb{R}^3$. Given $\varepsilon > 0$, our algorithms achieve relative $\ell^1$ - $\ell^\infty$ accuracy $\varepsilon$ in time $O(N^3 (\log N)^2 + N^3 |\log \varepsilon|^2)$, while the na\"{i}ve direct application of the expansion operators has time complexity $O(N^6)$. We illustrate our methods on numerical examples.