Fast expansion into harmonics on the ball

TL;DR

This work develops provably accurate, fast algorithms to convert between Cartesian voxel data on the unit ball and its ball-harmonics expansion, achieving relative error $\varepsilon$ in $O\big(V(\log V)^2 + V|\log \varepsilon|^2\big)$ operations for $V$ voxels. The method leverages plane-wave expansions, a carefully chosen bandlimit, and a discrete analogue of continuous identities, implemented via NUFFT, fast spherical-harmonic transforms, and interpolation. A precise discretization bound and extensive numerical experiments confirm tight $\ell^1$-$\ell^\infty$ accuracy and substantial speedups over naive transforms, with robust performance on real-world 3D datasets. The approach enables efficient, rotation-aware 3D data representations and has potential impact on high-throughput 3D imaging pipelines, such as cryo-electron microscopy.

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 naïve direct application of the expansion operators has time complexity $O(N^6)$. We illustrate our methods on numerical examples.

Figures (4)

• Figure 1: Illustration of low-pass filtering in the ball harmonics basis $\psi_{k,\ell,m}$ for volumes of size $128\times 128 \times 128$. The ground truth volume (leftmost panel) is a 3D density map of the SARS-CoV-2 Omicron spike glycoprotein complex guo2022structures downloaded from the online electron microscopy data bank lawson2016emdatabank. Subsequent panels expand the ground truth volume in ball harmonics and decrease the number of basis functions by dividing the bandlimit by successive factors of $2$, so by retaining the basis functions with corresponding $\lambda_{\ell k}$ at most 201.06, 100.53, 50.21, 25.10, respectively, and use $564641$, $69545$, $8253$ and $1007$ basis functions, respectively. The volumes are rendered using UCSF ChimeraX goddard2018ucsf.
• Figure 2: Illustration of the algorithm for applying $B^*$, where the arrows correspond to the steps of the algorithm.
• Figure 3: Singular values of $B$ when using the bandlimit $\lambda$ from \ref{['approxlambdan']} (left) and condition number $\kappa(B)$ as a function of the number of basis functions $n$ (right). We mark the number of basis functions resulting from using the bandlimit from \ref{['approxlambdan']} as a dashed vertical line.
• Figure 4: Timings of Algorithm \ref{['algoBH']} and Algorithm \ref{['algoB']}. "Variant 1" uses bonev2023spherical for the fast spherical harmonics transform in steps 2 of Algorithm \ref{['algoBH']} and Algorithm \ref{['algoB']}, and "Variant 2" uses slevinsky2019fast.

Theorems & Definitions (34)

• Remark 2.1
• Theorem : Informal Statement
• Theorem 3.1
• Remark 3.1: Other grids
• Remark 3.2: Choice of norms
• Remark 3.3: Dense matrix operators
• Remark 3.4: No smoothness assumptions
• Remark 3.5
• Lemma 3.1
• proof