FSH3D: 3D Representation via Fibonacci Spherical Harmonics

TL;DR

This paper introduces FSH3D, a novel 3D representation method using Fibonacci Spherical Harmonics that improves uniformity, reduces errors, and enhances stability in spherical harmonic transforms compared to traditional equiangular grids.

Contribution

The paper proposes a new spherical harmonic transform method on Fibonacci spherical grids, providing more uniform sampling and improved accuracy over equiangular grids.

Findings

FSG captures more features with the same number of points.

Root mean square error reduced by 34.6% for 32-degree coefficients.

More stable frequency representations under rotation.

Abstract

Spherical harmonics are a favorable technique for 3D representation, employing a frequency-based approach through the spherical harmonic transform (SHT). Typically, SHT is performed using equiangular sampling grids. However, these grids are non-uniform on spherical surfaces and exhibit local anisotropy, a common limitation in existing spherical harmonic decomposition methods. This paper proposes a 3D representation method using Fibonacci Spherical Harmonics (FSH3D). We introduce a spherical Fibonacci grid (SFG), which is more uniform than equiangular grids for SHT in the frequency domain. Our method employs analytical weights for SHT on SFG, effectively assigning sampling errors to spherical harmonic degrees higher than the recovered band-limited function. This provides a novel solution for spherical harmonic transformation on non-equiangular grids. The key advantages of our FSH3D method include: 1) With the same number of sampling points, SFG captures more features without bias compared to equiangular grids; 2) The root mean square error of 32-degree spherical harmonic coefficients is reduced by approximately 34.6% for SFG compared to equiangular grids; and 3) FSH3D offers more stable frequency domain representations, especially for rotating functions. FSH3D enhances the stability of frequency domain representations under rotational transformations. Its application in 3D shape reconstruction and 3D shape classification results in more accurate and robust representations.