FSH3D: 3D Representation via Fibonacci Spherical Harmonics

2024-06-12

Computer graphics forum (Print)

AI Summary

Existing spherical harmonic transforms (SHTs) rely on equi-angular sampling grids, which suffer from non-uniform spherical distribution and local anisotropy, leading to reconstruction distortion and poor rotational robustness. This paper proposes Fibonacci Spherical Harmonics 3D (FSH3D), the first SHT framework incorporating the spherical Fibonacci grid (SFG)—a quasi-uniform sampling scheme—combined with an analytical weighting mechanism that deliberately suppresses sampling errors into high-frequency components. FSH3D significantly enhances reconstruction stability and rotational invariance for band-limited functions: it reduces the mean squared error of 32-order spherical harmonic coefficients by 34.6%. In 3D shape reconstruction and classification tasks, FSH3D achieves higher accuracy and superior rotational robustness compared to conventional methods. The implementation is publicly available.

Technology Category

• Computer Vision: 3D Computer Vision
• Machine Learning: Learning with Manifolds
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Security and Privacy: Data transparency and provenance

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. Our code is publicly available at https://github.com/Miraclelzk/Fibonacci-Spherical-Harmonics.

Problem

• Improves 3D representation uniformity.
• Reduces spherical harmonic transformation errors.
• Enhances frequency domain stability.

Innovation

• Fibonacci Spherical Harmonic transform
• Uniform sampling with Fibonacci grid
• Reduced error in harmonic coefficients