Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

FSH3D: 3D Representation via Fibonacci Spherical Harmonics

Zikuan Li, Anyi Huang, Wenru Jia, Qiaoyun Wu, Mingqiang Wei, Jun Wang

TL;DR

The paper tackles nonuniform spherical sampling in spherical harmonic transforms by introducing Fibonacci Spherical Harmonics (FSH3D) with a Spherical Fibonacci Grid (SFG). It derives analytic SHT weights for SFG, redistributing sampling errors to higher spherical-harmonic degrees and achieving improved coefficient accuracy and rotation stability. Empirical results show significant improvements in 3D shape reconstruction and in rotation-invariant shape descriptors, with RMSE/MAE/VE reductions and robust performance across high degree expansions. The approach offers a robust, rotation-insensitive framework for 3D reconstruction and classification, with potential to integrate into learning-based 3D reasoning systems.

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.

FSH3D: 3D Representation via Fibonacci Spherical Harmonics

TL;DR

The paper tackles nonuniform spherical sampling in spherical harmonic transforms by introducing Fibonacci Spherical Harmonics (FSH3D) with a Spherical Fibonacci Grid (SFG). It derives analytic SHT weights for SFG, redistributing sampling errors to higher spherical-harmonic degrees and achieving improved coefficient accuracy and rotation stability. Empirical results show significant improvements in 3D shape reconstruction and in rotation-invariant shape descriptors, with RMSE/MAE/VE reductions and robust performance across high degree expansions. The approach offers a robust, rotation-insensitive framework for 3D reconstruction and classification, with potential to integrate into learning-based 3D reasoning systems.

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.
Paper Structure (17 sections, 1 theorem, 14 equations, 10 figures, 2 tables)

This paper contains 17 sections, 1 theorem, 14 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related Works
  3. 3D Representations
  4. Spherical Harmonics Transform
  5. Sampling on Sphere
  6. Method
  7. Overview
  8. Background of spherical harmonics
  9. Spherical Fibonacci Grids on Sphere
  10. Discretization Problem of SHT
  11. Determine the Weight of SHT
  12. Experiment and Analysis
  13. Accuracy Analysis of Analytic Weights
  14. Quantitative Analysis of SFG
  15. Reconstructing Shapes with Star-shaped Sets
  16. ...and 2 more sections

Key Result

Theorem 1

where $C_{{{\mu }_{1}},{{\mu }_{2}},\mu }^{{{\lambda }_{1}},{{\lambda }_{2}},\lambda }$ are Wigner symbols. When $\left| {{m}_{1}}+{{m}_{2}} \right| > L$, the value of $Y_{L}^{{{m}_{1}}+{{m}_{2}}}$ is 0.

Figures (10)

  • Figure 1: The FSH3D pipeline for 3D representation. Spherical parameterization can be achieved via conformal parameterization or multi-shell voxels, followed by sampling using SFG, and subsequent decomposition over the spherical domain through SHT. Two typical application scenarios are illustrated.
  • Figure 2: The distribution of different grid methods. (a-e) equiangular grid, SFG, HEALPix grid, Cube gird and Icosahedral grid.
  • Figure 3: The values of analytic weights at 16 and 32 degrees, and the deviations between analytic weights and DH weights.
  • Figure 4: The distribution of the DH weights of equiangular (left), the analytic weights of SFG (middle) and the area weights of SFG (right).
  • Figure 5: Error in reconstructing the unit sphere using analytic weights, equal weights, and area weights. Both equal weights and area weights exhibit deviations at the pole positions and introduce ripples. The SHT achieves optimal performance only when analytic weights are employed.
  • ...and 5 more figures

Theorems & Definitions (1)

  • Theorem 1