Spheroidal harmonics for generalizing the morphological decomposition of closed parametric surfaces

Abstract

Spherical harmonics (SH) have been extensively used as a basis for analyzing the morphology of particles in granular mechanics. The use of SH is facilitated by mapping the particle coordinates onto a unit sphere, in practice often a straightforward rescaling of the radial coordinate. However, when applied to oblate- or prolate-shaped particles the SH analysis quality degenerates with significant oscillations appearing after the reconstruction. Here, we propose a spheroidal harmonics (SOH) approach for the expansion and reconstruction of prolate and oblate particles. This generalizes the SH approach by providing additional parameters that can be adjusted per particle to minimize geometric distortion, thus increasing the analysis quality. We propose three mapping techniques for handling both star-shaped and non-star-shaped particles onto spheroidal domains. The results demonstrate the ability of the SOH to overcome the shortcomings of SH without requiring computationally expensive solutions or drastic changes to existing codes and processing pipelines.

Figures (25)

• Figure 1: A benchmark shape [see the online documentation of libigl] analyzed and reconstructed using the radial-based spherical harmonics approach. The top row (A) shows the original benchmark surface with an aspect ratio $\text{AR} = 1.0$, whereas the bottom row (B) repeats the analysis for the same surface stretched to an aspect ratio $\text{AR} = 2$. Each row shows, from left to right, the input meshes, the radial parameterization results, and the reconstruction results. The example shows how changing the AR away from unity distorts the mesh after the mapping, which limits the maximum reconstruction frequency and affects the orthogonality of the bases resulting in oscillations after reconstruction (bottom right).
• Figure 2: Oblate (A) and prolate (B) spheroids $\mathcal{E}$ embedded in $\mathbb{R}^3$. (C) The elliptic coordinates where the solid lines represent the confocal ellipses while the dashed lines are the confocal hyperbolas. When the elliptic system revolves about the vertical axis (red) it generates an oblate surface, while the prolate one can be generated by revolving about the horizontal axis (blue).
• Figure 3: Examples of the oblate and prolate harmonic basis. (A) $\to$ (D) contain oblate (left) and prolate (right) harmonics that correspond for $(n, m)$ pairs (below). To construct the spheroids, we used $\zeta = 0.4$ and $e = 5$. The color represents the normalized real part of the basis $N_m^n \ \Re\{\Theta_m^n(\eta) \ \Phi_m(\phi)\}$.
• Figure 4: Computing the mapping of 2D closed contours, constituted by equidistant segments, via two different coordinate inversions. The 2D contours were used to simplify the visualization of the mapping process as a cross-section of a given stone. (A) mapping the 2D contour using hyperbolic mapping onto several target coordinates as a function of the focal distance $e$. For this set the choice of $\zeta$ was arbitrary as the mapping is a function of $e$ only. (B) mapping of the same contour using the rescaled polar coordinates approach as a function of the aspect ratio $\text{AR} = a/c$. The radial bar chart underneath each inset shows the radial distribution of the arclength between two consecutive points onto the target elliptic domain and was annotated with the average $\pm$ standard deviation (STD). The confocal red dashed circle in the radial charts marks the corresponding average arclength. Minimal STD (boldfaced) of the illustrated examples represents a more uniform mapping that typically corresponds to a better reconstruction accuracy. The mapped points (red crosses), in the lower insets, have been visualized when $\zeta \to \infty$ for the hyperbolic mapping (A), and rescaled circles with $\text{AR} \to 1$ for the radial one (B).
• Figure 5: Comparison between the least-squares fit and the inscribed spheroids as a function of $\theta_c$ to find the target spheroid $\mathcal{E}$ for the stone FP_C_684_1 [the stone was retrieved from dataset_1]. (A) shows the results of the least squares fit with $\text{AR} = 0.4630$ (and the here redundant $e = 0.0805$) (prolate spheroid). (B) is the scaled spheroid results (using $\theta_c = \pi/18$) with an optimal $e_{opt} = 0.0626$. (C) same as (B) but with $\theta_c = \pi/6$ and a corresponding $e_{opt} = 0.0423$.