Spherical Cap Harmonic Analysis (SCHA) for Characterising the Morphology of Rough Surface Patches

TL;DR

A novel one-to-one conformal mapping algorithm with minimal area distortion for parameterising a surface onto a polar spherical cap with a prescribed half angle and it is shown that as a generalisation of the hemispherical harmonic analysis, the SCH analysis provides the most added value for small half angles.

Abstract

We use spherical cap harmonic (SCH) basis functions to analyse and reconstruct the morphology of scanned genus-0 rough surface patches with open edges. We first develop a novel one-to-one conformal mapping algorithm with minimal area distortion for parameterising a surface onto a polar spherical cap with a prescribed half angle. We then show that as a generalisation of the hemispherical harmonic analysis, the SCH analysis provides the most added value for small half angles, i.e., for nominally flat surfaces where the distortion introduced by the parameterisation algorithm is smaller when the surface is projected onto a spherical cap with a small half angle than onto a hemisphere. From the power spectral analysis of the expanded SCH coefficients, we estimate a direction-independent Hurst exponent. We also estimate the wavelengths associated with the orders of the SCH basis functions from the dimensions of the first degree ellipsoidal cap. By windowing the spectral domain, we limit the bandwidth of wavelengths included in a reconstructed surface geometry. This bandlimiting can be used for modifying surfaces, such as for generating finite or discrete element meshes for contact problems. The codes and data developed in this paper are made available under the GNU LGPLv2.1 license.

Figures (23)

• Figure 1: Representation of a polar spherical cap $\mathbb{S}^2_{~\theta \leq \theta_c}$ with half-angle $\theta_c$, where the spherical cap harmonics analysis (SCHA) is defined over.
• Figure 2: The associated Legendre functions (ALFs) for $\theta_c = \pi/18$, $m = 0$ and $k \in \{0, 1, \dots, 8\}$; (A) the even basis and the corresponding eigenvalues from Eq. (\ref{['EQN:BC_1']}); (B) the odd basis and the corresponding eigenvalues from Eq. (\ref{['EQN:BC_2']}).
• Figure 3: The normalised spherical cap harmonic (SCH) basis functions up to $k = 4$ at $\theta_c = 5\pi/18$. The figures show $\mathfrak{R}(^{5\pi/18} C_{k}^{m} (\theta, \phi) )$. Note that at the breathing mode when $k = m = 0$, the surface inflation is unity and is therefore not included in the colorbar. The complex part of the basis are similar but rotated $\pi/(2|m|)$ about the $z$-axis.
• Figure 4: Parameterisation of the monkey head model. A) The monkey head (Suzanne) benchmark mesh REF:37, with the heat map (texture) illustrating the radial distance from the model's centroid; B) The back view of the model reveals the open boundary location; C) The conformal mapping onto the unit disk obtained using REF:2; D) The convergence curve for finding the optimal Möbius transformation that minimises the area distortion on the spherical cap using the PSS algorithm $f:\mathcal{S} \to \mathbb{S}^2_{~\theta \leq \theta_c}$ for $\theta_c = 40^{\circ}$ (i.e. $r = 0.3640$).
• Figure 5: Spherical cap parameterisation of the Stanford bunny. A) The Stanford Bunny benchmark model REF:Stanford_Bunny with an open base, where the heat map (texture) illustrates the radial distance from the model's centroid; B) The conformal mapping onto the unit disk obtained using REF:2; C) The spherical cap parameterisation $f:\mathcal{S} \to \mathbb{S}^2_{~\theta \leq \theta_c}$ for $\theta_c = 120^{\circ}$ (i.e. $r = 1.7321$); D) The spherical cap parameterisation $f:\mathcal{S} \to \mathbb{S}^2_{~\theta \leq \theta_c}$ for $\theta_c = 90^{\circ}$ (a hemisphere, i.e. $r = 1.0$); E) The spherical cap parameterisation $f:\mathcal{S} \to \mathbb{S}^2_{~\theta \leq \theta_c}$ for $\theta_c = 40^{\circ}$ (i.e. $r = 0.3640$).