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

TL;DR

This paper introduces a novel spherical cap harmonic analysis method for characterising rough surface patches, including a new surface parameterisation algorithm and spectral analysis techniques for surface morphology estimation.

Contribution

It develops a conformal mapping algorithm for surface parameterisation and extends harmonic analysis to small spherical caps, improving analysis of flat surfaces.

Findings

SCH analysis is most effective for small half angles.

Estimated Hurst exponent from spectral analysis.

Bandwidth limiting aids in surface modification.

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.