Hemispheroidal parameterization and harmonic decomposition of simply connected open surfaces

TL;DR

This paper introduces a novel spectral analysis method for open surfaces using customizable hemispheroidal parameterizations and harmonic bases, enabling improved surface representation and analysis in engineering and medical fields.

Contribution

It proposes a new parameterization approach with various mappings and introduces hemispheroidal harmonic bases for spectral expansion of open surfaces.

Findings

Effective surface representation using hemispheroidal parameterization

Numerically stable basis functions for spectral analysis

Optimization-based mappings for surface reconstruction

Abstract

Spectral analysis of open surfaces is gaining momentum for studying surface morphology in engineering, computer graphics, and medical domains. This analysis is enabled using proper parameterization approaches on the target analysis domain. In this paper, we propose the usage of customizable parameterization coordinates that allow mapping open surfaces into oblate or prolate hemispheroidal surfaces. For this, we proposed the usage of Tutte, conformal, area-preserving, and balanced mappings for parameterizing any given simply connected open surface onto an optimal hemispheroid. The hemispheroidal harmonic bases were introduced to spectrally expand these parametric surfaces by generalizing the known hemispherical ones. This approach uses the radius of the hemispheroid as a degree of freedom to control the size of the parameterization domain of the open surfaces while providing numerically stable basis functions. Several open surfaces have been tested using different mapping combinations. We also propose optimization-based mappings to serve various applications on the reconstruction problem. Altogether, our work provides an effective way to represent and analyze simply connected open surfaces.