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.

翻译：开曲面的谱分析在工程学、计算机图形学和医学领域研究曲面形态学中日益受到重视。该分析通过在目标分析域上采用适当的参数化方法实现。本文提出使用可定制的参数化坐标，将开曲面映射至扁球或长球半球面。为此，我们提出采用Tutte映射、共形映射、保面积映射及平衡映射的组合，将任意给定单连通开曲面参数化至最优半球体。通过推广已知的半球调和基函数，引入半球体调和基函数对这些参数化曲面进行谱展开。该方法以半球体半径作为自由度来控制开曲面参数化域的大小，同时提供数值稳定的基函数。我们已使用不同映射组合测试了多种开曲面。针对重建问题中的各类应用，本文还提出了基于优化的映射方法。总体而言，我们的工作为单连通开曲面的表示与分析提供了一种有效途径。