Hemispheroidal parameterization and harmonic decomposition of simply connected open surfaces

TL;DR

This work introduces a hemispheroidal parameterization framework for simply connected open surfaces, enabling a size DOF c for the hemispheroidal domain and enabling four parameterization styles (Tutte, conformal, area-preserving, balanced). It pairs these parameterizations with hemispheroidal harmonics (HSOH), yielding spectrally stable bases on the hemispheroidal domain and enabling accurate, hierarchical reconstructions of complex surfaces. Critical innovations include a conformal correction pipeline using Beltrami coefficients and LBS, a density-diffusion-based area-preserving approach, and a Beltrami-weighted balancing scheme to tailor distortion. The results demonstrate improved parameterization quality and reconstruction accuracy on diverse geometries (e.g., brain, face, and bunny), with practical implications for shape analysis in engineering, graphics, and medical domains.

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.

Figures (12)

• Figure 1: Schematic of the surface alignment process of a given input surface as part of the registration. We start by fitting a plane to the boundary points. Then, we align the points via the SVD algorithm and accordingly align the whole surface, to finally go to the sizing process of a hemispheroid (find the radius $c$ given that $a = 1$).
• Figure 2: An illustration of our hemispheroidal Tutte parameterization method. We first utilize the Tutte embedding tutte1963draw to construct a disk flattening map. Then, we apply the inverse spheroidal projection to obtain a hemispheroidal parameterization.
• Figure 3: An illustration of our hemispheroidal conformal parameterization method. We start by computing an initial conformal flattening map onto the unit disk followed by an optimal Möbius transformation. Then, we compose the map with a quasi-conformal map that can effectively correct the conformal distortion caused by the subsequent inverse spheroidal projection. Finally, applying the inverse spheroidal projection gives the desired hemispheroidal conformal parameterization.
• Figure 4: An illustration of our hemispheroidal area-preserving parameterization method. We start by computing an initial surface flattening map onto the unit disk. Then, we perform an iterative update on the planar disk domain based on the disk density-equalizing mapping (DEM) method in choi2018density with certain modifications. Specifically, at each iteration, the mapping update is done on the planar disk domain but the updating criterion is based on the area distortion of the corresponding hemispheroidal map.
• Figure 5: Hemispheroidal spaces generated as a revolution of elliptic coordinates. Revolving about the z-axis will result in an oblate space (left inset), while revolving about the x-axis will result in an oblate space (right inset).