Hemispheroidal parameterization and harmonic decomposition of simply connected open surfaces

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.

References in corpus (12)

• Fast Disk Conformal Parameterization of Simply-connected Open Surfaces
• Parallelizable global conformal parameterization of simply-connected surfaces via partial welding
• A Linear Formulation for Disk Conformal Parameterization of Simply-Connected Open Surfaces
• Density-equalizing maps for simply-connected open surfaces
• Area-preserving mapping of 3D ultrasound carotid artery images using density-equalizing reference map
• Adaptive area-preserving parameterization of open and closed anatomical surfaces
• Bijective Density-Equalizing Quasiconformal Map for Multiply-Connected Open Surfaces
• Spherical Density-Equalizing Map for Genus-0 Closed Surfaces
• Spherical Cap Harmonic Analysis (SCHA) for Characterising the Morphology of Rough Surface Patches
• Disk Harmonics for Analysing Curved and Flat Self-affine Rough Surfaces and the Topological Reconstruction of Open Surfaces
• Spheroidal harmonics for generalizing the morphological decomposition of closed parametric surfaces
• Fast ellipsoidal conformal and quasi-conformal parameterization of genus-0 closed surfaces

Cited by in corpus (3)

• Toroidal density-equalizing map for genus-one surfaces
• Morphology-Preserving Remeshing Approach to Particulate Microstructures via Harmonic Decomposition
• Learning-based density-equalizing map