Adaptive Volumetric Parameterization of Simply Connected 3-Manifolds with Applications

2026-08-09

AI Summary

This work addresses the limitations of traditional volumetric parameterization methods, which rely on fixed canonical domains—such as the unit ball—and often neglect the global geometric structure of the input 3-manifold, leading to significant distortion. To overcome this, the authors propose an adaptive volumetric parameterization framework that dynamically adjusts the target domain and jointly optimizes local shape preservation and quality distortion for high-fidelity parameterization of simply connected 3-manifolds. The approach innovatively introduces three progressively refined target domains: a prescribed ellipsoid, a volume-normalized adaptive ellipsoid, and a free-boundary domain embedded in hyperbolic space. It integrates 3D quasi-conformal shape updates, diffusion-driven density equalization, and fold-over-free geometric correction, enabling multi-resolution and locally adaptive remeshing. Experiments demonstrate substantial reductions in geometric distortion, enhanced parameterization quality, and successful applications in volumetric registration, deformation, and adaptive remeshing.

Abstract

Volumetric parameterization, the process of mapping a 3-manifold onto a simplified volumetric domain, is important for many tasks in computer graphics and imaging science. However, most prior volumetric parameterization approaches have only utilized standardized domains such as a solid ball regardless of the overall shape of the given 3-manifolds, which introduces significant geometric distortion and affects the subsequent shape processing and analysis tasks. To overcome this issue, in this work we propose a novel volumetric parameterization framework for simply connected 3-manifolds. Specifically, the proposed framework jointly controls local shape and mass distortions, while adapting the target domain during the optimization process. It enables three progressively more flexible target-domain settings for the parameterization: a prescribed solid ellipsoid, a volume-normalized adaptive ellipsoid with variable radii, and a sea-embedded free-boundary domain. For each setting, the parameterization algorithm consists of a 3D quasi-conformality shape update, a diffusion-based density-equalizing update, and a geometric correction procedure for removing element foldings, thereby allowing for volumetric parameterizations with different desired effects. Experimental results are presented to demonstrate the effectiveness of our proposed framework. Moreover, our framework can be easily applied to multiresolution and localized adaptive volumetric remeshing, volumetric registration, and volumetric morphing. Altogether, our work provides a new way for the representation, processing, and analysis of 3-manifolds.

Problem

• volumetric parameterization
• geometric distortion
• 3-manifolds
• adaptive domain
• shape processing

Innovation

• volumetric parameterization
• adaptive domain
• quasi-conformal mapping
• density equalization
• 3-manifold processing