AI 中文总结
本文针对现有体积参数化方法不匹配三维流形形状导致畸变的问题，提出自适应体积参数化框架，含三类目标域设置与三步算法，可应用于多分辨率重网格化等任务，验证了其有效性。
AI 中文摘要
体积参数化是将三维流形映射到简化体积域的过程，在计算机图形学和成像科学的诸多任务中具有重要意义。然而，大多数现有的体积参数化方法仅使用实心球等标准化域，而不考虑给定三维流形的整体形状，这会引入显著的几何畸变，影响后续的形状处理与分析任务。为解决该问题，本文提出了一种针对单连通三维流形的新型体积参数化框架。该框架在优化过程中联合控制局部形状畸变与质量畸变，同时自适应调整目标域，为参数化提供三种灵活度递增的目标域设置：指定的实心椭球、体积归一化的可变半径自适应椭球，以及嵌入海面的自由边界域。对于每种设置，参数化算法包含三个步骤：三维拟共形形状更新、基于扩散的密度均衡更新，以及消除单元折叠的几何校正步骤，从而实现具有不同期望效果的体积参数化。实验结果验证了所提框架的有效性，且该框架可便捷应用于多分辨率与局部自适应体积重网格化、体积配准及体积形变任务。总体而言，本研究为三维流形的表示、处理与分析提供了新方法。
英文摘要
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.
