发表机构
Southeast University; Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS); The Chinese University of Hong Kong; Fudan University; Tsinghua University（东南大学; 上海数学与交叉学科研究院; 香港中文大学; 复旦大学; 清华大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对n维流形提出统一参数化框架，结合保角与保体积能量，利用余切拉普拉斯型表示实现高效计算，通过多种方式保证双射性，可应用于球边界及类球域参数化。
AI 中文摘要
我们提出了一个用于n维流形的平衡双射参数化统一框架。所提出的能量结合了保角项和保体积项，以控制局部各向异性和体积畸变。在连续层面上，两种能量均非负，且刻画了其零能量映射；在离散层面上，在定向单纯形流形上构建了保角、保体积及对数障碍能量。关键结果是，所有梯度均具有统一余切拉普拉斯型表示，支持稀疏且与维度无关的计算。双射性通过带符号单纯形雅可比矩阵、可行性恢复及严格保定向的对数障碍来实现。该框架统一适用于球边界参数化及离散n维流形到类球标准域的参数化。
英文摘要
We propose a unified framework for bijective parameterizations of $n$-dimensional manifolds that jointly control angular and volumetric distortion through a weighted combination of conformal and volume-preserving energies. A logarithmic barrier based on signed simplex Jacobians prevents element inversion and degeneration during energy minimization. On oriented simplicial manifolds, the gradients of all three discrete energies admit a unified cotangent Laplacian-type representation that preserves the sparsity of the mesh connectivity. Based on these formulations, we develop a three-stage algorithm that constructs an initial parameterization, repairs foldings to restore strict feasibility, and subsequently minimizes the weighted distortion energy while maintaining local injectivity. The framework is formulated in a general setting, with particular attention to parameterizations onto the unit sphere $\mathbb S^n$ and the unit ball $\mathbb B^n$. We derive sufficient conditions for discrete spherical maps to have and retain degree one, and apply classical topological criteria to establish global bijectivity. Numerical experiments demonstrate that the proposed framework achieves simultaneous control of angular and volumetric distortion, while maintaining bijectivity and eliminating surface and volumetric folds. In a brain-tumor label-transfer study, the method improves tumor-label reconstruction compared with a non-bijective ablation.
Comments30 pages
