基于离散高斯曲率感知的融合网格参数化方法

A Gaussian Curvature-Based Weight Fusion Mesh Parameterization Method

摘要: 三维网格参数化是计算机图形学和数字几何处理中的基础问题，其目标是将三维曲面映射到二维参数域，同时尽可能降低角度畸变，面积畸变以及三角形翻转等问题。传统保角参数化方法能够较好地保持局部角度结构，但在高曲率区域容易产生面积拉伸。保面积参数化方法能够改善面积分布，却可能引入局部形状变形。针对单一参数化方法难以兼顾角度保持与面积保持的问题本文提出一种基于离散曲率感知的加权融合网格参数化方法。该方法先分别得到构造共形能量函数的保角参数化坐标和基于迭代拉伸方法的保面积参数化坐标。随后利用面积归一化角亏量刻画顶点局部曲率特征，并构造曲率驱动的自适应融合权重。为提高融合结果的稳定性，本文对两类参数化结果进行相似变换对齐，在统一边界约束下对内部顶点进行加权融合后结合三角形有向面积检测与无翻转投影策略减少局部翻转现象。实验结果表明，所提出方法在部分具有复杂局部几何特征的模型上能够在角度畸变，面积畸变和翻转控制之间取得较为稳定的折中效果。本文方法在保持局部形状特征的同时改善面积分布，为低复杂度、几何感知的三维网格参数化提供了一种可行思路。该方法也适用于具有复杂局部几何特征的三角网格参数化处理，但在复杂拓扑模型，自适应切割以及严格全局双射性约束方面仍有进一步研究空间。

Abstract: 3D mesh parameterization is a fundamental problem in computer graphics and digital geometry processing. Its objective is to map a three-dimensional surface onto a two-dimensional parameter domain while minimizing angular distortion, area distortion, and triangle flips. Traditional conformal parameterization methods can effectively preserve local angular structures, but they tend to produce area stretching in high-curvature regions. Area-preserving parameterization methods can improve area distribution, but may introduce local shape deformation. To address the difficulty of simultaneously preserving angles and areas using a single parameterization method, this paper proposes a weighted fusion mesh parameterization method based on discrete curvature awareness. The proposed method first obtains conformal parameterization coordinates by constructing a conformal energy function and area-preserving parameterization coordinates based on an iterative stretching method. Then, an area-normalized angle defect is used to characterize the local curvature features of vertices, and a curvature-driven adaptive fusion weight is constructed. To improve the stability of the fusion result, the two parameterization results are aligned by a similarity transformation. Under consistent boundary constraints, the interior vertices are weighted and fused, followed by the use of oriented triangle area detection and a no-flip projection strategy to reduce local triangle inversions. Experimental results show that the proposed method can achieve a relatively stable trade-off among angular distortion, area distortion, and flip control on some models with complex local geometric features. While preserving local shape features, the proposed method improves area distribution and provides a feasible approach for low-complexity, geometry-aware 3D mesh parameterization. This method is also applicable to the parameterization of triangular meshes with complex local geometric features; however, further research is still needed for complex topological models, adaptive cutting, and strict global bijectivity constraints.