This paper introduces a new approach for generating globally consistent
normals for point clouds sampled from manifold surfaces. Given that the
generalized winding number (GWN) field generated by a point cloud with globally
consistent normals is a solution to a PDE with jump boundary conditions and
possesses harmonic properties, and the Dirichlet energy of the GWN field can be
defined as an integral over the boundary surface, we formulate a boundary
energy derived from the Dirichlet energy of the GWN. Taking as input a point
cloud with randomly oriented normals, we optimize this energy to restore the
global harmonicity of the GWN field, thereby recovering the globally consistent
normals. Experiments show that our method outperforms state-of-the-art
approaches, exhibiting enhanced robustness to noise, outliers, complex
topologies, and thin structures. Our code can be found at
https://github.com/liuweizhou319/BIM.

提出了一种新的方法来为从多重曲面采样的点云生成全局一致的法线。通过边界能量最优化，恢复全局调和性以恢复全局一致的法线。与现有方法相比，在噪声、异常值、复杂拓扑和薄结构方面表现出更强的鲁棒性。

基于边界积分的流形曲面一致点定向

Consistent Point Orientation for Manifold Surfaces via Boundary  Integration

Vector fields are widely used to represent and model flows for many science
and engineering applications. This paper introduces a novel neural network
architecture for learning tangent vector fields that are intrinsically defined
on manifold surfaces embedded in 3D. Previous approaches to learning vector
fields on surfaces treat vectors as multi-dimensional scalar fields, using
traditional scalar-valued architectures to process channels individually, thus
fail to preserve fundamental intrinsic properties of the vector field. The core
idea of this work is to introduce a trainable vector heat diffusion module to
spatially propagate vector-valued feature data across the surface, which we
incorporate into our proposed architecture that consists of vector-valued
neurons. Our architecture is invariant to rigid motion of the input, isometric
deformation, and choice of local tangent bases, and is robust to
discretizations of the surface. We evaluate our Vector Heat Network on triangle
meshes, and empirically validate its invariant properties. We also demonstrate
the effectiveness of our method on the useful industrial application of
quadrilateral mesh generation.

介绍了一种用于在三维嵌入式曲面上学习切向量场的新型神经网络体系结构，通过引入可训练的向量热扩散模块在曲面上空间传播向量值特征数据，该架构具有刚性运动不变性、等距变形不变性，以及局部切向基的选择不变性，对曲面的离散化也具有鲁棒性。在三角网格上评估了该 Vector Heat 网络，并从实证上验证了其不变性质，同时展示了该方法在四边形网格生成方面的有用工业应用。