Consistent Point Orientation for Manifold Surfaces via Boundary Integration

TL;DR

This work targets globally consistent normal orientation for unoriented point clouds on manifold surfaces by leveraging the generalized winding number (GWN). It formulates a boundary energy derived from the Dirichlet energy of the GWN and optimizes it to enforce global harmonicity, starting from random normals and using Voronoi-based sampling of the GWN field. The approach achieves strong robustness to noise, outliers, complex topology, and thin structures, and it demonstrates favorable runtime and memory characteristics against state-of-the-art methods, culminating in high-quality orientation and watertight reconstructions. This boundary-integral framework offers a scalable alternative to volume-based discretizations, enabling reliable point orientation for large and intricate geometries with practical downstream benefits in surface processing tasks.

Abstract

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 \url{https://github.com/liuweizhou319/BIM}.

Figures (11)

• Figure 1: Illustration of GWN sampling strategies. Input points are represented as large grey circles, with Voronoi vertices shown as small yellow circles. The interior $\Omega^-$ is shaded blue and the exterior is in gray. A representative point $\mathbf{p}_i$ is highlighted in black, with its associated samples $\mathbf{p}_i^+$ and $\mathbf{p}_i^-$ shown as red and blue triangles, respectively. (a) The normal displacement strategy, though straightforward and easy to implement, is ineffective for models with thin structures. In this example, it positions both $\mathbf{p}_i^+~(=\mathbf{p}_i+\mathbf{n}_i)$ and $\mathbf{p}_i^-~(=\mathbf{p}_i-\mathbf{n}_i)$ on the same side of the target surface (specifically, both inside the surface). (b) Using Voronoi diagrams, we identify $\mathbf{p}_i^\pm$ by selecting two Voronoi vertices associated with $\mathbf{p}_i$ that best align with the current normal $\mathbf{n}_i$. This approach ensures that for the majority of the input points, their chosen $\mathbf{p}^\pm$ are consistently positioned on opposite sides of $\partial\Omega$, thereby providing a reliable sampling of the GWN field around the target surface.
• Figure 2: Visualization of the GWN field $w$ and its gradient $\|\nabla w\|^2$ for globally consistent normals (top) and random normals (bottom). The decomposed boundary energies $f(\mathbf{n}_{n})$ and $f(\mathbf{n}_{t})$ are (167.53, -3.21$\times 10^{-15}$) and ($4.77\times 10^{-6}$, $2.11\times 10^{-5}$), respectively.
• Figure 3: The ambiguities of jump boundary conditions in Case 1 and Case 2. For each case, we plot the boundary energy (BE) and the Dirichlet energy (DE) with and without the constraints $0\leq w\leq 1$. By restricting the range of the winding numbers, DE and BE converge when the associated normals are globally consistent. We also visualize the GWN field using cut views and indicate the normals for the cross sections.
• Figure 4: Illustration of our algorithm on a 2D toy model. In different iterations, we visualize the normals, $\mathbf{p}^+$, $\mathbf{p}^-$ (row 1), winding number field (row 2), $w(\mathbf{p}^+)$ and $w(\mathbf{p}^-)$ (row 3). We highlight the regions with significant variations in normals between adjacent iterations using gray lines and red points.
• Figure 5: Reconstruction results for the 18 test models.