Robust Zero Level-Set Extraction from Unsigned Distance Fields Based on Double Covering

TL;DR

Robust zero level-set extraction from unsigned distance fields is challenging because zero is not a regular value and UDFs can represent open/topology-rich surfaces. The authors introduce DoubleCoverUDF (DCUDF), which builds the boundary of the $r$-offset volume to obtain a dilated double cover that is orientable, then learns a covering map $\\pi$ to project back to the target surface $S$, preserving topology. For orientable manifolds, a min-cut post-processing separates the double layers into a single surface; otherwise, the double layer is kept. Through extensive experiments on synthetic data, ShapeNet, and Deep Fashion3D, DCUDF yields meshes with fewer non-manifold artifacts and competitive Chamfer distances compared with MeshUDF and MeshCAP, while also offering memory efficiency and scalability up to MC resolutions of $1024^3$.

Abstract

In this paper, we propose a new method, called DoubleCoverUDF, for extracting the zero level-set from unsigned distance fields (UDFs). DoubleCoverUDF takes a learned UDF and a user-specified parameter $r$ (a small positive real number) as input and extracts an iso-surface with an iso-value $r$ using the conventional marching cubes algorithm. We show that the computed iso-surface is the boundary of the $r$-offset volume of the target zero level-set $S$, which is an orientable manifold, regardless of the topology of $S$. Next, the algorithm computes a covering map to project the boundary mesh onto $S$, preserving the mesh's topology and avoiding folding. If $S$ is an orientable manifold surface, our algorithm separates the double-layered mesh into a single layer using a robust minimum-cut post-processing step. Otherwise, it keeps the double-layered mesh as the output. We validate our algorithm by reconstructing 3D surfaces of open models and demonstrate its efficacy and effectiveness on synthetic models and benchmark datasets. Our experimental results confirm that our method is robust and produces meshes with better quality in terms of both visual evaluation and quantitative measures than existing UDF-based methods. The source code is available at https://github.com/jjjkkyz/DCUDF.

Figures (16)

• Figure 1: Algorithmic pipeline. Given a learned UDF of the target surface $S$ as input, the algorithm first extracts an iso-surface with iso-value $r$ using the marching cubes algorithm. The iso-surface, denoted by $\partial(S\oplus r)$, is a closed and orientable 2-manifold and represents the boundary of a $r$-offset volume. Next, the algorithm computes a covering map $\pi:\partial(S\oplus r)\rightarrow S$ that projects the double covering of the iso-surface back to $S$, yielding a mesh with the same topology as $\partial(S\oplus r)$ and with two identical layers overlapping each other. If $S$ is an orientable manifold, the algorithm applies a min-cut algorithm to separate the two layers into a single-layer mesh. Otherwise, it retains the double-layered mesh as the output. For visualization purposes, the two layers are distinguished by different colors.
• Figure 2: $r$-offset volume. In the 2D illustration (a), the solid curve represents the surface $S$, and the dotted curve is the boundary of the $r$-offset volume. The thickness parameter $r$ should be chosen to be less than half of the gap size $\eta$ to prevent the filling of the gap between the two ends. Regardless of the type of the input surface, the dilated double covering (i.e., the boundary surface of the $r$-offset volume) is always an orientable, closed 2-manifold. (b) shows an $r$-offset volume example. The model is scaled into a unit box. We use a relatively large offset value of $r=0.05$ for better visualization.
• Figure 3: Effects of the Laplacian term in the objective function for learning the covering map $\pi$. (a) Without the Laplacian term, the resulting mesh is poorly tessellated and contains many flipped and self-intersecting triangles (highlighted in red). (b) When using a constant weight $w(p_i)\equiv 1$ for the Laplacian term, undesired "ridges" or "valleys" often appear in certain relatively flat regions. The issue occurs because zero is not a regular value of the UDF, and the local minima of the UDF in the region are smaller than those of the neighboring points. Consequently, the vertices become overly concentrated in the region, resulting in an overcrowded appearance. (c) By employing an adaptive weight $w(p_i)=\sqrt{A_{\mathrm{max}}/A(p_i)}$ for the Laplacian term, the resulting triangles exhibit better quality. A marching cubes resolution $256^3$ is used in all three results.
• Figure 4: Taking the triangle centroids as additional constraints into the optimization is effective in improving the accuracy in learning the covering map $\pi$. In this 2D illustration, we use the solid dots to indicate mesh vertices and the thin lines for the triangular faces, the hollowed dots for the centroids, and the dashed arrows for the update of the centroids. The centroids are encouraged to move along the normal direction of the corresponding triangle in the fine-tune step, while there is no such requirement in the coarse step.
• Figure 5: Effects of using centroids in learning the covering map $\pi$. Without centroids, the extracted mesh may not fully recover regions with high curvature, such as the tip in this example. This limitation arises because the triangle vertices are already on the zero level set and cannot be further updated. However, by incorporating centroids as additional constraints, we can provide additional forces to guide the triangle vertices to move along the zero level set, leading to further reduction of the objective function. We present the visual results with marching cubes resolutions of $512^3$ and the thickness parameter $r=0.0025$.