DUDF: Differentiable Unsigned Distance Fields with Hyperbolic Scaling

TL;DR

Open-surface representations with unsigned distance fields are hindered by non-differentiability at the zero level set. The authors propose DUDF, which applies a hyperbolic scaling to obtain a differentiable proxy $t_\mathcal{S}$ and solves a modified Eikonal PDE with surface-boundary constraints within continuously differentiable implicit networks. They introduce a curvature-alignment term based on the Hessian to stabilize the solution and enable direct computation of normals and curvatures, improving rendering and downstream tasks. Across ShapeNet, Multi-Garment, and DeepFashion, DUDF delivers improved reconstruction quality and significant speed-ups over baselines, demonstrating practical impact for non-watertight geometry processing.

Abstract

In recent years, there has been a growing interest in training Neural Networks to approximate Unsigned Distance Fields (UDFs) for representing open surfaces in the context of 3D reconstruction. However, UDFs are non-differentiable at the zero level set which leads to significant errors in distances and gradients, generally resulting in fragmented and discontinuous surfaces. In this paper, we propose to learn a hyperbolic scaling of the unsigned distance field, which defines a new Eikonal problem with distinct boundary conditions. This allows our formulation to integrate seamlessly with state-of-the-art continuously differentiable implicit neural representation networks, largely applied in the literature to represent signed distance fields. Our approach not only addresses the challenge of open surface representation but also demonstrates significant improvement in reconstruction quality and training performance. Moreover, the unlocked field's differentiability allows the accurate computation of essential topological properties such as normal directions and curvatures, pervasive in downstream tasks such as rendering. Through extensive experiments, we validate our approach across various data sets and against competitive baselines. The results demonstrate enhanced accuracy and up to an order of magnitude increase in speed compared to previous methods.

Figures (5)

• Figure 1: Distance field cross-sections comparative. First row displays ground truth unsigned distance fields. Methods based on signed distances, like SIREN sitzmann2020implicit, mistakenly fill gaps in open surfaces. While methods based on unsigned distances such as CAP-UDF zhou2022learning can represent non-watertight surfaces, the learned distance field do not approximate the true function. DUDF's differentiable formulation successfully overcomes these challenges.
• Figure 2: In (a), view of different distance fields for a 2D slice of a torus. Note the effect of hyperbolic scaling near the isosurface (bottom). In (b), sign and gradient for each distance field $f$. In signed distance, the gradient $\nabla f$ at the isosurface is equal to the surface's normal field $\mathbf{n}_\mathcal{S}$ (top). In unsigned distance, the gradient is undefined at the isosurface due to the change in orientation (middle). Our maximum curvature field vectors $\mathbf{v}_1$ align with the surface's unoriented normals (bottom).
• Figure 3: Comparisons on DeepFashion zhu2020deepfashion (top row), Multi-Garment bhatnagar2019mgn (middle row), and ShapeNet cars shapenet2015 (bottom row) data sets. DUDF preserves fine details and accurately represents complex geometries without closing holes, outperforming SIREN (SDF), which tends to smooth and round models, and CAP-UDF, which captures sharp features but often closes open surfaces. Reconstructions for CAP-UDF and DUDF performed with MC1 zhou2022cap-udf.
• Figure 4: Rendering examples using sphere tracing. CAP-UDF struggles with non-linear growth of their unsigned distance fields, causing the sphere tracing marching step to miss the surface. Conversely, our method demonstrates precision in direct rendering scenarios.
• Figure 5: Mean and Gaussian curvatures computed with our method for closed surfaces (left) and open surfaces (right).