DUDF: Differentiable Unsigned Distance Fields with Hyperbolic Scaling (2402.08876v2)

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.

• The paper presents a novel differentiable approach for unsigned distance fields using hyperbolic scaling to solve the Eikonal problem.
• It refines UDFs for open surface representation, achieving lower Chamfer distances and faster training compared to CAP-UDF and SIREN.
• The method enables smooth neural implicit reconstructions, enhancing applications in graphics rendering, simulation, and real-world 3D processing.

Differentiable Unsigned Distance Fields (DUDF) with Hyperbolic Scaling: Summary and Analysis

The paper "DUDF: Differentiable Unsigned Distance Fields with Hyperbolic Scaling" addresses the challenge of 3D reconstruction of open surfaces using implicit neural representations. This publication presents a novel methodology that leverages hyperbolic scaling of unsigned distance fields to achieve a differentiable solution to the Eikonal problem, ensuring continuous and accurate surface representation.

Introduction and Problem Statement

Traditional surface representation methods, such as Signed Distance Functions (SDFs), are efficient for closed surfaces but fall short in representing open surfaces due to their inside/outside sign distinction. Unsigned Distance Functions (UDFs), while extending capabilities to open surfaces, suffer from non-differentiability at the zero level set, leading to inaccuracies in distances and gradients. This paper proposes a framework that overcomes these limitations by introducing a hyperbolic scaling function $$t_\mathcal{S} = d_\mathcal{S} \tanh(\alpha d_\mathcal{S})$$, where $d_\mathcal{S}$ is the unsigned distance, to approximate the distance field. The hyperbolic scaling ensures differentiability near the surface, facilitating the integration with continuously differentiable implicit neural networks.

Methodology

The proposed approach formulates the learning of UDFs as a solution to a new Eikonal problem:

$$\left\{\begin{matrix} \left \| \nabla f \right \| = \phi \ f \rvert_\mathcal{S} = 0 \ \nabla f \rvert_\mathcal{S} = \mathbf{0} \end{matrix}\right.$$

Here, the function $t_\mathcal{S}$ is a weak solution to the Eikonal equation. The method also introduces a boundary condition ensuring alignment between the maximum curvature direction and the surface normal. This is accomplished by training a neural network $f_{\theta}$ with periodic activation functions, penalizing deviations from the Eikonal constraint and the boundary conditions through a composite loss function.

The paper further details a refinement step to minimize oscillations at the isosurface, enhancing the surface consistency and precision. The use of the maximum curvature direction facilitates the computation of normals and curvatures, improving downstream tasks like rendering.

Results and Evaluation

The paper showcases rigorous experiments on several benchmarks, including ShapeNet cars, DeepFashion, and Multi-Garment datasets. The results demonstrate significant improvements in both reconstruction quality and computational efficiency compared to state-of-the-art methods such as CAP-UDF and SIREN. Quantitative metrics in terms of L1 and L2 Chamfer distances, as well as training time, underscore the advantages of the proposed method:

• ShapeNet Cars (Closed Surfaces): The method achieves significantly lower L1 and L2 Chamfer distances while maintaining competitively low training times against CAP-UDF and SIREN.
• DeepFashion and Multi-Garment (Open Surfaces): DUDF significantly outperforms CAP-UDF by accurately preserving open surface details and reducing training times by an order of magnitude. SIREN, designed for closed surfaces, fails in these scenarios.

The ability to render surfaces using sphere tracing without intermediate mesh reconstruction marks a notable improvement, providing smoother visual results and robust differentiation at the surfaces.

Implications and Future Work

The proposed framework opens new avenues for the practical application of neural implicit representations in 3D geometry processing. By resolving the differentiability issue in UDFs, this method enables accurate computation of essential topological properties such as normals and curvatures. This capability is crucial for applications in graphics rendering, simulation, and potentially in fields requiring high precision 3D reconstructions such as medical imaging and cultural heritage preservation.

Future developments may focus on optimizing the method to handle larger and more complex datasets, improving the efficiency of computing higher-order derivatives for real-time applications, and exploring integration with advanced surface reconstruction algorithms.

In conclusion, the "DUDF: Differentiable Unsigned Distance Fields with Hyperbolic Scaling" paper presents a well-founded and effective solution to the challenges inherent in open surface representation using neural networks. The method's balance of accuracy, efficiency, and differentiability sets a new standard for future research and applications in implicit neural representations of 3D surfaces.