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.

翻译：近年来，训练神经网络逼近无符号距离场（UDF）以表示三维重建中的开放表面引起了广泛关注。然而，UDF在零水平集处不可微，导致距离与梯度存在显著误差，通常产生碎片化和不连续的表面。本文提出学习无符号距离场的双曲缩放，通过定义具有独特边界条件的新Eikonal问题，使我们的公式能够与文献中广泛用于表示有符号距离场的最先进的连续可微隐式神经表示网络无缝集成。该方法不仅解决了开放表面表示难题，还显著提升了重建质量与训练性能。此外，解锁的距离场可微性使得法线方向和曲率等关键拓扑属性的精确计算成为可能，这些属性在渲染等下游任务中至关重要。通过大量实验，我们在多个数据集上验证了该方法，并与强基线方法进行对比。结果表明，与先前方法相比，本方法在精度上有所提升，速度提升可达一个数量级。