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以表示三维重建中开放曲面的方法日益受到关注。然而，无符号距离场在零水平集处不可微，导致距离与梯度出现显著误差，通常造成曲面碎片化与不连续。本文提出学习无符号距离场的双曲缩放，该缩放定义了具有独特边界条件的新程函问题。这使得我们的公式能够无缝集成当前最先进的连续可微隐式神经表示网络（此类网络在文献中广泛用于表示有符号距离场）。本方法不仅解决了开放曲面表示的挑战，还显著提升了重建质量与训练性能。此外，解锁场的可微性使得能够精确计算法向方向与曲率等关键拓扑属性——这些属性在渲染等下游任务中普遍存在。通过大量实验，我们在多个数据集上验证了该方法，并与竞争性基线进行了比较。结果表明，相较于先前方法，本方法在精度上有所提升，速度上可提高一个数量级。