  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.
