Abstract Neural implicit surfaces are a powerful tool for encoding a surface as the zero level set of a neural function. Trained using gradient‐descent based optimizers, these methods however suffer from a low‐frequency bias that prevents them to fit fine details of the surface. Solutions to break this bias exist but often damage the regularity of the long range implicit function, making geometrical queries more difficult. In this work, we combine a 1‐Lipschitz neural function with a detail field made of compactly supported radial basis functions. While the neural function handles long‐range queries with low surface accuracy, the detail field is fitted to perturb the neural function so that its zero level set precisely goes through sampled points of the surface and locally accounts for small‐scale details. As it is compactly supported, regions where the 1‐Lipschitz property breaks are finely controlled. We therefore achieve the best of both worlds by greatly increasing the representation power of the implicit function near the zero level set while maintaining robustness far from the zero level set.

Abstract Neural implicit surfaces are a promising tool for geometry processing that represent a solid object as the zero level set of a neural network. Usually trained to approximate a signed distance function of the considered object, these methods exhibit great visual fidelity and quality near the surface, yet their properties tend to degrade with distance, making geometrical queries hard to perform without the help of complex range analysis techniques. Based on recent advancements in Lipschitz neural networks, we introduce a new method for approximating the signed distance function of a given object. As our neural function is made 1‐Lipschitz by construction, it cannot overestimate the distance, which guarantees robustness even far from the surface. Moreover, the 1‐Lipschitz constraint allows us to use a different loss function, called the hinge‐Kantorovitch‐Rubinstein loss, which pushes the gradient as close to unit‐norm as possible, thus reducing computation costs in iterative queries. As this loss function only needs a rough estimate of occupancy to be optimized, this means that the true distance function need not to be known. We are therefore able to compute neural implicit representations of even bad quality geometry such as noisy point clouds or triangle soups. We demonstrate that our methods is able to approximate the distance function of any closed or open surfaces or curves in the plane or in space, while still allowing sphere tracing or closest point projections to be performed robustly.

We propose a new method, dubbed One Class Signed Distance Function (OCSDF), to perform One Class Classification (OCC) by provably learning the Signed Distance Function (SDF) to the boundary of the support of any distribution. The distance to the support can be interpreted as a normality score, and its approximation using 1-Lipschitz neural networks provides robustness bounds against $l2$ adversarial attacks, an under-explored weakness of deep learning-based OCC algorithms. As a result, OCSDF comes with a new metric, certified AUROC, that can be computed at the same cost as any classical AUROC. We show that OCSDF is competitive against concurrent methods on tabular and image data while being way more robust to adversarial attacks, illustrating its theoretical properties. Finally, as exploratory research perspectives, we theoretically and empirically show how OCSDF connects OCC with image generation and implicit neural surface parametrization.

This article introduces a new representation of surface global parametrization based on Cartan’s method of moving frames . We show that a system of structure equations , characterizing the local coordinates changes with respect to a local frame system, completely characterizes the set of possible cone parametrizations. The discretization of this system provably provides necessary and sufficient conditions for the existence of a valid mapping. We are able to derive a versatile algorithm for surface parametrization, allowing feature constraints and singularities. As the first structure equation is independent of the global coordinate system, we do not require prior knowledge of cuts or cone positions. So, a single non-linear least-square problem is enough to place quantized cones while minimizing a given distortion energy. We are therefore able to take full advantage of the link between the parametrization geometry and the topology of its cone metric to solve challenging constrained parametrization problems.