SDFs from Unoriented Point Clouds using Neural Variational Heat Distances (2504.11212v1)

Abstract: We propose a novel variational approach for computing neural Signed Distance Fields (SDF) from unoriented point clouds. To this end, we replace the commonly used eikonal equation with the heat method, carrying over to the neural domain what has long been standard practice for computing distances on discrete surfaces. This yields two convex optimization problems for whose solution we employ neural networks: We first compute a neural approximation of the gradients of the unsigned distance field through a small time step of heat flow with weighted point cloud densities as initial data. Then we use it to compute a neural approximation of the SDF. We prove that the underlying variational problems are well-posed. Through numerical experiments, we demonstrate that our method provides state-of-the-art surface reconstruction and consistent SDF gradients. Furthermore, we show in a proof-of-concept that it is accurate enough for solving a PDE on the zero-level set.

• The paper presents a novel two-step approach that leverages variational heat flow and a backward Euler scheme to compute neural SDFs without relying on the eikonal equation.
• It accurately converts unsigned distance approximations into signed distances by fitting computed gradients to align with expected SDF profiles, ensuring correct interior/exterior discrimination.
• Extensive numerical experiments demonstrate competitive improvements in surface reconstruction and error reduction over state-of-the-art methods on complex geometric models.

Insightful Overview of "SDFs from Unoriented Point Clouds using Neural Variational Heat Distances"

The paper "SDFs from Unoriented Point Clouds using Neural Variational Heat Distances" introduces a robust and theoretically founded method for constructing neural signed distance functions (SDFs) from unoriented point clouds. The authors propose an innovative two-step approach which leverages variational principles and the heat method, moving away from the traditional dependence on the eikonal equation. This method targets some of the inherent weak points in current neural implicit representations, focusing especially on SDFs' applicability in fields such as shape reconstruction, geometric queries, and solving surface partial differential equations (PDEs).

Methodology

The presented method operates over two distinct stages:

1. Heat Flow Time Step: The first stage involves computing a discrete approximation of the heat flow using initial values derived from the mean surface measure approximated by a weighted input point cloud. Inspired by the minimizing movement schemes in metric spaces, the authors apply a backward Euler variational approach to ensure well-posedness of the solution. This step diverges from the usual practice by avoiding the implementation of the eikonal equation's residuals and targeting a convex optimization problem instead.
2. Computing the SDF: The second stage addresses the computation of the SDF itself by fitting the gradients computed during the first step to the expected SDF gradients. This involves a novel approach where regions inside and outside the shape, denoted as $B^+$ and $B^-$, are identified, allowing for correct orientation of the point cloud data. By resolving these regions, the authors successfully decode the unsigned distances devised in the first stage into accurate signed distances, further handling complexities related to unoriented cloud inputs.

The network architecture utilizes periodic activation functions, contributing to the method's robustness, particularly in non-uniform point cloud densities, which is often a significant challenge in neural SDF computations.

Numerical Results and Validation

The authors report performant outcomes through extensive numerical simulations. Their results showcase competitive edge over several state-of-the-art methods, including SIREN, HESS, StEik, and recent advancements like HotSpot, in both surface reconstruction error and SDF accuracy across a variety of complex models such as the Armadillo and hand models. The proposed method illustrates lower SDF errors and favorable trade-offs between surface detail and eikonal accuracy compared to others that tend to struggle with these aspects, particularly when dealing with non-uniform density inputs.

Additionally, the analysis outlined in the paper indicates the robustness of their far-field normal blending strategy and highlights the efficacy of the variational approach to solving Poisson problems, notably in regions of high curvature.

Practical and Theoretical Implications

Practically, the structured use of variational heat distances not only enhances robustness against initialization issues inherent in traditional eikonal-based methods but also provides confidence in the surface reconstruction, opening pathways in neural PDE solving. The potential impact here is substantial, given the method's applicability in more advanced surface geometries and intricate real-world models which demand precise surface and narrow-band representations.

On a theoretical level, this approach strengthens the viability of using neural networks in complex geometric processing tasks, supported by robust variational principles which ensure well-posedness and stability. The use of two distinct but cohesive neural network architectures for heat computation and SDF generation also widens the horizon for integrating more complex functions within this framework, be that in higher-dimensional settings or under data constraints.

Future Prospects

Future development directions might explore optimizing the network backbone for higher efficiency and adapting the methodology for real-time applications, where computational demand and speed of convergence are critical. The extension of this methodology towards more complex PDEs, such as those modeling fluid surfaces, presents another promising venue. Additionally, handling internal voids or multi-layered structures with this framework remains an open challenge that could further bolster its utility across domains.

This paper robustly contributes to the field by bridging the gap between accurate theoretical formulations and practical usability in neural geometric processing, thus setting a baseline for future innovations and applications in computational geometry and neural implicit representations.