SDFs from Unoriented Point Clouds using Neural Variational Heat Distances

2025-04-15

AI Summary

This work addresses neural signed distance field (SDF) reconstruction from unoriented point clouds, proposing a differentiable variational framework grounded in the heat equation—thereby abandoning the conventional Eikonal constraint. The method proceeds in two stages: first, a neural network regresses the unsigned gradient field of the SDF; second, the SDF is recovered via heat-diffusion regularization. It marks the first integration of heat-based methods into neural implicit modeling, formulating two convex variational problems whose well-posedness is rigorously established. A key innovation enables partial differential equation (PDE) solving directly on the zero-level set, fundamentally alleviating gradient inconsistency in SDFs. Extensive evaluation on synthetic and real-world point clouds demonstrates state-of-the-art reconstruction accuracy, with continuous, stable SDFs and gradients—achieving both high-fidelity geometric reconstruction and robust PDE solvability on implicit surfaces.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Diffusion Models for Vision
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications

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.

Problem

• Computing neural SDFs from unoriented point clouds
• Replacing eikonal equation with heat method for distances
• Ensuring well-posed variational problems and accurate gradients

Innovation

• Neural variational approach for SDF computation
• Heat method replaces eikonal equation
• Convex optimization with neural networks