SDFs from Unoriented Point Clouds using Neural Variational Heat Distances

TL;DR

This work tackles building accurate neural signed distance fields (SDFs) from unoriented point clouds. It introduces HeatSDF, a two-step variational framework that lifts the classical heat method to neural representations: first diffusing a short-time heat from a density-weighted point cloud to estimate unoriented normals, then fitting a signed distance whose gradient aligns with those normals while enforcing correct inside/outside orientation. The authors prove existence of minimizers for both variational problems and demonstrate state-of-the-art surface reconstruction and robust SDF gradients, with the learned SDF enabling PDEs on neural surfaces and reliable geometric queries. Overall, HeatSDF provides a robust, convex-optimization-based alternative to eikonal-based neural SDFs, especially effective for nonuniform and unoriented data.

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.

Figures (19)

• Figure 1: We compute neural SDFs from unoriented point clouds (left) by first computing a small time step of heat flow (middle) and then using its gradient directions to solve for a neural SDF (right).
• Figure 2: Point cloud input data (left), a sliced color coding of two different time steps of the heat flow (middle), equispaced isosurfaces for positive (red) and negative (blue) values of the SDF (right).
• Figure 3: Comparison of gradient directions between a signed distance of a (hyper)surface in 2D (left), and the solution of a small time step of the heat equation with mean surface measure as initial data (right).
• Figure 4: Sketch of the heat solution $u^\tau$ and the SDF $\phi$ on a one dimensional cross section. Inside the (hyper)surface, the SDF gradient $\nabla \phi$ and the normalized gradient $n^\tau$ of the heat time step point in opposite directions; outside, they point in the same direction.
• Figure 5: Left: non-uniform input point cloud on a square (qualitative visualization), middle: heat solution $u^\tau$ in two dimensions using uniform weights for the mean (hyper)surface integral \ref{['eq:weights']} and right: heat solution using the locally adapted weights \ref{['eq:uniformweights']}.

Theorems & Definitions (6)

• Proposition 2.1: Existence and uniqueness of a heat time step
• Theorem 2.2: Existence of minimizers for ${\mathcal{E}_{\mathrm{SDF}}}$
• proof : \ref{['prop:ex']}
• Lemma A.1
• proof
• proof : \ref{['thm:sdl']}