Voronoi-Assisted Diffusion for Computing Unsigned Distance Fields from Unoriented Points

TL;DR

This work addresses the challenge of computing unsigned distance fields (UDFs) from unoriented point clouds, especially for open, non-manifold, and non-orientable geometries. It introduces Voronoi-Assisted Diffusion (VAD), a network-free pipeline that first optimizes bi-directional normals using a projection distance field tied to Voronoi bisectors, then diffuses the normals to produce a coherent UDF gradient which is finally recovered via a Poisson solve. The method demonstrates robust normal alignment, smooth UDFs, and accurate surface reconstructions across watertight, open, and non-standard topologies, while remaining efficient and controllable. By leveraging Voronoi geometry and heat-based diffusion, VAD provides a practical alternative to neural UDF methods, with strong performance on sparse, noisy, and complex data, and with explicit user control over boundary orientations. The approach also shows potential extensions to signed distance fields with orientation-aware Voronoi cues and scalable solvers for large-scale scenes.

Abstract

Unsigned Distance Fields (UDFs) provide a flexible representation for 3D shapes with arbitrary topology, including open and closed surfaces, orientable and non-orientable geometries, and non-manifold structures. While recent neural approaches have shown promise in learning UDFs, they often suffer from numerical instability, high computational cost, and limited controllability. We present a lightweight, network-free method, Voronoi-Assisted Diffusion (VAD), for computing UDFs directly from unoriented point clouds. Our approach begins by assigning bi-directional normals to input points, guided by two Voronoi-based geometric criteria encoded in an energy function for optimal alignment. The aligned normals are then diffused to form an approximate UDF gradient field, which is subsequently integrated to recover the final UDF. Experiments demonstrate that VAD robustly handles watertight and open surfaces, as well as complex non-manifold and non-orientable geometries, while remaining computationally efficient and stable.

Figures (30)

• Figure 1: VAD is particularly effective in handling open surfaces, non-manifold structures, and non-orientable geometries—cases that are challenging for conventional SDF- or occupancy field-based methods. Shown here is the reconstructed golden fish model, which contains both open boundaries and non-manifold structures. We visualize the computed UDFs using cut views: black curves indicate open boundaries, red dots represent non-manifold features, and the zero level set is shown in blue.
• Figure 2: With identical input points at relatively low density (a), different normal orientations produce markedly different reconstructed surface topologies and geometries (b-d), illustrating that without reliable orientation the reconstruction problem is inherently ill-posed at low sampling rates.
• Figure 3: 2D illustrations of four typical behaviors of projection fields. All fields are computed from the same point set $\mathcal{P}$ but under different bi-directional vector configurations $\mathcal{V}$. Except in case (a), where the vectors align with the true surface normals, either the field value $F$ or its gradient $\nabla F$ exhibits discontinuities across bisectors.
• Figure 4: Illustration of VAD on a 2D toy example. At each iteration, we visualize the bi-directional vectors at input points and the resulting projection field. As optimization progresses, the vectors become better aligned and the field more consistent.
• Figure 5: Conceptual illustration of the VAD pipeline. Starting from randomly initialized bi-directional vectors $\widetilde{\mathcal{V}}^{(0)}$, we optimize them by minimizing the energy $E$ to obtain well-aligned bi-directional normals $\widetilde{\mathcal{N}}$. The aligned normals are then diffused across the domain to construct a smooth vector field that approximates the UDF gradient. The final UDF is recovered by integrating the approximated gradient field via solving a Poisson equation.

Theorems & Definitions (3)

• Definition 1
• Definition 2
• Theorem 1