VoroMesh Algorithm: Voronoi Mesh Generation

• VoroMesh Algorithm is a family of techniques that use Voronoi diagrams to produce volumetric and surface meshes with provable geometric and topological guarantees.
• It employs surface sampling, ball unions, and adaptive seeding to accurately capture complex geometries and preserve sharp features.
• The algorithm supports distributed computation and differentiable pipelines, enabling high-performance, learning-based mesh reconstruction and scalable simulations.

The term "VoroMesh Algorithm" encompasses a family of algorithms and representations for generating volumetric and surface meshes using Voronoi diagrams, either for high-fidelity geometric modeling, robust mesh generation with theoretical guarantees, distributed computation, or differentiable surface reconstruction pipelines. Key contributions are found in algorithms such as VoroCrust, learning-based VoroMesh, planar-domain VoroMesh, high-performance distributed 3D Voronoi frameworks, and differentiable Voronoi networks.

1. Foundational Principles and Definitions

Voronoi-based mesh generation decomposes a domain $\Omega \subset \mathbb{R}^3$ (or $\mathbb{R}^2$ for planar variants) into convex polyhedral cells $V_i$ associated with a set of "sites" or "generators" $\{q_i\}$:

$$V_i = \{ x \in \mathbb{R}^d\ :\ \|x-q_i\| \leq \|x-q_j\|,\ \forall j \neq i \}$$

A conforming VoroMesh must approximate the boundary $\partial \Omega$ without clipping cells, maintain convexity and aspect-ratio guarantees, and support complex input features such as sharp edges and corners.

Key structural components:

• Medial axis $\mathcal{A}$: set of points with more than one closest point on the boundary, crucial for defining sampling conditions.
• Local feature size (lfs): $lfs(x) = \operatorname{dist}(x, \mathcal{A})$ governs the sampling density required for geometric fidelity near features.
• $\epsilon$-sample and weak $\sigma$-sparsity: ensures sampling adequacy and separation, formally $$\forall x \in M, \exists p \in P: \|x-p\| \leq \epsilon\,lfs(x)$$, and for $P$ satisfying $\|p_i-p_j\| \geq \sigma\,\epsilon\,lfs(p_j)$ if $lfs(p_i)\geq lfs(p_j)$ (Abdelkader et al., 2018).
• Dual Delaunay triangulation: leveraged for Voronoi/Delaunay conversion in both serial and distributed contexts (Mizrachi et al., 20 Feb 2025).

2. Meshing Algorithms and Construction Pipelines

2.1 VoroCrust/VoroMesh in 3D

The VoroCrust algorithm is the canonical VoroMesh approach with provable correctness for generic smooth surfaces and the ability to handle sharp features (Abdelkader et al., 2018, Abdelkader et al., 2019). The method proceeds as follows:

Step 1: Surface Sampling

• Generate an $\epsilon$-sample $P$ of the boundary $M=\partial \Omega$, enforcing weak $\sigma$-sparsity.

Step 2: Ball Union and Guide Point Extraction

• For each $p_i \in P$, define a ball $B_i$ of radius $r_i = \delta\,lfs(p_i)$ (typically $\delta=2\epsilon$).
• Compute the union $U = \cup_i B_i$.
• For every triple $(i, j, k)$ where $\partial B_i \cap \partial B_j \cap \partial B_k$ gives two intersection points, retain those lying on $\partial U$, reflecting the surface geometry.

Step 3: Surface Mesh via Voronoi

• Compute the Voronoi diagram of retained points (classified as inside/outside).
• The mesh surface is the set of facets between opposite-side Voronoi cells, yielding a manifold that is provably isotopic to $M$.

Step 4: Volume Meshing

• Adaptive octree refinement or Poisson-disk sampling inserts interior seeds according to local sizing.
• Final mesh is extracted as Voronoi cells whose seeds are within $\bar{U}$, with boundary exactly matching the reconstructed surface.

Pseudocode (condensed from (Abdelkader et al., 2018)):

For each surface sample p_i: B_i = Ball(p_i, δ * lfs(p_i)) U = union of all B_i For each triple (i,j,k): For each intersection guide point g on ∂U: if g is exterior: add to S^↑ if g is interior: add to S^↓ Extract Voronoi diagram of S^↑ ∪ S^↓ Construct surface as facets between S^↑ and S^↓ Add interior seeds in octree leaf boxes with no surface seeds Final mesh: Voronoi diagram of all seeds in U

This yields meshes with bounded aspect ratio, minimum angle guarantees, Hausdorff distance $O(\epsilon^2\,lfs)$, and avoids non-convex cells associated with clipping-based methods (Abdelkader et al., 2018, Abdelkader et al., 2019).

2.2 Learning-Based Differentiable VoroMesh

"VoroMesh: Learning Watertight Surface Meshes with Voronoi Diagrams" introduces a differentiable Voronoi-based surface representation for learning pipelines (Maruani et al., 2023).

• Sites ("generators") $\{q_i\}$ are assigned binary occupancy $\{o_i\}$.
• Boundary is formed as the union of all Voronoi faces $F_{i,j}$ separating inside/outside cells: $S = \cup_{o_i \neq o_j} F_{i,j}$.
• The VoroLoss function optimizes site positions by minimizing the squared distance from ground-truth samples $\{x_k\}$ to the closest relevant Voronoi face, which can be efficiently computed via $d(x, H_{i,j}) = |(x-m_{i,j})\cdot n_{i,j}|$ with $m_{i,j} = (q_i+q_j)/2$ and $n_{i,j} = (q_j-q_i)/\|q_j-q_i\|$.
• Fully differentiable, enabling direct optimization (Adam) and integration with deep learning networks to map from SDFs to Voronoi generator sets.

The learned VoroMesh representation guarantees watertight, manifold, non-self-intersecting surface extraction and is highly efficient in terms of geometric representation and computational cost, with mesh extraction times $<0.2$ s for $N\simeq 50$K generators (Maruani et al., 2023).

2.3 Planar Domain VoroMesh via Elastic Self-Organization

For planar domains with piecewise-smooth boundaries, VoroMesh algorithms minimize an elastic energy functional to self-organize boundary and interior generator points (Garanzha et al., 2018):

• Composite energy $W(P) = \theta_r W_r + \theta_s W_s + \theta_a W_a$ accounts for repulsion, boundary sharpening, and attraction toward $\Gamma$.
• Delaunay triangulation and its dual Voronoi mesh are updated in each iteration; boundary topologies are repaired via local insertion of new points.
• Special handling of sharp corners ensures dual Voronoi fans with correct combinatorics and convexity.

3. Theoretical Guarantees and Mesh Quality

Voronoi-based meshing admits strong theoretical guarantees that distinguish it from clipping-based and non-conforming schemes.

• Surface isotopy and approximation: For parameters $\epsilon\leq 1/20$ and $\delta=2\epsilon$, the reconstructed surface is ambient isotopic to $M$; projection is a homeomorphism (Abdelkader et al., 2018).
• Facet and cell quality: Minimum surface facet angles bounded below (e.g. $>7^\circ$), surface edge ratio $<4$, volumetric aspect ratio $<14.2$ (Abdelkader et al., 2018). For VoroCrust, all surface facets and volume cells are true, convex Voronoi elements.
• Feature preservation: Meshes preserve both smooth surface patches and sharp features through stratified sampling and co-smoothness-based ball union logic (Abdelkader et al., 2019).
• Optimal complexity: Total number of sites $$|S_\mathrm{all}| = O(\epsilon^{-3} \int_\Omega lfs(x)^{-3} dx)$$, matching the optimal scaling in $\epsilon$ (Abdelkader et al., 2018).

For learning-based VoroMesh, watertightness and absence of self-intersections are enforced by construction. Empirical Chamfer and F1 scores indicate geometric fidelity exceeding classical Marching Cubes, Dual Contouring, and recent mesh-prediction networks under fixed generator budgets (Maruani et al., 2023).

4. Distributed and High-Performance VoroMesh Construction

MadVoro (Mizrachi et al., 20 Feb 2025) addresses large-scale, distributed-memory 3D Voronoi mesh construction:

• Uses space-filling curve partitioning for load balancing.
• Local Delaunay triangulation is followed by iterative ghost-point discovery using small/large search balls, communicated via MPI_Alltoall exchanges, to ensure correctness across partition boundaries.
• Once no site requires additional ghosts, the Voronoi diagram is extracted globally via the dual of the (stitched) Delaunay structure.
• Demonstrated point rates up to 15 million/sec per rank and outperforms state-of-the-art distributed codes (AREPO, ParVoro++) in weak and strong scaling scenarios.

5. Differentiable Pipelines and Extensions

Recent research advances leverage differentiable Voronoi representations for neural surface and volume modeling:

• VoroLight (Lu et al., 15 Dec 2025) generalizes differentiable Voronoi meshing to diverse data types (SDF, point cloud, silhouettes, images). The pipeline comprises:
  1. Differentiable Voronoi surface initialization from paired boundary reflection seeds.
  2. Sphere-based surface refinement enforcing intersection/exclusion constraints for geometric regularity.
  3. Volumetric optimization of deep-interior generators under CVT (centroidal Voronoi tessellation) losses.

• Shape-fitting losses include SDF, Chamfer distance, and multi-view differentiable rendering terms. Regularization ensures uniform area, normal consistency, circularity, and Laplacian smoothness.

• The result is a volumetric mesh with near-equilateral faces and high topological consistency, suitable for inverse problems and neural implicit/surface hybrid representations.

6. Applications and Comparative Perspective

VoroMesh algorithms are employed across:

• Geometry processing and CAD analysis for boundary-conforming mesh generation.
• Scientific computing (finite-volume and finite-element methods) requiring convex, high-quality polyhedral cells with dual Delaunay structure (Springel, 2011).
• Machine learning pipelines for 3D surface/volume prediction from diverse sensory input (Maruani et al., 2023, Lu et al., 15 Dec 2025).
• Large-scale distributed simulation environments (astrophysical hydrodynamics, mesh decomposition) (Mizrachi et al., 20 Feb 2025).

Distinct advantages over power-diagram (Power Crust), $\alpha$-shape, and clipping-based approaches include unweighted Voronoi tessellation, true convex cells, absence of filtering/postprocessing, flexibility in interior grading, and provable geometric/topological guarantees.

7. Practical Implementation, Complexity, and Performance

Common features across VoroMesh methods:

• Mesh construction time scales as $O(N \log N)$ for both serial and multi-core scenarios.
• For distributed-memory settings (MadVoro), communication volume per host scales as $O((n/P)^{2/3})$, and near-linear strong/weak scaling is demonstrated up to $512$ ranks (Mizrachi et al., 20 Feb 2025).
• Memory requirements dominated by generator storage, SDF/feature field representations, and spatial data structures (KD-trees/grid).
• Learning-based pipelines utilize PyTorch, sparse 3D CNNs, and CGAL/Voro++ for mesh operations; batch-optimized k-NN and plane-distance computations yield GPU- or CPU-efficient implementations (Maruani et al., 2023, Lu et al., 15 Dec 2025).

Empirical benchmarks demonstrate that VoroMesh approaches achieve lower Chamfer errors and higher F1 scores than traditional isosurfacing, with consistent watertightness and absence of self-intersections, and comparable or better speed.

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References

• A. Abdelkader et al., "Sampling Conditions for Conforming Voronoi Meshing by the VoroCrust Algorithm," SoCG 2018 (Abdelkader et al., 2018).
• F. Maruani et al., "VoroMesh: Learning Watertight Surface Meshes with Voronoi Diagrams" (Maruani et al., 2023).
• Y. Lu et al., "VoroLight: Learning Quality Volumetric Voronoi Meshes from General Inputs" (Lu et al., 15 Dec 2025).
• M. Mizrachi et al., "MadVoro: Parallel Construction of Voronoi Diagrams in Distributed Memory Systems" (Mizrachi et al., 20 Feb 2025).
• Y.A.A. Shashkov et al., "Construction of near-boundary Voronoi mesh layers for planar domains" (Garanzha et al., 2018).
• V. Springel, "Hydrodynamic simulations on a moving Voronoi mesh" (Springel, 2011).
• A. Abdelkader et al., "VoroCrust: Voronoi Meshing Without Clipping" (Abdelkader et al., 2019).