Vertex-Based Mesh Representation

• Vertex-based mesh representations are discrete models that encode 3D surfaces using vertices and ordered connectivity, ensuring explicit control and high geometric fidelity.
• They enable efficient denoising and remeshing via methods like nonlinear diffusion and curvature-adaptive tessellation, preserving critical features during processing.
• Recent advances integrate neural tokenization and continuous latent embeddings to support scalable, editable, and high-fidelity 3D modeling in graphics, simulation, and medical applications.

A vertex-based mesh representation encodes a 3D surface as a discrete structure comprised primarily of vertices (points in $\mathbb{R}^3$), with connectivity data (edges and faces) specified in terms of ordered tuples of these vertices. This paradigm is foundational in geometry processing, computer vision, graphics, simulation, and modern neural modeling. Vertex-based approaches enable direct manipulation, analysis, and learning of surface geometry, yielding explicit, topologically controlled, and often efficient representations that are especially suited to tasks demanding feature preservation, high geometric fidelity, and mesh-editing capabilities.

1. Mathematical Foundations and Data Structures

The central abstraction is the set of $N$ vertices $V = \{v_i \in \mathbb{R}^3\}$, with faces defined as index sequences over $V$ (commonly triangles: $f = (v_{i_1}, v_{i_2}, v_{i_3})$). Various data structures, including adjacency matrices, halfedge data structures, and more recent continuous latent embeddings, are employed for encoding connectivity, enabling both explicit surface traversal and manifoldness guarantees (Shen et al., 30 Sep 2024).

The discrete-to-continuous mapping is formalized in several recent works. Spherical implicit surface models map continuous coordinates on the sphere to vertex positions via neural functions: $SIS: (\theta, \phi) \to \mathbb{R}^3$, where vertices need not be fixed but can be sampled at arbitrary resolution (Gao, 2023). Conversely, continuous connectivity can be represented by assigning an adjacency embedding $x_v \in \mathbb{R}^k$ per vertex and predicting edges via a thresholded pairwise “spacetime” distance $st(x_i, x_j)$ (Shen et al., 30 Sep 2024).

2. Vertex-Based Mesh Denoising and Optimization

Vertex-based mesh denoising directly updates vertex coordinates to suppress noise while preserving sharp features. This is cast as a discrete nonlinear diffusion process: $$P_i^{\text{new}} = P_i^{\text{old}} + \lambda \sum_{j\in N(P_i)} g(\|P_i - P_j\|)\, (P_j - P_i)$$ where $g(\cdot)$ is a diffusivity function—Laplace, reduced centered Gaussian, or Rayleigh PDFs have demonstrated improved accuracy and feature preservation over previous Cauchy-based schemes (Hassouni et al., 2010). Controlled diffusion based on geometric structure ensures that noise is suppressed without indiscriminate smoothing of critical high-curvature or edge features.

The effectiveness of such denoising is quantitatively assessed by L$^2$ vertex-position error and face-normal error metrics compared to ground-truth meshes. These metrics directly evaluate geometric and visual fidelity of denoised surfaces (Hassouni et al., 2010).

Beyond denoising, vertex optimization is central to remeshing. Adaptive Centroidal Voronoi Tessellation (CVT) methods iteratively optimize the vertex distribution for uniformity and quality, using curvature-adaptive original-facet-clipping to ensure fidelity to high-curvature features. The angular relationship between facet normals governs the number of clipping passes, adaptively regularizing the mesh geometry (Fei et al., 20 May 2025).

3. Learning, Generative Models, and Tokenization

Recent advances encode meshes for learning and generation as sequences or continuous embeddings centered on vertices. Transformer-based autoencoders, diffusion models, and autoregressive decoders operate over either vertex tokens or learned connectivity spaces:

• Tokenization and Compression: Compact mesh representations, such as Mesh Silksong and MeshAnything V2, map each vertex to a unique sequence position, minimizing repeated tokens and redundancy; innovations like Adjacent Mesh Tokenization (AMT) further reduce sequence length by encoding only new vertices per face, reaching compression ratios $\sim$22% of previous methods (Song et al., 3 Jul 2025, Chen et al., 5 Aug 2024). Such efficiency is crucial for training large-scale generative models.
• Hierarchical, Coarse-to-Fine Decoding: PivotMesh structures the generation process by first emitting “pivot vertices” (high-degree, structurally informative) to guide subsequent detailed mesh synthesis, using transformers for both face- and vertex-level decoding (Weng et al., 27 May 2024).
• Continuous and Differentiable Representations: SpaceMesh introduces continuous latent connectivity spaces, assigning each vertex a $k$-dimensional embedding used to reconstruct both adjacency and halfedge ordering (cyclic neighbor permutation); these can be efficiently learned via gradient-based methods and ensure manifoldness by construction (Shen et al., 30 Sep 2024). Such representations are suitable for both direct mesh generation and mesh repair tasks.

4. Editing, Feature Representation, and Hybrid Methods

Vertex-based mesh representations naturally support explicit, spatially localized editing of both geometry and appearance:

• Disentangled Neural Codes: NeuMesh assigns geometry and texture codes (and a learnable sign indicator) to each vertex, with inverse-distance-weighting interpolation of latent codes at query points. This enables mesh-guided geometry editing (e.g., as-rigid-as-possible deformations) as well as local texture swaps, fill, or painting, all realized consistently within the neural implicit field (Yang et al., 2022).
• Hybrid Mesh-Neural Models: Hybrid schemes combine explicit mesh vertices for global shape with neural networks (e.g., local surface-based MLPs for vertex displacement fields). For transparent object reconstruction, piecewise two-layer local MLPs predict per-vertex displacements, with smoothness ensured via geodesically weighted fusion layers (Xu et al., 2022). This separation allows efficient optimization and accurate fine-scale detail while easing the constraints in classical light path correspondence problems.
• Compact Neural Feature Storage: Vertex features for neural global illumination attach learnable features directly to mesh vertices, interpolated by barycentric coordinates during rendering. This results in dramatic memory efficiency gains (down to one-fifth of grid-based methods), sharp detail preservation, and effective support for dynamic scene changes (Su et al., 11 Aug 2025).

5. Applications in Human Shape, Medical Modeling, and Scene Representation

Vertex-based representations underpin a range of advanced applications:

• Human Pose and Mesh Reconstruction: Learnable templates and vertex-vertex relationship modeling via transformers enable reconstruction of 3D human meshes in the wild with strong generalizability (e.g., low mean per vertex error and resistance to catastrophic forgetting) (Tran-Quang et al., 2022). Volumetric vertex heatmaps, autoencoder-learned body priors, and latent code transfer reinforce data-driven reconstructions of SMPL-based meshes from multi-view images (Chun et al., 2023).
• Cardiac Motion Tracking: DeepMesh models the heart as a surface mesh, estimating per-vertex displacement fields over cardiac cycles from CMR images. A differentiable mesh-to-image rasterizer enables supervision from 2D anatomical segmentations, yielding vertex-correspondent, temporally coherent motion fields—allowing precise functional evaluation and model transfer across patient populations (Meng et al., 2023).
• Urban-scale Scene Synthesis: Deformable Neural Mesh Primitives (DNMP) parameterize urban radiance fields as collections of connected mesh vertices paired with radiance features. This enables high-fidelity, photo-realistic neural rendering with rasterization-based efficiency, exploiting low-dimensional latent codes for both geometry and appearance interpolation (Lu et al., 2023).

6. Generative and Polygonization Techniques for Complex Topology

Methods to represent and generate complex, high-fidelity meshes have focused on scalable auto-regressive strategies and direct topology modeling:

• Auto-Regressive Generation: Compact, redundancy-reducing tokenizations (e.g., Mesh Silksong, MeshAnything V2) enable generation of watertight, manifold, and normal-consistent polygon meshes, with large generative models managing intricate topologies without exceeding memory constraints (Song et al., 3 Jul 2025, Chen et al., 5 Aug 2024).
• Dual Contouring and Ambiguity Resolution: GenUDC uses Unsigned Dual Contouring (UDC), discretizing meshes over regular grids into separate face and vertex components. Face “activity” flags define global mesh topology, while local vertex coordinates are regressed conditionally; this separation eliminates the ambiguity found in deformable grid-based methods and improves both memory and geometric fidelity (Wang et al., 23 Oct 2024).
• Isosurface Extraction: Polygonization of implicit surfaces via methods like Marching Cubes, Dual Contouring, or adaptive cell-based extraction continues to be refined for quality and efficiency (Shen et al., 2023), while continuous representations and continuous-to-discrete conversions bridge neural and explicit geometric domains.

7. Comparative Analysis and Practical Considerations

Various vertex-based methodologies have demonstrated superiority in different tasks according to quantitative and qualitative benchmarks:

Approach	Efficiency	Fidelity/Editing	Topology
Explicit vertex-based PDE filtering (aniso. diffusion) (Hassouni et al., 2010)	Iterative, local	Feature preservation	Fixed connectivity
Transformer tokenization/coarse-to-fine gen. (Weng et al., 27 May 2024, Song et al., 3 Jul 2025)	Scalable, compressed	Sharp, compact	Complex, general
Continuous embedding/SpaceMesh (Shen et al., 30 Sep 2024)	Fast convergence	Manifold structure	Arbitrary polyfaces
Volumetric heatmaps (Chun et al., 2023)	Data-efficient	Robust generalization	SMPL topology
DNMP/urban radiance (Lu et al., 2023)	Real-time, low-mem	Rasterization-compat	Modular primitives

Performance trade-offs vary: explicit approaches offer direct control but less flexibility for variable topology; continuous neural approaches afford efficiency and learning suitability but may require careful mapping to discrete topologies; autoregressive and tokenized models enable large-scale generation and editing but depend on efficient compression and ordering to maintain tractability.

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Vertex-based mesh representation thus encompasses a spectrum of classical, discrete methodologies and emerging continuous or learned paradigms. Across denoising, editing, learning, remeshing, and generative modeling, the direct manipulation and encoding of mesh vertices—often coupled with advanced connectivity representations—remains essential for producing high-quality, semantically controllable, and computationally efficient 3D models.