Canonical Reference Mesh in Geometry Processing

• Canonical reference mesh is a fixed topology representation standardizing vertices, edges, and faces to enable consistent parameterization and correspondence across varying geometries.
• It supports applications like texture mapping, shape matching, and dynamic mesh compression by providing a unified template for both static and time-varying models.
• Advanced construction methods, including homotopy-basis algorithms, deep learning deformations, and ARAP volume tracking, balance accuracy with computational efficiency.

A canonical reference mesh is a mesh representation that provides a fixed, consistent topology (vertex, edge, and face structure) used as a template for parameterization, shape correspondence, mesh deformation, or time-varying mesh processing. This mesh serves as a global or object-centric coordinate domain, enabling tasks such as isomorphic mesh generation, parameterization via polygonal schemas, dynamic correspondence tracking, and robust compression. Across research domains, the canonical reference mesh concept underpins efficient workflows in geometry processing, graphics, and machine learning, as it allows mesh attributes and deformations to be indexed and processed in a unified manner even as the underlying geometry varies over time or object instances.

1. Formal Definitions and Conceptual Foundations

A canonical reference mesh possesses a fixed combinatorial structure (set of vertices, edges, and faces), typically watertight and genus-aware, serving as a template for establishing correspondences or parameterizations. For a closed orientable 2-manifold $\mathcal{M}$ of genus $g$, this includes schemas such as the canonical polygonal schema—obtained by cutting $\mathcal{M}$ along a homotopy basis of $2g$ loops based at a root, yielding a disk topologically mapped to a $4g$-gon with labeled sides. In time-varying mesh sequences, a canonical reference mesh $M_\text{ref}$ is constructed so all observed meshes $M(t)$ can be aligned or deformed from/to $M_\text{ref}$, yielding consistent surface or volumetric correspondence (Livesu, 2020, Chen et al., 2024).

2. Construction Methodologies

2.1. Homotopy-Basis and Polygonal Schema

For arbitrary genus-$g$ surfaces, the canonical polygonal schema is a foundational construction. It requires computing a homotopy basis: a set $L = \{\ell_1, \dots, \ell_{2g}\}$ of $2g$ simple loops, all based at a common root, generating $\pi_1(\mathcal{M})$ such that cutting along $L$ yields a disk. A greedy algorithm (Erickson–Whittlesey) generates an approximately shortest basis in $O(n\log n)$ for fixed root, but the resulting loops may overlap except at the base. Mesh refinement (e.g., vertex-split, edge-split, or hybrid-split operators) is then required to obtain mutually disjoint loops, enabling the extraction of the canonical $4g$-gon parameterization (Livesu, 2020). Choice of refinement strategy dictates the trade-off between mesh complexity and geometric fidelity (see Section 4).

2.2. Deep Learning–Driven Canonicalization

In isomorphic mesh generation from point clouds, a fixed genus-zero canonical mesh $R^{(0)}$ (often a subdivided icosahedral sphere) is nonrigidly deformed via neural mapping functions such that its vertices are mapped to a target geometry while preserving mesh connectivity. The mapping progresses in stages—global, coarse-local, fine-local—each parameterized by an MLP, producing a final output mesh with the same vertex and face structure as $R^{(0)}$. This yields isomorphic meshes that can be directly used in DNN workflows, with robust topology preservation and efficient batched processing (Miyauchi et al., 2022).

2.3. Dynamic Scene and Time-Varying Mesh Tracking

For time-varying meshes (TVMs), canonical reference meshes are constructed to capture the union envelope of the object’s volumetric occupancy. Methods employ ARAP (as-rigid-as-possible) volume tracking to determine interior “volume centers” over time, followed by multidimensional scaling (MDS) based on maximum pairwise separations to embed reference centers in a self-contact-free configuration. Landmarks from each frame are mapped onto this reference via RBF deformation, producing a combined, well-aligned point cloud. Poisson surface reconstruction is then applied to generate the watertight reference mesh, which can be further deformed via ARAP to individual frames, supporting efficient compression and correspondence (Chen et al., 2024).

In dynamic Gaussian-based approaches, objects are represented in canonical space as sets of 3D Gaussians, from which a canonical mesh is extracted via differentiable Poisson solvers and marching cubes. Learned forward and backward deformations enable tracking of the canonical mesh to all frames, ensuring persistent correspondence and enabling temporally consistent mesh processing (Liu et al., 2024).

3. Applications and Downstream Uses

Canonical reference meshes serve a central role in:

• Parameterization and Flattening: Polygonal schemas provide the basis for mapping high-genus surfaces to plane domains (e.g., regular $4g$-gons), facilitating conformal flattening, texture mapping, or shape matching (Livesu, 2020).
• Statistical Shape Analysis: Fixed-topology meshes allow direct application of statistical and learning models across object collections.
• Dynamic Scene Processing: Reference meshes are crucial for compressing TVMs, as deformation parameters can be encoded relative to the template, substantially reducing storage and transmission costs (Chen et al., 2024).
• Neural Architecture Input: Deep neural networks benefit from fixed-structure mesh input for geometry-based tasks, as in isomorphic mesh generation (Miyauchi et al., 2022).
• Physics and Animation: Canonical meshes tracked through deformation allow consistent simulation, retargeting, and animation with exact vertex correspondence (Liu et al., 2024).
• Collision Detection and Shape Retrieval: Unified volumetric envelopes derived from canonical reference meshes facilitate stable object comparisons and spatial analyses.

4. Topology, Refinement, and Trade-offs

Mesh refinement is essential when the loops or cut graphs defining the canonical schema are not disjoint. The data highlights three key strategies with quantitative trade-offs (Livesu, 2020):

Strategy	Mesh Complexity Growth	Geometric Fidelity	High-Genus Scalability
Edge-Split	Super-quadratic in $g$	Exact (0 deviation)	Poor (OOM beyond $g=50$)
Vertex-Split	Linear in loop count ($g$)	Locally unbounded	Excellent
Hybrid-Split	Linear/interpolated, bounded	Bounded by $\theta_0$	Excellent

Hybrid-split uses vertex-split in nearly planar regions (angle between normals < $\theta_0$), otherwise edge-split for geometry. This controls mesh growth while limiting geometric deviation (Hausdorff distance empirically $<10^{-4}\cdot$ mesh diameter for $\theta_0=5^\circ$).

A plausible implication is that, for extreme-genus or high-resolution models, the hybrid-split is the only practical refinement method that preserves both memory efficiency and surface quality.

5. Robustness, Noise, and Correspondence

Canonical reference meshes enable robust processing in the presence of noise, hole regions, and/or varying connectivity. In data-free neural methods, the fully connected MLP prior acts as a smooth interpolator, filtering noise and providing plausible fill-in where point data is sparse. Coarse/fine patch schemes and geometry-prior loss terms further stabilize deformations, avoiding self-intersections and producing valid templates even from incomplete or noisy inputs (Miyauchi et al., 2022). In dynamic mesh tracking, the use of maximum-distance MDS embeddings and ARAP regularization ensures self-contact avoidance and stable volumetric correspondence across time frames (Chen et al., 2024, Liu et al., 2024).

6. Computational Considerations and Efficiency

Canonical reference meshes enable high-throughput and memory-efficient processing. Storage and computation can be arranged in fixed-size contiguous arrays (vertex/face lists), ideal for GPU-based pipelines. Isomorphic mesh generation from point clouds using MLPs demonstrates construction of a 36k-vertex mesh from a few-thousand points in under one minute on commodity GPU hardware. In contrast, classical methods (e.g., Poisson reconstruction plus remeshing) require larger memory footprints and processing time (Miyauchi et al., 2022). For TVM compression, the canonical mesh ensures deformation and compression costs are drastically reduced versus full connectivity updates per frame (Chen et al., 2024).

7. Comparative Assessment and Limitations

Canonical reference mesh methodologies outperform template-mesh or keyframe-based pipelines, particularly in scenarios involving self-contact, large topological change, or correspondence ambiguity. Keyframe selection methods may fail catastrophically when the chosen mesh contains self-intersecting regions, impairing deformation operators and downstream tasks. Volume-tracking plus MDS ensures reference geometry remains self-contact-free across the full temporal extent, preserving correspondence quality and compression robustness (Chen et al., 2024). Limitations include dependence on number and quality of ARAP-tracked centers, Poisson reconstruction fidelity, and lack of capacity to handle object disintegration or dramatic topological changes not covered by initial samples.

• (Livesu, 2020) Obtaining a Canonical Polygonal Schema from a Greedy Homotopy Basis with Minimal Mesh Refinement
• (Miyauchi et al., 2022) Isomorphic Mesh Generation from Point Clouds with Multilayer Perceptrons
• (Liu et al., 2024) Dynamic Gaussians Mesh: Consistent Mesh Reconstruction from Dynamic Scenes
• (Chen et al., 2024) Volume Tracking Based Reference Mesh Extraction for Time-Varying Mesh Compression