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Matérn Noise for Triangulation-Agnostic Flow Matching on Meshes

Tianshu Kuai, Arman Maesumi, Daniel Ritchie, Noam Aigerman

arXiv:2605.19305v1cs.GRcs.CVcs.LG

TL;DR

Triangle-mesh generation still lacks practical, high-quality methods that remain insensitive to triangulation. This paper combines spectrally defined Matérn noise with flow matching and a triangulation-agnostic denoiser, achieving plausible generations on meshes exceeding one million vertices.

  • Problem

    Triangle meshes have yet to achieve practical generative capabilities of comparable quality, motivating distributions consistent across triangulations and mesh refinement.

  • Method

    The method defines triangulation-agnostic noise spectrally, samples a Matérn process through finite-element discretization, and uses it in flow matching with a triangulation-agnostic denoiser.

  • Results

    The framework achieves state-of-the-art results at resolutions far exceeding prior mesh-generative methods, producing plausible humanoid deformations and elastic rest states on meshes over one million vertices.

  • Takeaways & Limitations

    The authors present the framework as a step toward practical mesh generation and a viable option for real-world applications.

  • Takeaways & Limitations

    The PoissonNet denoiser backbone limits the method to tasks without extreme high-frequency content, preventing generation of crisp RGB signals.

Abstract

from arXiv · show

This paper tackles the task of learning to generate signals over triangle meshes in a triangulation-agnostic manner, meaning the trained model can be applied to different meshes and triangulations effectively. Practically, the paper adapts the flow matching (FM) paradigm to a mesh-based, triangulation-agnostic setting. Theoretically, it proposes a specific noise distribution which is triangulation agnostic, to be used inside the FM model's denoising process. While noise distributions are usually trivial to devise for, e.g., images, devising a triangulation-agnostic distribution proves to be a much more difficult task. We formulate a mathematical definition of triangulation agnosticism of distributions, via their spectrum. We then show that a discretization of a specific Gaussian random field called a Matérn process holds these desired properties, and provides a simple and efficient sampling algorithm. We use it as our noise model, and adapt FM to the triangulation-agnostic setting by using a state-of-the-art approach for learning signals on meshes in the gradient domain -- PoissonNet -- as the denoiser. We conduct experiments on elaborate tasks such as sampling elastic rest states, and generating poses of humanoids. Our method is shown to be capable of producing highly realistic results for meshes of over one million triangles, significantly exceeding the state-of-the-art in quality and diversity.

1 Introduction

The paper develops a triangulation-agnostic generative framework for signals over triangle meshes, addressing the difficulty of achieving practical mesh generation that is insensitive to surface discretization. Its central solution combines a spectrally defined Matérn noise distribution with flow matching and PoissonNet, enabling high-resolution generation across arbitrary well-behaved triangulations.

  • Motivation: Triangle meshes have not yet achieved generative capabilities comparable in quality to those available for images and voxel grids.The paper motivates mesh generation as producing values over the vertices of an existing mesh, with applications spanning graphics, engineering, and biomedical imaging.
  • Problem and goal: The framework is triangulation agnostic: it can operate on arbitrary well-behaved triangulations of a 3D model with near-identical behavior.This property supports heterogeneous mesh datasets whose models may use varying triangulations.
  • Technical challenge: An appropriate triangulation-agnostic noising scheme is required because iid Gaussian vertex noise changes with discretization and can prevent sampling on unseen triangulations.The noise distribution must preserve its characteristics under superficial changes to the surface discretization so samples remain compatible with the trained denoiser.
  • Noise model: The paper defines triangulation-agnostic distributions through probabilistic spectral analysis and uses a finite-element discretization of a Matérn process to sample from distributions with the desired properties.The resulting sampling scheme is described as highly simple and efficient.
  • Generative pipeline and results: The complete pipeline combines Matérn noise, flow matching, and PoissonNet, producing plausible humanoid deformations and elastic rest states on meshes with over one million vertices.Prior methods are described as unable to generalize beyond their training meshes because of triangulation sensitivity, while this method attains state-of-the-art results at a resolution exceeding prior mesh generative methods.

2 Related Work

The paper situates its contribution within flow-matching generative modeling, mesh-signal generation, spectral analysis, and Matérn-process noise. Unlike prior manifold-flow methods that sample points on manifolds, it defines flows in the functional space of a manifold to generate signals on existing meshes triangulation-agnostically.

  • Generation through denoising: Flow matching interpolates between noise and data while predicting a time-dependent vector field, and is related to diffusion, score-based, and rectified-flow methods.The paper adopts FM as its denoising framework and contributes a triangulation-agnostic noising and denoising scheme for mesh signals.
  • Generation over existing meshes: Existing-mesh generation preserves mesh structure, but prior state-of-the-art denoising-based methods are not triangulation agnostic.The paper compares against these prior methods and reports exceeding them in performance.
  • Non-Euclidean generative modeling: The paper defines flows in a manifold’s functional space to sample functions on the manifold, rather than flows over the manifold to sample points.This distinguishes its setting from many non-Euclidean generative methods that compute flows directly over a manifold.
  • Spectral mesh analysis: Spectral mesh analysis based on mesh-Laplacian eigenvectors is a foundational geometry-processing tool used in applications including parameterization, shape signatures, and matching.The paper uses this spectral perspective to formulate triangulation agnosticism of distributions.
  • Matérn processes: The proposed noise belongs to the Matérn Gaussian random-field class and uses its discretization via linear finite elements.Matérn processes are established statistical tools with applications in statistical modeling and machine learning.

3 Triangulation-Agnostic Distributions

The section defines triangulation-agnostic signal distributions through spectral statistics that remain consistent across triangulations and mesh refinement. It identifies a Matérn process whose FEM discretization satisfies these properties, while noting that the screening parameter requires care across different shapes and scales.

  • Definition: Triangulation-agnostic distributions should produce similar signal distributions on different triangulations of the same surface.Direct comparison is difficult because meshes can have different vertex counts and no correspondence, motivating a spectral comparison.
  • Definition: The definition requires independent spectral coefficients, mesh-invariant frequency statistics, and negligible high-frequency tail contribution.Each coefficient distribution depends only on its eigenvalue and varies Lipschitz-continuously with frequency.
  • Matérn process: A Matérn process satisfies the desired properties in the continuous case by filtering white Gaussian noise through a screened Poisson equation.The screening term τ controls the process, while the screened Poisson solve acts as a spectral low-pass filter.
  • FEM discretization: The discretized Matérn construction is presented as a new noise-sampling approach for generative tasks, graphics, and geometry processing.The authors motivate the FEM review by proving that the discretized distribution satisfies the triangulation-agnostic properties.
  • FEM discretization: The FEM-discretized Matérn distribution is triangulation agnostic because its spectral covariance is diagonal, coefficient variance depends only on eigenvalue, and variance decays with frequency.The construction yields iid Gaussian white-noise coefficients before frequency-dependent filtering.
  • Limitations: The screening parameter τ does not guarantee consistent noise across different shapes or scaled versions of the same mesh.These inconsistencies arise because scaling and changing surfaces alter the Laplacian spectrum and eigenvectors.

4 Triangulation-Agnostic Flow Matching

The method adapts rectified flow to triangulation-agnostic mesh generation by connecting Matérn noise to data samples along linear paths and denoising with PoissonNet. Training supervises vertex and gradient-domain Jacobian velocities, while sampling integrates the resulting vertex-velocity ODE with a midpoint solver.

  • Flow construction: The conditional flow path linearly interpolates between a Matérn noise sample f0 and a data sample f1 as f_t = (1 − t)·f0 + t·f1.Here, t∈[0,1], f1 is drawn from the dataset, and f0 from the Matérn noise distribution.
  • Denoiser: PoissonNet predicts gradient-domain Jacobians, which are converted into vertex velocities through the Poisson equation.The denoiser is a state-of-the-art triangulation-agnostic architecture, denoted F_θ(f_t,t).
  • Training objective: The standard conditional flow-matching loss is applied to both vertex velocities and Jacobian velocities.The formulation uses diagonal vertex-mass and face-area matrices, the ground-truth velocity δ = f1 − f0, and the spatial gradient operator.
  • Generalization: The trained model generalizes to humanoid meshes unseen during training, while generated samples approximate the ground-truth deformation distribution and can be nearly perfectly matched by SMPL regression.The closest training sample is reported as far away, supporting generalization beyond memorized examples.
  • Sampling: Sampling numerically integrates the predicted vertex-velocity ODE with the midpoint ODE solver.The integration follows Equation (1).

5 Experiments

Experiments show that the method generates diverse, plausible mesh deformations across resolutions and triangulations, including yoga poses and elastic states. It outperforms non-triangulation-agnostic baselines and remains effective under topology or triangulation changes, though accuracy can deteriorate in extreme cases.

  • Experimental setup: All experiments train on one triangulation and validate equivalent performance on different, often higher-resolution triangulations where direct training is infeasible.The evaluation focuses on generated mesh deformations and uses experiment-specific errors alongside MMD and COV to assess distributional similarity.
  • Human pose generation: On SMPL yoga poses, the method generates diverse, plausible deformations across different triangulations up to 1.2M triangles.Generated samples are also shown to conform well to the SMPL deformation space while remaining far from the closest training example, indicating generalization.
  • Elastic deformation generation: For elastic deformations, the method produces diverse plausible equilibrium states on a 70k-face bunny trained only at 9k resolution and handles thin fish fins.Generation remains comparable across resolutions, with timings reported as fast enough for real-world applications.
  • Human pose generation: A single trained model transfers successfully across human-like meshes with varied shapes while maintaining similar diversity and quality.This demonstrates application beyond the source mesh geometry used during training.
  • Comparisons and ablations: The method outperforms MDF and DoubleDiffusion on dataset matching and deformation-space matching, while alternative noise models fail on non-uniform triangulations.The reported comparisons use MMD and COV for dataset matching and MSE for deformation-space matching; Matérn noise without screening also deteriorates results.
  • Robustness and limitations: The method remains plausible after introducing holes or severely degrading triangulation quality, but accuracy drops near new boundaries and under extremely poor triangulations.Topology changes also cause generated results to drift from the SMPL parametric model, and the method is not intended as neural physical simulation.

6 Conclusion

The method produces plausible, intricate mesh generations and is presented as a viable practical option, while remaining limited by its denoiser architecture and Matérn noise assumptions. Future work targets broader generative applications, architectural extensions, and uses of Matérn processes beyond meshes.

  • The experiments produce highly plausible results, emulate intricate datasets, and support the method’s viability for real-world applications.
  • Limitations: PoissonNet limits the method to tasks without extreme high-frequency content, single connected-component meshes, and 2-manifolds, excluding crisp RGB image generation.The noise itself can be defined in any dimension and may be incorporated into future architectures addressing these constraints.
  • Limitations: Matérn noise is sensitive to topological changes and relies on sufficiently well-behaved mesh approximations, although tested models produced triangulation-agnostic noise.
  • Future work: Future applications include generating variations from templates, retargeting poses or physical simulations, adding mesh details, temporal animations, and volumetric medical-imaging data.
  • Future work: Matérn processes may support graphics applications such as noise modeling and non-mesh generative contexts such as pixel-agnostic image generators.

A Proof of the Properties from Section 3.3

The section proves the stated properties of the proposed covariance and spectrum-based construction. It establishes independent coefficient sampling, invokes Wasserstein convergence for mesh coefficients, and uses Weyl’s law with a convergent tail bound.

  • The covariance matrix in Equation (11) is diagonal, so each coefficient is sampled independently.
  • For two meshes, the proof analyzes coefficient distributions using the 2-Wasserstein distance for univariate normal distributions as ε goes to zero.
  • The eigenvalues satisfy Weyl’s law for a 2-manifold with area A under the assumed continuous-spectrum approximation.
  • The resulting series converges as n tends to infinity, allowing a k such that the expression is below ε for any mesh resolution.

B Implementation Details · B.1 Shape Deformation Network · B.2 Deformation Autoencoder

The implementation uses PoissonNet-based deformation networks conditioned on noisy meshes, source geometry, and time, with separate schedules for different tasks. A deformation autoencoder further encodes deformations into 128-dimensional latent codes for PoissonNet decoding.

  • B.1 Shape Deformation Network: The shape deformation network comprises five PoissonNet blocks with hidden dimension 128 and three-layer VectorMLPs.The network inputs the noisy mesh xyz and conditions each block on the source mesh xyz and timestep t.
  • B.1 Shape Deformation Network: The network uses the same NJF head to predict flow velocity in Jacobian space, which can be transformed into vertex-space velocity.
  • B Implementation Details: Single-source generation and elastic-deformation models train for 160k iterations, using learning rates 0.0005 for the first 40k and 0.0001 thereafter.Training used a single NVIDIA H100 GPU; single-source generation at 18k faces took around 42 hours.
  • B.1 Shape Deformation Network: The eigenvector predictor is trained on randomly posed and shaped SMPL meshes to predict normalized eigenvectors corresponding to the smallest 64 Laplacian eigenvalues.Its predictions provide consistent positional encoding on arbitrary 3D human-like meshes, including SMPL meshes in arbitrary poses and shapes.
  • B.2 Deformation Autoencoder: The deformation autoencoder uses five DiffusionNet blocks with spectral size 128 and compresses each input deformation into a 128-dimensional latent code via global mean pooling.The latent code conditions a five-block PoissonNet decoder.

C Datasets · C.1 SMPL datasets

The paper constructs deformed SMPL human-mesh datasets from MOYO yoga-pose motion captures, supporting single-source and arbitrary-human source models. It uses separate datasets for model training and source-target pairing.

  • C.1 SMPL datasets: The datasets consist of deformed SMPL human meshes in yoga poses derived from the MOYO dataset.
  • C.1 SMPL datasets: 64k deformed human meshes are generated for the single source generation model and the eigenvector predictor.
  • C.1 SMPL datasets: The same 64k-mesh dataset supports the eigenvector predictor in the arbitrary human source model.
  • C.1 SMPL datasets: 32k pairs of deformed human meshes are generated as source-target pairs for the arbitrary human source model.
  • C.1 SMPL datasets: The pose samples are drawn from MOYO motion captures using a procedure following Maesumi et al. 2025.
  • C.1 SMPL datasets: A greedy farthest point sampling strategy is used to obtain diverse deformed poses from the motion captures.

C.2 Elastic deformations datasets · C.3 GMM datasets

The datasets comprise physically simulated elastic-object resting states and random Gaussian mixture models on triangulated meshes. Elastic samples are generated by randomized dropping simulations, while GMM samples use mass-weighted centers on the Stanford Bunny.

  • C.2 Elastic deformations datasets: Elastic-object resting states are generated with IPC physical simulations on downsampled high-resolution assets converted to tetrahedral meshes using fTetWild.Objects are simulated on the floor until reaching equilibrium.
  • C.2 Elastic deformations datasets: Objects are dropped with random initial orientation and angular speed sampled between π and 3π.Each equilibrium state is collected as one deformed training sample and converted back to a triangle mesh.
  • C.2 Elastic deformations datasets: The elastic simulations assign Young’s modulus, Poisson’s ratio, and density values of 1e4, 0.4, and 1e3, respectively.The simulation continues until the object reaches an equilibrium state on the floor.
  • C.3 GMM datasets: The GMM dataset follows MDF settings on a uniformly triangulated Stanford Bunny containing 70k faces.The dataset construction uses random Gaussian mixture models with three non-overlapping centers.
  • C.3 GMM datasets: Each GMM contains three non-overlapping centers sampled on the mesh, with vertex selection weighted by vertex mass.Mass weighting ensures a uniform distribution of Gaussians on the surface.
  • C.3 GMM datasets: The GMM training dataset contains 32k randomly generated mixtures.A test mesh is also created by downsampling the Stanford Bunny.

D Evaluation Metrics

The paper evaluates generation quality using complementary fidelity and coverage metrics, and introduces a differentiable SMPL-based metric for realism of generated deformations.

  • MMD and COV: MMD measures the average distance from generated samples to their closest reference samples, providing a fidelity measure.The paper uses average l2 distance weighted by the corresponding vertex mass.
  • MMD and COV: COV measures the percentage of reference samples for which generated data provide the closest sample, complementing MMD in assessing generation quality.Together, MMD and COV represent generation quality.
  • SMPL deformation metrics: The SMPL deformation metric regresses SMPL pose parameters that best match each generated shape using gradient descent, then reconstructs a deformed mesh with SMPL-X.Realism is quantified using mean squared error weighted by the vertex mass.

E Comparison to Point Cloud Generation · F FEM Discretization

The mesh-based method is evaluated against point-cloud generation and shown to preserve geometric detail through intrinsic mesh connectivity. Its FEM discretization represents vertex signals with piecewise-linear functions and uses mass and stiffness matrices to define the mesh operators.

  • E Comparison to Point Cloud Generation: Mesh-based and point-cloud generative methods target different representations and therefore cannot be compared directly without accounting for their objectives.Mesh methods generate vertex details on an existing mesh, whereas point-cloud methods pursue a different representation.
  • E Comparison to Point Cloud Generation: TIGER slightly outperforms the mesh-based method on most MMD, COV, and 1-NNA metrics under Chamfer distance and Earth Mover’s Distance.The comparison treats the mesh method’s generated samples as point clouds and uses commonly adopted point-cloud metrics.
  • E Comparison to Point Cloud Generation: Point-cloud generation can minimize Chamfer distance while omitting small geometric details because Chamfer distance is insensitive to them.The passage contrasts TIGER’s valid-SMPL approximation with its loss of connectivity-dependent detail.
  • E Comparison to Point Cloud Generation: The mesh-based network preserves fingers and facial details by operating intrinsically on a triangulation-agnostic mesh framework.Connectivity information enables minute geometric detail that the compared point cloud omits.
  • F FEM Discretization: A vertex signal is interpreted as a piecewise-linear function over triangles that interpolates the vertex values using standard linear finite elements.For signal values f_i, the resulting function is linear on each triangle and uniquely determined by the vertex interpolation conditions.
  • F FEM Discretization: The finite-element inner product is expressed as f⊤Mg, where M is the symmetric consistent mass matrix assembled from Lagrange hat basis functions.The matrix M belongs to R^n×n and represents the integral inner product of the corresponding piecewise-linear functions.
  • F FEM Discretization: The lumped mass matrix diagonalizes the system by summing rows of the consistent mass matrix, while its diagonal entries correspond geometrically to dual-cell areas.In practice, each diagonal entry is the barycentric area, equal to one-third of the areas of all incident triangles.
  • F FEM Discretization: The cotangent Laplacian is the stiffness matrix discretizing Dirichlet energy, with cotangent edge weights and diagonal entries chosen so every row sums to zero.For off-diagonal entries, the weights depend on the angles opposite each edge; diagonal entries satisfy L_ii = −Σ_j∈N(i) L_ij.
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