Matérn Noise for Triangulation-Agnostic Flow Matching on Meshes

2026-05-18

AI Summary

This work addresses the sensitivity of signal generation on triangular meshes to mesh triangulation by proposing a triangulation-invariant generative approach. The method integrates flow matching into mesh signal generation, employing PoissonNet as a denoiser in the gradient domain and introducing, for the first time, a noise model grounded in spectral theory that satisfies triangulation invariance: a discretized Matérn Gaussian random field. An efficient sampling algorithm is also provided for this model. The proposed framework enables high-quality generation of diverse human poses and elastic rest shapes on meshes with millions of triangles, significantly outperforming existing methods.

Abstract

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.

Problem

• triangulation-agnostic
• mesh generation
• flow matching
• signal generation on meshes
• Matérn noise

Innovation

• triangulation-agnostic
• flow matching
• Matérn noise
• mesh generation
• Gaussian random field