Mat\'ern Noise for Triangulation-Agnostic Flow Matching on Meshes

TL;DR

This paper introduces a triangulation-agnostic flow matching method for mesh signals, utilizing a Matérn process-based noise distribution to enable high-quality, diverse mesh generation across different triangulations.

Contribution

It proposes a mathematically grounded, triangulation-agnostic noise distribution based on the Matérn process, and adapts flow matching for mesh signals with state-of-the-art results.

Findings

Capable of generating realistic mesh signals on meshes with over one million triangles.

Outperforms previous methods in quality and diversity of generated meshes.

Successfully applies to tasks like elastic rest states and humanoid pose generation.

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 Mat\'ern 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.