面向网格的三角剖分无关流匹配的Matérn噪声

摘要

本文针对三角网格上的信号生成任务，提出了一种与三角剖分无关的算法，即训练后的模型可有效适用于不同的网格和三角剖分方式。在实践层面，本文创新性地将流匹配（FM）范式适配至基于网格的三角剖分无关场景；在理论层面，提出了一种用于FM模型去噪过程的特定噪声分布，该分布具有三角剖分无关性。尽管为图像等数据设计噪声分布通常较为简单，但构建三角剖分无关的分布却极具挑战性。我们通过谱分析给出了分布三角剖分无关性的数学定义，并证明一种名为Matérn过程的高斯随机场离散化符合这些理想性质，同时提供了简洁高效的采样算法。我们将其作为噪声模型，并采用当前最先进的网格梯度域信号学习方法——PoissonNet作为去噪器，将FM适配至三角剖分无关场景。实验涵盖弹性静止态采样、人形机器人姿态生成等复杂任务。结果表明，该方法能够对包含百万级三角面的网格生成高度逼真的结果，在质量和多样性上显著超越现有技术水平。

English

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.