網格上三角剖分無關流匹配的Matérn噪聲

摘要

本文探討如何在三角形網格上以「三角化無關」的方式學習生成訊號，意即訓練後的模型能有效應用於不同的網格與三角剖分。在實務上，本文將流匹配（flow matching, FM）範式調整為適用於基於網格且三角化無關的設定；在理論上，則提出一種特定的雜訊分佈（具備三角化無關特性），用於FM模型的去噪過程。雖然對於如影像等領域，設計雜訊分佈通常很簡單，但要設計出三角化無關的分佈卻是一大挑戰。我們透過頻譜對分佈的三角化無關性給出數學定義，接著證明一種稱為馬特恩過程（Matérn process）的特定高斯隨機場之離散化具有這些理想性質，並提供簡潔高效的取樣演算法。我們以此作為雜訊模型，並採用當前在網格上於梯度域學習訊號的尖端方法——PoissonNet——作為去噪器，將FM調整至三角化無關的設定。我們在複雜任務（如取樣彈性靜止狀態、生成人形姿態）上進行實驗，結果顯示本方法能為超過一百萬個三角形的網格產生高度逼真的結果，在品質與多樣性上均大幅超越當前最佳技術。

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.