基于域可变格林函数的 cage 变形
Domain-Varying 2D Green' s Functions for Cage-based Deformation
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中文总结 AI 辅助
本研究提出域可变格林坐标（DVGC）框架，调和并统一经典 cage 变形方法 HC 与 GC，可通过调整格林函数域生成多样化变形效果，且无需有限元离散或数值积分。
中文摘要 AI 辅助
在本研究中，我们提出了一种基于域可变格林函数的 cage 变形的新理论视角，并将该域视为变形效果的新控制空间。调和坐标（Harmonic Coordinates, HC）和格林坐标（Green Coordinates, GC）是 cage 变形领域的经典方法，也是众多实用变形工具中形状编辑的理论基础。我们的方法重新审视了这两种经典技术，具体而言，我们提出了一个基于跨不同域的格林函数（独立于 cage 包围域）的框架，以统一这两种技术。据我们所知，这是近二十年来首次开展此类尝试。基于该视角，我们提出了一种新颖的 cage 变形技术，引入了新的控制空间，并利用域可变格林函数生成多样化的变形效果。当格林函数域 Θ 从 cage 区域 Ω 扩展至整个二维平面 ℝ² 时，我们的方法还建立了从 HC 到 GC 的效果连续过渡，我们将该方法命名为域可变格林坐标（Domain-Varying Green Coordinates, DVGC）。当 Θ 为圆盘或矩形时，格林函数分别具有解析或半解析表达式，这使得 DVGC 无需有限元离散即可计算；此外，当 Θ 为圆盘时，DVGC 对于二维单纯形 cage 具有闭式表达式，从而无需数值积分。实验表明，我们的方法提供了一个新的控制空间，其范围从更贴合 cage 到更保形，通过改变格林函数域可生成多样化的变形效果。
英文摘要
In this work, we propose a novel theoretical view of cage-based deformation based on domain-varying Green' s functions and treat this domain as a new control space for the deformation effects. Harmonic Coordinates (HC) and Green Coordinates (GC) are classic methods in cage-based deformation and serve as the theoretical foundation for shape editing in a range of practical deformation tools. Our method revisits these two classical approaches. Specifically, we propose a framework based on Green' s functions across diverse domains (independent of the cage-enclosed domain) to unify these two techniques. To our knowledge, this represents the first such attempt in nearly two decades. Based on this perspective, we propose a novel cage-based deformation technique that introduces a new control space and utilizes domain-varying Green' s functions to yield varying deformation effects. Our method also establishes a continuous transition of effects from HC to GC as the Green' s function domain $Θ$ expands from the cage region $Ω$ to the entire $\mathbb{R}^2$. We call our method Domain-Varying Green Coordinates (DVGC). When $Θ$ is a disk or a rectangle, the Green' s function possesses analytic or semi-analytic expressions, respectively, enabling the DVGC to be computed without finite element discretization. Furthermore, when $Θ$ is a disk, the DVGC admit a closed-form expression for 2D simplicial cages, thereby eliminating the need for numerical integration. Experiments demonstrate that our method provides a novel control space ranging from more consistent with the cage to more shape-preserving, generating diverse deformation effects by varying the Green' s function domains.
