Symmetry Analysis of 3D Shapes and its Applications in Computer Graphics

三维形状的对称性分析及其在计算机图形学中的应用托马斯·方克豪斯对称性是三维形状的一个基本性质。了解物体的对称性有助于洞察物体的整体形状、分解成几个部分、质量平衡以及运动的可能性。与此同时，对称性在我们的世界中无处不在。几乎所有人造物体都表现出某种完美的对称性，许多有机结构是近乎对称的和/或由近乎对称的部分(例如，动物的身体)组成。然而，几十年来，计算机图形学研究人员在设计几何处理算法时忽略了对称性，而是在操作曲面时专注于局部形状特征和/或差异曲面属性。因此，例如，曲面重建算法为对称对象生成非对称网格，网格压缩技术无法利用近似对称性，网格完成算法通过外推边界附近的曲面属性来填充空洞，而不是从对称部分复制形状特征。显然，分析和利用对象的对称性的方法可以极大地改进这些和其他表面处理应用。该研究正在完成以下工作：(1)研究3D形状的对称性理论，(2)开发用于表征3D形状对称性的分析算法，以及(3)演示对称性分析在几个计算机图形学应用中的应用。预期的研究结果包括对象近似对称性的新的多分辨率描述，以及使用这些描述重新划分网格、压缩、完成、逆向工程和编辑3D网格的新算法。对计算机图形学的直接影响不仅是形状分析的新算法，而且可能还是一种新的思维方式--在编辑和处理表面时理解和保持形状的全局属性(例如，对称性)，而不是只关注局部几何属性。从改进的对称性分析和综合教育活动中受益的其他领域将感受到更广泛的影响。

Symmetry Analysis of 3D Shapes and its Applications in Computer GraphicsThomas FunkhouserSymmetry is a fundamental property of 3D shape. Understanding an object's symmetries provides insight into its overall shape, its decomposition into parts, its balance of mass, and its possibilities for motion. Meanwhile, symmetry is ubiquitous in our world. Almost all man-made objects exhibit some perfect symmetry, and many organic structures are nearly symmetric and/or composed of nearly symmetric parts (e.g., the bodies of animals). For decades, however, computer graphics researchers have ignored symmetries when designing geometric processing algorithms, instead focusing upon local shape features and/or differential surface properties when manipulating surfaces. As a result, for example, surface reconstruction algorithms produce asymetric meshes for symmetric objects, mesh compression techniques fail to take advantage of approximate symmetries, and mesh completion algorithms fill holes by extrapolating surface properties near their boundaries rather than by copying shape features from symmetric parts. Clearly, methods for analyzing and exploiting the symmetries of an object could greatly improve these and other surface processing applications.The research is accomplishing the following: (1) investigating the theory of symmetry for 3D shapes, (2) developing analysis algorithms for characterizing symmetries of 3D shapes, and (3) demonstrating the utility of symmetry analysis in several computer graphics applications. Expected research results include new multiresolution descriptions of the approximate symmetries of an object and new algorithms that use these descriptions for remeshing, compression, completion, reverse engineering, and editing of 3D meshes. The immediate impact on computer graphics is not only new algorithms for shape analysis, but perhaps also a new way of thinking - understanding and preserving global properties of shape (e.g., symmetry) when editing and processing surfaces, rather than focusing only on local geometric properties. The broader impacts will be felt in other fields that benefit from improved symmetry analysis and in integrated educational activities.