CGV: Small: Collaborative Research: Theories, algorithms, and applications of medial forms for shape analysis

形状分析是计算机图形学和视觉中的一个基本问题。在过去的几十年里，大量的研究都集中在中轴上，这是Blum在1967年引入的一种经典的几何结构。中轴不仅提供了捕捉对象形状和拓扑的低维表示，而且还创建了用于测量有用形状属性(例如厚度)的派生形状描述符。这一涉及两个伙伴机构的合作项目建立在这一研究机构的基础上，并将其扩展到两个重要和相关的方向。首先，PI将概括中轴的定义，以系统地定义连续较低维度的中轴表示序列(例如，3D对象的中轴曲线和中点)，它们都继承了中轴的基本属性。这些表示被称为中间形式，在形状匹配和分解等各种应用中很有用。虽然计算低维骨架结构的算法很丰富，但还没有开发出合理的数学定义。其次，基于中间形式，PI将引入一组形状描述符来表征形状的非均匀(或各向异性)扩展。这些描述符不仅能够区分板状和管状零件，而且还可以测量板状拉伸和管状伸长的量。形状各向异性在3D对象中很常见，对于从数字模型中获取知识非常重要，特别是在生物研究中。然而，现有的描述符都不能描述3D形状的这一重要方面。PI将建立在他们最近对2D对象的研究的基础上，以开发3D对象中间形式的数学基础和计算算法，并将通过几个派生形状描述符来探索它们在分析形状各向异性中的应用。他们将正式定义3D对象的中间形式，并将检查它们的属性，包括它们的维度、拓扑和对边界扰动的敏感性。根据这些定义，PI将开发有效的算法，在给定对象的离散样本的情况下，创建可证明良好的中间形式的近似。PI将探索各种几何骨架或表面特征函数形式的衍生各向异性感知描述符如何在形状建模应用(例如，分割和匹配)和生物形状分析中补充现有的描述符。广泛的影响：新的数学公式和形状描述符将对计算机图形学的理论和应用方面做出重大贡献。表征形状各向异性的能力将直接转化为在各种应用领域中对形状的更准确和更有效的分析，PI将特别探索它们在理解生物形状方面的用途。该团队将开发和分发实现新算法的在线软件。将积极招募本科生和研究生，强调多样性，此外，项目成果将用于当地初中和高中的推广计划。

Shape analysis is a fundamental problem in computer graphics and vision. A significant body of research over the past few decades has centered around the medial axis, a classical geometric structure introduced by Blum in 1967. Not only does the medial axis provide a low-dimensional representation that captures both the shape and topology of the object, it also creates derivative shape descriptors that measure useful shape properties, such as thickness. This collaborative project that involves two partner institutions builds on that body of research and extends it in two important and related directions. First, the PIs will generalize the definition of medial axis to systemically define a sequence of medial representations of successively lower dimensionality (e.g., medial curve and medial point of a 3D object) that all inherit the essential properties of the medial axis. These representations, called medial forms, are useful in various applications such as shape matching and decomposition. While algorithms for computing low-dimensional skeletal structures are abundant, sound mathematical definitions have not yet been developed. Second, based on the medial forms, the PIs will introduce a set of shape descriptors that can characterize non-uniform (or anisotropic) expansion of shape. These descriptors will not only be able to differentiate plate-like and tubular parts, but also to measure the amount of plate-like stretching and tube-like elongation. Shape anisotropy is common in 3D objects, and is important for deriving knowledge from digital models, particularly in biological research. However, none of the existing descriptors can characterize this important aspect of 3D shape. The PIs will build on their recent study of 2D objects to develop the mathematical foundation and computational algorithms of medial forms of 3D objects, and will explore their use in analyzing shape anisotropy through several derivative shape descriptors. They will formalize the definitions of medial forms of a 3D object, and will examine their properties including their dimensionality, topology, and sensitivity to boundary perturbations. Based on the definitions, the PIs will develop efficient algorithms that create provably good approximations of the medial forms given discrete samples of an object. The PIs will explore how a variety of derivative anisotropy-aware descriptors, in the forms of either geometric skeletons or surface signature functions, complement existing descriptors in shape modeling applications (e.g., segmentation and matching) and in biological shape analysis.Broader Impacts: The new mathematical formulations and shape descriptors will contribute significantly to both the theoretical and applied side of computer graphics. The ability to characterize shape anisotropy will directly translate to more accurate and efficient analysis of shapes in various application domains, and the PIs will particularly explore their use in understanding biological shapes. The team will develop and distributable online software that implements the new algorithms. Students, both at the undergraduate and graduate level, will be actively recruited with an emphasis on diversity, and in addition project outcomes will be used in outreach programs at local middle- and high-schools.