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，一个重要的研究机构一直围绕着中轴线，这是Blum在1967年提出的一种经典的几何结构。中轴不仅提供了一个低维的表示，捕捉物体的形状和拓扑结构，它还创建了衍生的形状描述符，测量有用的形状属性，如厚度。 这一涉及两个伙伴机构的合作项目以这一研究为基础，并在两个重要和相关的方向上加以扩展。 首先，PI将概括中轴的定义，以系统地定义一系列连续较低维度的中轴表示（例如，3D对象的中间曲线和中间点），它们都继承了中间轴的基本属性。这些表示，称为中间形式，在各种应用中非常有用，例如形状匹配和分解。 虽然计算低维骨架结构的算法是丰富的，健全的数学定义尚未开发。 其次，基于内侧形式，PI将引入一组形状描述符，可以表征形状的非均匀（或各向异性）扩展。 这些描述符将不仅能够区分板状和管状部件，而且能够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，现有的描述符都无法表征3D形状的这一重要方面。 该PI将建立在他们最近的研究二维对象开发的数学基础和计算算法的三维对象的中间形式，并将探索他们的使用分析形状各向异性通过几个衍生形状描述符。 他们将正式定义一个三维物体的中间形式，并将检查其属性，包括其维度，拓扑结构和边界扰动的敏感性。 基于这些定义，PI将开发有效的算法，这些算法可以在给定对象的离散样本的情况下创建可证明的良好近似的中间形式。 PI将探索几何骨架或表面签名函数形式的各种衍生各向异性感知描述符如何补充形状建模应用中的现有描述符（例如，更广泛的影响：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对各种应用领域中的形状进行更准确和有效的分析，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。 学生，无论是在本科和研究生水平，将积极招募，强调多样性，此外，项目成果将用于在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.