CDS&E: Geometrical Regression Models Involving Complex Shape Variables

量化物体形状的需求出现在许多科学工作中，在解剖学、生物学、物理学和计算机视觉中都有突出的例子。这些物体可以是解剖部分、生物细胞、道路网络、面部表面或恐龙骨骼。在统计形状分析中，人们使用数学表示法来测量和分析受试者群体内和受试者之间形状的统计可变性。此外，我们还研究了形状与其他相关变量之间的相互作用。例如，在医学成像中，人们利用肿瘤的形状来诊断和治疗疾病，或者研究衰老对细胞结构形状的影响，以开发合适的药物。这样的研究被广泛地称为形状回归，其中一个人形成统计模型来分析形状与其他感兴趣变量的交互作用。形状既可以用来预测，也可以用来回应，这取决于问题的背景。迫切需要开发正式的统计工具，特别是回归模型，用于分析许多学科中的形状数据。虽然近年来在形状表示的黎曼方法方面取得了巨大的进步，但形状回归的统计模型的发展相对有限。最大的两个挑战是形状表示的非欧几里得性质和给定对象数据中缺乏配准。过去的方法一方面限于使用预先登记的数据和全局线性近似的统计模型，另一方面限于缺乏可解释解决方案的机器学习解决方案。拟议的研究将提供详细的可解释的解决方案，能够正式测试形状和其他变量之间的关系，即使在数据稀疏的情况下也是如此。主要的创新是：(1)使用形状流形的局部线性近似来减少失真；(2)在回归模型中加入对讨厌的(与配准相关的)变换的优化而不是作为预处理。该项目将开发基本的估计理论和有效的计算解决方案，以在科学学科中实施这些方法。这些模型还将服从于实时、可扩展算法的开发，用于分析大数据集，用于估计、预测和测试涉及形状变量的模型。该项目汇集了计算黎曼几何、统计方法和科学应用等不同领域的广泛专业知识，以取得新的进展。这一研究方向虽然具有明显的挑战性，但代表了一个新的视角和一个开发新统计数据并为重大科学进步做出贡献的绝佳机会。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The need for quantifying shapes of objects arises in many scientific endeavors, with prominent examples in anatomy, biology, physics, and computer vision. These objects can be anatomical parts, biological cells, road networks, facial surfaces, or dinosaur bones. In statistical shape analysis, one uses mathematical representations to measure and analyze statistical variability of shapes within and across subject populations. Furthermore, one studies interactions of shapes with other related variables of interest. For examples, in medical imaging one uses shapes of tumors to diagnose and treat diseases or one studies the effects of aging on shapes of cellular structures to develop appropriate drugs. Such studies are broadly termed shape regression, where one forms statistical models for analyzing interactions of shapes with other variables of interests. Shape can either be used a predictors or responses depending upon the problem context. There is an urgent need to develop formal statistical tools, especially regression models, for analyzing shape data in many disciplines. While recent years have seen tremendous progress in Riemannian approaches to shape representations, the development of statistical models for shape regressions has been relatively limited. The two biggest challenges are non-Euclidean nature of shape representations and lack of registrations in given object data. Past approaches are restricted to statistical models that use pre-registered data and globally linear approximations on one hand, and machine learning solutions that lack interpretable solutions on the other. The proposed research will provide detailed interpretable solutions with ability to formally test relationships between shapes and other variables, even when data is sparse. The key innovations are: (1) use of locally linear approximations of shape manifolds to reduce distortions, and (2) incorporate optimizations over nuisance (registration-related) transformations inside regression models rather than as pre-processing. The project will develop underlying estimation theory and efficient computational solutions for implementing these methods across scientific disciplines. These models will also be amenable to development of real-time, scalable algorithms for analyzing large datasets, for estimation, prediction and testing of models involving shape variables. This project brings together a broad expertise from diverse areas such as computational Riemannian geometry, statistical methodology, and scientific applications, to make new inroads. This research direction, while clearly challenging, represents a fresh perspective and a great opportunity to develop new statistics and to contribute in major scientific advances.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.