A General and Efficient Framework for Computational Shape Analysis Through Geometric Distributions

形状及其可变性的分析已经成为数据科学多个领域中日益重要的问题。在计算机视觉领域，形状识别和分类通常是自动驾驶汽车等机器学习系统的关键组成部分。在自然科学中，计算解剖学的最新发展，即通过数值算法自动分析解剖结构，为理解和诊断各种病理和疾病提供了富有成效的方法。沿着这些不同的科学问题，可用数据的数量和种类从未停止增长。因此，形状本身的概念已经大大扩展，可以指各种类型的几何对象，这构成了重要的挑战，构建和计算所有这些不同形态的形状之间的相关相似性度量。该研究项目的目的是开发一个集成的数学模型和相关的数值管道，允许以灵活有效的方式进行几何结构的形态分析，并探索其在计算解剖学和计算机视觉中的可能应用。它还将包括一个实质性的教育组成部分，培训一名研究生，支持在会议和研讨会上的演讲，并向科学界传播一个开源代码。统计形状分析的主要挑战是对象所属的非标准和不同的数学空间，无论所讨论的形状是原始图像，手动或自动提取的地标，曲线，曲面、矢量场或多模态对象。虽然Grenander提出的开创性模型引入了通过估计最佳变形（通过某个同构群上的度量测量）来比较任何两个形状的想法，但该模型的通用性福尔斯在许多真实的应用中不足，其中需要一定量的残差相异度来考虑其他可变性来源（如噪声）。该项目旨在通过引入一种灵活的方法来量化形状相似性来填补这一空白，该方法依赖于形状空间作为广义分布的统一嵌入，遵循几何测度理论的原则。除了这些表示曲线和曲面配准问题的过去的成功，目标将是在数学和计算层面上展示它如何扩展到更广泛的几何数据结构，并允许跨模态分析，同时将应用范围推到其他问题，如聚类，分类和稀疏表示形状。这个新框架的快速数值方法也是该项目的一个重要方面，其目标是使实现可扩展到当前的数据集维度，例如在医学成像中。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估来支持。

The analysis of shapes and their variability has become an increasingly central problem in multiple areas of data science. In the field of computer vision, shape recognition and classification is often a crucial component of machine learning systems such as self-driving cars. In natural sciences, the recent development of computational anatomy, that is the automatic analysis of anatomical structures by numerical algorithms, provides a fruitful approach in understanding and diagnosing a wide range of pathologies and disorders. Along these different scientific questions, the amount and variety of available data has never ceased to grow. As a result, the concept of shape itself has considerably expanded and may refer to various types of geometric objects, which poses the important challenge of constructing and computing relevant similarity metrics between shapes across all these different modalities. The purpose of this research project is to develop an integrated mathematical model and associated numerical pipeline that allows for morphological analysis of geometric structures in a flexible and efficient way, and explore its possible applications to computational anatomy and computer vision. It will also include a substantial educational component with the training of a graduate student, support for presentations in conferences and workshops, and dissemination of an open-source code to the scientific community.The primal challenge of statistical shape analysis is the rather non-standard and disparate mathematical spaces in which objects belong, whether the shapes in question are raw images, manually or automatically extracted landmarks, curves, surfaces, vector fields or multi-modal objects. While the seminal model proposed by Grenander introduced the idea of comparing any two shapes through the estimation of an optimal deformation (measured by a metric on a certain diffeomorphism group), this model's generality falls short in many real applications where a certain amount of residual dissimilarity is necessary to account for other sources of variability (like noise). This project intends to fill this current gap by introducing a flexible approach to quantify shape similarity which relies on a unified embedding of shape spaces as generalized distributions, following the principles of geometric measure theory. Beyond the past success of these representations for curve and surface registration problems, the objective will be to demonstrate on a mathematical and computational level how it extends to a much wider class of geometric data structures and allows for cross-modality analysis, while pushing the scope of applications to other problems like clustering, classification and sparse representations on shapes. Fast numerical methods for this new framework is also an important aspect of the project, with the objective of making implementations scalable to the current dimensionality of datasets e.g. in medical imaging.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.