CAREER: Role of geometry in dynamical modeling of human movement: Applications to activity quality assessment across Euclidean, non-Euclidean, and function spaces

人体运动是由各种肌肉和关节之间复杂的非线性相互作用以及与外力和物体的相互作用而产生的复杂的多维动态过程。这些物理因素通常对从外部传感器获得的运动数据施加一定的分析限制，通常带有相关的几何表示。这种表征的例子在人类活动分析领域比比皆是，其中一些当代和新兴的人类形状和运动表征，如轮廓、简笔画序列、方向信号和光流，具有复杂的信号空间，通常用黎曼几何语言描述。研究重点集中在提取有意义的动态属性的基本方法上，这些方法将与应用推力相吻合。应用的中心是低保真感测基础设施的身体活动质量的计算建模，这是技术介导的身体康复和预防干预的重要组成部分。经典的向量值信号处理和机器学习在理解和模拟人类活动方面产生了很大的影响，但是当涉及到非欧几里得几何的信号空间时，它们的适用性受到很大的限制。该研究项目旨在推进一类新的鲁棒非参数动态建模方法，该方法将从经典的向量值观测空间扩展到有限维黎曼流形，以及无限维函数空间。为了足够普遍地解释这里提到的各种几何空间，需要一个没有对动力学参数形式的限制性假设的框架。此外，将动态分析与黎曼几何理论相结合，可以将分析扩展到多个特征空间，包括深度图、形状序列、简写图和方向数据，而无需重新定义活动分析的基本模型。

Human movement is a complex multi-dimensional dynamical process arising out of complex non-linear interactions between various muscles and joints as well as interactions with external forces and objects. These physical factors often impose certain analytical constraints on movement data obtained from external sensors, often with associated geometric representations. Examples of such representations abound in the area of human activity analysis, where several contemporary and emerging human shape and movement representations such as silhouettes, stick-figure sequences, orientation signals, and optical- flow have intricate signal spaces, often described in the language of Riemannian geometry. The research thrusts focus on foundational methods for extracting meaningful dynamical attributes that will dovetail into the application thrust. The application centers on computational modeling of qualities of physical activities from low-fidelity sensing infrastructure, which is an important component for technology-mediated physical rehabilitation and preventive interventions. Classical vector-valued signal processing and machine learning has had a large impact in the area of understanding and modeling human activities, but their applicability is significantly limited when it comes to signal-spaces with non-Euclidean geometry. This research project aims to advance a new class of robust, non-parametric dynamical modeling approaches, which will extend from classical vector-valued observation spaces, to finite-dimensional Riemannian manifolds, as well as to infinite-dimensional function-spaces. In order to be general enough to account for various geometric spaces as mentioned here, a framework that is free of restrictive assumptions on parametric forms of dynamics is needed. Further, integrating dynamical analysis with Riemannian geometric theory allows the analysis to extend to several feature spaces including depth maps, shape sequences, stick-figures, and orientation data, without re-defining the fundamental models for activity analysis.