A New Paradigm in Joint Registration, Analysis and Modeling of Function Data

函数数据配准和分析的双重问题在统计分析中是非常重要的。它们在实值函数的相幅分离、曲线和曲面的形状分析、2D和3D图像的配准以及具有非线性流形上的值的纵向数据分析等应用中发挥着重要的作用。PI开发了一种新的框架，将导致统计上一致的功能分析，并将大大提高算法性能超过当前的方法。主要的创新是使用黎曼方法和基于距离的目标函数：(1)设计用于明确地测量函数的配准水平，(2)在以配准群为模的函数的商空间中进行研究，以及(3)为随后的统计分析/建模而制定。这里，人们用函数空间上的区域翘曲群的作用来表示配准的可变性，选择适当的黎曼度量(使得群的作用是等距的)，并研究能够在这种度量下进行统计分析的新的数学表示。主要优点是配准和分析，例如，平均值、协方差、主成分分析的计算，都是在相同的度量下联合执行的，而不是当前使用连续和不相交步骤的做法。在实值函数配准和曲线形状分析等子问题上的初步结果表明，在经验和理论上都优于现有的方法。其目标是将这项研究扩大到具有类似基本解决方案的更大类别的注册问题。高维函数数据在当今社会变得无处不在，人们需要在时间和空间上非线性地对齐这些观测，以便改进数据分析、统计建模和推理。该项目是一项多学科的努力，将开发基本的统计科学和功能数据登记的计算工具。这反过来将影响数据丰富的应用程序，如开发儿童准确的生长图表、基因表达分析、使用图像和视频的人脸识别、使用医学图像检测和评估大脑疾病(例如阿尔茨海默氏症)，以及使用监控视频数据识别人类活动。这个项目的新颖性和潜在的高回报来自于所使用的各种工具-从微分几何和统计学到成像科学。

The dual problems of registration and analysis of functional data are very important in statistical analysis. They play important roles in applications involving phase-amplitude separation of real-valued functions, shape analysis of curves and surfaces, registration of 2D and 3D images, and analysis of longitudinal data with values on nonlinear manifolds. The PIs develop a novel framework that will lead to statistically consistent functional analysis and will substantially improve algorithmic performances over the current methods. The key novelty is to use Riemannian methods and distance-based objective functions that are: (1) designed for measuring registration levels of functions explicitly, (2) studied in the quotient spaces of functions modulo the registration groups, and (3) formulated for ensuing statistical analysis/modeling. Here one represents registration variability by actions of the domain-warping groups on function spaces, chooses an appropriate Riemannian metric (such that the group actions are by isometries) and studies novel mathematical representations that enable statistical analysis under such metrics. The main advantage is that both registration and analysis, e.g, computation of mean, covariance, PCA, are performed jointly under the same metric rather than the current practice of using sequential and disjoint steps. Preliminary results on some subproblems including real-valued function registration and shape analysis of curves are shown to be superior, both empirically and theoretically, to the current approaches. The goal is to broaden this research to a larger class of registration problems with similar fundamental solutions. High-dimensional functional data is becoming omnipresent in today's society and one needs to nonlinearly align such observations in time and space, in order to improve data analysis, statistical modeling, and inferences. This project represents a multi-disciplinary effort that will develop both basic statistical science and computational tools for registration of functional data. This, in turn, will impact such data-rich applications as development of accurate growth charts for children, gene expression analysis, face recognition using images and videos, detection and evaluation of brain disorders (e.g. Alzheimer) using medical images, and human activity recognition using surveillance video data. The novelty and potential high returns of this project come from the variety of tools utilized---from differential geometry and statistics to imaging science.