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

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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.