RI: Small: Efficient Statistical Computing on Riemannian Manifolds with Applications to Medical Imaging and Computer Vision

这个项目开发了高效的增量算法，用于从增量获取的数据池中计算平均值和其他感兴趣的统计量。许多现有的数据采集和处理方法已经达到了复杂的水平，以便能够采集和/或合成位于球面、双曲面等曲面空间中的数据。随着这些数据在许多科学和工程领域变得无处不在，对这些数据进行有效的统计分析的需求已经成为一个非常重要的领域。此外，在这个海量和连续流数据的时代，数据样本是随着时间的推移而顺序获取的。因此，从应用程序和计算效率的角度来看，期望的平均算法应该服从增量更新，以适应随着时间的推移新获得的数据。所开发的算法可以应用于不同的应用，如视频中的人脸识别、动作识别、视频中的轨迹平均和聚类、图像和视频恢复、模式聚类和分类等。在诊断医学成像的背景下，本项目开发的方法可以用于自动区分不同类型的运动障碍的各种疾病类别，如帕金森和特发性震颤。本研究研究了几种常见流形如对称正定矩阵流形、Grassmann流形、Stiefel流形、超球面流形、特殊正交矩阵流形等递归计算本征均值和主测地线分析的一般框架。研究小组将开发的从多值数据计算统计的递归框架应用于几项任务，即医学成像中的扩散MRI计算图谱，使用扩散MRI对神经退行性疾病的子类型进行类间区分，面部和动作识别，计算机视觉应用中的图像和形状检索。

This project develops efficient incremental algorithms are proposed for computing averages and other statistical quantities of interest from pools of data incrementally acquired. Many existing data acquisition and processing methods have reached a level of sophistication so as to be able to acquire and/or synthesize data that reside in curved spaces such as spheres, hyperboloids etc. As such data have become ubiquitous in many Science and Engineering fields, need for efficient statistical analysis of these data has emerged as an area of significant importance. Further, in this era of massive and continuous streaming data, samples of data are acquired sequentially over time. Hence, from an applications and computational efficiency perspective, the desired averaging algorithm ought to be amenable to incremental updates to accommodate the newly acquired data over time. The developed algorithms can be applied different applications, such as face recognition from videos, action recognition, trajectory averaging and clustering from videos, image and video restoration, pattern clustering and classification, etc. In the context of diagnostic medical imaging, methods developed in this project can be used to automatically discriminate between various disease classes, such as Parkinson's and Essential Tremor which are distinct types of movement disorders.This research investigates a general framework for recursive computation of the intrinsic mean and the principal geodesic analysis on several commonly encountered manifolds such as the manifold of symmetric positive definite matrices, the Grassmann, the Stiefel manifolds, the hypersphere, the manifold of special orthogonal matrices, and several others. The research team applies the developed recursive framework of computing statistics from manifold-valued data to several tasks namely, atlas computation from diffusion MRI in Medical Imaging, inter-class discrimination between sub-types of a neuro-degenerative disorder using diffusion MRI, face and action recognition, image and shape retrieval in Computer Vision applications.