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

这个项目开发了有效的增量算法，用于从增量获取的数据池中计算平均值和其他统计量。许多现有的数据采集和处理方法已经达到了一定的复杂程度，以便能够采集和/或合成的数据，驻留在弯曲的空间，如球体，双曲面等，因为这样的数据已经成为无处不在的许多科学和工程领域，需要有效的统计分析这些数据已经成为一个领域的显着重要性。此外，在这个大规模和连续流数据的时代，数据样本随时间顺序地获取。因此，从应用和计算效率的角度来看，所需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，例如从视频中进行面部识别，动作识别，从视频中进行轨迹平均和聚类，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，该项目中开发的方法可以用于自动区分各种疾病类别，如帕金森氏症和特发性震颤，这是不同类型的运动障碍。本研究探讨了递归计算的一般框架的内在平均和主要测地线分析几个常见的流形，如流形的对称正定矩阵，格拉斯曼，Stiefel流形，超球面，特殊正交矩阵的流形，以及其他几个。研究团队将开发的从流形值数据计算统计数据的递归框架应用于几项任务，即医学成像中的扩散MRI的图谱计算，使用扩散MRI的神经退行性疾病亚型之间的类间区分，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.