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

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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.