CDS&E: Computational Riemannian Approaches for Statistical Analysis and Modeling of Complex Structures

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Associating structures or morphologies of biological parts to their functionalities in larger, complex biosystems is a grand challenge with widespread applications. Approaches to this challenge require efficient tools for quantifying shape differences in structures of interest, extracting normal modes of variability, and using shape as a predictor in regression studies. This project aims to develop computational techniques for analyzing shapes of two types of biological parts: (1) subcortical structures in human brain, represented as 2D surfaces inside MRI-imaged brain volumes, and (2) neurons, represented as tree-like structures obtained using high-resolution microscopy. While recent research, including the Human Connectome Project and the NeuroMorpho database, has focused on imaging-based generation of large databases of such structures, the techniques for systematically analyzing them lag far behind. This interdisciplinary research aims to develop fundamental solutions for characterizing shapes of biological structures using tools from several different areas. It is anticipated that the techniques under development will also be applicable in broader scientific contexts.Current techniques for comparing morphological properties are either purely topological (generally focusing on part counts while ignoring their shapes, e.g. the tree-edit distance) or purely geometrical (focusing on geometries of curves and surfaces). This project aims to bridge the gap between these two approaches by comparing geometries of objects but allowing certain topological variability. It will also address the most challenging issue of shape analysis -- registration of parts across objects -- using Riemannian methods. These tools will also naturally extend to analysis of trees with temporally-evolving structures. The nature of the framework under development -- automatic registration and comparison of parts across brain surfaces and neurons -- makes this approach both novel and challenging, and the tools needed are interdisciplinary. The project will use elements from differential geometry (especially the geometry of Hilbert manifolds), algebra, computational statistics, and imaging sciences to develop efficient solutions. A significant portion of this project will concern development of real-time, scalable algorithms for analyzing large datasets available for studying subcortical anatomy and neuronal morphology.