Collaborative Research: Data-Driven Elastic Shape Analysis with Topological Inconsistencies and Partial Matching Constraints

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This research project focuses on data structures that are represented as curves or surfaces. Such structures occur in applications ranging from brain anatomy, computer vision, and molecular biology to meteorological and financial data. Data can be either directly acquired by devices such as laser scans or indirectly reconstructed from microscopy or magnetic resonance imaging. Such images and analyses appear for example in the study of human anatomy and motion or in applications to computer graphics and motion. The project lies in the broad area of statistical shape analysis, in which one tries to quantify geometric and/or topological variability within and across populations. The work aims to develop practical methods to characterize the shape of data objects in order to ascertain their roles in larger systems. The project will involve graduate students and produce open source software. The project relies on the paradigms of elastic shape analysis, which is traditionally concerned with analyzing the variability in the geometries of the objects under consideration. At its core is the notion of a distance between two shapes, which stems from a Riemannian setting using a metric that is invariant to the action of certain shape-preserving transformations and embeds both the global nonlinearity of the space as well as its local linearity. The first goal is to develop a comprehensive theoretical and numerical framework for elastic shape analysis of curves and surfaces that allows for topological inconsistencies and partial matching constraints. This framework combines elastic shape analysis with methods from geometric measure theory and topological data analysis. The second goal is to apply the framework to a wide variety of synthetic and real data, in particular to morphological analysis of high-resolution orthopedic surface complexes.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.