A General and Efficient Framework for Computational Shape Analysis Through Geometric Distributions

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The analysis of shapes and their variability has become an increasingly central problem in multiple areas of data science. In the field of computer vision, shape recognition and classification is often a crucial component of machine learning systems such as self-driving cars. In natural sciences, the recent development of computational anatomy, that is the automatic analysis of anatomical structures by numerical algorithms, provides a fruitful approach in understanding and diagnosing a wide range of pathologies and disorders. Along these different scientific questions, the amount and variety of available data has never ceased to grow. As a result, the concept of shape itself has considerably expanded and may refer to various types of geometric objects, which poses the important challenge of constructing and computing relevant similarity metrics between shapes across all these different modalities. The purpose of this research project is to develop an integrated mathematical model and associated numerical pipeline that allows for morphological analysis of geometric structures in a flexible and efficient way, and explore its possible applications to computational anatomy and computer vision. It will also include a substantial educational component with the training of a graduate student, support for presentations in conferences and workshops, and dissemination of an open-source code to the scientific community.The primal challenge of statistical shape analysis is the rather non-standard and disparate mathematical spaces in which objects belong, whether the shapes in question are raw images, manually or automatically extracted landmarks, curves, surfaces, vector fields or multi-modal objects. While the seminal model proposed by Grenander introduced the idea of comparing any two shapes through the estimation of an optimal deformation (measured by a metric on a certain diffeomorphism group), this model's generality falls short in many real applications where a certain amount of residual dissimilarity is necessary to account for other sources of variability (like noise). This project intends to fill this current gap by introducing a flexible approach to quantify shape similarity which relies on a unified embedding of shape spaces as generalized distributions, following the principles of geometric measure theory. Beyond the past success of these representations for curve and surface registration problems, the objective will be to demonstrate on a mathematical and computational level how it extends to a much wider class of geometric data structures and allows for cross-modality analysis, while pushing the scope of applications to other problems like clustering, classification and sparse representations on shapes. Fast numerical methods for this new framework is also an important aspect of the project, with the objective of making implementations scalable to the current dimensionality of datasets e.g. in medical imaging.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.