CAREER: Shape Analysis in Submanifold Spaces: New Directions for Theory and Algorithms

Shape analysis has now become an integral component of data science as it is key to modelling and analyzing quantitatively the geometric variability within datasets for applications as diverse as computer vision, speech/motion recognition, morphogenesis or computational anatomy. Among the variety of geometric structures that are studied in this field, curves, surfaces and more generally manifolds are both very natural objects but also particularly challenging to process and analyze due to the non-canonical structure of the corresponding shape spaces. This has in part hindered the development and effectiveness of shape analysis frameworks for such data, if compared for instance to the more widely studied case of images. This project attempts to bridge a few of these important gaps, both on the theoretical and computational side and develop new scalable algorithms for morphological analysis adapted to the growing size and complexity of real datasets. The project will also promote those research topics among students at various levels of the educational system, with the creation of an upper-level undergraduate course on differential and computational geometry, training of PhD students and K-12 outreach activities through the Women in Science and Engineering (WISE) program in particular. Building up on several prior works on shape spaces and metrics, the specific research objectives of this project are (1) to advance the analysis and comparison of relaxed shape matching problems deriving from Riemannian metrics on spaces of manifolds; (2) to investigate supervised and unsupervised deep learning approaches to improve the efficiency of manifold registration algorithms; and (3) to study novel extensions of those models to account for partial or incomplete data and model joint shape/topological variations across shapes. As part of this project's outcome, Python pipelines will be developed and made openly accessible to the scientific community with the long term goal of expanding the potential scope of applications of those methods. 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.