Large-Scale Models and Algorithms in Diffeomorphic Shape and Image Registration

Data in various fields including medical imaging, computer vision or physics often have geometric properties that are essential for their understanding, and these types of data are increasing in sample size, volume per sample and modalities. An example of such data sets, and a special focus of this project, are images resulting from new acquisition techniques that render biological tissues at the micron scale while providing high-dimensional information on each cell contained in the tissue. The analysis of geometric data has seen considerable progress in the recent past with, in particular, the mathematical formulation of the notion of shape space and the design of associated computational methods. However, these methods do not scale yet to the massive data size that are associated with the new modalities. This project, which focuses on shape and image data, develops new data science approaches inspired by recent advances over the past decade and develops new models to handle multi-scale representations. Shape analysis provides major elements for the understanding of biological and medical data, and becomes increasingly relevant with the refinement of data acquisition technology. It has been extensively applied in the context of brain diseases or dysfunction, such as schizophrenia, depression, ADHD, autism, Huntington or Alzheimer. This project will extend the range of tools available for such studies, allowing for the analysis of datasets captured at finer resolution, accommodating recently introduced modalities, and enabling statistical investigations working at multiple scales. The project develops new concepts and models for shape analysis through diffeomorphic mapping. It describes three main research themes: introducing new approaches for multiscale analysis, developing randomized optimization strategies for the large deformation diffeomorphic metric mapping algorithm, and investigating registration methods for images that may have discontinuities or singularities. The proposed work involves a combination of theoretical analyses, numerical developments, and experimental exploration. The multiscale models studied in the first theme provide an appealing and previously unexplored paradigm, with notable challenges, especially on the numerical side. Results are expected to significantly impact statistical shape analysis by providing an enriched decomposition of shape changes, and enabling the separation of various effects affecting the data when they occur at different scales. The second theme revisits some of the bases underlying diffeomorphic registration to allow for randomized implementations using stochastic gradient descent. Image varifolds, studied in the third theme, have the ability to model shapes carrying spatial information and enable new data modalities to be handled by registration methods. The project will create algorithms, with numerical representation adapted to these modalities, and explore cross-modality registration. The project will support the education of a graduate student and summer internships for undergraduate students. 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.