Geometric numerical methods for computational anatomy

Research Project, 2018 – 2021

Computational anatomy (CA) is a state-of-the-art mathematical framework for deformation based registration of medical images and other shapes, for example of human organs acquired by magnetic resonance imaging (MRI). Registration consists in warping a template image into a target image, thereby obtaining a mapping between anatomical correspondences for meaningful voxel-by-voxel comparisons.

CA is rigorously founded in geometric PDE analysis and infinite-dimensional Riemannian geometry, allowing analysis of robustness and accuracy by well-developed tools. Consequently, CA produces better registration results than heuristically engineered algorithms, especially for heavily non-linear warps.

In this project we combine geometric numerical integration (GNI) with CA to address 3 numerical challenges. (1) Convergence analysis of spectral methods for CA, where powerful results in geometric analysis are used for the first time in a numerical context. (2) New CA algorithms based on finite elements, including adaptivity, in combination with geometric integrators. (3) A new approach to tomographic shape reconstruction, where CA techniques are used for regularization of the inverse problem. The developed algorithms are implemented in a high-quality, open-source software library.

In addition to CA and its accompanying mathematical fields, the project will have a significant impact on the availability of efficient and stable registration algorithms for medical and other applications.