Klas Modin: Computational Anatomy and Shape Analysis: From Fields Medalists to Medical Doctors

Computational anatomy (CA), also called Shape Analysis, is an interdisciplinary area combining mathematical imaging, differential geometry, statistics, theoretical physics, numerical analysis and other branches of mathematics. CA is an emerging field of mathematics that harbours a wealth of challenging problems concerning both theoretical developments and practical applications; it engages researchers ranging from Fields Medalists to medical doctors.

The mathematical foundation of CA is a class of partial differential equations (PDEs) called Euler-Arnold equations. They describe geodesic curves on the infinite-dimensional manifold of diffeomorphisms of an underlying finite-dimensional configuration manifold. An example is the remarkable discovery by Vladimir Arnold that Euler's equations for a perfect incompressible fluid can be cast as a geodesic equation on the infinite-dimensional group of volume preserving diffeomorphisms. The objective of CA is to find a geodesic path on the full group of diffeomorphisms that warps a template shape into a target shape. Here, 'shape' refers to any object on which the diffeomorphism group acts, for example images, probability densities, Riemannian metrics, or embedded surfaces.

In this talk I will outline the framework of CA. I will discuss connections to fluid mechanics, optimal transport, and information geometry. A new efficient numerical method is based on the Fisher-Rao metric and the framework of 'optimal information transport'.