Geometry of Shape spaces and related infinite dimensional spaces

Nowadays infinite dimensional manifolds naturally arise in several fields of applied mathematics, such as image analysis, computational anatomy, or hydrodynamics. An example for an infinite dimensional space, that is of particular interest, is the space of all shapes of a certain type, e.g. the space of all curves, surfaces or images. Since this space is inherently nonlinear, the usual methods of linear statistics cannot be applied. For example the addition of two surfaces cannot be meaningfully defined. One way to overcome this difficulty is to introduce a Riemannian structure on the space of shapes, which locally linearizes the space and allows the development of statistical methods that are analogous to the linear case. In the Riemannian setting, the average of two shapes may be defined as the middle point of a geodesic joining these two shapes. In a similar way one may define the corresponding geodesic mean of a collection of n shapes. Still, to do meaningful statistics on these spaces one needs a better understanding of the geometry of the resulting spaces. In my project I want to study Riemannian metrics on various types of shape spaces and of other related infinite dimensional spaces. In particular I would like to investigate the following set of problems: 1. Riemannian metrics on manifolds of immersions: Fractional order Sobolev type metrics, geodesic distance, well - posedness of the geodesic equation, curvature bounds and completeness. 2. Metrics on shape spaces of unparametrized surfaces that are induced by metrics of the same type as in (1): Applications to image analysis and computational anatomy. 3. Shape descriptors: Differential invariants and Integral invariants. 4. (Fractional order) Sobolev type metrics on diffeomorphism groups: Relations of the geodesic equation to several prominent PDE`s, the induced geodesic distance and well-posedness of the geodesic equation. 5. Riemannian metrics on the space of all Riemannian metrics.

Infinite-dimensional spaces naturally appear in various applications of mathematics. Examples include the field of shape analysis, computational anatomy, or even mathematical physics. The spaces that appear in these contexts include the space of planar curves, the space of regular surfaces in 3 , as well as the group of diffeomorphisms.The emphasis of this project was the research of new Riemannian structures on these spaces. The main achievements of the research can be found both in the theoretical fundaments as well as in applications in shape analysis and medical imaging. The results of the project include contributions in the following areas: Characterization of the induced geodesic distance Geodesic and metric completeness Development of numerical algorithms for shape analysis of planar curves Development of numerical algorithms for registration of probability densities with applications in thoracic image registration