Whats an infinite dimensional manifold and how can it be useful in hospitals

1. What's an infinite dimensional manifold and how can it be useful in hospitals? Department of Mathematics, University of Coimbra, Sept., 2007

2. There has been a huge explosion in medical imaging, in spatial and temporal resolution, in imaging many processes as well as organs New mathematical tools have entered the game, which are not so elementary. In particular, infinite-dimensional manifolds of shapes and infinite-dimensional groups of diffeomorphisms with associated Riemannian metrics have been used. This talk is an elementary step-by-step intro to what this is all about.

3. Outline The ingredients of differential geometry The set of closed plane curves S as an infinite dimensional manifold The 3D version and its application to anatomy 4. Exploratory data analysis and geodesics 5. BUT geodesics are not always act like straight lines: curvature The outlook

4. I. What is a manifold?

5. The 4 ingredients of differential geometry

6. Some history All this (and more to come) is due to Gauss in the 2-D case. Riemann put together the n-D case in his Habilitation Lecture in 1854. This is not so well known, but here he also imagined the infinite dimensional version: “There are however manifolds in which the fixing of position requires not a finite number but either an infinite series or a continuous manifold of determinations of quantity. Such manifolds are constituted for example by … the possible shapes of a figure in space, etc.”

8. Think of S geometrically

9. The geometric heat eqn is the gradient of curve length! is a function on S To form a gradient, we need an inner prod: The simplest inner product of 2 vectors: What makes it work: Move a curve normally proportional to its curvature: this is the geometric heat eqn.

10. The geometric heat equation Starting with one plane curve C0, it generates a family Ct: What does this mean? Describe the curves as polygons:

11. Example of the geometric heat equation In traditional terms, let Ct be y=f(x,t). Then (after some work) it becomes: which is the usual heat equation if the x-deriv of f is small.

12. III. The 3-D version and applications to anatomy

14. What do we want to match up? The body has organs, bones, vessels with boundaries, well defined points (e.g. traditional points on the skull, like the nasion, menton or gonion), muscle fibre and nerve fibre tracts (giving line fields). The diffeomorphism should be constrained to respect these boundaries, points, orientations, maybe even some densities.

15. What is optimal? I The group G of diffeomorphisms is also a manifold and we can put a metric on it. Actually, consider the diffeomorphisms from the domain W of the scan to the domain W’ of the ideal human. Its tangent space is the linear space of vector fields on W and paths are described by: We can consider all such diffeomorphisms which respect all these structures: call this S, and consider the set of all such diffeomorphisms R which are rigid maps.

16. What is optimal? II Consider all paths in G from R to S. Take the inf of their lengths: this measures the distortion. The endpoints of the path of least length is the optimal diffeomorphism. But what is length? Many choices for norm How many derivatives – at least one is needed or metric collapses, Lp for some p. These choises have a huge effect on what optimal means.

19. Heart Mapping via Diffusion Tensor Magnetic Resonance Imaging. The diffeo must respect the fibre structure too.

20. IV. Exploratory data analysis and geodesics

21. What is a geodesic?

22. Geodesics come from differential equations

23. Data analysis via geodesics Given a dataset { Pi } on a manifold M, its Karcher mean is a point Q minimizing Once you have the mean, take the shortest geodesics from each Pi to Q and let be the tangent vector to this geodesic at Q. Then take the principal components via the linear theory on { ti }. k-means can also be done via Karcher means. This approach has been applied, e.g. to the shape of the hippocampus and the diagnosis of schizophrenia and Alzeimer’s; to the shape of the heart in various conditions; to the shape of the prostate; etc.

24. V. BUT geodesics are not always act like straight lines: curvature

25. Gravitational lensing: positive curvature in space-time

26. Curvature in infinite diml spaces

27. Geometry behind curvature Get curvature at each point in each 2-plane: Riemann’s sectional curvature and curvature tensor Rijkl, Ricci and scalar curvatures. When positive, beyond cut locus, geodesics are not unique. Datasets may not have means. When negative, easy to get lost, space is big – but datasets do have means, geodesics are unique.

28. VI. The outlook Curvature is a big obstacle to doing analysis and statistics on non-linear spaces. It affects discrimination and clustering, e.g. by fitting Gaussian models to data. But it reflects the non-linear nature of these big spaces: shapes and diffeos don’t live in vector spaces but have their own geometry. These infinite dimensional spaces have their own geometry which is only partially understood A huge challenge is to define probability measures on S and G: this is essential to e.g. maximum likelihood tests (is this shape a cat or a dog?).