Research
It was such a relief to be right, even though you knew you'd only got there by trying every possible way to be wrong.
—Terry Pratchett, Feet of Clay
- Discretization of Sobolev metrics on the space of curves
- Information geometry
- Riemannian geometry of diffeomorphism groups
- Sobolev metrics on the space of curves
- Shape analysis
- Geometry of image registration
- Vanishing geodesic distance
Discretization of Sobolev Metrics on the Space of Curves
Defining a Riemannian metric on the space of curves is a first step in shape analysis. In order for the metric to be useful however one needs to discretize this infinite-dimensional space. For certain first order Sobolev metrics there exist special transformations – the square root velocity transform for example – which greatly simplify the numerical treatment. We wanted to explore the behaviour of second order metrics in order to provide an alternative choice of metrics.
Together with Martin Bauer, Philipp Harms and Jakob Møller-Andersen we are indeveloping fast and stable disretizations of second order Sobolev metrics.
Publication
- M. Bauer, M. Bruveris, P. Harms, J. Møller-Andersen. A numerical framework for Sobolev metrics on the space of curves. Accepted for publication in SIAM J. Imaging Sci., 2016.
- M. Bauer, M. Bruveris, P. Harms, J. Møller-Andersen Second order elastic metrics on the shape space of curves. 1st Workshp on Differential Geometry in Computer Vision for Analysis of Shapes, Images and Trajectories, 2015.
- M. Bauer, M. Bruveris, P. Harms, J. Møller-Andersen Curve matching with applications in medical imaging. 5th MICCAI workshop on Mathematical Foundations of Computational Anatomy, 2015.
Code
Our code is written in Matlab and was published under a GPL licence here.
Information Geometry
Information geometry is a branch of mathematics that studies manifolds of probability distributions. For example all Gaussian distributions having mean \( \mu \) and standard deviation \( \sigma \) form a two-dimensional manifold. Instead of finite-dimensional families of probability densities, one can also consider the space of all probability densities. Under certain regularity assumptions the space of all probabilty densities becomes an infinite-dimensional manifold.
The Fisher–Rao metric is a Riemannian metric on the space of probability densities. When restricted to finite-dimensional submanifolds, it is called the Fisher information metric. An important property is that the Fisher–Rao metric is invariant under different representations of the underlying space. Together with M. Bauer and P. W. Michor we have shown that under suitable assumptions the Fisher–Rao metric is the unique Riemannian metric with this property.
Publication
- M. Bruveris, P. W. Michor. Geometry of the Fisher-Rao metric on the space of smooth densities on a compact manifold, 2016.
- M. Bauer, M. Bruveris, P. W. Michor. Uniqueness of the Fisher-Rao metric on the space of smooth densities. B. Lond. Math. Soc. , 48(3), 499-506, 2016.
Video
This work was presented at the conference GSI 2015 in October 2015 at the École Polytechnique. A video recording of the talk as well as slides are available.
Riemannian Geometry of Diffeomorphism Groups
Interest in the Riemannian geometry of diffeomorphism groups originates in Arnold's discovery of its connection to hydrodynamics: solutions of Euler's equation, which models the motion of an incompressible fluid correspond to geodesics on the group of volume preserving diffeomorphisms with respect to a suitable right-invariant Riemannian metric. Since then other Riemannian metrics on the diffeomorphism group and its subgroups have also been connected to PDEs arising in mathematical physics.
For finite-dimensional manifolds questions of completeness are settled by the Hopf–Rinow theorem. It states that on a Riemannian manifold metric completeness is equivalent to geodesic completeness and both imply the existence of minimal geodesics. Due to the lack of local compactness this theorem does not hold in infinite dimensions. With F. X. Vialard we studied completeneness properties of Sobolev type metrics on the diffeomorphism group. We showed that smooth right-invariant metrics of sufficiently high Sobolev order satisfy an infinite-dimensional version of the Hopf–Rinow theorem.
Another example of a PDE that is a geodesic equation is the Hunter–Saxton equation, \[ u_{tx} = -u u_{xx} - \frac 12 u_x^2\,. \] The geometry underlying the Hunter‐Saxton equation is related to the group \( \operatorname{Diff}_c(\mathbb R) \) of diffeomorphisms on the real line. With M. Bauer and P. W. Michor we studied the Riemannian geometry on this group and finite-dimensional extensions of it and we found the – to our knowledge – first example of a weak Riemannian manifold, whose Levi-Civita connection does not exist.
Publications
- M. Bruveris. Regularity of maps between Sobolev spaces, 2016. arXiv:1602.06558
- M. Bruveris, F.-X. Vialard. On completeness of groups of diffeomorphisms, 2015. To appear in the Journal of the EMS.
- M. Bauer, M. Bruveris, P. W. Michor. The homogeneous Sobolev metric of order one on diffeomorphism groups on the real line. J. Nonlinear Sci, 24(5), 769-808, 2014.
Sobolev Metrics on the Space of Curves
Shape analysis leads naturally to an investigation of Riemannian metrics on the space of curves. The arguably simplest metric on the space of embedded curves is the \( L^2 \)-metric \[ G_c(u,v) = \int_{S^1} \langle u, v \rangle |c'| \,\mathrm{d}\theta\,.\] Because the integration is performed with respect to arc length \( |c'| \,\mathrm{d} \theta \), the resulting metric is invariant under reparametrisations of the curve and it naturally induces a Riemannian metric on the quotient space \( \operatorname{Emb}(S^1, \mathbb R^2) / \operatorname{Diff}(S^1) \). Understanding the latter space is the real aim of shape analysis, the space \( \operatorname{Emb}(S^1,\mathbb R^2) \) is mostly regarded as a convenient approach to it.
The \( L^2 \)-metric is not suitable for applications to shape analysis, mainly because the geodesic distance induced by this metric on the space of embeddings is identically zero. This means that any two embedded curves can be joined by a path of arbitrary small \( L^2 \)-length.
To address this problem several approaches have been proposed. I have been particulary interested in Riemannian metrics of Sobolev type. These are metrics of the form \[ G_c(u,v) = \int_{S^1} \langle u,v \rangle + \langle D_s u, D_s v \rangle + \dots \langle D_s^n u, D_s^n v \rangle \,\mathrm{d}s\,, \] that is they involve derivatives of the tangent vectors. I have been studying the mathematical properties of these metrics, in particular how the Hopf–Rinow theorem generalised from finite dimensions to this space.
Publications
- M. Bruveris. Optimal reparametrizations in the square root velocity framework. 2015. arXiv:1507.02728
- M. Bruveris. Completeness properties of Sobolev metrics on the space of curves. J. Geom. Mech, 7(2), 125-150, 2015.
- M. Bruveris, P. W. Michor, D. Mumford. Geodesic completeness for Sobolev metrics on the space of immersed plane curves. Forum Math. Sigma, 2, e19, 38 pages, 2014.
- M. Bauer, M. Bruveris, S. Marsland and P. W. Michor. Constructing reparametrization invariant metrics on spaces of plane curves. Differ. Geom. Appl., 34C, 139-165, 2014.
- M. Bauer, M. Bruveris, P. W. Michor. \(R\)-transforms for \(H^2\)-metrics on spaces of plane curves. Geom. Imaging Comput., 1(1), 1-56, 2014.
Shape Analysis
How does one measure similarity of shapes and why would one want to do it? The term "shape" has many meanings; it can, for example, denote the information contained in the geometric form of an object. Shape analysis deals with questions, like how to define and measure similarity between shapes; how to sort a collection of shapes into groups; how to encode the information contained in the collection to make predictions about future shapes?
The answer to these questions depends on the mathematical representation one chooses for a shape. A natural representation of planar shapes is as the boundary of a simply connected region or equivalently as the image of an embedded curve. The collection of all shapes is the shape space.
Mathematically shape space is the quotient \( \operatorname{Emb}(S^1, \mathbb R^2) / \operatorname{Diff}(S^1) \) of embedded curves modulo diffeomorphisms. This space is an infinite-dimensional manifold and one can consider distances that arise as geodesic distances from Riemannian metrics. This approach provides more than just a metric structure: the exponential map is a preferred local linearization of the space, minimal geodesics are the "straigth lines" connecting two shapes and the curvature tensor measures how nonlinear the space is. Because shape space is infinite-dimensional, studying any of these objects requires a combination of differential geometry and functional analysis and leads to interesting mathematics. The figure shows examples of geodesics between shapes. An overview of shape analysis with an emphasis on the Riemannian point of view can be found in the papers below.
Publications
- M. Bauer, M. Bruveris, P. W. Michor. Why use Sobolev metrics on the space of curves. Riemannian Computing in Computer Vision, Springer-Verlag, 233–255, 2016.
- M. Bauer, M. Bruveris, P. W. Michor. Overview of the geometries of shape spaces and diffeomorphism groups. J. Math. Imaging Vis., 50(1-2), 60–97, 2014.
Geometry of Image Registration
The aim of computational anatomy is to study the relationship between anatomical shape and physiological function. In order to compare anatomical shapes, it is first necessary to find point-to-point correspondences between them. This is the problem of pair-wise registration: given two objects, e.g. volumetric grey-scale images representing MRI scans of the brain, to find a diffeomorphism deforming one object into the other. The most geometric registration method is the large deformation diffeomorphic metric matching (LDDMM) framework: the paths of deformations in LDDMM are geodesics on the diffeomorphism group for certain right-invariant Riemannian metrics.
With F. X. Vialard we also showed that the diffeomorphism groups used in the mathematical formulation of LDDMM coincide under certain conditions with the classical groups of Sobolev diffeomorphisms, used, e.g., in the work of Ebin and Marsden. This provides among other things a rigorous differential geometric formulation of the LDDMM framework.
Publications
- M. Bruveris, D. D. Holm. Geometry of image registration: the diffeomorphism group and momentum maps. Geometry, Mechanics, and Dynamics. Fields Insitute Communications Volume 73, 19-56, 2015.
- M. Bruveris, L. Risser and F.-X. Vialard. Mixture of kernels and iterated semi-direct product of diffeomorphism groups. Multiscale Model. Simul., 10(4), 1344-1368, 2012.
- M. Bruveris, F. Gay-Balmaz., D. D. Holm and T. S. Ratiu. The momentum map representation of images. J. Nonlinear Sci, 21(1), 115-150, 2011.
Vanishing Geodesic Distance
On a finite-dimensional Riemannian manifold \((M, g)\) the induced geodesic distance function \(\operatorname{dist}\) is point-separating, i.e. if \( x \neq y \), then \(\operatorname{dist}(x,y) > 0\); even more, the topology induced by the geodesic distance coincides with the manifold topology. This does not necessarily hold for infinite-dimensional Riemannian manifolds.
Together with M. Bauer, P. Harms and P. W. Michor we studied the behaviour of the geodesic distance on the group \( \operatorname{Diff}_c(M) \) of compactly supported diffeomorphisms of a manifold \( M \) and the Virasoro–Bott group; the Riemannian metrics we considered are right-invariant metrics of Sobolev type.
We obtain a complete classification for the group \( \operatorname{Diff}(S^1) \): the geodesic distance vanishes identically for metrics of order \( \frac 12 \) and lower and is point-separating for order larger than \( \frac 12 \). For other diffeomorphism groups we obtain partial results.
Publications
- M. Bauer, M. Bruveris, P. W. Michor. Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group. II. Ann. Glob. Anal. Geom., 44(4), 361–368, 2013.
- M. Bauer, M. Bruveris, P. Harms and P. W. Michor. Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group. Ann. Glob. Anal. Geom., 44(1), 5–21, 2013.
- M. Bruveris. The energy functional on the Virasoro-Bott group with the \(L^2\)-metric has no local minima. Ann. Glob. Anal. Geom., 43(4), 385–395, 2012.
- M. Bauer, M. Bruveris, P. Harms and P. W. Michor. Vanishing geodesic distance for the Riemannian metric with geodesic equation the KdV-equation. Ann. Glob. Anal. Geom., 41(4), 461–472, 2012.