An Example of Banach and Hilbert manifold: the universal Teichm\"uller space

For $s >\frac{3}{2}$, the group of Sobolev class s diffeomorphisms of the circle is a smooth manifold modeled on the space of Sobolev class s sections of the tangent bundle of the circle. It is a topological group in the sense that multiplication given by the composition of applications is well-defined and continuous, the inverse is continuous, left translation is continuous and right translation is smooth. These results are consequences of the Sobolev Lemma. For the same reasons, the subgroup of Sobolev class s diffeomorphisms of the circle preserving three points, is, for $s >\frac{3}{2}$ a smooth manifold and a topological group modeled on the space of Sobolev class s vector fields vanishing at these three points. One may ask what happens for the critical value $\frac{3}{2}$ and look for a group with some regularity and a manifold structure such that the tangent space at the identity is isomorphic to the space Sobolev class $\frac{3}{2}$ vector fields vanishing at three given points. The universal Teichm\"uller space verify these conditions

Banach Poisson-Lie groups and Bruhat-Poisson structure of the restricted Grassmannian

The first part of this paper is devoted to the theory of Poisson-Lie groups in the Banach setting. Our starting point is the straightforward adaptation of the notion of Manin triples to the Banach context. The difference with the finite-dimensional case lies in the fact that a duality pairing between two non-reflexive Banach spaces is necessary weak (as opposed to a strong pairing where one Banach space can be identified with the dual space of the other). The notion of generalized Banach Poisson manifolds introduced in this paper is compatible with weak duality pairings between the tangent space and a subspace of the dual. We investigate related notion like Banach Lie bialgebras and Banach Poisson-Lie groups, suitably generalized to the non-reflexive Banach context. The second part of the paper is devoted to the treatment of particular examples of Banach Poisson-Lie groups related to the restricted Grassmannian and the KdV hierarchy. More precisely, we construct a Banach Poisson-Lie group structure on the unitary restricted Banach Lie group which acts transitively on the restricted Grassmannian. A "dual" Banach Lie group consisting of (a class of) upper triangular bounded operators admits also a Banach Poisson-Lie group structure of the same kind. We show that the restricted Grassmannian inherits a generalized Banach Poisson structure from the unitary Banach Lie group, called Bruhat-Poisson structure. Moreover the action of the triangular Banach Poisson-Lie group on it is a Poisson map. This action generates the KdV hierarchy, and its orbits are the Schubert cells of the restricted Grassmannian

On canonical parameterizations of 2D-shapes

This paper is devoted to the study of unparameterized simple curves in the plane. We propose diverse canonical parameterizations of a 2D-curve. For instance, the arc-length parameterization is canonical, but we consider other natural parameterizations like the parameterization proportionnal to the curvature of the curve. Both aforementionned parameterizations are very natural and correspond to a natural physical movement: the arc-length parameterization corresponds to travelling along the curve at constant speed, whereas parameterization proportionnal to curvature corresponds to a constant-speed moving frame. Since the curvature function of a curve is a geometric invariant of the unparameterized curve, a parameterization using the curvature function is a canonical parameterization. The main idea is that to any physically meaningful stricktly increasing function is associated a natural parameterization of 2D-curves, which gives an optimal sampling, and which can be used to compare unparameterized curves in a efficient and pertinent way. An application to point correspondance in medical imaging is given

Temporal Alignment of Human Motion Data: A Geometric Point of View

Temporal alignment is an inherent task in most applications dealing with videos: action recognition, motion transfer, virtual trainers, rehabilitation, etc. In this paper we dive into the understanding of this task from a geometric point of view: in particular, we show that the basic properties that are expected from a temporal alignment procedure imply that the set of aligned motions to a template form a slice to a principal fiber bundle for the group of temporal reparameterizations. A temporal alignment procedure provides a reparameterization invariant projection onto this particular slice. This geometric presentation allows to elaborate a consistency check for testing the accuracy of any temporal alignment procedure. We give examples of alignment procedures from the literature applied to motions of tennis players. Most of them use dynamic programming to compute the best correspondence between two motions relative to a given cost function. This step is computationally expensive (of complexity $O(NM)$ where $N$ and $M$ are the numbers of frames). Moreover most methods use features that are invariant by translations and rotations in $\mathbb{R}^3$, whereas most actions are only invariant by translation along and rotation around the vertical axis, where the vertical axis is aligned with the gravitational field. The discarded information contained in the vertical direction is crucial for accurate synchronization of motions. We propose to incorporate keyframe correspondences into the dynamic programming algorithm based on coarse information extracted from the vertical variations, in our case from the elevation of the arm holding the racket. The temporal alignment procedures produced are not only more accurate, but also computationally more efficient

Gauge Invariant Framework for Shape Analysis of Surfaces

This paper describes a novel framework for computing geodesic paths in shape spaces of spherical surfaces under an elastic Riemannian metric. The novelty lies in defining this Riemannian metric directly on the quotient (shape) space, rather than inheriting it from pre-shape space, and using it to formulate a path energy that measures only the normal components of velocities along the path. In other words, this paper defines and solves for geodesics directly on the shape space and avoids complications resulting from the quotient operation. This comprehensive framework is invariant to arbitrary parameterizations of surfaces along paths, a phenomenon termed as gauge invariance. Additionally, this paper makes a link between different elastic metrics used in the computer science literature on one hand, and the mathematical literature on the other hand, and provides a geometrical interpretation of the terms involved. Examples using real and simulated 3D objects are provided to help illustrate the main ideas.Comment: 15 pages, 11 Figures, to appear in IEEE Transactions on Pattern Analysis and Machine Intelligence in a better resolutio

Shape spaces of nonlinear flags

The gauge invariant elastic metric on the shape space of surfaces involves the mean curvature and the normal deformation, i.e. the sum and the difference of the principal curvatures $\kappa_1,\kappa_2$. The proposed gauge invariant elastic metrics on the space of surfaces decorated with curves involve, in addition, the geodesic and normal curvatures $\kappa_g,\kappa_n$ of the curve on the surface, as well as the geodesic torsion $\tau_g$. More precisely, we show that, with the help of the Euclidean metric, the tangent space at $(C,\Sigma)$ can be identified with $C^\infty(C)\times C^\infty(\Sigma)$ and the gauge invariant elastic metrics form a 6-parameter family that we give explicitly