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Alice Barbara Tumpach

Publications and source records attributed to Alice Barbara Tumpach.

13 recordsLinked to original sources
arXiv2025

Geometric Learning of Canonical Parameterizations of $2D$-curves

Most datasets encountered in computer vision and medical applications present symmetries that should be taken into account in classification tasks. A typical example is the symmetry by rotation and/or scaling in object detection. A common way to build neural networks that learn the symmetries is to use data augmentation. In order to avoid data augmentation and build more sustainable algorithms, we present an alternative method to mod out symmetries based on the notion of section of a principal fiber bundle. This framework allows the use of simple metrics on the space of objects in order to measure dissimilarities between orbits of objects under the symmetry group. Moreover, the section used can be optimized to maximize separation of classes. We illustrate this methodology on a dataset of contours of objects for the groups of translations, rotations, scalings and reparameterizations. In particular, we present a $2$-parameter family of canonical parameterizations of curves, containing the constant-speed parameterization as a special case, which we believe is interesting in its own right. We hope that this simple application will serve to convey the geometric concepts underlying this method, which have a wide range of possible applications. The code is available at the following link: $\href{https://github.com/GiLonga/Geometric-Learning}{https://github.com/GiLonga/Geometric-Learning}$. A tutorial notebook showcasing an application of the code to a specific dataset is available at the following link: $\href{https://github.com/ioanaciuclea/geometric-learning-notebook}{https://github.com/ioanaciuclea/geometric-learning-notebook}$

cs.CV↗
arXiv2024

Totally geodesic submanifolds in the manifold SPD of symmetric positive-definite real matrices

This paper is a self-contained exposition of the geometry of symmetric positive-definite real $n\times n$ matrices $\operatorname{SPD}(n)$, including necessary and sufficent conditions for a submanifold $\mathcal{N} \subset\operatorname{SPD}(n)$ to be totally geodesic for the affine-invariant Riemannian metric. A non-linear projection $x\mapsto π(x)$ on a totally geodesic submanifold is defined. This projection has the minimizing property with respect to the Riemannian metric: it maps an arbitrary point $x \in\operatorname{SPD}(n)$ to the unique closest element $π(x)$ in the totally geodesic submanifold for the distance defined by the affine-invariant Riemannian metric. Decompositions of the space $\operatorname{SPD}(n)$ follow, as well as variants of the polar decomposition of non-singular matrices known as Mostow's decompositions. Applications to decompositions of covariant matrices are mentioned.

math.DG↗
arXiv2023

An Example of Banach and Hilbert manifold: the universal Teichmüller 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 Teichmüller space verify these conditions.

math-ph↗
arXiv2023

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 $κ_1,κ_2$. The proposed gauge invariant elastic metrics on the space of surfaces decorated with curves involve, in addition, the geodesic and normal curvatures $κ_g,κ_n$ of the curve on the surface, as well as the geodesic torsion $τ_g$. More precisely, we show that, with the help of the Euclidean metric, the tangent space at $(C,Σ)$ can be identified with $C^\infty(C)\times C^\infty(Σ)$ and the gauge invariant elastic metrics form a 6-parameter family that we give explicitly.

math.DG↗
arXiv2023

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.

math.DG↗
arXiv2023

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.

math.DG↗
arXiv2019

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.

math-ph↗
arXiv2017

Quotient elastic metrics on the manifold of arc-length parameterized plane loops

We study the pull-back of the 2-parameter family of quotient elastic metrics introduced in Mio-Srivastava-Joshi on the space of arc-length parameterized loops. This point of view has the advantage of concentrating on the manifold of arc-length parameterized curves, which is a very natural manifold when the analysis of un-parameterized curves is concerned, pushing aside the tricky quotient procedure detailed in Lahiri-Robinson-Klassen of the preshape space of parameterized curves by the reparameterization (semi-)group. In order to study the problem of finding geodesics between two given arc-length parameterized loops under these quotient elastic metrics, we give a precise computation of the gradient of the energy functional in the smooth case as well as a discretization of it, and implement a path-straightening method. This allows us to have a better understanding of how the landscape of the energy functional varies with respect to the parameters.

math.DG↗
arXiv2015

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.

math.DG↗
arXiv2008

Infinite-dimensional hyperkaehler manifolds associated with Hermitian-symmetric affine coadjoint orbits

This paper is devoted to the construction of a hyperkaehler structure on the complexification of any Hermitian-symmetric affine coadjoint orbit O of a semi-simple L*-group of compact type, which is compatible with the complex symplectic form of Kirillov-Kostant-Souriau and restricts to the Kaehler structure of O. By the identification of the complexification of O with the cotangent space of O induced by Mostow's Decomposition Theorem, this leads to the existence of a hyperkaehler structure on the cotangent space of O compatible with Liouville's symplectic form and whose restriction to the zero section is the Kaehler structure of O. Explicit formulas of the metric in terms of the complexified orbit and of the cotangent space are given.

math-ph↗
arXiv2007

Classification of infinite-dimensional irreducible Hermitian-symmetric affine coadjoint orbits

In the finite-dimensional setting, every Hermitian-symmetric space of compact type is a coadjoint orbit of a finite-dimensional Lie group. It is natural to ask whether every infinite-dimensional Hermitian-symmetric space of compact type, which is a particular example of an Hilbert manifold, is transitively acted upon by a Hilbert Lie group of isometries. In this paper we give the classification of infinite-dimensional irreducible Hermitian-symmetric affine coadjoint orbits of L*-groups of compact type using the notion of simple roots of non-compact type. The key step is, given an infinite-dimensional symmetric pair (g, k), where g is a simple L*-algebra and k a subalgebra of g, to construct an increasing sequence of finite-dimensional subalgebras g_n of g together with an increasing sequence of finite-dimensional subalgebras k_n of k such that g (resp. k) is the closure of the union of g_n (resp. k_n), and such that the pairs (g_n, k_n) are symmetric. Comparing with the classification of Hermitian-symmetric spaces given by W. Kaup, it follows that any Hermitian-symmetric space of compact type is an affine-coadjoint orbit of an Hilbert Lie group.

math-ph↗
arXiv2006

The restricted Grassmannian, Banach Lie-Poisson spaces, and coadjoint orbits

We investigate some basic questions concerning the relationship between the restricted Grassmannian and the theory of Banach Lie-Poisson spaces. By using universal central extensions of Lie algebras, we find that the restricted Grassmannian is symplectomorphic to symplectic leaves in certain Banach Lie-Poisson spaces, and the underlying Banach space can be chosen to be even a Hilbert space. Smoothness of numerous adjoint and coadjoint orbits of the restricted unitary group is also established. Several pathological properties of the restricted algebra are pointed out.

math.DG↗