arXiv subjects

Thomas Pierron

Publications and source records attributed to Thomas Pierron.

7 recordsLinked to original sources

arXiv2026

A Framework for Joint Affine and Diffeomorphic Image Registration

Anatomical image registration commonly relies on a sequential pipeline where an affine alignment is estimated first and then held fixed while a non-rigid diffeomorphic deformation is applied. This two-step process often leads to suboptimal results, as the initial stage can absorb local deformations, biasing the residual passed to the diffeomorphic registration. To address this, we introduce a Joint Affine-Diffeomorphic framework, based on the large deformations model, that estimates both global affine and local diffeomorphic motions simultaneously within a single optimization. We propose two models: a Full Affine (FA) model that combines affine and diffeomorphic deformations, and a Decomposed Affine (DA) model that restricts the affine part from FA to rotations, translations, and anisotropic scalings. To numerically implement these models for image registration tasks, we develop a tailored optimization strategy that combines progressive affine enrichment, gradually increasing the complexity of the affine component, with a variational weighting scheme that smoothly manages the coarse-to-fine handover between the affine and diffeomorphic components. Evaluated on 2D synthetic datasets and 3D brain MRIs from the IXI cohort, our unsupervised approach avoids the pathological deformations of sequential baselines. We demonstrate that our joint formulation outperforms in Dice overlap two state-of-the-art deep learning foundation models, CARL and uniGradI-CON, as well as a sequential baseline using FLIRT for the classical affine registration followed by LDDMM. Our implementation is publicly available.

math.DG↗

arXiv2026

Surface reconstruction-driven band folding and spin-orbit enhancement at the $\alpha$-antimonene/Au(111) interface

The electronic properties of the two-dimensional (2D) $\alpha$ phase of antimonene are unique, featuring unpinned Dirac cones that can be moved with strain. Here we investigate the structural and electronic properties of an epitaxial 2D $\alpha$-antimonene, grown on Au(111). Using angle-resolved photoemission spectroscopy and density-functional theory, we reveal a strong hybridization at the Sb/Au interface, which imprints a rectangular reconstruction in the Au states, producing a band folding and hybrid bands exhibiting trigonal pockets. Additionally, hybridization displaces part of the Au wavefunction in regions of large electrostatic potential gradient, thereby enhancing spin-orbit splitting. Our work underscores that the pristine electronic properties of $\alpha$-antimonene may be deeply modified by its substrate, and even overwhelmed by the bands of the latter, and also shows that spin-orbit interaction in a heavy metal (Au) can be substantially enhanced by a lighter element (Sb).

cond-mat.mtrl-sci↗

arXiv2025

Decoupling actions of finite-dimensional Lie groups and of groups of diffeomorphisms in the large deformation framework

In computational anatomy, the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework has become a central tool for modeling smooth, invertible transformations between shapes such as curves or landmarks. In this paper, we extend this framework by enriching diffeomorphic deformations with transformations induced by finite-dimensional Lie groups (e.g. isometries, scalings), and we develop a registration model that decouples the actions of these two types of deformation on the shape during the matching process. To achieve this, we consider semidirect products between finite-dimensional groups and groups of diffeomorphisms, endowed with a right-invariant sub-Riemannian structure that give rise to new variational problems for shape registration. By exploiting symmetries and reduction theory, we decouple the contributions of each group throughout the matching process. We further extend the framework to incoroporate anisotropic deformations that preferentially favor certain directions during registration. On the numerical side, we propose an algorithm based on a joint optimization over both deformation groups, in contrast to the standard twostage approach that optimizes first over the finite-dimensional component and then over the diffeomorphic one. Experiments on curves and landmarks demonstrate that the proposed joint optimization improves registration accuracy and more effectively disentangles the contributions of the two deformation groups.

math.DG↗

arXiv2025

Three-dimensional deformations in single-layer $\alpha$ antimonene and interaction with a Au(111) surface from first principles

Using density functional theory, we investigate the electronic structure of the alpha phase of an antimony monolayer in its isolated form and in contact to the (111) surface of gold. We demonstrate that the isolated single-layer actually displays a slightly modulated puckering that stabilizes the monolayer, not a uniform one as often assumed. Moreover, it has dramatic consequences on the electronic band structure: the material is a semiconductor with low-dispersing bands near the Brillouin zone center. By further application of about 12% strain on the armchair direction, a double-cone features develops wherein an electronic bandgap of about 21~meV is found. When in contact with a Au(111) surface, a strong interaction with gold arises, as it appears clearly from (i) substantial atomic displacements compared to the isolated form, and (ii) hybridization of Sb and Au orbitals. The latter profoundly modifies the electronic band structure by strengthening the spin-orbit splitting of hybridized bands and spoiling the double-cone feature whose manipulation through substrate-induced strain appears therefore questionable, at least in the simulated epitaxial implementation.

cond-mat.mtrl-sci↗

arXiv2024

The graded group action framework for sub-riemannian orbit models in shape spaces

In the standard orbit model on shape analysis, a group of diffeomorphism on the ambient space equipped with a right invariant sub-riemannian metric acts on a space of shapes and induces a sub-riemannian structure on various spaces. An important example is given by the Large Deformation Diffeomorphic Metric Mapping (LDDMM) theory that has been developed initially in the context of medical imaging and image registration. However, the standard theory does not cover many interesting settings emerging in applications. We provide here an extended setting, the graded group action (GGA) framework, specifying regularity conditions to get most of the well known results on the orbit model for general groups and shape spaces equipped with a smooth structure of Banach manifold with application to multi-scale shape spaces. A specific study of the Euler-Poincar{\'e} equations inside the GCA framework leads to a uniqueness result for the momentum map trajectory lifted from shape spaces with different complexities deciphering possible benefits of over-parametrization in shooting algorithms.

math.DG↗

arXiv2022

Elastic Metrics on Spaces of Euclidean Curves: Theory and Algorithms

A main goal in the field of statistical shape analysis is to define computable and informative metrics on spaces of immersed manifolds, such as the space of curves in a Euclidean space. The approach taken in the elastic shape analysis framework is to define such a metric by starting with a reparameterization-invariant Riemannian metric on the space of parameterized shapes and inducing a metric on the quotient by the group of diffeomorphisms. This quotient metric is computed, in practice, by finding a registration of two shapes over the diffeomorphism group. For spaces of Euclidean curves, the initial Riemannian metric is frequently chosen from a two-parameter family of Sobolev metrics, called elastic metrics. Elastic metrics are especially convenient because, for several parameter choices, they are known to be locally isometric to Riemannian metrics for which one is able to solve the geodesic boundary problem explictly -- well-known examples of these local isometries include the complex square root transform of Younes, Michor, Mumford and Shah and square root velocity (SRV) transform of Srivastava, Klassen, Joshi and Jermyn. In this paper, we show that the SRV transform extends to elastic metrics for all choices of parameters, for curves in any dimension, thereby fully generalizing the work of many authors over the past two decades. We give a unified treatment of the elastic metrics: we extend results of Trouv\'{e} and Younes, Bruveris as well as Lahiri, Robinson and Klassen on the existence of solutions to the registration problem, we develop algorithms for computing distances and geodesics, and we apply these algorithms to metric learning problems, where we learn optimal elastic metric parameters for statistical shape analysis tasks.

math.DG↗

arXiv2020

On length measures of planar closed curves and the comparison of convex shapes

In this paper, we revisit the notion of length measures associated to planar closed curves. These are a special case of area measures of hypersurfaces which were introduced early on in the field of convex geometry. The length measure of a curve is a measure on the circle $\mathbb{S}^1$ that intuitively represents the length of the portion of curve which tangent vector points in a certain direction. While a planar closed curve is not characterized by its length measure, the fundamental Minkowski-Fenchel-Jessen theorem states that length measures fully characterize convex curves modulo translations, making it a particularly useful tool in the study of geometric properties of convex objects. The present work, that was initially motivated by problems in shape analysis, introduces length measures for the general class of Lipschitz immersed and oriented planar closed curves, and derives some of the basic properties of the length measure map on this class of curves. We then focus specifically on the case of convex shapes and present several new results. First, we prove an isoperimetric characterization of the unique convex curve associated to some length measure given by the Minkowski-Fenchel-Jessen theorem, namely that it maximizes the signed area among all the curves sharing the same length measure. Second, we address the problem of constructing a distance with associated geodesic paths between convex planar curves. For that purpose, we introduce and study a new distance on the space of length measures that corresponds to a constrained variant of the Wasserstein metric of optimal transport, from which we can induce a distance between convex curves. We also propose a primal-dual algorithm to numerically compute those distances and geodesics, and show a few simple simulations to illustrate the approach.

math.DG↗