Rayane Mouhli

In computational anatomy, analyzing morphological variability across shape populations often requires multi-component deformation models that combine structured motions and unconstrained diffeomorphisms. However, a major challenge arises during the registration process, as high-dimensional deformations tend to absorb lower-dimensional components, altering the true geometric variability and preventing accurate statistical analysis. To address this issue, we introduce a novel coupling score designed to decouple distinct deformation modes during registration. This score is defined using first variation of varifolds which is a varifold representation of the infinitesimal action of vector fields on shapes. The proposed score quantifies the extent to which the action of a given vector field on a shape can be replicated by another subspace of vector fields. We provide a theoretical analysis of this coupling score, illustrating its behavior on finite-dimensional spaces. Finally, we integrate this score as a penalization term in registration problems. Numerical experiments illustrate its efficiency in various use cases such as enforcing or preventing specific motions, and iteratively correcting complex matching scenarios by enforcing structured directional priors.

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 two-stage 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.