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  1. Free, publicly-accessible full text available January 1, 2024
  2. Free, publicly-accessible full text available October 1, 2023
  3. Free, publicly-accessible full text available March 1, 2023
  4. This paper studies a longitudinal shape transformation model in which shapes are deformed in response to an internal growth potential that evolves according to an advection reaction diffusion process. This model extends prior works that considered a static growth potential, i.e., the initial growth potential is only advected by diffeomorphisms. We focus on the mathematical study of the corresponding system of coupled PDEs describing the joint dynamics of the diffeomorphic transformation together with the growth potential on the moving domain. Specifically, we prove the uniqueness and long time existence of solutions to this system with reasonable initial and boundary conditions as well as regularization on deformation fields. In addition, we provide a few simple simulations of this model in the case of isotropic elastic materials in 2D.
    Free, publicly-accessible full text available March 1, 2023
  5. Motivated by applications from computer vision to bioinformatics, the field of shape analysis deals with problems where one wants to analyze geometric objects, such as curves, while ignoring actions that preserve their shape, such as translations, rotations, scalings, or reparametrizations. Mathematical tools have been developed to define notions of distances, averages, and optimal deformations for geometric objects. One such framework, which has proven to be successful in many applications, is based on the square root velocity (SRV) transform, which allows one to define a computable distance between spatial curves regardless of how they are parametrized. This paper introduces a supervised deep learning framework for the direct computation of SRV distances between curves, which usually requires an optimization over the group of reparametrizations that act on the curves. The benefits of our approach in terms of computational speed and accuracy are illustrated via several numerical experiments on both synthetic and real data.
  6. This paper introduces an extension of diffeomorphic registration to enable the morphological analysis of data structures with inherent density variations and imbalances. Building on the framework of Large Diffeomorphic Metric Matching (LDDMM) registration and measure representations of shapes, we propose to augment previous measure deformation approaches with an additional density (or mass) transformation process. We then derive a variational formulation for the joint estimation of optimal deformation and density change between two measures. Based on the obtained optimality conditions, we deduce a shooting algorithm to numerically estimate solutions and illustrate the practical interest of this model for several types of geometric data such as fiber bundles with inconsistent fiber densities or incomplete surfaces.