Apparaît dans la collection : Energy-Based Modeling, Simulation, and Control of Complex Constrained Multiphysical Systems / Modélisation structurée, intégration géométrique et commande de systèmes multiphysiques contraints

Finding the optimal reparametrization in shape analysis of curves or surfaces is a computationally demanding task. The problem can be phrased as an optimisation problem on the infinite dimensional group of orientation preserving diffeomorphisms $\mathrm{Diff}^+(\Omega)$, where $\Omega$ is the domain where the curves or surfaces are defined. We consider the composition of a finite number of elementary diffeomprphisms \begin{equation} \label{elem_diff} \varphi_{\ell}:=\mathrm{id}+\sum_{j=1}^M \lambda_j^{\ell} f_j,\qquad \ell=1,\dots , L,\qquad \varphi\approx \varphi_L\circ \cdots \circ\varphi_1, \end{equation} where ${f_i}_{i=1}^{\infty}$, in $T_{\mathrm{id}}\mathrm{Diff}^+(\Omega)$ is an orthonormal basis, and we optimise simultaneously on all the parameters ${\lambda_j^{\ell}}$ for $j=1,\dots ,M$ and $\ell=1,\dots, L$. The obtained algorithm is similar to a deep neural network and its implementation can be carried out using PyTorch. Properties and analysis of the method will be discussed as well as numerical results.