Abstract : 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.

Keywords : shape analysis; deep learning of diffeomorphisms; infinite dimensional Lie groups