  The Trouv\'e group $\mathcal G_{\mathcal A}$ from image analysis consists of
the flows at a fixed time of all time-dependent vectors fields of a given
regularity $\mathcal A(\mathbb R^d,\mathbb R^d)$. For a multitude of regularity
classes $\mathcal A$, we prove that the Trouv\'e group $\mathcal G_{\mathcal
A}$ coincides with the connected component of the identity of the group of
orientation preserving diffeomorphims of $\mathbb R^d$ which differ from the
identity by a mapping of class $\mathcal A$. We thus conclude that $\mathcal
G_{\mathcal A}$ has a natural regular Lie group structure. In many cases we
show that the mapping which takes a time-dependent vector field to its flow is
continuous. As a consequence we obtain that the scale of Bergman spaces on the
polystrip with variable width is stable under solving ordinary differential
equations.
