  In computational anatomy, the statistical analysis of temporal deformations
and inter-subject variability relies on shape registration. However, the
numerical integration and optimization required in diffeomorphic registration
often lead to important numerical errors. In many cases, it is well known that
the error can be drastically reduced in the presence of a symmetry. In this
work, the leading idea is to approximate the space of deformations and images
with a possibly non-metric symmetric space structure using an involution, with
the aim to perform parallel transport. Through basic properties of symmetries,
we investigate how the implementations of a midpoint and the involution compare
with the ones of the Riemannian exponential and logarithm on diffeomorphisms
and propose a modification of these maps using registration errors. This leads
us to identify transvections, the composition of two symmetries, as a mean to
measure how far from symmetric the underlying structure is. We test our method
on a set of 138 cardiac shapes and demonstrate improved numerical consistency
in the Pole Ladder scheme.
