  In his beautiful book [66], Jean Petitot proposes a sub-Riemannian model for
the primary visual cortex of mammals. This model is neurophysiologically
justified. Further developments of this theory lead to efficient algorithms for
image reconstruction, based upon the consideration of an associated
hypoelliptic diffusion. The sub-Riemannian model of Petitot and Citti-Sarti (or
certain of its improvements) is a left-invariant structure over the group
$SE(2)$ of rototranslations of the plane. Here, we propose a semi-discrete
version of this theory, leading to a left-invariant structure over the group
$SE(2,N)$, restricting to a finite number of rotations. This apparently very
simple group is in fact quite atypical: it is maximally almost periodic, which
leads to much simpler harmonic analysis compared to $SE(2).$ Based upon this
semi-discrete model, we improve on previous image-reconstruction algorithms and
we develop a pattern-recognition theory that leads also to very efficient
algorithms in practice.
