A semidiscrete version of the Petitot model as a plausible model for anthropomorphic image reconstruction and pattern recognition

Preface In his beautiful book [54], Jean Petitot proposes a subriemannian 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 subriemannian model of Petitot (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 the image-reconstruction algorithms and we develop a pattern-recognition theory that leads also to very efficient algorithms in practice.