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