Image Processing in the Semidiscrete Group of Rototranslations

It is well-known, since [12], that cells in the primary visual cortex V1 do much more than merely signaling position in the visual field: most cortical cells signal the local orientation of a contrast edge or bar – they are tuned to a particular local orientation. This orientation tuning has been given a mathematical interpretation in a sub-Riemannian model by Petitot, Citti, and Sarti [6, 14]. According to this model, the primary visual cortex V1 lifts grey-scale images, given as functions \(f:{\mathbb R}^2\rightarrow [0,1]\), to functions Lf defined on the projectivized tangent bundle of the plane \(PT\mathbb R^2 = \mathbb R^2\times \mathbb P^1\). Recently, in [1], the authors presented a promising semidiscrete variant of this model where the Euclidean group of rototranslations SE(2), which is the double covering of \(PT\mathbb R^2\), is replaced by SE(2, N), the group of translations and discrete rotations. In particular, in [15], an implementation of this model allowed for state-of-the-art image inpaintings.