Image Processing in the Semidiscrete Group of Rototranslations

Abstract

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 $f:\mathbb{R}^2\to [0,1]$, 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.

Publication
Geometric Science of Information