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 : ℝ2→ [0, 1], to functions Lf defined on the projectivized tangent bundle of the plane PTℝ2= ℝ2× ℙ1. 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 PTℝ2, 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. In this work, we review the inpainting results and introduce an application of the semidiscrete model to image recognition. We remark that both these applications deeply exploit the Moore structure of SE(2,N) that guarantees that its unitary representations behaves similarly to those of a compact group. This allows for nice properties of the Fourier transform on SE(2,N) exploiting which one obtains numerical advantages.