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Image Processing in the Semidiscrete Group of Rototranslations

  • Dario Prandi
  • Ugo Boscain
  • Jean-Paul Gauthier
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9389)

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 \(f:{\mathbb R}^2\rightarrow [0,1]\), 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.

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.

Keywords

Primary Visual Cortex Morse Function Irreducible Unitary Representation Hexagonal Grid Amodal Completion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Dario Prandi
    • 1
  • Ugo Boscain
    • 2
    • 3
  • Jean-Paul Gauthier
    • 3
    • 4
  1. 1.CNRSCEREMADE, Univ. Paris-DauphineParisFrance
  2. 2.CNRSCMAP, Ecole PolytechniqueSaclayFrance
  3. 3.INRIA Team GECOSaclayFrance
  4. 4.LSISUniversité de Toulon USTVLa Garde CedexFrance

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