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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 [14,6]. According to this model, the primary visual cortex V1 lifts grey-scale images, given as functions f : R 2 → [0, 1], to functions Lf defined on the projectivized tangent bundle of the plane P T R 2 = R 2 × P 1. Recently, in [1], the authors presented a promising semidiscrete variant of this model where the Euclidean group of roto-translations SE(2), which is the double covering of P T 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. 1 The semi-discrete model The starting point of our work is the sub-Riemannian model of the primary visual cortex V1 [14,6], and our recent contributions [3,1,2,4]. This model has also been deeply studied in [8,11]. In the sub-Riemannian model, V1 is modeled as the projective tangent bundle P T R 2 ∼ = R 2 × P 1 , whose double covering is the roto-translation group SE(2) = R 2 S 1 , endowed with a left-invariant sub-Riemannian structure that mimics the connections between neurons. In particular , grayscale visual stimuli f : R 2 → [0, 1] feeds V1 neurons N = (x, θ) ∈ P T R 2 with an extracellular voltage Lf (ξ) that is widely accepted to be given by Lf (ξ) = f, Ψ ξ. The functions {Ψ ξ } ξ∈P T R 2 are the receptive fields. A good fit is Ψ (x,θ) = π(x, θ)Ψ where Ψ is the Gabor filter (a sinusoidal multiplied by a Gaussian function) and π(x, θ)Ψ (y) := Ψ (R −θ (x − y)).
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Submitted on : Friday, March 2, 2018 - 2:42:46 PM
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Dario Prandi, Ugo Boscain, Jean-Paul Gauthier. Image processing in the semidiscrete group of rototranslations. GSI 2015: Geometric Science of Information, Dec 2015, Palaiseau, France. ⟨hal-01721740⟩



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