Highly corrupted image inpainting through hypoelliptic diffusion

We present a new image inpainting algorithm, the Averaging and Hypoelliptic Evolution (AHE) algorithm, inspired by the one presented in [SIAM J. Imaging Sci., vol. 7, no. 2, pp. 669--695, 2014] and based upon a semi-discrete variation of the Citti-Petitot-Sarti model of the primary visual cortex V1. The AHE algorithm is based on a suitable combination of sub-Riemannian hypoelliptic diffusion and ad-hoc local averaging techniques. In particular, we focus on reconstructing highly corrupted images (i.e. where more than the 80% of the image is missing), for which we obtain reconstructions comparable with the state-of-the-art.Comment: 15 pages, 10 figure

A semidiscrete version of the Citti-Petitot-Sarti model as a plausible model for anthropomorphic image reconstruction and pattern recognition

In his beautiful book [66], Jean Petitot proposes a sub-Riemannian model for the primary visual cortex of mammals. This model is neurophysiologically justified. Further developments of this theory lead to efficient algorithms for image reconstruction, based upon the consideration of an associated hypoelliptic diffusion. The sub-Riemannian model of Petitot and Citti-Sarti (or certain of its improvements) is a left-invariant structure over the group $SE(2)$ of rototranslations of the plane. Here, we propose a semi-discrete version of this theory, leading to a left-invariant structure over the group $SE(2,N)$, restricting to a finite number of rotations. This apparently very simple group is in fact quite atypical: it is maximally almost periodic, which leads to much simpler harmonic analysis compared to $SE(2).$ Based upon this semi-discrete model, we improve on previous image-reconstruction algorithms and we develop a pattern-recognition theory that leads also to very efficient algorithms in practice.Comment: 123 pages, revised versio

A sub-Riemannian model of the visual cortex with frequency and phase

In this paper we present a novel model of the primary visual cortex (V1) based on orientation, frequency and phase selective behavior of the V1 simple cells. We start from the first level mechanisms of visual perception: receptive profiles. The model interprets V1 as a fiber bundle over the 2-dimensional retinal plane by introducing orientation, frequency and phase as intrinsic variables. Each receptive profile on the fiber is mathematically interpreted as a rotated, frequency modulated and phase shifted Gabor function. We start from the Gabor function and show that it induces in a natural way the model geometry and the associated horizontal connectivity modeling the neural connectivity patterns in V1. We provide an image enhancement algorithm employing the model framework. The algorithm is capable of exploiting not only orientation but also frequency and phase information existing intrinsically in a 2-dimensional input image. We provide the experimental results corresponding to the enhancement algorithm

New Exact and Numerical Solutions of the (Convection-)Diffusion Kernels on SE(3)

We consider hypo-elliptic diffusion and convection-diffusion on $\mathbb{R}^3 \rtimes S^2$, the quotient of the Lie group of rigid body motions SE(3) in which group elements are equivalent if they are equal up to a rotation around the reference axis. We show that we can derive expressions for the convolution kernels in terms of eigenfunctions of the PDE, by extending the approach for the SE(2) case. This goes via application of the Fourier transform of the PDE in the spatial variables, yielding a second order differential operator. We show that the eigenfunctions of this operator can be expressed as (generalized) spheroidal wave functions. The same exact formulas are derived via the Fourier transform on SE(3). We solve both the evolution itself, as well as the time-integrated process that corresponds to the resolvent operator. Furthermore, we have extended a standard numerical procedure from SE(2) to SE(3) for the computation of the solution kernels that is directly related to the exact solutions. Finally, we provide a novel analytic approximation of the kernels that we briefly compare to the exact kernels.Comment: Revised and restructure

A sub-Riemannian model of the visual cortex with frequency and phase

International audienceIn this paper we present a novel model of the primary visual cortex (V1) based on orientation, frequency and phase selective behavior of the V1 simple cells. We start from the first level mechanisms of visual perception: receptive profiles. The model interprets V1 as a fiber bundle over the 2-dimensional retinal plane by introducing orientation, frequency and phase as intrinsic variables. Each receptive profile on the fiber is mathematically interpreted as a rotated, frequency modulated and phase shifted Gabor function. We start from the Gabor function and show that it induces in a natural way the model geometry and the associated horizontal connectivity modeling the neural connectivity patterns in V1. We provide an image enhancement algorithm employing the model framework. The algorithm is capable of exploiting not only orientation but also frequency and phase information existing intrinsically in a 2-dimensional input image. We provide the experimental results corresponding to the enhancement algorithm

Sub-Riemannian geometry and its applications to Image Processing

Master's Thesis in MathematicsMAT399MAMN-MA

Image processing in the semidiscrete group of rototranslations

International audienceIt 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))