A Semidiscrete Version of the Citti-Petitot-Sarti Model as a Plausible Model for Anthropomorphic Image Reconstruction and Pattern Recognition

Abstract

PrefaceIn 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 semidiscrete model, we improve on previous image-reconstruction algorithms and we develop a pattern-recognition theory that leads also to very efficient algorithms in practice.

A Cortical-Inspired Model for Orientation-Dependent Contrast Perception: A Link with Wilson-Cowan Equations

“…In the following, after introducing the functional lifting under consideration, which follows wellestablished ideas contained, e.g., in [10,12,19], we present the proposed cortical-inspired extension of the energy functional (2), and discuss the numerical implementation of the associated gradient descent. We remark that the resulting procedure thus consists of a linear lifting step combined with WC-type evolution, which is non-linear due to the presence of the sigmoid σ α , see (2).…”

“…Under mild continuity assumption on L, it has been shown in [19] that L is then a continuous wavelet transform. That is, there exists a mother wavelet (2), and such that…”

“…In this paper, we consider a generalisation of this modelling and introduce explicit orientation dependence via a lifting procedure inspired by neuro-physiological models of V1 [10,12,19]. Interestingly, the Euler-Lagrange equations associated with the proposed functional yield orientationdependent WC-type equations analogous to (1).…”

Visual illusions via neural dynamics: Wilson–Cowan-type models and the efficient representation principle

“…The study of visual illusions has always been key in the vision science community, as the mismatches between reality and perception provide insights that can be very useful to develop new models of visual perception [Kingdom, 2011] or of neural activity [Eagleman, 1959, Murray andHerrmann., 2013], and also to validate the existing ones. It is commonly accepted that visual illusions arise due to neurobiological constraints [Purves et al, 2008] that modify the underpinned mechanisms of the visual system.…”

Cortical-inspired image reconstruction via sub-Riemannian geometry and hypoelliptic diffusion

“…In view of the many well-studied examples of contour completion operated by the human brain (see Figure 1), one of the main problems that was attacked from this point of view has been that of image reconstruction, also known as image inpainting [3,12,15,28]. In this survey, we present the results we obtained in a series of works [4][5][6][7]33] in this context. The paper is divided in three parts, corresponding to different methods and purposes of image reconstruction.…”