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

Abstract

In this paper, we present a novel model of the primary visual cortex (V1) based on orientation, frequency, and phase selective behavior of 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 two-dimensional retinal plane by introducing orientation, frequency, and phase as intrinsic variables. Each receptive profile on the fiber is mathematically interpreted as 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 of 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 two-dimensional input image. We provide the experimental results corresponding to the enhancement algorithm.

Multi-Frequency Image Completion via a Biologically-Inspired Sub-Riemannian Model with Frequency and Phase

“…In this section, we explain the cortical model geometry in which our completion algorithm is defined. This model framework is composed of two mechanisms: feature value extraction and neural connectivity [55].…”

“…For this reason, the approach considered by Citti, Petitot and Sarti and used in our present paper is referred to as biologically inspired. The sub-Riemannian model geometry proposed in [33] was extended in a recent work to multi-frequency and multi-phase setting [55]. The extended model corresponds to a natural geometry which is derived [55,56] from one of the very first perceptual mechanisms of vision: receptive profile.…”

“…The sub-Riemannian model geometry proposed in [33] was extended in a recent work to multi-frequency and multi-phase setting [55]. The extended model corresponds to a natural geometry which is derived [55,56] from one of the very first perceptual mechanisms of vision: receptive profile. It is different from the classical approach, in which a suitable geometry is assigned to the neural responses represented in terms of receptive profiles.…”

“…This is not the case in the sub-Riemannian model proposed in [33], which uses the SE(2) geometry, and in which only orientation can be represented as a visual feature. The extended model geometry was applied to enhancement on images in which several spatial frequencies were equally present [55], such as texture images. In the present work, we employ the same extended multi-frequency sub-Riemannian setting presented in [55] and propose an image completion algorithm within, which is aimed for use with grayscale texture images containing multiple spatial frequencies.…”

“…In Section 2, we explain the model framework which was introduced in [55] and its relation to our completion algorithm. In Section 3, we present a specific family of integral curves defined in the model geometry, which are the models of the neural connections in V1.…”