A Cortical-Inspired Sub-Riemannian Model for Poggendorff-Type Visual Illusions

Abstract

We consider Wilson-Cowan-type models for the mathematical description of orientation-dependent Poggendorff-like illusions. Our modelling improves two previously proposed cortical-inspired approaches, embedding the sub-Riemannian heat kernel into the neuronal interaction term, in agreement with the intrinsically anisotropic functional architecture of V1 based on both local and lateral connections. For the numerical realisation of both models, we consider standard gradient descent algorithms combined with Fourier-based approaches for the efficient computation of the sub-Laplacian evolution. Our numerical results show that the use of the sub-Riemannian kernel allows us to reproduce numerically visual misperceptions and inpainting-type biases in a stronger way in comparison with the previous approaches.

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

“…Simple cell receptive profiles can be modeled in terms of Gabor functions [33,41,55,57]. In the orientation, frequency and phase selective model framework, the receptive profile of a simple cell is a Gabor function of the following type:…”

“…This framework was extended to higher dimensional geometries where scale [34] and velocity [35,36] were taken into account. Various biologically inspired models for optical illusions [37][38][39][40][41] and orientation preference maps [42], as well as frameworks for image processing [43][44][45][46][47][48][49][50], for pattern recognition [51] and for medical applications [52,53], were proposed in the sub-Riemannian geometry of SE (2).…”

Multi-frequency image completion via a biologically-inspired sub-Riemannian model with frequency and phase

“…Simple cell receptive profiles can be modeled in terms of Gabor functions [33,41,54,57]. In the orientation, frequency and phase selective model framework, receptive profile of a simple cell is a Gabor function of the following type:…”

“…This framework was extended to higher dimensional geometries where scale [34], and velocity [35,36] were taken into account. Various biologically-inspired models for optical illusions [37][38][39][40][41] and orientation preference maps [42], as well as frameworks for image processing [43][44][45][46][47][48][49], for pattern recognition [50] and for medical applications [51,52], were proposed in the sub-Riemannian geometry of SE (2).…”

ChebLieNet: Invariant Spectral Graph NNs Turned Equivariant by Riemannian Geometry on Lie Groups

“…First, the organization of visual data based on their location and orientation [Hubel and Wiesel, 1962] is modeled by Lie group convolutions [Bekkers, 2019], in which feature maps encode response for every position and every orientation. Second, long-range interactions between aligned neurons [Bosking et al, 1997] are modeled by building graphs with affinity matrices based on (approximate) sub-Riemannian distances on the Lie groups, inspired by sub-Riemannian image analysis methods such as [Franken and Duits, 2009, Favali et al, 2016, Mashtakov et al, 2017, Boscain et al, 2018, Duits et al, 2018, Baspinar et al, 2021. Defferrard et al [2020] showed how to construct powerful graph NNs that are faithful to the manifolds on which they are defined.…”