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

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## Abstract

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## 1. Introduction

#### 1.1. The Functional Architecture of the Primary Visual Cortex

#### 1.2. Mean-Field Neural Dynamics & Visual Illusions

#### 1.3. Main Contributions

## 2. Cortical-Inspired Modelling

#### 2.1. Receptive Profiles

#### 2.2. Horizontal Connectivity and Sub-Riemannian Diffusion

#### 2.3. Reconstruction on the Retinal Plane

## 3. Describing Neuronal Activity via Wilson-Cowan-Type Models

#### 3.1. Wilson-Cowan (WC) Model

#### 3.2. Local Histogram Equalisation (LHE) Model

#### 3.3. A Sub-Riemannian Choice of the Interaction Kernel ${\omega}_{\xi}$

## 4. Discrete Modelling and Numerical Realisation

#### 4.1. Discrete Modelling and Lifting Procedure via Cake Wavelets

#### 4.2. Sub-Riemannian Heat Diffusion

#### 4.3. Discretisation via Gradient Descent

#### 4.4. Pseudocode

Algorithm 1: sR-WC and sr-LHE pseudocode. |

## 5. Numerical Experiments

#### 5.1. Poggendorff Gratings

#### 5.2. Dependence on Parameters: Inpainting vs. Perceptual Completion

#### 5.3. Poggendorff Illusion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**The original Poggendorff illusion: the red colored line is aligned with the black line although the blue one is falsely perceived as its continuation. Source: Wikipedia.

**Figure 3.**Greyscale Poggendorff-type illusions. (

**a**) is the standard 200 × 200 Poggendorff illusion with a 30 pixel-wide central and an incidence angle of π/3 drawn by the black lines with the central bar. (

**b**) is a variation of the classical Poggendorff illusion where a further background grating is present.

**Figure 6.**Sensitivity to the parameter for τ for (sR-LHE) model for the visual perception of Figure 3b. The completion inside the central grey bar changes from geometrical (inpainting type) to illusory (perception type). Parameters: τ varies from 0.1 to 5, α = 6, σ

_{μ}= 1, Δt = 0.15, Δτ = 0.01.

**Figure 7.**Model output for Poggendorff gratings in Figure 3b via LHE models. (

**a**) result of the LHE model proposed in [44,45] (with parameters σ

_{μ}= 2, σ

_{ω}= 12, λ = 0.7, α = 5). (

**b**) result of (sR-LHE) with parameters α = 8, τ = 2.5, λ = 0.5, σ

_{μ}= 2.5, Δt = 0.15, Δτ = 0.1. (

**d**) (resp. Figure 7c): zoom and renormalization on [0,1] of the central region of the result in (

**b**) (resp. Figure 7a).

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**MDPI and ACS Style**

Baspinar, E.; Calatroni, L.; Franceschi, V.; Prandi, D.
A Cortical-Inspired Sub-Riemannian Model for Poggendorff-Type Visual Illusions. *J. Imaging* **2021**, *7*, 41.
https://doi.org/10.3390/jimaging7030041

**AMA Style**

Baspinar E, Calatroni L, Franceschi V, Prandi D.
A Cortical-Inspired Sub-Riemannian Model for Poggendorff-Type Visual Illusions. *Journal of Imaging*. 2021; 7(3):41.
https://doi.org/10.3390/jimaging7030041

**Chicago/Turabian Style**

Baspinar, Emre, Luca Calatroni, Valentina Franceschi, and Dario Prandi.
2021. "A Cortical-Inspired Sub-Riemannian Model for Poggendorff-Type Visual Illusions" *Journal of Imaging* 7, no. 3: 41.
https://doi.org/10.3390/jimaging7030041