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Geometry of the Visual Cortex with Applications to Image Inpainting and Enhancement

Francesco Ballerin, Erlend Grong

TL;DR

This work proposes algorithms for image inpainting and enhancement based on hypoelliptic diffusion based on sub-Riemannian structure inspired by the visual cortex V1 and exploits the sub-Riemannian structure to define a completely new unsharp filter analogous to the classical unsharp filter for 2D image processing.

Abstract

Equipping the rototranslation group $SE(2)$ with a sub-Riemannian structure inspired by the visual cortex V1, we propose algorithms for image inpainting and enhancement based on hypoelliptic diffusion. We innovate on previous implementations of the methods by Citti, Sarti, and Boscain et al., by proposing an alternative that prevents fading and is capable of producing sharper results in a procedure that we call WaxOn-WaxOff. We also exploit the sub-Riemannian structure to define a completely new unsharp filter using $SE(2)$, analogous to the classical unsharp filter for 2D image processing. We demonstrate our method on blood vessels enhancement in retinal scans.

Geometry of the Visual Cortex with Applications to Image Inpainting and Enhancement

TL;DR

This work proposes algorithms for image inpainting and enhancement based on hypoelliptic diffusion based on sub-Riemannian structure inspired by the visual cortex V1 and exploits the sub-Riemannian structure to define a completely new unsharp filter analogous to the classical unsharp filter for 2D image processing.

Abstract

Equipping the rototranslation group with a sub-Riemannian structure inspired by the visual cortex V1, we propose algorithms for image inpainting and enhancement based on hypoelliptic diffusion. We innovate on previous implementations of the methods by Citti, Sarti, and Boscain et al., by proposing an alternative that prevents fading and is capable of producing sharper results in a procedure that we call WaxOn-WaxOff. We also exploit the sub-Riemannian structure to define a completely new unsharp filter using , analogous to the classical unsharp filter for 2D image processing. We demonstrate our method on blood vessels enhancement in retinal scans.
Paper Structure (14 sections, 1 theorem, 26 equations, 13 figures, 2 algorithms)

This paper contains 14 sections, 1 theorem, 26 equations, 13 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Sub-Riemannian geometry and image processing
  3. Geometry of the rototranslation group
  4. The visual cortex V1 and curve completion
  5. Curve completion in image processing
  6. Sub-Riemannian application to image processing
  7. The classical inpainting algorithm
  8. New innovations in the sub-Riemannian method
  9. Lift as a normal distribution
  10. Preserving details through WaxOn, WaxOff
  11. Unsharp masking
  12. WaxOn-WaxOff in the AHE algorithm
  13. Geometric preliminaries
  14. Sub-Riemannian manifolds

Key Result

Theorem 4.1.1

Define an operator $\Pi_{\sigma}: C^\infty(\mathop{\mathrm{SE}}\nolimits(2),(0,1]) \to C^\infty(\mathbb{R}^2, (0,1])$ by Then $\Pi_{\sigma}(\mathcal{L}_{\sigma}(I)) = I$.

Figures (13)

  • Figure 1: Modeling $\mathop{\mathrm{SE}}\nolimits(2)$ as a car with orientation, where $X_1$ is forward movement and $X_2$ is counter-clockwise rotation. Translation in the direction $X_3$ can be obtained by combining infinitesimal movements along $X_1$ and $X_2$.
  • Figure 2: Visual Cortex V1 under a stimulus (red curve): the red orientation columns receive direct stimulus from the input, as opposed to the orange ones. Excitatory synapses for simple cells located in the same hypercolumn or that are spatially close and sensitive to the same orientation are indicated by cyan arrows.
  • Figure 3: Examples of lifted images. In the first column are the original images, in the second column the orientation of the level lines, and in the third column, the images lifted to $\mathop{\mathrm{SE}}\nolimits(2)$, suppressing the trivial zero values for visualization purposes.
  • Figure 4: Integral lines of the vector fields $X_1^2+\beta X_2^2$ (red) and $X_3^2+\beta X_2^2$ (green) for a polynomial curve, at point $\left(\frac{1}{2},\frac{1}{2}\right)$, varying the coefficient $\beta$.
  • Figure 5: Application of the classic restoration algorithm to a basic example of a broken circle. We see that an increase in $\beta$ produces a more spread-out diffusion.
  • ...and 8 more figures

Theorems & Definitions (6)

  • Remark 3.1.1
  • Theorem 4.1.1
  • proof
  • Remark 4.2.1
  • Definition A.1.1
  • Definition A.1.2