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.

翻译：受视觉皮层V1区启发，在旋转平移群$SE(2)$上赋予次黎曼结构，我们提出基于亚椭圆扩散的图像修复与增强算法。针对Citti、Sarti及Boscain等人此前提出的方法实现，我们通过提出一种能够防止图像退化并产生更锐利结果的替代方案进行创新，该方案称为"蜡涂-蜡脱"（WaxOn-WaxOff）过程。同时利用次黎曼结构，我们定义了一种全新的基于$SE(2)$的非锐化掩膜滤波器，该滤波器类似于二维图像处理中的经典非锐化掩膜滤波。我们通过视网膜扫描图像中的血管增强实验验证了该方法。