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区启发的旋转平移群$SE(2)$子黎曼结构，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等人此前实现的同类方法，我们提出改进方案——通过名为WaxOn-WaxOff的流程防止纹理淡化并生成更清晰的修复结果。同时利用子黎曼结构，我们定义了基于$SE(2)$的完全新型非锐化掩蔽滤波器，该滤波器类似于经典二维图像处理中的非锐化掩蔽技术。我们在视网膜扫描图像血管增强任务中验证了本方法的有效性。