Cortical-inspired image reconstruction via sub-Riemannian geometry and hypoelliptic diffusion

Abstract

In this paper we review several algorithms for image inpainting based on the hypoelliptic diffusion naturally associated with a mathematical model of the primary visual cortex. In particular, we present one algorithm that does not exploit the information of where the image is corrupted, and others that do it. While the first algorithm is able to reconstruct only images that our visual system is still capable of recognize, we show that those of the second type completely transcend such limitation providing reconstructions at the state-of-the-art in image inpainting. This can be interpreted as a validation of the fact that our visual cortex actually encodes the first type of algorithm.

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

“…The discretisation of such anisotropic operators can be done in several ways, see for example [ 29 , 30 , 39 , 65 ]. In our implementation, we follow the method presented in [ 26 ], which is tailored around the group structure of , the universal cover of , and based on the non-commutative Fourier transform, see also [ 9 ].…”

“…We remark that the presented model for neural activity is a phenomenological model that provides a mathematical understanding of early perceptual mechanisms at the cortical level by starting from very structure of receptive profiles. Nevertheless, it has been very useful for many image-processing applications, see, for example, [ 9 , 10 ].…”

“…In recent years, the CPS model has been exploited as a framework for several cortical-inspired image processing problems by various researchers. We mention the large corpus of literature by Duits et al, see, for example, [ 28 , 29 , 30 ] and the state-of-the-art image inpainting and image recognition algorithms developed by Boscain, Gauthier, et al [ 9 , 31 ]. Some extensions of the CPS model geometry and its applications to other image processing problems can be found in [ 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 ].…”

DIP-VBTV: A Color Image Restoration Model Combining a Deep Image Prior and a Vector Bundle Total Variation

“…There is a growing interest in designing human vision-inspired mathematical models in image processing and computer vision (see e.g. [1], [3], [4], [6], [7], [9], [17], [25]). Dealing with restoration of natural images, this approach is justified by the fact that one aims to maintain the perception of the original scene rather than reproducing its light intensity.…”

A cortical-inspired sub-Riemannian model for Poggendorff-type visual illusions

“…In recent years, the CPS model have been exploited as a framework for several cortical-inspired image processing problems by various researchers. We mention the large corpus of literature by Duits et al, see e.g., [28][29][30] and the state-of-the-art image inpainting and image recognition algorithms developed by Boscain, Gauthier, et al [31,32]. Some extensions of the CPS model geometry and its applications to other image processing problems can be found in [33][34][35][36][37][38][39][40].…”