In this talk, we explain the role of sub--Riemannian (SR) geometry in digital image processing and modelling of human visual system. In recently proposed mathematical models of human vision (J. Petitot, G. Citti, A. Sarti), it was shown that SR geodesics appear as natural curves caused by a mechanism of the primary visual cortex V1 of a human brain for completion of contours that are partially corrupted or hidden from observation. We extend the model by including data adaptivity via a suitable external cost in the SR metric and show the advantages of such extension: 1) it leads to a powerful method of extraction the information from digital images; 2) it provides a refined model of V1 that takes into account a presence of visual stimulus.
We start from explanation of basic concepts of SR geometry and then show how they provide brain inspired methods in digital image processing. We discuss how considering of SR structures on 2D and 3D images (or more precisely on their lift to the extended space of positions and directions) helps to detect some features, e.g. salient curves. We consider several particular examples: tracking of blood vessels in planar and spherical images of human retina, tracking of neural fibers in MRI images of human brain. Afterwards we show how a proper choice of the external cost based on a response of simple cells to the visual stimulus provides a model for geometrical optical illusions.
The talk is based on joint works with coauthors.
International Conference on Geometric Analysis in honor of the 90th anniversary of academician Yu.G. Reshetnyak, Novosibirsk, Russia