The lecture is devoted to usage of sub-Riemannian (SR) geometry in image processing and modelling of human visual system. In recent research in psychology of vision it was shown (J. Petitot, G. Citti, A. Sarti) that SR geodesics appear as natural curves that model 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. We show that data-driven SR geodesics are useful in real image analysis applications and provide 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 computer vision. We discuss how to construct the SR structures on 2D and 3D images (or more precisely on their lift to the extended space of positions and directions) in order 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 provide a model for geometrical optical illusions.