Sub-Riemannian Geometry in Image Processing and Modeling of the Human Visual System

Abstract

This paper summarizes results of a sequence of works related to usage of sub-Riemannian (SR) geometry in image processing and modeling of the human visual system. In recent research in psychology of vision (J. Petitot, G. Citti, A. Sarti) it was shown 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 to include data adaptivity via a suitable external cost in the SR metric. We show that data adaptive SR geodesics are useful in real image analysis applications and provide a refined model of V1 that takes into account the presence of a visual stimulus.

Attainable set for rank 3 step 2 free Carnot group with positive controls

“…, r, ẏij = x i u j − x j u i , i < j (2) is invariant under the left action of the group G. Control systems on Carnot groups are some kind of cornerstones in geometric control theory [1] due to existence of a nilpotent approximation for general control systems [2]. In particular, such control systems appear in several robotic systems [3] and in some models for contour reconstruction without cusps in image processing [4]. We consider system (2) with controls u 1 , .…”

“…Note that endpoints of bang-bang trajectories with periodical control sweep non-trivial faces of the boundary of the attainable set. We give the answer in terms of the set B and the coordinates p = p 12 , q = p 23 , r = p 31 on this set, see (4).…”