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AbstractThe paper is devoted to an approach for image inpainting developed on the basis of neurogeometry of vision and sub-Riemannian geometry. Inpainting is realized by completing damaged isophotes (level lines of brightness) by optimal curves for the left-invariant sub-Riemannian problem on the group of rototranslations (motions) of a plane SE(2). The approach is considered as anthropomorphic inpainting since these curves satisfy the variational principle discovered by neurogeometry of vision. A parallel algorithm and software to restore monochrome binary or halftone images represented as series of isophotes were developed. The approach and the algorithm for computation of completing arcs are presented in detail.

On the Engel group a nilpotent sub-Riemannian problem is considered, a 4-dimensional optimal control problem with a 2-dimensional linear control and an integral cost functional. It arises as a nilpotent approximation to nonholonomic systems with 2-dimensional control in a 4-dimensional space (for example, a system describing the navigation of a mobile robot with trailer). A parametrization of extremal trajectories by Jacobi functions is obtained. A discrete symmetry group and its fixed points, which are Maxwell points, are described. An estimate for the cut time (the time of the loss of optimality) on extremal trajectories is derived on this basis. Bibliography: 25 titles.

This article is concerned with developing an intelligent system for the control of a wheeled robot. An algorithm for training an artificial neural network for path planning is proposed. The trajectory ensures steering optimal motion from the current position of the mobile robot to a prescribed position taking its orientation into account. The proposed control system consists of two artificial neural networks. One of them serves to specify the position and the size of the obstacle, and the other forms a continuous trajectory to reach it, taking into account the information received, the coordinates, and the orientation at the point of destination. The neural network is trained on the basis of samples obtained by modeling the equations of motion of the wheeled robot which ensure its motion along trajectories in the form of Euler’s elastica.

Let be the Engel group and let be a rank-two left-invariant distribution with Lorentzian metric on . The sub-Lorentzian problem is stated as the problem of maximizing the sub-Lorentzian distance. A parametrization of timelike and spacelike normal extremal trajectories is obtained in terms of Jacobi elliptic functions. Discrete symmetry groups are described in the cases of timelike and spacelike trajectories; in both cases the fixed points and the corresponding Maxwell points are calculated for each symmetry. These calculations underlie estimates for the cut time (when the trajectory ceases to be globally optimal).

On the Engel group a nilpotent sub-Riemannian problem is considered, a 4-dimensional optimal control problem with a 2-dimensional linear control and an integral cost functional. It arises as a nilpotent approximation to nonholonomic systems with 2-dimensional control in a 4-dimensional space (for example, a system describing the navigation of a mobile robot with trailer). A parametrization of extremal trajectories by Jacobi functions is obtained. A discrete symmetry group and its fixed points, which are Maxwell points, are described. An estimate for the cut time (the time of the loss of optimality) on extremal trajectories is derived on this basis.

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