This paper presents a semidiscrete alternative to the theory of neurogeometry of vision, due to Citti, Petitot, and Sarti. We propose a new ingredient, namely, working on the group of translations and discrete rotations $SE(2,N)$. The theoretical side of our study relates the stochastic nature of the problem with the Moore group structure of $SE(2,N)$. Harmonic analysis over this group leads to very simple finite dimensional reductions. We then apply these ideas to the inpainting problem which is reduced to the integration of a completely parallelizable finite set of Mathieu-type diffusions (indexed by the dual of $SE(2,N)$ in place of the points of the Fourier plane, which is a drastic reduction). The integration of the the Mathieu equations can be performed by standard numerical methods for elliptic diffusions and leads to a very simple and efficient class of inpainting algorithms. We illustrate the performances of the method on a series of deeply corrupted images.

An important question in the literature focusing on motor control is to determine which laws drive biological limb movements. This question has prompted numerous investigations analyzing arm movements in both humans and monkeys. Many theories assume that among all possible movements the one actually performed satisfies an optimality criterion. In the framework of optimal control theory, a first approach is to choose a cost function and test whether the proposed model fits with experimental data. A second approach (generally considered as the more difficult) is to infer the cost function from behavioral data. The cost proposed here includes a term called the absolute work of forces, reflecting the mechanical energy expenditure. Contrary to most investigations studying optimality principles of arm movements, this model has the particularity of using a cost function that is not smooth. First, a mathematical theory related to both direct and inverse optimal control approaches is presented. The first theoretical result is the Inactivation Principle, according to which minimizing a term similar to the absolute work implies simultaneous inactivation of agonistic and antagonistic muscles acting on a single joint, near the time of peak velocity. The second theoretical result is that, conversely, the presence of non-smoothness in the cost function is a necessary condition for the existence of such inactivation. Second, during an experimental study, participants were asked to perform fast vertical arm movements with one, two, and three degrees of freedom. Observed trajectories, velocity profiles, and final postures were accurately simulated by the model. In accordance, electromyographic signals showed brief simultaneous inactivation of opposing muscles during movements. Thus, assuming that human movements are optimal with respect to a certain integral cost, the minimization of an absolute-work-like cost is supported by experimental observations. Such types of optimality criteria may be applied to a large range of biological movements.

The authors deal with the radar ambiguity functions and their symmetries. It is well known that Hermite functions give rise to elliptic symmetries. Hermite functions are eigenvectors of the harmonic-oscillator Schrodinger operator. It is shown that the situation is essentially the same for hyperbolic symmetries: the signals are eigenvectors of the Schrodinger operator associated to the hyperbolic oscillator. Since this operator has continuous spectrum, these symmetries can only be reached approximately. It is also shown how to construct such signals and their ambiguity functions, which fall in two classes, are given. This work is essentially abstract nilpotent harmonic analysis, and is based on the well-known main fact that the Woodward ambiguity function is a positive-type function on the real Heisenberg Lie group.>

Fourier-like transforms that are defined on several groups of motions on the plane and on the plane itself (viewed as the homogeneous space of these motion groups) are presented. These transforms should have many applications in the area of pattern recognition, detection, and the representation of motions in pattern analysis. The method is based upon group-representation theory and abstract harmonic analysis on semidirect product groups. However, most of the ideas are generalizations and interpretations of classical concepts and methods such as the Fourier descriptor method used for boundary recognition. Computational aspects are investigated, and examples are given.>