Abstract. This paper is devoted to present an algorithm implementing the theory of neurogeometry of vision, described by Jean Petitot in his book. We propose a new ingredient, namely working on the group of translations and discrete rotations SE(2, N ). We focus on the theoretical and numerical aspects of integration of an hypoelliptic diffusion equation on this group. Our main tool is the generalized Fourier transform. We provide a complete numerical algorithm, fully parallellizable. The main objective is the validation of the neurobiological model.
Here we shall present a brief chronology of the appearance of the concepts discussed in this book. The development of mathematical ideas generally proceeds in such a way that some concepts gradually emerge from others. Therefore, it is generally impossible to fix accurately the appearance of some particular idea. We shall only point out the important milestones and, it goes without saying, shall do so only roughly. In particular, we shall limit our view to Western European mathematics.The principal stimulus was, of course, the creation of analytic geometry by Fermat and Descartes in the seventeenth century. This made it possible to specify points (on the line, in the plane, and in three-dimensional space) using numbers (one, two, or three), to specify curves and surfaces by equations, and to classify them according to the algebraic nature of their equations. In this regard, linear transformations were used frequently, especially by Euler, in the eighteenth century.Determinants (particularly as a symbolic apparatus for finding solutions of systems of n linear equations in n unknowns) were considered by Leibniz in the seventeenth century (even if only in a private letter) and in detail by Gabriel Cramer in the eighteenth. It is of interest that they were constructed on the basis of the rule of "general expansion" of the determinant, that is, on the basis of the most complex (among those that we considered in Chap. 2) way of defining them. This definition was discovered "empirically," that is, conjectured on the basis of the formulas for the solution of systems of linear equations in two and three unknowns. The broadest use of determinants occurred in the nineteenth century, especially in the work of Cauchy and Jacobi.The concept of "multidimensionality," that is, the passage from one, two, and three coordinates to an arbitrary number, was stimulated by the development of mechanics, where one considered systems with an arbitrary number of degrees of freedom. The idea of extending geometric intuition and concepts to this case was developed systematically by Cayley and Grassmann in the nineteenth century. At the same time, it became clear that one must study quadrics in spaces of arbitrary dimension (Jacobi and Sylvester in the nineteenth century). In fact, this question had already been considered by Euler.
We present a new bio-mimetic image inpainting algorithm, the Averaging and Hypoelliptic Evolution (AHE) algorithm, inspired by the one presented in (U. Boscain et al. SIAM J. Imaging Sci., 7(2):669-695, 2014) and based upon a semi-discrete variation of the Citti-Petitot-Sarti model of the primary visual cortex V1. The AHE algorithm is based on a suitable combination of sub-Riemannian hypoelliptic diffusion and ad-hoc local averaging techniques. In particular, we focus on highly corrupted images (i.e., where more than the 80% of the image is missing), for which we obtain high-quality reconstructions.
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