Abstract
This final chapter describes the transition to the mathematical themes that will be developed in the second volume. (i) The explicit calculation of the elements of the sub-Riemannian geometry of the (mathbb {V}_mathrm{J}) model using the tools of control theory: geodesics , unit sphere, wave front, caustic , cut locus, conjugate points, and so on. (ii) The more natural model (mathbb {V}_mathrm{S}), constructed on SE(2) itself (which is the principal bundle associated with (mathbb {V}_mathrm{J})). SE(2) is no longer nilpotent. Its "nilpotentization', which defines its "tangent cone' at the origin, is isomorphic to the polarized Heisenberg group, , but globally it has a very different sub-Riemannian geometry. (iii) As far as the models model a functional architecture of connections between neurons which act as filters, the natural mathematical framework for low-level visual perception is the one in which non-commutative harmonic analysis on the group SE(2) is related to its sub-Riemannian geometry. (iv) The stochastic interpretation of the variational models leads to advection–diffusion algorithms described by a Fokker–Planck equation which can be calculated explicitly for the (mathbb {V}_mathrm{J}) model (while the calculation in (mathbb {V}_mathrm{S}) remains very complicated). Such techniques belong to the general theory of the heat kernel for the hypoelliptic Laplacians of sub-Riemannian manifolds. (v) One can interpolate between (mathbb {V}_mathrm{J}) and (mathbb {V}_mathrm{S}) using a continuous family of sub-Riemannian models.