Transition to Volume II

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.