I will consider a model of the primary visual cortex in terms of Lie groups equipped with a sub-Riemannian metric. The shape of receptive profiles as well as the patterns of short range and long range connectivity will have a precise geometric meaning. After showing examples of contour completion in the sub-Riemannian structure, I will consider the coupling of heterogeneous cells to model amodal completion (Kanitza triangle) as well as contrast-costancy image reconstruction in V1. The reconstruction involves a new type of Poisson problem with heterogeneous differential operators. (Joint work with Giovanna Citti)