In this paper we will consider mathematical models of the functional architecture of the primary visual cortex based on Lie groups equipped with sub- Riemannian metrics. We will critically review and clarify our line of work, joining together within an integrative point of view geometric, statistical, and harmonic models. The neurogeometry of the cortex in the SE(2) groups introduced recalling the original paper [12]. Amodal perceptual completion is reconsidered in terms of constitution of minimal surfaces in the geometric space of the functional architecture, and a new Lagrangian field model is introduced to afford the problem of modal perceptual completion [14] of the Kanizsa triangle. The neurogeometric structure is considered also from the a probabilistic point of view and compared with the statistics of co-occurence of edges in natural images following [48]. Finally the problem of perceptual units constitution is introduced by means of a neurally based non linear PCA technique able to perform a spectral decomposition of the neurogeometrical operator and produce the perceptual gestalten [52, 53].