Invariants for primary visual cortex
The authors consider a 2-dimensional manifold \(M\) as a model for the retina. Simple neurons detect any oriented line in the tangent or cotangent plane in an arbitrary point \(a\) of the retina. In the paper under review the authors work with oriented lines in cotangent planes to \(M\), which form a 3-dimensional contact manifold \(S (M) = (T^*M\setminus 0) /R^+\), called the spherization of the cotangent bundle, replaced by the projectivization of the cotangent bundle \(P(M)=(T^*M\setminus 0) /\mathbb{R}^*\), when the lines are nonoriented. According to [\textit{D. Alekseevsky}, ``Light, eye, brain. Geometric problems in vision'', in: Proceedings of the 8th seminar on geometry and topology, GTS. Teheran: Amirkabir University of Technology. 10--32 (2015); \textit{J. Petitot}, Chaos Solitons Fractals 50, 75--92 (2013; Zbl 1310.92012)], simple neurons operate as filters and the optic signal can be considered as convolution with the Gaussian or Gabor filters. Thus, the manifold \(M\) has not only a contact structure, but also a metric or, at least, a conformal structure. These structures naturally related to primary visual cortex, as well as rational differential invariants of distributions under the action of the structure group, are treated in the paper under review. For example, it is shown that Euclidean differential invariants of (non)oriented distributions on the plane are generated by basic differential invariants \(J_1 = u_x \cos u + u_y \sin u , J_2 = -u_x \sin u + u_y \cos u\) and invariant derivations \(\nabla_1 = \cos u\frac{d}{dx} + \sin u\frac{d}{dy}, \nabla_2 = -\sin u\frac{d}{dx} + \cos u\frac{d}{dy}\), satisfying the relation \(J_{12} - J_{21} + J^2_1 + J^2_2 = 0,\) where \(J_{kl}=\nabla_k(J_l), k, l=1, 2\). In other words, any Euclidean differential invariant is a rational function of invariants \(\nabla^\mu(J_1)\) and \(\nabla^\mu(J_2)\), where \(\mu=(m_1,m_2)\) is a multi-index and \(\nabla^\mu=\nabla_1^{m_1}\nabla_2^{m_2}\). Moreover, the authors find the explicite factor equations, which characterize orbits of regular distributions and provide their classification.
scientific article; zbMATH DE number 6914488