In surveillance applications, head and body orientation of people is of primary importance for assessing many behavioral traits. Unfortunately, in this context people are often encoded by a few, noisy pixels so that their characterization is difficult. We face this issue, proposing a computational framework which is based on an expressive descriptor, the covariance of features. Covariances have been employed for pedestrian detection purposes, actually a binary classification problem on Riemannian manifolds. In this paper, we show how to extend to the multiclassification case, presenting a novel descriptor, named weighted array of covariances, especially suited for dealing with tiny image representations. The extension requires a novel differential geometry approach in which covariances are projected on a unique tangent space where standard machine learning techniques can be applied. In particular, we adopt the Campbell-Baker-Hausdorff expansion as a means to approximate on the tangent space the genuine (geodesic) distances on the manifold in a very efficient way. We test our methodology on multiple benchmark datasets, and also propose new testing sets, getting convincing results in all the cases.
The present article delves into some symplectic features arising in basic knot theory. An interpretation of the writhing number of a knot (with reference to a plane projection thereof) is provided in terms of a phase function analogous to those encountered in geometrical optics, its variation upon switching a crossing being akin to the passage through a caustic, yielding a knot theoretical analogue of Maslov's theory, via classical fluidodynamical helicity. The Maslov cycle is given by knots having exactly one double point, among those having a fixed plane shadow and lying on a semi-cone issued therefrom, which turn out to build up a Lagrangian submanifold of Brylinski's symplectic manifold of (mildly) singular knots. A Morse family (generating function) for this submanifold is determined and can be taken to be the Abelian Chern–Simons action plus a source term (knot insertion) appearing in the Jones–Witten theory. The relevance of the Bohr–Sommerfeld conditions arising in geometric quantization are investigated and a relationship with the Gauss linking number integral formula is also established, together with a novel derivation of the so-called Feynman–Onsager quantization condition. Furthermore, an additional Chern–Simons interpretation of the writhe of a braid is discussed and interpreted symplectically, also making contact with the Goldin–Menikoff–Sharp approach to vortices and anyons. Finally, a geometrical setting for the ground state wave functions arising in the theory of the Fractional Quantum Hall Effect is established.
A differential geometric approach to Milnor-Massey higher order linking numbers for generic links is devised, via Chen's theory of iterated path integrals. Massey linking numbers arise from curvature forms of nilpotent "topological" connections, determined by the link structure, and interpreted in terms of intersection theory, leading to a fairly easy computation thereof. A version of the Turaev-Porter theorem expressing equality of Milnor and Massey linking numbers is also exhibited along the same lines, by computing suitable flat connection parallel transport operators in two different ways.
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