The paper is concerned with the properties of the distance function from a closed subset of a Riemannian manifold, with particular attention to the set of singularities.

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We define a novel metric on the space of closed planar curves which decomposes into three intuitive components. According to this metric centroid translations, scale changes and deformations are orthogonal, and the metric is also invariant with respect to reparameterizations of the curve. While earlier related Sobolev metrics for curves exhibit some general similarities to the novel metric proposed in this work, they lacked this important three-way orthogonal decomposition which has particular relevance for tracking in computer vision. Another positive property of this new metric is that the Riemannian structure that is induced on the space of curves is a smooth Riemannian manifold, which is isometric to a classical well-known manifold. As a consequence, geodesics and gradients of energies defined on the space can be computed using fast closed-form formulas, and this has obvious benefits in numerical applications.The obtained Riemannian manifold of curves is ideal to address complex problems in computer vision; one such example is the tracking of highly deforming objects. Previous works have assumed that the object deformation is smooth, which is realistic for the tracking problem, but most have restricted the deformation to belong to a finite-dimensional group -such as affine motions -or to finitely-parameterized models. This is too restrictive for highly deforming objects such as the contour of a beating heart. We adopt the smoothness assumption implicit in previous work, but we lift the restriction to finite-dimensional motions/deformations. We define a dynamical model in this Riemannian manifold of curves, and use it to perform filtering and prediction to infer and extrapolate not just the pose (a finitely parameterized quantity) of an object, but its deformation (an infinite-dimensional quantity) as well. We illustrate these ideas using a simple first-order dynamical model, and show that it can be effective even on image sequences where existing methods fail.

Previous studies have suggested that female facial attractiveness is associated with exaggerated sex-specific facial traits and averageness. Here we applied geometric morphometrics, a method for multivariate statistical analysis of shape, to measure geometric averageness and geometric sexual dimorphism of natural female face profiles. Geometric averageness and geometric sexual dimorphism correlate with attractiveness ratings. However, principal component analysis extracted a shape component robustly correlated with attractiveness but independent of sexual dimorphism. The shape differences between attractive- and hyperfeminine traits are localised: attractive facial shape and sexual dimorphism are similar in the upper face, but are markedly distinct in the jaw and chin.

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