Shape dynamics

Abstract: We present an approach for comparing two sequences of deforming shapes using both parametric models and nonparametric methods. In our approach, Kendall's definition of shape is used for feature extraction. Since the shape feature rests on a non-Euclidean manifold, we propose parametric models like the autoregressive model and autoregressive moving average model on the tangent space and demonstrate the ability of these models to capture the nature of shape deformations using experiments on gait-based human recognition. The nonparametric model is based on dynamic time-warping. We suggest a modification of the dynamic time-warping algorithm to include the nature of the non-Euclidean space in which the shape deformations take place. We also show the efficacy of this algorithm by its application to gait-based human recognition. We exploit the shape deformations of a person's silhouette as a discriminating feature and provide recognition results using the nonparametric model. Our analysis leads to some interesting observations on the role of shape and kinematics in automated gait-based person authentication.

Abstract: The Korteweg-de Vries, modified Korteweg-de Vries, and Harry Dym hierarchies of integrable systems are shown to be equivalent to a hierarchy of chiral shape dynamics of closed curves in the plane. These purely local dynamics conserve an infinite number of global geometric properties of the curves, such as perimeter and enclosed area. Several techniques used to study these integrable systems are shown to have simple differential-geometric interpretations. Connections with incompressible, inviscid fluid flow in two dimensions are suggested

Abstract: Shape dynamics is a completely background-independent universal framework of dynamical theories from which all absolute elements have been eliminated. For particles, only the variables that describe the shapes of the instantaneous particle configurations are dynamical. In the case of Riemannian three-geometries, the only dynamical variables are the parts of the metric that determine angles. The local scale factor plays no role. This leads to a shape-dynamic theory of gravity in which the four-dimensional diffeomorphism invariance of general relativity is replaced by three-dimensional diffeomorphism invariance and three-dimensional conformal invariance. Despite this difference of symmetry groups, it is remarkable that the predictions of the two theories – shape dynamics and general relativity – agree on spacetime foliations by hypersurfaces of constant mean extrinsic curvature. However, the two theories are distinct, with shape dynamics having a much more restrictive set of solutions. There are indications that the symmetry group of shape dynamics makes it more amenable to quantization and thus to the creation of quantum gravity. This introduction presents in simple terms the arguments for shape dynamics, its implementation techniques, and a survey of existing results.

Abstract: The aim is to model "activity" performed by a group of moving and interacting objects (which can be people, cars, or different rigid components of the human body) and use the models for abnormal activity detection. Previous approaches to modeling group activity include co-occurrence statistics (individual and joint histograms) and dynamic Bayesian networks, neither of which is applicable when the number of interacting objects is large. We treat the objects as point objects (referred to as "landmarks") and propose to model their changing configuration as a moving and deforming "shape" (using Kendall's shape theory for discrete landmarks). A continuous-state hidden Markov model is defined for landmark shape dynamics in an activity. The configuration of landmarks at a given time forms the observation vector, and the corresponding shape and the scaled Euclidean motion parameters form the hidden-state vector. An abnormal activity is then defined as a change in the shape activity model, which could be slow or drastic and whose parameters are unknown. Results are shown on a real abnormal activity-detection problem involving multiple moving objects.