Phase-Amplitude Separation and Modeling of Spherical Trajectories

Abstract

This paper studies the problem of separating phase-amplitude components in sample paths of a spherical process (longitudinal data on a unit two-sphere). Such separation is essential for efficient modeling and statistical analysis of spherical longitudinal data in a manner that is invariant to any phase variability. The key idea is to represent each path or trajectory with a pair of variables, a starting point and a Transported Square-Root Velocity Curve (TSRVC). A TSRVC is a curve in the tangent (vector) space at the starting point and has some important invariance properties under the L 2 norm. The space of all such curves forms a vector bundle and the L 2 norm, along with the standard Riemannian metric on S 2 , provides a natural metric on this vector bundle. This invariant representation allows for separating phase and amplitude components in given data, using a template-based idea. Furthermore, the metric property is useful in deriving computational procedures for clustering, mean computation, principal component analysis (PCA), and modeling. This comprehensive framework is demonstrated using two datasets: a set of bird-migration trajectories and a set of hurricane paths in the Atlantic ocean.

A Discrete Framework to Find the Optimal Matching Between Manifold-Valued Curves

“…These curves can be closed or open, and take their values in a Euclidean space or more generally in a Riemannian manifold. To name a few examples, closed plane curves are central in shape analysis of objects [22]; the study of trajectories on the Earth requires to deal with open curves on the sphere [24]; and in signal processing, locally stationary Gaussian processes can be represented by open curves in the hyperbolic plane, seen as the statistical manifold of Gaussian densities [9], [10].…”

“…Concerning manifold-valued curves, the geodesic equations for Sobolev metrics in the space of curves and in the shape space were given in [1] in terms of the gradient of the metric with respect to itself. A generalization of the SRV framework to manifold-valued curves was introduced in [23] and used in [24], while another one was proposed in [10]. Extension to curves in a Lie group or a homogeneous space can also be found in [4], [5], [18].…”

“…Both metrics in [23] and [10] coincide with the metric of [17] in the flat case, however they define different Riemannian structures when the base space has curvature. The difference lies in the way computations are made -in [23] and [24] they are moved to the tangent spaces at the origins of the curves, resulting in simpler computations, whereas in [10] they are done directly in the base manifold M , transporting data pointwise across M from one curve to the other, thus making the comparison more sensitive to the local "geography" of M . When comparing the geodesic distances, this difference is measured by a curvature term (see [10], Prop.…”

“…In addition, unlike the metric of [23], the one in [10] can be written as an elastic metric (with a = 2b = 1). The work of [23] is applied to curves in the space of symmetric positive definite matrices, while the case of spherical trajectories is investigated in [24], where the authors exhibit simplifications. In both cases, optimal matching between curves is achieved through dynamic programming.…”

“…Regarding the geometry of H 2 and the useful algorithms such as the exponential map and the logarithm map, we refer the reader to [10]. Concerning the geometry of S 2 , we have used the same formulas as those given in appendix in [24]. We start by comparing Algorithm 5 to the popular dynamic programming (DP) method, presented e.g.…”