Statistical analysis of trajectories on Riemannian manifolds: Bird migration, hurricane tracking and video surveillance

By Jingyong Su, Sebastian N. Kurtek, Eric Klassen, Anuj Srivastava. 2014

132 citations recorded by the source corpus.

We consider the statistical analysis of trajectories on Riemannian manifolds that are observed under arbitrary temporal evolutions. Past methods rely on cross-sectional analysis, with the given temporal registration, and consequently may lose the mean structure and artificially inflate observed variances. We introduce a quantity that provides both a cost function for temporal registration and a proper distance for comparison of trajectories. This distance is used to define statistical summaries, such as sample means and covariances, of synchronized trajectories and "Gaussian-type" models to capture their variability at discrete times. It is invariant to identical time-warpings (or temporal reparameterizations) of trajectories. This is based on a novel mathematical representation of trajectories, termed transported square-root vector field (TSRVF), and the $\mathbb{L}^2$ norm on the space of TSRVFs. We illustrate this framework using three representative manifolds---$\mathbb{S}^2$, $\mathrm {SE}(2)$ and shape space of planar contours---involving both simulated and real data. In particular, we demonstrate: (1) improvements in mean structures and significant reductions in cross-sectional variances using real data sets, (2) statistical modeling for capturing variability in aligned trajectories, and (3) evaluating random trajectories under these models. Experimental results concern bird migration, hurricane tracking and video surveillance.

Subfields: Artificial Intelligence, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f801b1e1ca070a40b087c3c2e0cc654351f9c414

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. Boulbaba Ben Amor
2. Hassen Drira
3. Amor Ben Tanfous
4. Jingyong Su
5. Anuj Srivastava
6. Sudeep Sarkar

In Computational Theory and Mathematics

1. Boulbaba Ben Amor
2. Anuj Srivastava
3. Jingyong Su

In Geometry and Topology

1. Boulbaba Ben Amor
2. Jingyong Su
3. Anuj Srivastava
4. Hassen Drira
5. Amor Ben Tanfous
6. Sudeep Sarkar
7. Fillipe D. M. de Souza

Citations in the corpus

• Action Recognition Using Rate-Invariant Analysis of Skeletal Shape Trajectories (2016, methodology): “They use a Gaussian mixture model (GMM) to capture the statistical distribution of the postures and demonstrate the superiority of their approach over the 2D silhouette-based approach.”
• Coding Kendall's Shape Trajectories for 3D Action Recognition (2018, methodology): “[1] used the framework of Transported Square-Root Velocity Fields (TSRVF) [34] to encode trajectories lying on Lie groups.”
• Rate-Invariant Analysis of Trajectories on Riemannian Manifolds with Application in Visual Speech Recognition (2014, methodology): “The standard algorithm to compute the Karcher mean in [8] [15] is adapted to this problem as follows: Algorithm 1 Karcher mean of multiple trajectories: Compute the Karcher mean of {α i (0)}s and set it to be µ(0).”