Rate-invariant analysis of covariance trajectories

Zhang, Zhengwu and Su, Jingyong and Klassen, Eric and Le, Huiling and Srivastava, Anuj (2018) Rate-invariant analysis of covariance trajectories. Journal of Mathematical Imaging and Vision . ISSN 1573-7683

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Abstract

Statistical analysis of dynamic systems, such as videos and dynamic functional connectivity, is often translated into a problem of analyzing trajectories of relevant features, particularly covariance matrices. As an example, in video-based action recognition, a natural mathematical representation of activity videos is as parameterized trajectories on the set of symmetric, positive-definite matrices (SPDMs). The variable execution-rates of actions, implying arbitrary parameterizations of trajectories, complicates their analysis and classification. To handle this challenge, we represent covariance trajectories using transported square-root vector fields (TSRVFs), constructed by parallel translating scaled-velocity vectors of trajectories to their starting points. The space of such representations forms a vector bundle on the SPDM manifold. Using a natural Riemannian metric on this vector bundle, we approximate geodesic paths and geodesic distances between trajectories in the quotient space of this vector bundle. This metric is invariant to the action of the reparameterization group, and leads to a rate-invariant analysis of trajectories. In the process, we remove the parameterization variability and temporally register trajectories during analysis. We demonstrate this framework in multiple contexts, using both generative statistical models and discriminative data analysis. The latter is illustrated using several applications involving video-based action recognition and dynamic functional connectivity analysis.

Item Type: Article
Additional Information: This is a post-peer-review, pre-copyedit version of an article published in Journal of Mathematical Imaging and Vision. The final authenticated version is available online at: http://dx.doi.org/10.1007/s10851-018-0814-0
Schools/Departments: University of Nottingham, UK > Faculty of Science > School of Mathematical Sciences
Identification Number: https://doi.org/10.1007/s10851-018-0814-0
Depositing User: Eprints, Support
Date Deposited: 18 Apr 2018 13:55
Last Modified: 24 Apr 2019 04:30
URI: http://eprints.nottingham.ac.uk/id/eprint/51245

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