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Rate-invariant analysis of covariance trajectories

Zhang, Zhengwu; Su, Jingyong; Klassen, Eric; Le, Huiling; Srivastava, Anuj


Zhengwu Zhang

Jingyong Su

Eric Klassen

Professor of Probability

Anuj Srivastava


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.


Zhang, Z., Su, J., Klassen, E., Le, H., & Srivastava, A. (2018). Rate-invariant analysis of covariance trajectories. Journal of Mathematical Imaging and Vision, 60(8), 1306–1323. doi:10.1007/s10851-018-0814-0

Journal Article Type Article
Acceptance Date Mar 31, 2018
Online Publication Date Apr 24, 2018
Publication Date 2018-10
Deposit Date Apr 18, 2018
Publicly Available Date Apr 25, 2019
Journal Journal of Mathematical Imaging and Vision
Print ISSN 0924-9907
Electronic ISSN 1573-7683
Publisher Springer Verlag
Peer Reviewed Peer Reviewed
Volume 60
Issue 8
Pages 1306–1323
Public URL
Publisher URL
Copyright Statement Copyright information regarding this work can be found at the following address: http://eprints.nottingh.../end_user_agreement.pdf
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:


Rate-Invariant Analysis (5.2 Mb)

Copyright Statement
Copyright information regarding this work can be found at the following address:

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