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Rate-Invariant Analysis of Covariance Trajectories

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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 execution rates of actions, implying arbitrary parameterizations of trajectories, complicate their analysis. To handle this challenge, we represent covariance trajectories using transported square-root vector fields, 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 space of this vector bundle. This metric is invariant to the action of the re-parameterization group, and leads to a rate-invariant analysis of trajectories. In the process, we remove the parameterization variability and temporally register trajectories. 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.

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Author information

Authors and Affiliations

  1. Department of Biostatistics and Computational Biology, University of Rochester, Rochester, NY, USA

    Zhengwu Zhang

  2. Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX, USA

    Jingyong Su

  3. Department of Mathematics, Florida State University, Tallahassee, FL, USA

    Eric Klassen

  4. School of Mathematical Sciences, University of Nottingham, Nottingham, UK

    Huiling Le

  5. Department of Statistics, Florida State University, Tallahassee, FL, USA

    Anuj Srivastava

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  1. Zhengwu Zhang
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  2. Jingyong Su
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  3. Eric Klassen
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  4. Huiling Le
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  5. Anuj Srivastava
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Correspondence to Zhengwu Zhang.

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Zhang, Z., Su, J., Klassen, E. et al. Rate-Invariant Analysis of Covariance Trajectories. J Math Imaging Vis 60, 1306–1323 (2018). https://doi.org/10.1007/s10851-018-0814-0

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  • DOI: https://doi.org/10.1007/s10851-018-0814-0

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