TY - JOUR
T1 - Rate-Invariant Analysis of Covariance Trajectories
AU - Zhang, Zhengwu
AU - Su, Jingyong
AU - Klassen, Eric
AU - Le, Huiling
AU - Srivastava, Anuj
N1 - Publisher Copyright:
© 2018, Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2018/10/1
Y1 - 2018/10/1
N2 - 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.
AB - 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.
KW - Covariance trajectories
KW - Invariant metrics
KW - Rate-invariant classification
KW - SPDM Riemannian structure
KW - SPDM parallel transport
KW - Vector bundles
UR - http://www.scopus.com/inward/record.url?scp=85045943528&partnerID=8YFLogxK
U2 - 10.1007/s10851-018-0814-0
DO - 10.1007/s10851-018-0814-0
M3 - Article
AN - SCOPUS:85045943528
SN - 0924-9907
VL - 60
SP - 1306
EP - 1323
JO - Journal of Mathematical Imaging and Vision
JF - Journal of Mathematical Imaging and Vision
IS - 8
ER -