Video-Based Action Recognition Using Rate-Invariant Analysis of Covariance Trajectories

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

doi:10.48550/arxiv.1503.06699

Cited by 11 publications

(35 citation statements)

References 29 publications

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“…The SRV framework can be extended to curves in a Lie groupe using translations [11], and to curves in a general manifold using parallel transport. For manifoldvalued curves, this can be done in a way that enables to move the computations to the tangent space to the origin of one of the two curves under comparison [17], [29], [34]. In [17] the authors consider the general elastic metric G a,b , but no Riemannian framework is given.…”

Section: Introductionmentioning

confidence: 99%

“…In [17] the authors consider the general elastic metric G a,b , but no Riemannian framework is given. In [34], a Riemannian framework is given for the case a = 1, b = 1/2, and the geodesic equations are derived. In this paper we also restrict to this particular choice of coefficients a and b for simplicity, but we propose another generalization of the SRV framework to manifold-valued curves.…”

Section: Introductionmentioning

confidence: 99%

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Computing distances and geodesics between manifold-valued curves in the SRV framework

Brigant¹

2017

Journal of Geometric Mechanics

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This paper focuses on the study of open curves in a Riemannian manifold M , and proposes a reparametrization invariant metric on the space of such paths. We use the square root velocity function (SRVF) introduced by Srivastava et al. in [28] to define a Riemannian metric on the space of immersions M = Imm([0, 1], M ) by pullback of a natural metric on the tangent bundle TM. This induces a first-order Sobolev metric on M and leads to a distance which takes into account the distance between the origins in M and the L 2 -distance between the SRV representations of the curves. The geodesic equations for this metric are given and exploited to define an exponential map on M. The optimal deformation of one curve into another can then be constructed using geodesic shooting, which requires to characterize the Jacobi fields of M. The particular case of curves lying in the hyperbolic half-plane H is considered as an example, in the setting of radar signal processing.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Computing distances and geodesics between manifold-valued curves in the SRV framework

Brigant¹

2017

Journal of Geometric Mechanics

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“…Given any two points [q 0 ] and [q 1 ] in S o , a tangent representative (Absil et al, 2008) in the tangent space T [q 0 ] (S o ) can be calculated in the following way, as suggested in (Zhang et al, 2015c;Su and Srivastava, 2014) based on (4),…”

Section: Then All the Orbitsmentioning

confidence: 99%

“…where q 1 is the representative of [q 1 ] given by the well-defined algorithm in (Zhang et al, 2015c;Su and Srivastava, 2014) and θ = cos −1 ( q 0 , q 1 ). In fact, v is the lifting representation of abstract tangent vector log…”

Section: Then All the Orbitsmentioning

confidence: 99%

Tractable Clustering of Data on the Curve Manifold

Tierney¹,

Gao²,

Guo³

et al. 2017

Preprint

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In machine learning it is common to interpret each data point as a vector in Euclidean space. However the data may actually be functional i.e. each data point is a function of some variable such as time and the function is discretely sampled. The naive treatment of functional data as traditional multivariate data can lead to poor performance since the algorithms are ignoring the correlation in the curvature of each function. In this paper we propose a tractable method to cluster functional data or curves by adapting the Euclidean Low-Rank Representation (LRR) to the curve manifold. Experimental evaluation on synthetic and real data reveals that this method massively outperforms prior clustering methods in both speed and accuracy when clustering functional data.

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“…This family of metrics is actually a subfamily of the general class of reparameterization invariant Sobolev metrics, as studied in [5,6,4]. It is also a natural generalization of the family of elastic G a,b -metrics on the space of curves [22], which has been proven efficient and successful in numerous shape analysis applications [24,30,31,25,26,32,9,27].…”

Section: Introductionmentioning

confidence: 99%

Shape Analysis of Surfaces Using General Elastic Metrics

Su¹,

Bauer²,

Preston³

et al. 2019

Preprint

Self Cite

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In this article we introduce a family of elastic metrics on the space of parametrized surfaces in 3D space using a corresponding family of metrics on the space of vector valued one-forms. We provide a numerical framework for the computation of geodesics with respect to these metrics. The family of metrics is invariant under rigid motions and reparametrizations; hence it induces a metric on the "shape space" of surfaces. This new class of metrics generalizes a previously studied family of elastic metrics and includes in particular the M.

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Video-Based Action Recognition Using Rate-Invariant Analysis of Covariance Trajectories

Cited by 11 publications

References 29 publications

Computing distances and geodesics between manifold-valued curves in the SRV framework

Computing distances and geodesics between manifold-valued curves in the SRV framework

Tractable Clustering of Data on the Curve Manifold

Shape Analysis of Surfaces Using General Elastic Metrics

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