Shape Analysis of Elastic Curves in Euclidean Spaces

By Anuj Srivastava, Eric Klassen, Shantanu H. Joshi, Ian H. Jermyn. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2011

672 citations recorded by the source corpus.

This paper introduces a square-root velocity (SRV) representation for analyzing shapes of curves in euclidean spaces under an elastic metric. In this SRV representation, the elastic metric simplifies to the IL2 metric, the reparameterization group acts by isometries, and the space of unit length curves becomes the unit sphere. The shape space of closed curves is the quotient space of (a submanifold of) the unit sphere, modulo rotation, and reparameterization groups, and we find geodesics in that space using a path straightening approach. These geodesics and geodesic distances provide a framework for optimally matching, deforming, and comparing shapes. These ideas are demonstrated using: 1) shape analysis of cylindrical helices for studying protein structure, 2) shape analysis of facial curves for recognizing faces, 3) a wrapped probability distribution for capturing shapes of planar closed curves, and 4) parallel transport of deformations for predicting shapes from novel poses.

Subfields: Computer Vision and Pattern Recognition, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/377415d1b8e328c7cf42cd5016dedcac09beb992

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Citations in the corpus

Rolling Rotations for Recognizing Human Actions from 3D Skeletal Data (2016, background): “We note that the recently proposed transported squareroot vector field [30] representation of curves, which is an extension of the earlier square-root velocity representation [29] to Riemannian manifolds, provides a distance metric that is invariant to temporal warping (i,.”
Human Action Recognition by Shape Analysis of Motion Trajectories on Riemannian Manifold (2014, neutral): “As shown in [37], the SRVF of γ ◦β is given by √γ̇ (t)(q◦γ )(t).”
Statistical analysis of trajectories on Riemannian manifolds: Bird migration, hurricane tracking and video surveillance (2014, methodology): “Srivastava et al. (2011b) proposed a solution that applies to curves in arbitrary Rn.”
Space-Time Pose Representation for 3D Human Action Recognition (2013, neutral): “As shown in [14], the SRVF of γ ◦ β is given by √ γ̇(t)(q ◦ γ)(t).”
Action Recognition Using Rate-Invariant Analysis of Skeletal Shape Trajectories (2016, methodology): “Some of these datasets are also used in the experimental studies described later.”
Functional data analysis for density functions by transformation to a Hilbert space (2016, methodology): “In the context of functional data and shape analysis, such time warping functions have been represented as square roots of the corresponding densities [42, 43, 44], and these square root densities reside in the Hilbert sphere, about which much is known.”
Accurate 3D action recognition using learning on the Grassmann manifold (2015, methodology): “In the rest of the section, we present a statistical computation of the mean in the Grassmann manifold [55].”
Rate-Invariant Analysis of Trajectories on Riemannian Manifolds with Application in Visual Speech Recognition (2014, methodology): “If M = R n with the Euclidean metric then h is exactly the square-root velocity function defined in [13].”