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Issue No.07 - July (2011 vol.33)
pp: 1415-1428
Anuj Srivastava , Florida State University, Tallahassee
Eric Klassen , Florida State University, Tallahassee
Shantanu H. Joshi , UCLA School of Medicine, Los Angeles
Ian H. Jermyn , Durham University, Durham
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 {\hbox{\rlap{I}\kern 2.0pt{\hbox{L}}}}^2 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.
Elastic curves, Riemannian shape analysis, elastic metric, Fisher-Rao metric, square-root representations, path straightening method, elastic geodesics, parallel transport, shape models.
Anuj Srivastava, Eric Klassen, Shantanu H. Joshi, Ian H. Jermyn, "Shape Analysis of Elastic Curves in Euclidean Spaces", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.33, no. 7, pp. 1415-1428, July 2011, doi:10.1109/TPAMI.2010.184
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