Title:
A New Geometric Metric in the Space of Curves, and Applications To Tracking Deforming Objects by Prediction and Filtering

dc.contributor.author Sundaramoorthi, Ganesh en_US
dc.contributor.author Mennucci, Andrea C. en_US
dc.contributor.author Soatto, Stefano en_US
dc.contributor.author Yezzi, Anthony en_US
dc.contributor.corporatename University of California, Los Angeles. Computer Science Dept. en_US
dc.contributor.corporatename Scuola normale superiore (Italy) en_US
dc.contributor.corporatename Georgia Institute of Technology. School of Electrical and Computer Engineering en_US
dc.date.accessioned 2013-09-04T20:36:47Z
dc.date.available 2013-09-04T20:36:47Z
dc.date.issued 2011-02
dc.description ©2011 Society for Industrial and Applied Mathematics. Permalink: http://dx.doi.org/10.1137/090781139 en_US
dc.description DOI: 10.1137/090781139 en_US
dc.description.abstract We define a novel metric on the space of closed planar curves which decomposes into three intuitive components. According to this metric, centroid translations, scale changes, and deformations are orthogonal, and the metric is also invariant with respect to reparameterizations of the curve. While earlier related Sobolev metrics for curves exhibit some general similarities to the novel metric proposed in this work, they lacked this important three-way orthogonal decomposition, which has particular relevance for tracking in computer vision. Another positive property of this new metric is that the Riemannian structure that is induced on the space of curves is a smooth Riemannian manifold, which is isometric to a classical well-known manifold. As a consequence, geodesics and gradients of energies defined on the space can be computed using fast closed-form formulas, and this has obvious benefits in numerical applications. The obtained Riemannian manifold of curves is ideal for addressing complex problems in computer vision; one such example is the tracking of highly deforming objects. Previous works have assumed that the object deformation is smooth, which is realistic for the tracking problem, but most have restricted the deformation to belong to a finite-dimensional group—such as affine motions—or to finitely parameterized models. This is too restrictive for highly deforming objects such as the contour of a beating heart. We adopt the smoothness assumption implicit in previous work, but we lift the restriction to finite-dimensional motions/deformations. We define a dynamical model in this Riemannian manifold of curves and use it to perform filtering and prediction to infer and extrapolate not just the pose (a finitely parameterized quantity) of an object but its deformation (an infinite-dimensional quantity) as well. We illustrate these ideas using a simple first-order dynamical model and show that it can be effective even on image sequences where existing methods fail. en_US
dc.identifier.citation Sundaramoorthi, Ganesh; Mennucci, Andrea; Soatto, Stefano; Yezzi, Anthony, “A New Geometric Metric in the Space of Curves, and Applications to Tracking Deforming Objects by Prediction and Filtering,” SIAM Journal on Imaging Sciences, 4 (1), 109-145 (February 2011) en_US
dc.identifier.doi 10.1137/090781139
dc.identifier.issn 1936-4954
dc.identifier.uri http://hdl.handle.net/1853/48781
dc.language.iso en_US en_US
dc.publisher Georgia Institute of Technology en_US
dc.publisher.original Society for Industrial and Applied Mathematics en_US
dc.subject Visual object tracking en_US
dc.subject Shape analysis en_US
dc.subject Filtering in the space of curves en_US
dc.subject Sobolev-type metrics en_US
dc.title A New Geometric Metric in the Space of Curves, and Applications To Tracking Deforming Objects by Prediction and Filtering en_US
dc.type Text
dc.type.genre Article
dc.type.genre Post-print
dspace.entity.type Publication
local.contributor.author Yezzi, Anthony
local.contributor.corporatename School of Electrical and Computer Engineering
local.contributor.corporatename College of Engineering
relation.isAuthorOfPublication 53ee63a2-04fd-454f-b094-02a4601962d8
relation.isOrgUnitOfPublication 5b7adef2-447c-4270-b9fc-846bd76f80f2
relation.isOrgUnitOfPublication 7c022d60-21d5-497c-b552-95e489a06569
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