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    Properties of Sobolev-type metrics in the space of curves

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    ifb_Properties_of_Sobolev-type_metrics.pdf (297.5Kb)
    Date
    2008
    Author
    Mennucci, Andrea C.
    Yezzi, Anthony
    Sundaramoorthi, Ganesh
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    Abstract
    We define a manifold M where objects c ϵ M are curves, which we parameterize as c : S¹ → R ⁿ (n ≥2, S¹ is the circle). Given a curve c, we define the tangent space TcM of M at c including in it all deformations h : S¹ → R ⁿ of c. We study geometries on the manifold of curves, provided by Sobolev–type Riemannian metrics H[superscript j]. We initially present some mathematical examples to show how the metrics H[superscript j] simplify or regularize gradient flows used in Computer Vision applications. We then provide some basilar results of Hj metrics; and, for the cases j = 1, 2, we characterize the completion of the space of smooth curves; we call this completion(s) “H¹ and H² Sobolev–type Riemannian Manifolds of Curves”. As a byproduct, we prove that the Fréchet distance of curves (see [MM06b]) coincides with the distance induced by the “Finsler L H [superscript ∞] metric” defined in §2.2 in [YM04b].
    URI
    http://hdl.handle.net/1853/48931
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    • Laboratory of Computational Computer Vision Publications [106]

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