We wish to endow the manifold M of smooth curves in Rn with a Riemannian metric that allows us to treat continuous morphs (homotopies) between two curves c0 and c1 as trajectories with computable lengths which are independent of the parameterization or representation of the two curves (and the curves making up the morph between them). We may then define the distance between the two curves using the trajectory of minimal length (geodesic) between them, assuming such a minimizing trajectory exists. At first we attempt to utilize the metric structure implied rather unanimously by the past twenty years or so of shape optimization literature in computer vision. This metric arises as the unique metric which validates the common references to a wide variety of contour evolution models in the literature as "gradient flows" to various formulated energy functionals. Surprisingly, this implied metric yields a pathological and useless notion of distance between curves. In this paper, we show how this metric can be minimally modified using conformal factors that depend upon a curve's total arclength. A nice property of these new conformal metrics is that all active contour models that have been called "gradient flows" in the past will constitute true gradient flows with respect to these new metrics under specific time reparameterizations.

Conformal metrics and true "gradient flows" for curves

MENNUCCI, Andrea Carlo Giuseppe
2005

Abstract

We wish to endow the manifold M of smooth curves in Rn with a Riemannian metric that allows us to treat continuous morphs (homotopies) between two curves c0 and c1 as trajectories with computable lengths which are independent of the parameterization or representation of the two curves (and the curves making up the morph between them). We may then define the distance between the two curves using the trajectory of minimal length (geodesic) between them, assuming such a minimizing trajectory exists. At first we attempt to utilize the metric structure implied rather unanimously by the past twenty years or so of shape optimization literature in computer vision. This metric arises as the unique metric which validates the common references to a wide variety of contour evolution models in the literature as "gradient flows" to various formulated energy functionals. Surprisingly, this implied metric yields a pathological and useless notion of distance between curves. In this paper, we show how this metric can be minimally modified using conformal factors that depend upon a curve's total arclength. A nice property of these new conformal metrics is that all active contour models that have been called "gradient flows" in the past will constitute true gradient flows with respect to these new metrics under specific time reparameterizations.
2005
IEEE International Conference on Computer Vision and Pattern Recognition
Beijing, PEOPLES R CHINA
OCT 17-20, 2005
Computer Vision, 2005. ICCV 2005. Tenth IEEE International Conference
9780769523347
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11384/6499
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