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ter Haar Romeny, FEV 2005 Curve Evolution Rein van den Boomgaard, Bart ter Haar Romeny Univ. van Amsterdam, Eindhoven University of technology

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ter Haar Romeny, FEV 2005

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Alvarez et al. introduced the following evolution equation: In Cartesian coordinates:

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Geometrical Reasoning Choose a deformation function β such that: The deformation is invariant under the symmetry group one is interested in (e.g. rotational invariant, scale invariance, affine invariance); The shape deformation relates to our goal (chosen a priori) like `smoothing', 'shape simplification' etc. Notes: Curve deformation has little to do with an observation scale-space. Diffusing curves (or even observing curves) is problematic from a perception point of view (but it makes perfectly sense as a mathematical model to capture shape properties).

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Sethian: Fast Level Sets

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Euclidean shortening flow: Working out the derivatives, this is L vv in gauge coordinates, i.e. the ridge detector. So: Because the Laplacian is we get We see that we have corrected the normal diffusion with a factor proportional to the second order derivative in the gradient direction (in gauge coordinates: ). This subtractive term cancels the diffusion in the direction of the gradient.

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ter Haar Romeny, FEV 2005 This diffusion scheme is called Euclidean shortening flow due to the shortening of the isophotes, when considered as a curve-evolution scheme. Advantage: there is no parameter k. Disadvantage: rounding of corners.

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