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

ter Haar Romeny, FEV 2005

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

ter Haar Romeny, FEV 2005

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).

ter Haar Romeny, FEV 2005

Sethian: Fast Level Sets

ter Haar Romeny, FEV 2005

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.

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.

ter Haar Romeny, FEV 2005

http://www.yoshii.com/zbrush/function03.html

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