Ter Haar Romeny, FEV 2005 Curve Evolution Rein van den Boomgaard, Bart ter Haar Romeny Univ. van...
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Transcript of Ter Haar Romeny, FEV 2005 Curve Evolution Rein van den Boomgaard, Bart ter Haar Romeny Univ. van...
ter Haar Romeny, FEV 2005
Curve EvolutionRein van den Boomgaard, Bart ter Haar Romeny
Univ. van Amsterdam, Eindhoven University of technology
ter Haar Romeny, FEV 2005
ter Haar Romeny, FEV 2005
Alvarez et al. introducedthe following evolutionequation:
Lt L LLIn Cartesian coordinates:
Lt Lx
2L y y 2LxL yLx y L y2Lx x
Lx2 L y2
ter Haar Romeny, FEV 2005
ter Haar Romeny, FEV 2005
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
ter Haar Romeny, FEV 2005
ter Haar Romeny, FEV 2005
ter Haar Romeny, FEV 2005
ter Haar Romeny, FEV 2005
ter Haar Romeny, FEV 2005
ter Haar Romeny, FEV 2005
ter Haar Romeny, FEV 2005
ter Haar Romeny, FEV 2005
ter Haar Romeny, FEV 2005
ter Haar Romeny, FEV 2005
ter Haar Romeny, FEV 2005
Sethian: Fast Level Sets
ter Haar Romeny, FEV 2005
ter Haar Romeny, FEV 2005
Ls
L.
LLEuclidean shortening flow:
Lyy Lx2 2 Lxy Ly Lx Lxx Ly
2
Lx2 Ly
2
Working out the derivatives, this is Lvv in gauge coordinates,i.e. the ridge detector.
So: Ls
Lv v
L Lx x Ly y Lv v Lww
Ls
L Lww
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
ter Haar Romeny, FEV 2005
ter Haar Romeny, FEV 2005
Name of
flow
Luminance
evolution
Curve
evolution
Timestep
N.N.
Timestep
Gauss der.
Linear Lt
L Ct
LL x22D
2s
Variable
conductance
Lt
.cL C
t
.cLL x2
2D2s
Normal or
constant motion
Lt
cLwCt
cN x
c
Euclidean
shortening
Lt
LvvCt
N x2
22s
Affine
shortening
Lt
Lvv13 Lw
23 C
t
13 N
Affine shortening
modified
Lt
Lvv 13 Lw
23
1Lwk23 2
Ct
13 Lw
k 2
3 N
Entropy Lt
0Lw 1LvvCt
0 1N,x2
2, 2s
ter Haar Romeny, FEV 2005http://www.yoshii.com/zbrush/function03.html