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