Computer Science
Department
Technion-Israel Institute of Technology
Planar Curve Evolution
Ron Kimmel
www.cs.technion.ac.il/~ron
Geometric Image Processing Lab
Planar Curves
C(p)={x(p),y(p)},
C(0.1)
p  [0,1]
C(0.2)
C(0.7)
C(0)
y
C(0.4)
C(0.95)
C(0.8)
C(0.9)
x
C p =tangent
Arc-length and Curvature
p

s(p)= | C p |dp
0

| Cs | 1,

Css   N

Css   N

1

C

Cp 
 Cs 

|
C
|
p 

Invariant arclength should be
1. Re-parameterization invariant
Geometric measure
w   F C , C p , C pp ,...dp   F C , Cr , Crr ,...dr
2. Invariant under the group of transformations
Transform
Euclidean arclength
Length is preserved, thus
dp
ds  dx  dy 
dx 2  dy 2  dp
dp
2
2
Cs  1,
dx dp2  dy dp2  C p dp
s   C p dp
ds
dy
dx
1
1
Total Length L   C p dp   C p , C p
0
0
1
L
2
dp   ds
0

Curvature flow Ct  n
 Euclidean geometric heat equation
Ct  Css

where Css  n
Euclidean
transform
flow

Curvature flow Ct  n


Takes any simple curve into a circular
point
in finite time proportional to the area
inside
the curve
Embedding is preserved (embedded
curves
keep their order along the evolution).
First becomes
convex
Given any simple
planar curve
Grayson
Vanish at a
Circular
point
Gage-Hamilton
Important property
Tangential components do not affect the
geometry of an evolving curve

  
Ct  V  Ct  V , n n

V
  
V,n n
Reminder: Equi-affine arclength
re-parameterization
invariance
Area is preserved, thus Cv , Cvv   1
v   C p , C pp  dp
1
3
v   Cs , Css  ds    ds
1
Cv Area  1
Cvv
1
3
dv   3 ds
1
3
Affine heat equation

Special (equi-)affine heat flow Ct   n
1
 

1
Ct  Cvv , n n where Cvv , n   3
Affine
transform
flow
3
Given any simple
planar curve
Sapiro
First becomes
convex
Vanish at an
elliptical
point
Constant flow
Offset curves
Level sets of distance map
Equal-height contours of
the distance transform
Envelope of all disks of
equal radius centered
along the curve
(Huygens principle)

Ct  n
Constant flow

Ct  n
Offset curves
Change in topology
Cusp
Area inside C
Area is defined via
A  12  C , C p dp
C
Cp
So far we defined
Constant flow
Curvature flow
Equi-affine flow

Ct  n

Ct  n
1 
Ct   3 n
We would like to explore evolution properties
of measures like curvature, length, and area
1
L   Cp ,Cp
0
1
2
dp
A  12  C , C p dp
For

Ct  Vn
1



2
L   C p , C p dp  
C p , C p C p dp  ...    Vds
t
t
t
0
L








A  12  C , C p dp    C , C p dp    C , C p dp  ...    Vds
t
t
 t

 t 
0



  C p , C pp  
2
 

...

V


V
3
ss
2 
t
t  C , C

p
p


L
Length
L
Lt    Vds
0
Area
Curvature
L
At    Vds
0
 t  Vss   2V
Constant flow ( V  1)
Length
Area
L
L
0
0
Lt    Vds    ds  2
L
L
0
0
At    Vds    ds   L
Curvature  t  Vss   2V   2
The curve vanishes at
Riccati eq.
Singularity (`shock’) at
t
L (0)
2
 ( p, t )  1t( p( ,p0,)0)
t   ( p,0)
Curvature flow (V   )
Length
Area
L
L
0
0
Lt    Vds     2 ds
L
L
0
0
At    Vds    ds  2
Curvature  t  Vss   2V   ss   3
The curve vanishes at
t
A( 0 )
2
Equi-Affine flow (V   )
1
L
Length
L
Lt    Vds     3 ds
4
0
Area
0
L
L
At    Vds     3 ds
1
0
Curvature
0
 t  Vss   V    ss    s  
2
1
3
2
3
2
9
5
3
2
7
3
3
Geodesic active contours
 
Ct  g ( x, y)  g ( x, y), n n
Goldenberg, Kimmel, Rivlin, Rudzsky,
IEEE T-IP 2001
Tracking in color movies
 
Ct  g ( x, y, )  g ( x, y, ), n n
Goldenberg, Kimmel, Rivlin, Rudzsky,
IEEE T-IP 2001
From curve to surface evolution
It’s a bit more than invariant measures…
Surface
A surface, S :   R 2  M n
For example, in 3D
n2
S (u, v)  x(u, v), y(u, v), z(u, v)
Normal
 Su  Sv
N
Su  Sv
Area element dA  Su  Sv dudv
Total area
A   Su  S v dudv

N
Sv
Su
Surface evolution
  
V,N N
Tangential velocity has
no influence on the geometry
  
S 
S
V 
 V,N N
t
t
Mean curvature flow,
area minimizing

S
 HN
t

V
Segmentation in 3D
 
St  g ( x, y, z ) H  g ( x, y, z ), n n
Change in topology
Caselles,Kimmel, Sapiro, Sbert, IEEE T-PAMI 97
Conclusions
Constant flow, geometric heat equations
Euclidean
Equi-affine
Other data dependent flows
Surface evolution
www.cs.technion.ac.il/~ron