Planar Curve Evolution Ron Kimmel www.cs.technion.ac.il/ ~ron Computer Science Department Technion-Israel Institute of Technolog Geometric Image Processing Lab
Dec 21, 2015
Planar Curve EvolutionRon Kimmel
www.cs.technion.ac.il/~ron
Computer Science Department Technion-Israel Institute of Technology
Geometric Image Processing Lab
Planar Curves C(p)={x(p),y(p)}, p [0,1]
y
x
C(0)
C(0.1) C(0.2)
C(0.4)
C(0.7)
C(0.95)
C(0.9)
C(0.8)
pC =tangent
Arc-length and Curvature
s(p)= | |dp 0
p
| | 1,sC pC | |p
sp
CC
C
ssC N
1
ssC N
C
Invariant arclength should be
1. Re-parameterization invariant
2. Invariant under the group of transformations
drCCCFdpCCCFw rrrppp ,...,,,...,,
Geometric measure
Transform
Euclidean arclength
Length is preserved, thus ,
dpCdpdydxdp
dpdydxds pdp
dydp
dx 222222
ds dy
dx
1sC
dpCs p
L
ppp dsdpCCdpCL0
1
0
1
0
21
,Length Total
Curvature flow
Euclidean geometric heat equation nCCC sssst
where
nCt
flow
Euclideantransform
Curvature flow
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).
nCt
Gage-Hamilton
Grayson
Given any simple planar curve
First becomes convex
Vanish at aCircular point
Important property
Tangential components do not affect the geometry of an evolving curve
nnVCVC tt
,
V
nnV
,
1Area
Reminder: Equi-affine arclength
Area is preserved, thus
vC
vvC
dpCCv ppp
31
,
1, vvv CC
dsdsCCv sss
31
31
,
dsdv 31
re-parameterizationinvariance
Affine heat equation
Special (equi-)affine heat flow
31
, where, nCnnCC vvvvt
nCt
31
Sapiro
Given any simple planar curve
First becomes convex
Vanish at anelliptical
pointflow
Affinetransform
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)
nCt
Constant flow
Offset curves
nCt
Change in topology
Shock
Cusp
Area inside C
dpCCA p,21 Area is defined via
C
pC
So far we defined
Constant flow Curvature flow Equi-affine flow
We would like to explore evolution properties of measures like curvature, length, and area
nCt
nCtnCt
31
1
0
21
, dpCCL pp dpCCA p,21
For
L
ppp VdsdpCt
CdpCCt
dpCCt
At 0
21 ...,,,
nVCt
VV
CC
CC
tt ss
pp
ppp 2...,
,2
3
12
0
, , ...L
p p p p pL C C dp C C C dp Vdstt t
0
0
2
L
t
L
t
t ss
L Vds
A Vds
V V
Length
Area
Curvature
Constant flow ( )1V
22
00
00
2
VV
LdsVdsA
dsVdsL
sst
LL
t
LL
tLength
Area
Curvature
The curve vanishes at 2)0(Lt
)0,(1)0,(),( pt
ptp Riccati eq.
Singularity (`shock’) at
)0,( pt
Curvature flow ( )V
32
00
0
2
0
2
sssst
LL
t
LL
t
VV
dsVdsA
dsVdsLLength
Area
Curvature
The curve vanishes at 2)0(At
Equi-Affine flow ( )31V
37
35
32
31
34
292
312
00
00
ssssst
LL
t
LL
t
VV
dsVdsA
dsVdsLLength
Area
Curvature
Geodesic active contours
nnyxgyxgCt
),,(),(
Goldenberg, Kimmel, Rivlin, Rudzsky,
IEEE T-IP 2001
Tracking in color movies
Goldenberg, Kimmel, Rivlin, Rudzsky, IEEE T-IP 2001
nnyxgyxgCt
),,,(),,(
From curve to surface evolution
It’s a bit more than invariant measures…
Surface
A surface, For example, in 3D
Normal
Area element Total area
2 M: 2 nS nR
),(),,(),,(),( vuzvuyvuxvuS
vu
vu
SS
SSN
N
uS
vS
u vdA S S dudv
dudvSSA vu
Surface evolution
Tangential velocity has no influence on the geometry
Mean curvature flow,area minimizing
NNVt
SV
t
S ,
NNV
, V
NHt
S
Segmentation in 3D
Change in topology
Caselles,Kimmel, Sapiro, Sbert, IEEE T-PAMI 97
nnzyxgHzyxgSt
),,,(),,(
Conclusions
Constant flow, geometric heat equations Euclidean Equi-affine Other data dependent flows
Surface evolution
www.cs.technion.ac.il/~ron