planar curve evolution ron kimmel ron computer science department technion-israel institute of...
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Planar Curve EvolutionRon Kimmel
www.cs.technion.ac.il/~ron
Computer Science Department Technion-Israel Institute of Technology
Geometric Image Processing Lab
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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
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Arc-length and Curvature
s(p)= | |dp 0
p
| | 1,sC pC | |p
sp
CC
C
ssC N
1
ssC N
C
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Invariant arclength should be
1. Re-parameterization invariant
2. Invariant under the group of transformations
drCCCFdpCCCFw rrrppp ,...,,,...,,
Geometric measure
Transform
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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
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Curvature flow
Euclidean geometric heat equation nCCC sssst
where
nCt
flow
Euclideantransform
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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
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Important property
Tangential components do not affect the geometry of an evolving curve
nnVCVC tt
,
V
nnV
,
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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
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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
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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
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Constant flow
Offset curves
nCt
Change in topology
Shock
Cusp
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Area inside C
dpCCA p,21 Area is defined via
C
pC
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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
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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
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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
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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
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Equi-Affine flow ( )31V
37
35
32
31
34
292
312
00
00
ssssst
LL
t
LL
t
VV
dsVdsA
dsVdsLLength
Area
Curvature
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Geodesic active contours
nnyxgyxgCt
),,(),(
Goldenberg, Kimmel, Rivlin, Rudzsky,
IEEE T-IP 2001
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Tracking in color movies
Goldenberg, Kimmel, Rivlin, Rudzsky, IEEE T-IP 2001
nnyxgyxgCt
),,,(),,(
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From curve to surface evolution
It’s a bit more than invariant measures…
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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
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Surface evolution
Tangential velocity has no influence on the geometry
Mean curvature flow,area minimizing
NNVt
SV
t
S ,
NNV
, V
NHt
S
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Segmentation in 3D
Change in topology
Caselles,Kimmel, Sapiro, Sbert, IEEE T-PAMI 97
nnzyxgHzyxgSt
),,,(),,(
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Conclusions
Constant flow, geometric heat equations Euclidean Equi-affine Other data dependent flows
Surface evolution
www.cs.technion.ac.il/~ron