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