Level Set Formulationfor Curve Evolution

Ron Kimmel

www.cs.technion.ac.il/~ron

Computer Science Department Technion-Israel Institute of Technology

Geometric Image Processing Lab

Consider a closed planar curve

The geometric trace of the curve can be alternatively represented implicitly as

Implicit representation21:)( RS pC

}0),(|),{( yxyxC

0

0

1

1

Properties of level sets

The level set normal

Proof. Along the level sets we have zero change,

that is , but by the chain rule

So,

||

N

N

0s

Tyxyx sysxs

,),(

T

||||0,

||

NTT

||

T

Properties of level sets

The level set curvature

Proof. zero change along the level sets, , also

So,

||

div

0ss

NTds

dT

ds

dyx

ds

dyx sysxss

,,,)(),(

...||

,||

,,||

,||

,,,,

||],,[||

||,

yxyx

syysxysxysxx

TT

yxyx

1

Optical flow

Problem: find the velocity also known as `optical flow’It’s an `inverse’ problem,Given I(t) find

VII t

,

),( yxV

),( yxV

VtxItxI

dtVdtdtVOVdttxItxdtItxItxI

dtVOVdtdttxIdttxItxI

dttVdtxItxI

t

t

),,(),(

0),,(),,(),(),(),(

0)(),,(),(),(

0),(),(

2222

22

Aperture Problem

Aperture Problem

`Normal’ vertical flow

Horizontal flow can not be computed differentially.

Normal flow

Due to the `aperture problem’only the `normal’ velocity can be locally computed

for the normal flow we have

V

||,

I

IVNNVNV NN

||||

,,, IVI

IIVNVIVII NNNt

NVN

|| IVI Nt

Level Set Formulation

implicit representation of CThen,

Proof. By the chain rule

Then,

Recall that , and

RR 2:, yx 0),(:, yxyxC

dC d

VN Vdt dt

ttytx yxt

tyx

);,(

0

y

x

C(t)

C(t) level set 0

, ,x y t

x

y

NVNVCyx ttytxt

,,,

||

N

||||

,,

VVNV

|| Vt

Level Set Formulation

Handles changes in topology

Numeric grid points never collide or drift apart.

Natural philosophy for dealing with gray level images.

Numerical Considerations

Finite difference approximation. Order of approximation, truncation error, stencil. (Differential) conservation laws. Entropy condition and vanishing viscosity. Consistent, monotone, upwind scheme. CFL condition (stability examples)

Numerical Considerations

Central derivative

Forward derivative

Backward derivative

)(ihuui )(xu

x

h

1 1 iii

h

uuuD ii

x 211

h

uuuD ii

x

1

h

uuuD ii

x1

Truncation Error

Taylor expansion about x=ih

)()('

)()('

)()('

)()('')(')()(

)()('')(')()(

2

32!2

11

32!2

11

hOihuuD

hOihuuD

hOihuuD

hOihuhihhuihuhihuu

hOihuhihhuihuhihuu

ix

ix

ix

i

i

21

21

1 1

11

Stencils

Numerical Approximations

2

1,11,11,11,1

4h

uuuuuD jijijiji

xy

211 2

h

uuuuD iii

xx

1 2- 1

41

41

41

41

Conservation Law

Rate of change of the amount in a fixed domain G =

Flux across the boundaries of G

Differential conservation law

GG

dSnfudxdt

d ,

G

u

0 div fut

nnf

,

f

Generalized Solution 1D

In 1D

Weak solution satisfies

u

0),(),(),(),(

0

0 div

1

0

1

0

1

0

1

0

1

0

1

0

0101

dttxftxfdxtxutxu

dtdxfu

dtdxfu

t

t

x

x

t

t

x

x

xt

t

t

x

x

t

)),(()),((),( 10

1

0

txuHtxuHdxtxudt

dx

x

t

x

1t

0t

0x 1x

ff u

u

Hamilton-Jacobi

In 1D: HJ=Hyperbolic conservation lawsIn 2D: just the `flavor’…

Vanishing viscosity, of

The `entropy condition’ selected the `weak solution’

that is the `vanishing viscosity solution’ also known

as `entropy solution’.

xxxt uuHu )(0

lim

Nεκ

NCt

Numerical Schemes

Conservation form

Numerical flux

The scheme is monotone, if F is non-decreasing.

Theorem: A monotone, consistent scheme, in conservation form converges to the entropy solution.

Yet, up to 1st order accurate ;-( …

x

gg

t

uu nj

nj

nj

nj

2

12

11

),...,( 11 n

qjn

pjnj uuFu

)(),...,( ),,...,( 1121 uHuuguugg qjpj

nj

Upwind Monotone

Upwind scheme

For we have upwind-monotone schemes

we define Then, and the final scheme is

)()( 2uhuH

0' )(

0' )(

12

1HuH

HuHg

j

jnj

)))0,,((min(),(

)))0,max()0,((min(),(2

11

21

1

nj

nj

nj

njM

nj

nj

nj

njHJ

uuhuug

uuhuug

xxdtxutx ~),~(),(

1

nj

nj

nj

nj

Du

Du

),(1 nj

nj

nj

nj DDtg

CFL Stability Condition

At the limitFor 3-point scheme of

we need for the numerical domain of dependenceto include the PDE domain of dependence

0)( xt uHu

0 0, tx t

x1x

tx ~,~

domain of dependence

0x

domain of influence

x̂

'1 Hx

t

CFL Stability Condition

At the limitFor 3-point scheme of

we need for the numerical domain of dependenceto include the PDE domain of dependence

0)( xt uHu

0 0, tx t

x1x

tx ~,~

0x

'1 Hx

t

x

1D Example

SolutionCharacteristics dx/dt=1CFL condition

Numeric scheme

xt uu

)0,(),( txutxu t

x

tx 0

0x

x

t

1

ni

xni

t uDuD

1D Example

where

Characteristics

Numeric scheme

CFL condition

xt uxau )(

1 1

1 1)(

x

xxa

t

x

ta

x

)2()(

))0,min()0,(max(

112||

1121 n

ini

ni

ani

ni

axtn

ini

ni

xi

ni

xi

ni

t

uuuuuuu

uDauDauDii

1 1

1 1

x

x

dx

dt

1

xxxxi uauax ||||

Numerical viscosity

2D Example

Numeric scheme

CFL condition

t

2

1

h

t

2,1,1

12

22

)0,,min(max)0,,max(

)0,,max()0,,max(

nji

nji

nijh

nij

xnij

x

nij

ynij

ynij

xnij

xnij

t

DD

DDDDD

tt NC

2D Examples

Some flows

Vt

31

31

31 22

22

22

2div

2div

xyyyxxyyxxtt

yx

xyyyxxyyxxtt

tt

NC

NC

NC

require upwind/monotoneschemes

,2

div ,

22

22

gg

gNNggC

yx

xyyyxxyyxx

tt