level set methods and fast marching methods · 2005-10-25 · conclusion: level set and fast...

Level Set Methods and Fast Marching Methods

ILyulinaScientific Computing Group

May 1 2002

Overview

Existing Techniques for Tracking Interfaces

Basic Ideas of Level Set Method and Fast Marching Method

Linking moving fronts and hyperbolic conservation laws

Tracking a moving boundaryLagrangian approach

s

x(st=0)y(st=0)

discreteparameterization of the curveparameterization of the curve

(x(st)y(st))

How to deal with topological changes

Tracking a moving boundaryEulerian approach

Volume-of-fluid method

5111110

7111120

8111199

8111111

7111111

6111111

2323795

Drawbacks-- approximation to the front is crude a large number of cells-- curvature and normal is difficult to derive-- in 3D very complicated to perform

Level set and Fast marching methods

Sethian J A Level Set Methods and Fast Marching Methods Evolving Interfaces in Geometry Fluid Mechanics Computer Vision and Material Science Cambridge University Press 1999

httpmathberkeleyedu~sethianlevel_sethtml

Level Set Methodan initial value formulation

y

x

x

y

φ(xyt)

φ=0

F=F(LGI)

original front level set function

How do you move the front

Why is this called an ldquoinitial value formulationrdquo

Level set equation

x(t) (x(t)t)= ( 0 ) 0t x txφφ φ part prime+ sdot =part

If front moves in normal direction

( )

0 ( 0 )t

n F n x t

F IC x t

φφ

φ φ φ

nabla prime= = sdotnabla

+ nabla = =

If front is advected by velocity field

( )0 0

( 0 )t t x y

F u vF u v

IC x tφ φ φ φ φ

φ

=

+ sdot nabla = + sdot + sdot =

=

Fast Marching Methoda boundary value formulation

1

1 0

d Td x F d T Fd x

F T T o n

= sdot =

nabla = = Γ

x

T(x)dx

dT

Construction of stationary level set solution

Summary

Boundary Value Formulation Initial Value Formulation

1

( ) ( ) ( )0

T FFront

t x y T x y tF

nabla =

Γ = =

gt

0

( ) ( ) ( ) 0

t FFront

t x y x y tF arbitrary

φ φ

φ

+ nabla =

Γ = =

Advantages of these perspectives

Unchanged in higher dimensionsTopological changes are handled naturallyGeometric properties are are easily determined

Tn or nT

k

φφ

φφ

nabla nabla= =nabla nabla

nabla= nabla sdot

nabla

normal vector

curvature

Both equations can be accurately solvedusing numerical schemes for hyperbolicconservation laws

Hamilton-Jacobi equation

Level set equation and stationary equation are particularcases of the more general Hamilton-Jacobi equation

( ) 0( ) (1 )

( )10

t

x y z

u H D u xH D u x F uD uH u u u x y z

αα

αα

+ =

= nabla minus minus

==

level set equation

stationary equation

partial derivatives of u in each variable

Hamiltonian

Example viscosity solutions

Smooth front constant speed function F=1

The swallowtail solution The leading wave solution

1 0F kε ε= minus sdot gtSpeed function in the form

0

( ) ( )

lim ( ) ( )cu rva tu re con st

cu rva tu re con st

X t X t

X t X t

ε

εε rarr =

two solutions then

Link between propagating fronts and hyperbolic conservation laws

( )t x x xu H u uα ε+ =Hamilton-Jacobi equation with viscosity

[ ]( ) 0t xu G u+ =Hyperbolic conservation law

0t x

t x xx

u uuu uu uε+ =+ =

Burgersrsquo equation

Burgersrsquo equation with viscosity

Conclusion Level set and Fast marching methods rely on viscosity solutions of the associated partial differential equations in order to guarantee that unique entropy-satisfying weak solution is obtained Both equations can be accurately solved using numerical schemes for hyperbolic conservation laws

Next lectures

Efficient numerical algorithms for the Level Set and Fast Marching methods

Applications of Level Set and Fast Marching methods


Overview


Tracking a moving boundary Eulerian approach







Summary





Next lectures

Overview

Existing Techniques for Tracking Interfaces

Basic Ideas of Level Set Method and Fast Marching Method

Linking moving fronts and hyperbolic conservation laws


s

x(st=0)y(st=0)


(x(st)y(st))




5111110

7111120

8111199

8111111

7111111

6111111

2323795






y

x

x

y

φ(xyt)

φ=0

F=F(LGI)




Level set equation



( )

0 ( 0 )t

n F n x t

F IC x t

φφ

φ φ φ


+ nabla = =


( )0 0

( 0 )t t x y

F u vF u v


φ

=


=


1

1 0

d Td x F d T Fd x

F T T o n

= sdot =

nabla = = Γ

x

T(x)dx

dT


Summary


1

( ) ( ) ( )0

T FFront

t x y T x y tF

nabla =

Γ = =

gt

0

( ) ( ) ( ) 0

t FFront


φ φ

φ

+ nabla =

Γ = =



Tn or nT

k

φφ

φφ


nabla= nabla sdot

nabla

normal vector

curvature




( ) 0( ) (1 )

( )10

t

x y z


αα

αα

+ =

= nabla minus minus

==

level set equation

stationary equation


Hamiltonian





0

( ) ( )


cu rva tu re con st

X t X t

X t X t

ε

εε rarr =

two solutions then




0t x

t x xx

u uuu uu uε+ =+ =




Next lectures




Overview









Summary





Next lectures


s

x(st=0)y(st=0)


(x(st)y(st))




5111110

7111120

8111199

8111111

7111111

6111111

2323795






y

x

x

y

φ(xyt)

φ=0

F=F(LGI)




Level set equation



( )

0 ( 0 )t

n F n x t

F IC x t

φφ

φ φ φ


+ nabla = =


( )0 0

( 0 )t t x y

F u vF u v


φ

=


=


1

1 0

d Td x F d T Fd x

F T T o n

= sdot =

nabla = = Γ

x

T(x)dx

dT


Summary


1

( ) ( ) ( )0

T FFront

t x y T x y tF

nabla =

Γ = =

gt

0

( ) ( ) ( ) 0

t FFront


φ φ

φ

+ nabla =

Γ = =



Tn or nT

k

φφ

φφ


nabla= nabla sdot

nabla

normal vector

curvature




( ) 0( ) (1 )

( )10

t

x y z


αα

αα

+ =

= nabla minus minus

==

level set equation

stationary equation


Hamiltonian





0

( ) ( )


cu rva tu re con st

X t X t

X t X t

ε

εε rarr =

two solutions then




0t x

t x xx

u uuu uu uε+ =+ =




Next lectures




Overview









Summary





Next lectures



5111110

7111120

8111199

8111111

7111111

6111111

2323795






y

x

x

y

φ(xyt)

φ=0

F=F(LGI)




Level set equation



( )

0 ( 0 )t

n F n x t

F IC x t

φφ

φ φ φ


+ nabla = =


( )0 0

( 0 )t t x y

F u vF u v


φ

=


=


1

1 0

d Td x F d T Fd x

F T T o n

= sdot =

nabla = = Γ

x

T(x)dx

dT


Summary


1

( ) ( ) ( )0

T FFront

t x y T x y tF

nabla =

Γ = =

gt

0

( ) ( ) ( ) 0

t FFront


φ φ

φ

+ nabla =

Γ = =



Tn or nT

k

φφ

φφ


nabla= nabla sdot

nabla

normal vector

curvature




( ) 0( ) (1 )

( )10

t

x y z


αα

αα

+ =

= nabla minus minus

==

level set equation

stationary equation


Hamiltonian





0

( ) ( )


cu rva tu re con st

X t X t

X t X t

ε

εε rarr =

two solutions then




0t x

t x xx

u uuu uu uε+ =+ =




Next lectures




Overview









Summary





Next lectures





y

x

x

y

φ(xyt)

φ=0

F=F(LGI)




Level set equation



( )

0 ( 0 )t

n F n x t

F IC x t

φφ

φ φ φ


+ nabla = =


( )0 0

( 0 )t t x y

F u vF u v


φ

=


=


1

1 0

d Td x F d T Fd x

F T T o n

= sdot =

nabla = = Γ

x

T(x)dx

dT


Summary


1

( ) ( ) ( )0

T FFront

t x y T x y tF

nabla =

Γ = =

gt

0

( ) ( ) ( ) 0

t FFront


φ φ

φ

+ nabla =

Γ = =



Tn or nT

k

φφ

φφ


nabla= nabla sdot

nabla

normal vector

curvature




( ) 0( ) (1 )

( )10

t

x y z


αα

αα

+ =

= nabla minus minus

==

level set equation

stationary equation


Hamiltonian





0

( ) ( )


cu rva tu re con st

X t X t

X t X t

ε

εε rarr =

two solutions then




0t x

t x xx

u uuu uu uε+ =+ =




Next lectures




Overview









Summary





Next lectures



Level set equation



( )

0 ( 0 )t

n F n x t

F IC x t

φφ

φ φ φ


+ nabla = =


( )0 0

( 0 )t t x y

F u vF u v


φ

=


=


1

1 0

d Td x F d T Fd x

F T T o n

= sdot =

nabla = = Γ

x

T(x)dx

dT


Summary


1

( ) ( ) ( )0

T FFront

t x y T x y tF

nabla =

Γ = =

gt

0

( ) ( ) ( ) 0

t FFront


φ φ

φ

+ nabla =

Γ = =



Tn or nT

k

φφ

φφ


nabla= nabla sdot

nabla

normal vector

curvature




( ) 0( ) (1 )

( )10

t

x y z


αα

αα

+ =

= nabla minus minus

==

level set equation

stationary equation


Hamiltonian





0

( ) ( )


cu rva tu re con st

X t X t

X t X t

ε

εε rarr =

two solutions then




0t x

t x xx

u uuu uu uε+ =+ =




Next lectures




Overview









Summary





Next lectures


1

1 0

d Td x F d T Fd x

F T T o n

= sdot =

nabla = = Γ

x

T(x)dx

dT


Summary


1

( ) ( ) ( )0

T FFront

t x y T x y tF

nabla =

Γ = =

gt

0

( ) ( ) ( ) 0

t FFront


φ φ

φ

+ nabla =

Γ = =



Tn or nT

k

φφ

φφ


nabla= nabla sdot

nabla

normal vector

curvature




( ) 0( ) (1 )

( )10

t

x y z


αα

αα

+ =

= nabla minus minus

==

level set equation

stationary equation


Hamiltonian





0

( ) ( )


cu rva tu re con st

X t X t

X t X t

ε

εε rarr =

two solutions then




0t x

t x xx

u uuu uu uε+ =+ =




Next lectures




Overview









Summary





Next lectures


Summary


1

( ) ( ) ( )0

T FFront

t x y T x y tF

nabla =

Γ = =

gt

0

( ) ( ) ( ) 0

t FFront


φ φ

φ

+ nabla =

Γ = =



Tn or nT

k

φφ

φφ


nabla= nabla sdot

nabla

normal vector

curvature




( ) 0( ) (1 )

( )10

t

x y z


αα

αα

+ =

= nabla minus minus

==

level set equation

stationary equation


Hamiltonian





0

( ) ( )


cu rva tu re con st

X t X t

X t X t

ε

εε rarr =

two solutions then




0t x

t x xx

u uuu uu uε+ =+ =




Next lectures




Overview









Summary





Next lectures



Tn or nT

k

φφ

φφ


nabla= nabla sdot

nabla

normal vector

curvature




( ) 0( ) (1 )

( )10

t

x y z


αα

αα

+ =

= nabla minus minus

==

level set equation

stationary equation


Hamiltonian





0

( ) ( )


cu rva tu re con st

X t X t

X t X t

ε

εε rarr =

two solutions then




0t x

t x xx

u uuu uu uε+ =+ =




Next lectures




Overview









Summary





Next lectures



( ) 0( ) (1 )

( )10

t

x y z


αα

αα

+ =

= nabla minus minus

==

level set equation

stationary equation


Hamiltonian





0

( ) ( )


cu rva tu re con st

X t X t

X t X t

ε

εε rarr =

two solutions then




0t x

t x xx

u uuu uu uε+ =+ =




Next lectures




Overview









Summary





Next lectures





0

( ) ( )


cu rva tu re con st

X t X t

X t X t

ε

εε rarr =

two solutions then




0t x

t x xx

u uuu uu uε+ =+ =




Next lectures




Overview









Summary





Next lectures




0t x

t x xx

u uuu uu uε+ =+ =




Next lectures




Overview









Summary





Next lectures

Next lectures




Overview









Summary





Next lectures

level set methods and fast marching methods · 2005-10-25 · conclusion: level set and fast...

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