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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Next lectures
Efficient numerical algorithms for the Level Set and Fast Marching methods
Applications of Level Set and Fast Marching methods