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Level Set methodsSandra Allaart-Bruin
sbruin@win.tue.nl
Level Set methods – p.1/24
Overview
Introduction
Boundary Value Formulation
Initial Value Formulation
Numerical schemes
Fast Marching Methods
Narrow Band Level Set
Level Set methods – p.2/24
Overview
Introduction
Boundary Value Formulation
Initial Value Formulation
Numerical schemes
Fast Marching Methods
Narrow Band Level Set
Level Set methods – p.2/24
Overview
Introduction
Boundary Value Formulation
Initial Value Formulation
Numerical schemes
Fast Marching Methods
Narrow Band Level Set
Level Set methods – p.2/24
Overview
Introduction
Boundary Value Formulation
Initial Value Formulation
Numerical schemes
Fast Marching Methods
Narrow Band Level Set
Level Set methods – p.2/24
Overview
Introduction
Boundary Value Formulation
Initial Value Formulation
Numerical schemes
Fast Marching Methods
Narrow Band Level Set
Level Set methods – p.2/24
Overview
Introduction
Boundary Value Formulation
Initial Value Formulation
Numerical schemes
Fast Marching Methods
Narrow Band Level Set
Level Set methods – p.2/24
Introduction
moving interface problem
Local propertiesGlobal properties
Independent properties
Level Set methods – p.3/24
Introduction
moving interface problemSpeed function � �� � � � �
Local propertiesGlobal properties
Independent properties
Level Set methods – p.3/24
Introduction
moving interface problemSpeed function � � � � � � �Local properties
Global properties
Independent properties
Level Set methods – p.3/24
Introduction
moving interface problemSpeed function � � � � � � �Local propertiesGlobal properties
Independent properties
Level Set methods – p.3/24
Introduction
moving interface problemSpeed function � � � � � � �Local propertiesGlobal properties
Independent properties
Level Set methods – p.3/24
Formulations
Boundary Value Formulation
Initial Value Formulation
Level Set methods – p.4/24
Boundary Value Formulation
Speed function � �
Arrival Time
interface is given by
Level Set methods – p.5/24
Boundary Value Formulation
Speed function � �
Arrival Time
interface is given by
��� � � �
Level Set methods – p.5/24
Boundary Value Formulation
distance � rate · time
� �
���
More dimensions:
|∇ | � �
with � �
on�
Level Set methods – p.6/24
Example Boundary Value Formulation
�
is circular, � �
:
Level Set methods – p.7/24
Initial Value Formulation
Level Set Function
��� � � �� � �
Interface is given by
��� � � �� � � � �
.
Level Set methods – p.8/24
Initial Value Formulation
level set function
�� � � �� � � � �
By the chain rule:
� ∇
��� � � �� � �
· � ′ � � � � �
is speed in outward normal direction, then′ · , ∇
|∇ |
|∇ |
given
Level Set methods – p.9/24
Initial Value Formulation
level set function
�� � � �� � � � �
By the chain rule:
� ∇
��� � � �� � �
· � ′ � � � � �
is speed in outward normal direction, then
� ′
� � �
· � � , � � ∇
�
|∇
�
|
|∇ |
given
Level Set methods – p.9/24
Initial Value Formulation
level set function
�� � � �� � � � �
By the chain rule:
� ∇
��� � � �� � �
· � ′ � � � � �
is speed in outward normal direction, then
� ′
� � �
· � � , � � ∇
�
|∇
�
|
� |∇ | � �
given
��� � � � � �
Level Set methods – p.9/24
Example Initial Value Formulation
Level Set methods – p.10/24
Summary
Boundary Value Formulation ( � �
)
|∇ | � �Front =
� � � �
= {
�� � ��
|��� � �� � �
}
Initial Value Formulation
|∇ |
Front = = { | }
Level Set methods – p.11/24
Summary
Boundary Value Formulation ( � �
)
|∇ | � �Front =
� � � �
= {
�� � ��
|��� � �� � �
}
Initial Value Formulation
� |∇ | � �
Front =� � � �
= {
��� � ��
|
��� � � � � � � �
}
Level Set methods – p.11/24
Numerical Scheme for BVP
|∇ |
��� � � �
front:
� � {� |
��� � � �
}
−
where − −− Backward Scheme
and − Forward Scheme.
Level Set methods – p.12/24
Numerical Scheme for BVP
|∇ |
��� � � �
front:
� � {� |
��� � � �
}
� � �� �
− �� � � � � � � � � �� � � � � �� � ����
where − �� � �� − ��
−
�� � Backward Scheme
and
� �� � ��� �− ��� � Forward Scheme.
Level Set methods – p.12/24
Numerical Scheme for IVP
� |∇ | � �
front:
� � {� |
��� � � � � �
}
− ∇ ∇−
where
∇ −
∇− −
Level Set methods – p.13/24
Numerical Scheme for IVP
� |∇ | � �
front:
� � {� |
��� � � � � �
}
� � �� � � �− � � � �� � �� � �∇
� � � �� � �
∇−
�
where
∇
� � � � � � � − �� � � � � � � � � �� � � � � �
� �
∇− � � � � � � � �� � � � � � � � − �� � � � � �
��
Level Set methods – p.13/24
CFL condition
Time Step restriction
� �� �
≤ �
Level Set methods – p.14/24
Notes
Extensions to higher order schemes
Schemes for non-convex speed function
These Numerical Schemes costs a lot ofcomputations
Level Set methods – p.15/24
Notes
Extensions to higher order schemes
Schemes for non-convex speed function
These Numerical Schemes costs a lot ofcomputations
Level Set methods – p.15/24
Notes
Extensions to higher order schemes
Schemes for non-convex speed function
These Numerical Schemes costs a lot ofcomputations
Level Set methods – p.15/24
Fast Marching Method
Beginning of Fast Marching MethodLevel Set methods – p.16/24
Fast Marching Method
Update “downwind”Level Set methods – p.16/24
Fast Marching Method
A
B
D
C
Compute new possible valuesLevel Set methods – p.16/24
Fast Marching Method
A
B
D
C
Choose smallest dark pink sphere (A)Level Set methods – p.16/24
Fast Marching Method
A
B
D
C
Freeze value at A, update neighboring downwindpoints
Level Set methods – p.16/24
Fast Marching Methods
Initialisation: Tag points as Known, Trial andFar .
Level Set methods – p.17/24
Fast Marching Methods
Begin Loop: Let A be the Trial point with thesmallest T value
Level Set methods – p.17/24
Fast Marching Methods
Begin Loop: Let A be the Trial point with thesmallest T value
Add the point A to Known ; remove it fromTrial
Level Set methods – p.17/24
Fast Marching Methods
Begin Loop: Let A be the Trial point with thesmallest T value
Add the point A to Known; remove it fromTrial
Tag as Trial all neighbors of A that are notKnown . If neighbor is in Far , remove andadd to the set Trial .
Level Set methods – p.17/24
Fast Marching Methods
Recompute the values of T at all Trialneighbors of A according to
� �� − ���� � −� �� �� � �
�
� �� − � � �� � −� � � �� � �
�
� � � − � � �� � −
� � � �� � �
� ���
�� ��
Level Set methods – p.17/24
Fast Marching Methods
Begin Loop: Let A be the Trialpoint with thesmallest T value
Add the point A to Known ; remove it fromTrial
Tag as Trial all neighbors of A that are notKnown . If neighbor is in Far , remove andadd to the set Trial .
Recompute the values of T at all Trialneighbors of A according to . . .
Return to top of loop.Level Set methods – p.17/24
Narrow Band Level Set
Level Set methods – p.18/24
Narrow Band Level Set
Level Set methods – p.18/24
Narrow Band Level Set
Level Set methods – p.18/24
Narrow Band Level Set
Tag Alive points in narrow band
Build Land Mines to indicate near edge
Initialize Far Away points outside the narrowband with large positive (negative) values ifvalues are outside (inside) the front itself
Solve level set equation until Land Mine hit
Rebuild and loop
Level Set methods – p.19/24
Narrow Band Level Set
Tag Alive points in narrow band
Build Land Mines to indicate near edge
Initialize Far Away points outside the narrowband with large positive (negative) values ifvalues are outside (inside) the front itself
Solve level set equation until Land Mine hit
Rebuild and loop
Level Set methods – p.19/24
Narrow Band Level Set
Tag Alive points in narrow band
Build Land Mines to indicate near edge
Initialize Far Away points outside the narrowband with large positive (negative) values ifvalues are outside (inside) the front itself
Solve level set equation until Land Mine hit
Rebuild and loop
Level Set methods – p.19/24
Narrow Band Level Set
Tag Alive points in narrow band
Build Land Mines to indicate near edge
Initialize Far Away points outside the narrowband with large positive (negative) values ifvalues are outside (inside) the front itself
Solve level set equation until Land Mine hit
Rebuild and loop
Level Set methods – p.19/24
Narrow Band Level Set
Tag Alive points in narrow band
Build Land Mines to indicate near edge
Initialize Far Away points outside the narrowband with large positive (negative) values ifvalues are outside (inside) the front itself
Solve level set equation until Land Mine hit
Rebuild and loop
Level Set methods – p.19/24
Narrow Band Level Set
Level Set methods – p.20/24
Narrow Band Level Set
Level Set methods – p.21/24
Advantages of Narrow Band Level Set
Speed
instead of in 3D
TimestepCFL condition on narrow band instead ofentire domain
≤
Level Set methods – p.22/24
Advantages of Narrow Band Level Set
Speed
� � � �
instead of
� � �
in 3D
TimestepCFL condition on narrow band instead ofentire domain
≤
Level Set methods – p.22/24
Advantages of Narrow Band Level Set
Speed
� � � �
instead of
� � �
in 3D
Timestep
CFL condition on narrow band instead ofentire domain
≤
Level Set methods – p.22/24
Advantages of Narrow Band Level Set
Speed
� � � �
instead of
� � �
in 3D
TimestepCFL condition on narrow band instead ofentire domain
� �� ≤ �
Level Set methods – p.22/24
Next lecture
Applications
Level Set methods – p.23/24
End
Level Set methods – p.24/24
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