Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods...
Transcript of Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods...
![Page 2: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/2.jpg)
Overview
Introduction
Boundary Value Formulation
Initial Value Formulation
Numerical schemes
Fast Marching Methods
Narrow Band Level Set
Level Set methods – p.2/24
![Page 3: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/3.jpg)
Overview
Introduction
Boundary Value Formulation
Initial Value Formulation
Numerical schemes
Fast Marching Methods
Narrow Band Level Set
Level Set methods – p.2/24
![Page 4: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/4.jpg)
Overview
Introduction
Boundary Value Formulation
Initial Value Formulation
Numerical schemes
Fast Marching Methods
Narrow Band Level Set
Level Set methods – p.2/24
![Page 5: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/5.jpg)
Overview
Introduction
Boundary Value Formulation
Initial Value Formulation
Numerical schemes
Fast Marching Methods
Narrow Band Level Set
Level Set methods – p.2/24
![Page 6: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/6.jpg)
Overview
Introduction
Boundary Value Formulation
Initial Value Formulation
Numerical schemes
Fast Marching Methods
Narrow Band Level Set
Level Set methods – p.2/24
![Page 7: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/7.jpg)
Overview
Introduction
Boundary Value Formulation
Initial Value Formulation
Numerical schemes
Fast Marching Methods
Narrow Band Level Set
Level Set methods – p.2/24
![Page 8: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/8.jpg)
Introduction
moving interface problem
Local propertiesGlobal properties
Independent properties
Level Set methods – p.3/24
![Page 9: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/9.jpg)
Introduction
moving interface problemSpeed function � �� � � � �
Local propertiesGlobal properties
Independent properties
Level Set methods – p.3/24
![Page 10: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/10.jpg)
Introduction
moving interface problemSpeed function � � � � � � �Local properties
Global properties
Independent properties
Level Set methods – p.3/24
![Page 11: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/11.jpg)
Introduction
moving interface problemSpeed function � � � � � � �Local propertiesGlobal properties
Independent properties
Level Set methods – p.3/24
![Page 12: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/12.jpg)
Introduction
moving interface problemSpeed function � � � � � � �Local propertiesGlobal properties
Independent properties
Level Set methods – p.3/24
![Page 13: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/13.jpg)
Formulations
Boundary Value Formulation
Initial Value Formulation
Level Set methods – p.4/24
![Page 14: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/14.jpg)
Boundary Value Formulation
Speed function � �
Arrival Time
interface is given by
Level Set methods – p.5/24
![Page 15: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/15.jpg)
Boundary Value Formulation
Speed function � �
Arrival Time
interface is given by
��� � � �
Level Set methods – p.5/24
![Page 16: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/16.jpg)
Boundary Value Formulation
distance � rate · time
� �
���
More dimensions:
|∇ | � �
with � �
on�
Level Set methods – p.6/24
![Page 17: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/17.jpg)
Example Boundary Value Formulation
�
is circular, � �
:
Level Set methods – p.7/24
![Page 18: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/18.jpg)
Initial Value Formulation
Level Set Function
��� � � �� � �
Interface is given by
��� � � �� � � � �
.
Level Set methods – p.8/24
![Page 19: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/19.jpg)
Initial Value Formulation
level set function
�� � � �� � � � �
By the chain rule:
� ∇
��� � � �� � �
· � ′ � � � � �
is speed in outward normal direction, then′ · , ∇
|∇ |
|∇ |
given
Level Set methods – p.9/24
![Page 20: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/20.jpg)
Initial Value Formulation
level set function
�� � � �� � � � �
By the chain rule:
� ∇
��� � � �� � �
· � ′ � � � � �
is speed in outward normal direction, then
� ′
� � �
· � � , � � ∇
�
|∇
�
|
|∇ |
given
Level Set methods – p.9/24
![Page 21: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/21.jpg)
Initial Value Formulation
level set function
�� � � �� � � � �
By the chain rule:
� ∇
��� � � �� � �
· � ′ � � � � �
is speed in outward normal direction, then
� ′
� � �
· � � , � � ∇
�
|∇
�
|
� |∇ | � �
given
��� � � � � �
Level Set methods – p.9/24
![Page 22: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/22.jpg)
Example Initial Value Formulation
Level Set methods – p.10/24
![Page 23: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/23.jpg)
Summary
Boundary Value Formulation ( � �
)
|∇ | � �Front =
� � � �
= {
�� � ��
|��� � �� � �
}
Initial Value Formulation
|∇ |
Front = = { | }
Level Set methods – p.11/24
![Page 24: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/24.jpg)
Summary
Boundary Value Formulation ( � �
)
|∇ | � �Front =
� � � �
= {
�� � ��
|��� � �� � �
}
Initial Value Formulation
� |∇ | � �
Front =� � � �
= {
��� � ��
|
��� � � � � � � �
}
Level Set methods – p.11/24
![Page 25: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/25.jpg)
Numerical Scheme for BVP
|∇ |
��� � � �
front:
� � {� |
��� � � �
}
−
where − −− Backward Scheme
and − Forward Scheme.
Level Set methods – p.12/24
![Page 26: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/26.jpg)
Numerical Scheme for BVP
|∇ |
��� � � �
front:
� � {� |
��� � � �
}
� � �� �
− �� � � � � � � � � �� � � � � �� � ����
where − �� � �� − ��
−
�� � Backward Scheme
and
� �� � ��� �− ��� � Forward Scheme.
Level Set methods – p.12/24
![Page 27: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/27.jpg)
Numerical Scheme for IVP
� |∇ | � �
front:
� � {� |
��� � � � � �
}
− ∇ ∇−
where
∇ −
∇− −
Level Set methods – p.13/24
![Page 28: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/28.jpg)
Numerical Scheme for IVP
� |∇ | � �
front:
� � {� |
��� � � � � �
}
� � �� � � �− � � � �� � �� � �∇
� � � �� � �
∇−
�
where
∇
� � � � � � � − �� � � � � � � � � �� � � � � �
� �
∇− � � � � � � � �� � � � � � � � − �� � � � � �
��
Level Set methods – p.13/24
![Page 29: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/29.jpg)
CFL condition
Time Step restriction
� �� �
≤ �
Level Set methods – p.14/24
![Page 30: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/30.jpg)
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
![Page 31: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/31.jpg)
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
![Page 32: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/32.jpg)
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
![Page 33: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/33.jpg)
Fast Marching Method
Beginning of Fast Marching MethodLevel Set methods – p.16/24
![Page 34: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/34.jpg)
Fast Marching Method
Update “downwind”Level Set methods – p.16/24
![Page 35: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/35.jpg)
Fast Marching Method
A
B
D
C
Compute new possible valuesLevel Set methods – p.16/24
![Page 36: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/36.jpg)
Fast Marching Method
A
B
D
C
Choose smallest dark pink sphere (A)Level Set methods – p.16/24
![Page 37: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/37.jpg)
Fast Marching Method
A
B
D
C
Freeze value at A, update neighboring downwindpoints
Level Set methods – p.16/24
![Page 38: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/38.jpg)
Fast Marching Methods
Initialisation: Tag points as Known, Trial andFar .
Level Set methods – p.17/24
![Page 39: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/39.jpg)
Fast Marching Methods
Begin Loop: Let A be the Trial point with thesmallest T value
Level Set methods – p.17/24
![Page 40: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/40.jpg)
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
![Page 41: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/41.jpg)
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
![Page 42: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/42.jpg)
Fast Marching Methods
Recompute the values of T at all Trialneighbors of A according to
� �� − ���� � −� �� �� � �
�
� �� − � � �� � −� � � �� � �
�
� � � − � � �� � −
� � � �� � �
� ���
�� ��
Level Set methods – p.17/24
![Page 43: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/43.jpg)
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
![Page 44: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/44.jpg)
Narrow Band Level Set
Level Set methods – p.18/24
![Page 45: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/45.jpg)
Narrow Band Level Set
Level Set methods – p.18/24
![Page 46: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/46.jpg)
Narrow Band Level Set
Level Set methods – p.18/24
![Page 47: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/47.jpg)
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
![Page 48: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/48.jpg)
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
![Page 49: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/49.jpg)
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
![Page 50: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/50.jpg)
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
![Page 51: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/51.jpg)
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
![Page 52: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/52.jpg)
Narrow Band Level Set
Level Set methods – p.20/24
![Page 53: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/53.jpg)
Narrow Band Level Set
Level Set methods – p.21/24
![Page 54: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/54.jpg)
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
![Page 55: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/55.jpg)
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
![Page 56: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/56.jpg)
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
![Page 57: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/57.jpg)
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
![Page 58: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/58.jpg)
Next lecture
Applications
Level Set methods – p.23/24
![Page 59: Level Set methods - win.tue.nl€¦ · Fast Marching Method Update “downwind” Level Set methods – p.16/24. Fast Marching Method A B D C Compute new possible values Level Set](https://reader034.fdocuments.in/reader034/viewer/2022050302/5f6af7a999f7582d7f58fe2f/html5/thumbnails/59.jpg)
End
Level Set methods – p.24/24