Biomedical Engineering Reference
In-Depth Information
proceeds, the particle locations are periodically checked to determine whether
they have strayed across the level set function interface.
When a particle is determined to have strayed sufficiently far across the level
set interface, the interface is reconstructed using the particle information. To do
this, each particle,
p
, located at the point
x
p
, is assigned a local signed distance
function
d
p
(
x
)
=
s
p
(
r
p
−
x
−
x
p
)
.
(4.79)
The level set function is now reconstructed in two steps. First, the functions
φ
+
and
φ
−
are computed where
φ
+
(
x
)
=
max
p
∈
P
+
d
p
(
x
)
,
(4.80)
φ
−
(
x
)
=
min
p
∈
P
−
d
p
(
x
)
,
(4.81)
and where
P
+
and
P
−
are the sets of points which were assigned positive and
negative
s
p
respectively. The final
φ
function is now recovered from
φ
+
and
φ
−
by the equation
φ
(
x
)
=
absmin(
φ
+
(
x
)
,φ
−
(
x
))
,
(4.82)
where
a
,
|
a
|
<
|
b
|
b
,
|
b
|≤|
a
|
.
absmin(
a
,
b
)
=
(4.83)
There is no guarantee that the resulting reconstructed level set function will
be a signed distance function, so if this is desired, a reinitialization step will be
applied to reform
φ
into a signed distance function.
What is novel about this approach is the use of the Lagrangian and Eulerian
methods to play against each other to ensure proper interface motion. However,
one must carefully determine when the particle solution is correct, versus the
level set evolution. This is determined by checking the local characteristics to
see if they are colliding or expanding. The level set evolution tends to be better
when characteristics are colliding, whereas the particle method will be more
reliable when the interface is moving tangentially or stretching. Nonetheless,
this combination tries to extract the positive capabilities of both the Lagrangian
and Eulerian types of approaches to interface motion, while discounting the
negatives.
