Biomedical Engineering Reference
In-Depth Information
and then compute its derivative which is also zero,
∂φ
∂
t
+
∂φ
∂
x
∂
x
∂
t
+
∂φ
∂
y
∂
y
∂
t
=
0
,
(9.51)
Converting the terms to the dot product form of the gradient vector and the
x
and
y
derivatives vector, we get
∂φ
∂
x
∂
t
,
∂
y
∂φ
∂
t
+
∂
x
,
∂φ
=
0
.
(9.52)
.
∂
y
∂
t
Multiplying and dividing by
|∇
φ
|
and takeing the other part to be
F
, we get the
following equation:
∂φ
∂
t
+
F
|∇
φ
|=
0
,
(9.53)
Where
F
, the speed function, is given by
∂φ
∂
x
∂
x
,
∂φ
∂
t
,
∂
y
F
=
.
/
|∇
φ
|
.
(9.54)
∂
y
∂
t
The selection of the speed function is very important to keep the change of
the front smooth and also it is application dependent. Equation 9.55 represents
speed function containing the mean curvature
k
. The positive sign means that
the front is shrinking and the negative sign means that the front is expanding
and
is selected to be a small value for smoothness. The curvature term allows
the front to merge and break and also handles sharp corners,
F
=±
1
−
k
,
(9.55)
Where
k
is given by
y
x
k
=
φ
xx
φ
−
2
φ
x
φ
y
φ
xy
+
φ
yy
φ
(9.56)
.
(
φ
x
y
)
3
/
2
+
φ
In 3D, the front will be an evolving surface rather than an evolving curve.
9.5.3 Stability and CFL Restriction
The numerical solution of the partial differential equation (PDE) describing the
front is very important to be accurate and stable. For simplicity, Taylor's series
expansion is used to handle the partial derivatives of
φ
as listed below,
φ
(
x
,
y
,
t
+
t
)
=
φ
(
x
,
y
,
t
)
−
tF
|∇
φ
|
,
(9.57)
φ
x
(
x
,
y
,
t
)
=
(
φ
(
x
+
x
,
y
,
t
)
−
φ
(
x
,
y
,
t
))
/
x
,
(9.58)
