Biomedical Engineering Reference
In-Depth Information
x
i−1, j+2
x
i, j+2
x
i+1, j+2
x
i+2, j+2
x
i−1, j+1
x
i, j+1
x
i+1, j+1
x
i+2, j+1
x
i−1, j
x
i, j
x
i+1, j
x
i+2, j
x
i−1, j−1
x
i, j−1
x
i+1, j−1
x
i+2, j−1
Figure 4.7: Sample portion of the mesh where a bicubic interpolation is used.
This figure reprinted from [22].
accuracy for the distance to the zero level set, subgrid resolution of the shape
of the interface, as well as subgrid resolution of the level set function
φ
(
x
)
itself.
We begin with a description of the bicubic interpolation for a level set func-
tion given on a rectangular mesh. The approximation is done locally in a box of
the mesh bounded by grid points, call them
x
i
,
j
,
x
i
+
1
,
j
,
x
i
,
j
+
1
, and
x
i
+
1
,
j
+
1
,asin
Fig. 4.7.
A bicubic interpolation
p
(
x
) of a function
φ
(
x
) is a function
3
3
a
m
,
n
x
m
y
n
p
(
x
)
=
p
(
x
,
y
)
=
,
(4.26)
m
=
0
n
=
0
which solves the following set of equations:
p
(
x
k
,
)
=
φ
(
x
k
,
)
∂
p
∂
x
(
x
k
,
)
=
∂φ
∂
x
(
x
k
,
)
∂
p
∂
y
(
x
k
,
)
=
∂φ
∂
y
(
x
k
,
)
2
p
2
∂
x
∂
y
(
x
k
,
)
=
∂
∂
φ
∂
x
∂
y
(
x
k
,
)
