Biomedical Engineering Reference
In-Depth Information
The evaluation of
g
T
included in coefficients (17) can be done in several ways.
First, we may replace the convolution by the weighted average to get
I
σ
:=
G
σ
∗
I
0
(see, e.g., [26]), and then relate discrete values of
I
σ
to voxel centers. Then,
as above, we may construct its piecewise linear representation on the grid and
get a constant value of
g
T
≡
g
(
|∇
I
σ
|
) on every tetrahedron
T
∈T
h
. Another
possibility is to solve numerically the linear heat equation for time
t
corresponding
to variance
σ
with the initial datum given by
I
0
(see, e.g., [30]) by the same method
as above. The convolution represents a preliminary smoothing of the data. It is also
a theoretical tool to have bounded gradients and thus a strictly positive weighting
coefficient
g
0
. In practice, the evaluation of gradients on a fixed discrete grid (e.g.,
described above) always gives bounded values. So, working on a fixed grid, one
can also avoid the convolution, especially if preliminary denoising is not needed or
not desirable. Then it is possible to work directly with gradients of the piecewise
linear representation of
I
0
in evaluation of
g
T
.
A change in the L
2
norm of numerical solutions in subsequent time steps is
used to stop the segmentation process. We check whether
u
n−
1
m
(
p
)(
u
p
−
)
2
<δ,
(23)
p
p
with a prescribed threshold
δ
. For our semi-implicit scheme and small
ε
, a good
choice of threshold is
δ
=10
−
5
.
We start all computations with an initial function given as a peak centered in
a “focus point” inside the segmented object. Such a function can be described at a
sphere with center
s
and radius
R
by
u
0
(
x
)=
1
|x−s|
+
v
, where
s
is the focus point
and
v
gives a maximum of
u
0
. Outside the sphere we take the value
u
0
equal to
1
R
+
v
.
R
usually corresponds to the halved inner diameter of the image domain.
For small objects, a smaller
R
can be used to speed up computations. Usually we
put the focus point
s
inside a small neighborhood of a center of the mass of the
segmented object.
3.3. Semi-Implicit 3DCo-Volume Scheme inFinite-DifferenceNotation
The presented co-volume scheme is designed for the specific mesh given by
the cubic voxel structure of a 3D image. For simplicity of implementation, reader
convenience, and due to the relation to the next section devoted to parallelization,
we will write the co-volume scheme (18) in a “finite-difference notation.” As is
usual for 3D rectangular grids, we associate co-volume
p
and its center (DF node)
with a triple (
i, j, k
),
i
representing the index in the
x
-direction,
j
in the
y
-direction,
and
k
in the
z
-direction (see Figure 7 for our convention of coordinate notation).
The unknown value
u
p
then can be denoted by
u
i,j,k
.IfΩ is a rectangular sub-
domain of the image domain (usually Ω is the image domain itself) and
N
1
+1,
N
2
+1,
N
3
+1are the numbers of voxels of Ω in the
x, y, z
-directions, and if we
