Biomedical Engineering Reference
In-Depth Information
Since the 3D spline is the tensor product of independent 1D polynomial
functions, its second-order derivative with respect to one variable,
x
, is easily
computed as follows:
d
2
du
2
B
l
(
u
)
B
m
(
v
)
B
n
(
w
)
φ
i
+
l
,
j
+
m
,
k
+
n
.
(11.6)
3
3
3
2
∂
x
2
T
(
x
,
y
,
z
)
=
∂
1
δ
x
l
=
0
m
=
0
n
=
0
Computation of the derivatives of
T
is in fact very similar to computing
T
itself.
Depending on the derivation variable, the spline polynomials
B
0
through
B
3
in the
respective dimension are simply replaced by their respective derivatives. These
derivatives are easily computed analytically. Mixed second-order derivatives
with respect to two different variables are computed by substituting two spline
polynomials with their respective first-order derivatives, e.g.,
2
∂
x
∂
y
T
(
x
,
y
,
z
)
∂
d
du
B
l
(
u
)
d
B
m
(
v
)
B
n
(
w
)
φ
i
+
l
,
j
+
m
,
k
+
n
.
3
3
3
1
δ
x
δ
y
=
(11.7)
d
v
l
=
0
m
=
0
n
=
0
Using the above derivative terms, the continuous integral in Eq. (11.5) is ap-
proximated as a discretely sampled sum over a set of points, for example, the
N
D
=
n
x
×
n
y
×
n
z
voxels in the reference image.
11.4
Atlas Selection Strategies
This section will take a closer look at possible choices for atlases in atlas-based
segmentation. Usually, this aspect of atlas-based segmentation receives little
attention. Yet, the decision about what atlas to use has a substantial impact on
the segmentation accuracy, and simple methods are not always the best as we
will see below.
We describe and compare here four different atlas-based segmentation
strategies with different atlas selections: segmentation with one single individ-
ual atlas, segmentation with varying single individual atlases, segmentation with
an average shape atlas, and simultaneous segmentation with multiple atlases.
These four strategies can be categorized according to the number of atlases used
per raw image (one or multiple), the type of atlas used (individual or average),
and the assignment of atlases to raw images (fixed, i.e., same atlas(es) for all
raw images, or variable, i.e., different atlas image selected for each raw image).
