Information Technology Reference
In-Depth Information
fV
ðÞ
ðÞ
V
i
V
fV
r
fV
ðÞ
i
¼
ð
7
Þ
j
j
The vector components in Eqs. (
5
)
(
7
) are relative to local orthonormal frames
-
de
ned at each vertex V. The gradients
rw
and
r
g in Eq. (
1
) are computed using
Eqs. (
5
)
rw
is then used to compute the mean curvature
j
.
The gradient
-
(
7
).
rw
is used to form two functions on the mesh:
w
x
V
ðÞ
and
w
y
V
ðÞ
,
which respectively associate the x and y components of
rw
to each vertex V.We
can then evaluate the gradients of
w
x
V
(
7
), which in
turn yields
w
xx
,
w
xy
and
w
yy
. Those are the quantities necessary to compute the mean
curvature
j
of the distance function
w
:
ðÞ
and
w
y
V
ðÞ
using Eqs. (
5
)
-
w
xx
w
y
2
w
x
w
y
w
xy
þ w
yy
w
x
w
x
þ w
y
j ¼
ð
8
Þ
3
2
The seeding for the mesh level set is also derived from the user placed seed for
the vertebral body segmentation. From that seed (roughly in the center of the
vertebral body), a vertical line cuts the upper and lower end plates in two points.
Those points are used as the seeds for the mesh level sets on the upper and lower
end plates. An alternative seeding technique without user input and that relies on
the clustering of vertices with low curvedness has recently been proposed [
48
].
Figure
5
shows an example of contour evolution on the upper end plate of a
vertebra. Figure
6
shows several examples of
final segmentation results.
2.3 Vertebral Body Registration
Ideally, ridgelines detected on different scans of the same vertebra should be located
at identical positions at the junction where the syndesmophytes merge with the end
plates. In reality, those positions can be subject to variations, especially for syn-
desmophytes that do not grow at a right angle in respect to the end plate but
laterally and merge with the end plate in a smooth gradual junction. In such cases,
the curvature at the junction can be low and the level set might stop at the syn-
desmophyte
s base or slightly leak into the syndesmophyte depending on differing
image resolution, sharpness or noise. Figure
7
shows such a discrepancy between
baseline and year 1 ridgelines, with a small leak at year 1 (red arrow). Bone above
the ridgeline will be labeled as syndesmophyte. If the syndesmophytes at baseline
and year 1 were cut from their respective ridgelines, the leak in the year 1 syn-
desmophyte would cause a de
'
cit in volume compared to baseline. This difference
would not be due to real syndesmophyte change. Because real growth may be
small, it is important to reduce the error coming from ridgeline discrepancies. We
use registration to correct such inconsistencies. Registration aligns the vertebral
