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two level sets based on edge features: the geodesic active contour (GAC) [
44
] and
what we call for convenience the classical level set (CLS) [
43
]. The GAC evolves
according to the equation [
44
]:
d
w
dt
¼ a
g
ðÞ
~
x
c
j
rw
j þ b
g
ðÞj rw
~
x
j
j þ cr
g
ðÞrw
~
x
ð
1
Þ
Contours encoded as the zero level set of a distance function
w ~
ðÞ
x
;
t
: points that
verify
w ~
0 form the contour. The three terms on the right-hand side of the
equation respectively control the expansion or contraction of the contour (velocity c),
the smoothness of the contour using the mean curvature
j
and the adherence of the
contour to the boundary of the object to be segmented. The last term, often called
advection term, is speci
ðÞ¼
x
;
t
c to the GAC and is responsible for its robustness to gaps in
s boundary. The parameters
a
,
b
and
c
allow the user to weight the
importance of each term. The spatial function g
'
an object
ðÞ
~
, often called speed function, is
derived from the images to be segmented and contains information about the objects
x
'
boundaries. The design of the speed function is crucial for the success of the seg-
mentation. Depending on the speci
c needs of the application, information on the
object
s boundary can be based on image gradient, Laplacian or any other relevant
feature. The CLS is equivalent to the GAC without the advection term. The omission
of the advection term makes the CLS more
'
flexible.
A vertebral body is composed of trabecular bone surrounded by denser cortical
bone. Syndesmophytes are made of cortical bone (Fig.
1
). To capture those different
components, we adopted a multistage strategy in which successive level sets seg-
ment the trabecular and cortical bone. Our algorithm is also multiscale. It was
originally uniscale [
45
] but we found that multiscaling made the segmentation not
only faster but also more robust and accurate. Our multiscale, multistage, 3D
segmentation algorithm is summarized in the
fl
first linearly
subsample our data (step 1). Then the original algorithm is applied to the obtained
half-scale volume. The preprocessing (described below) determines the parameters
of the sigmoid used to compute the speed function of the
fl
flowchart in Fig.
3
.We
rst GAC (step 2.1). The
first GAC roughly segments the interior of the vertebra (step 2.2). Its seed is the
result of a fast marching (FM) stage starting from a seed point roughly placed by the
user in the center of the vertebral body and lasting 20 iterations. The second level
set, also a GAC, re
nes this segmentation using a Laplacian convolution of the
image as the speed function (step 2.3). The third level set, a CLS, segments the
cortical bone (step 2.4). A postprocessing step
fills some remaining holes using a
dilation followed by an erosion (step 2.5). The resulting segmentation is then super-
sampled back to full scale (step 3) and re
ned using a CLS (step 4). A last hole-
filling postprocessing is performed (step 5).
The speed function g
ðÞ
should ideally have values close to 1 where there are no
boundaries (so that the level set can expand rapidly) and values close to 0 where
boundaries are present (so that the level set stops). This can be achieved for instance
by writing [
46
]:
