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The alignment essentially removes the feature variations from spatial transforma-
tion, hence, the learning task becomes easier.
Using the above strategy, we train detectors for anchor vertebrae, bundle ver-
tebrae and inter-vertebral discs as
A
i
ðFð
p
ÞÞ
,
B
j
ðFð
p
ÞÞ
,and
D
k
ðFð
p
ÞÞ
. Here,
Fð
p
Þ
A
i
,
B
j
and
denotes the over-complete Haar features extracted around voxel p, and
D
k
are the trained cascade Adaboost classi
ers, which select and combine a small
proportion of
to achieve best anatomy detection. The appearance terms in
Eq. (
3
) are eventually concretized as A
1
ð
Fð
p
Þ
Þ
¼
P
v
i
2
V
A
A
i
ðFðm
i
ÞÞ
V
A
j
I
, A
2
ð
V
B
j
I
Þ
¼
P
m
j
2
V
B
B
j
ðFðm
j
ÞÞ
Þ
¼
P
d
k
2
D
P
p
2
d
k
D
k
ðFð
and A
3
ð
D
j
I
p
ÞÞ
.
5 Local Articulated Spine Model
Recall the de
model the spatial
relations between anchor-bundle vertebrae and vertebrae-discs, respectively. In
our spine detection method, spatial relations are exploited in threefold. First, it
determines where the detectors should be invoked. For example, after the anchor
vertebrae are detected, we can predict the positions of bundle vertebrae and only
invoke the bundle vertebrae detectors in these local regions. Second, spatial rela-
tions can be used to verify the detection. For example, if a detected disc is almost
parallel to the line connecting its two neighboring vertebrae center, it is highly
probable that either vertebrae centers or disc are erroneously detected. Third, since
bundle vertebrae detectors only determines which bundle the vertebra belongs to,
spatial relations should be employed to assign exact labels to bundle vertebrae.
Accordingly, a proper modeling of spatial relations across vertebrae and inter-
vertebral discs becomes critical to the success of spine detection.
Various methodologies, including principal component analysis [
25
] and sparse
representation [
26
], have been investigated for anatomy shape/geometry modeling
and achieved tremendous success. However, since spine is a
nition of Eq. (
3
), S
1
ð
V
B
j
V
A
Þ
and S
2
ð
D
j
V
A
;
V
B
Þ
flexible structure
where each vertebra has freedom of local articulation (see Fig.
6
). those methods
that treat the object as a whole may not model the characteristics of spine geometry
properly. For example, for a patient with severe scoliosis (see Fig.
1
a), the spine
geometry appears as an outlier in eigen space. Hence, even when the vertebrae
detection is correct, they will be
fl
“
mis-corrected
”
to follow a normal spine geometry
using standard active shape model [
25
].
To deal with the specialty of spine geometry, local articulated model is designed
in [
27
,
28
]. In our study, we employ similar model to describe the spatial relations
across vertebrae. The key idea is to decompose the spatial transformation of a spine
into a set of local transformations between neighboring vertebrae. Instead of
enforcing constraints on the global transformation, we constrains local transfor-
mations based on learned statistics. In addition, smoothness across neighboring
local transformations is applied as another constraints. Note that this constraint still
holds for patients with severe scoliosis, since the spinal cord usually forms a
smoothing curve even for scoliosis patients.
