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Here, A
1
, A
2
and A
3
relate to the appearance characteristics of anchor, bundle
vertebrae and inter-vertebral discs. S
1
and S
2
describe the spatial relations of
anchor-bundle vertebrae and vertebrae-disc, respectively. It is worth noting that the
posterior of anchor vertebrae solely depends on the appearance term, while those of
bundle vertebrae and inter-vertebral discs depend on both appearance and spatial
relations. This is in accordance to the intuition: while anchor vertebrae can be
identi
ed based on its distinctive appearance, bundle vertebrae and inter-vertebral
discs have to be identi
ed using both appearance characteristics and the spatial
relations to anchor ones.
Figure
2
b gives a schematic explanation of Eq. (
3
). Our framework consists of
three layers of appearance models targeting to anchor, bundle vertebrae and
discs. The spatial relations across different anatomies
different layers (lines
in Fig.
2
). Note that this framework is completely different from the two-level model
of [
11
], which separates pixel- and object-level information. Instead, different layers
of our framework target to anatomies with different appearance distinctiveness.
“
bridge
”
4 Hierarchical Learning Framework
4.1 Learning-Based Anatomy Detection
Before detailing hierarchical learning framework, we first introduce the basic
learning modules for anatomy detection. Due to the complex appearance of verte-
brae and discs, particularly in MR images, we resort to learning-based approach to
model the appearance characteristics of vertebrae and inter-vertebral discs. Thanks
to its data-driven nature, learning-based approaches also provide the scalability to
extend our method on both CT and MR images. We formulate anatomy detection as
a voxel-wise classi
cally, voxels within the anatomy prim-
itive, i.e., vertebrae or inter-vertebral discs, are considered as positive samples and
voxels away from the anatomy primitive are regarded as negative samples. To learn
an anatomy detector, we
cation problem. Speci
first annotate vertebrae and inter-vertebral discs in training
images. For each training sample (voxel), a set of elementary features are extracted
in its neighborhood. Our elementary features are generated by a set of mother
functions,
extended from Haar wavelet basis. As shown in Eq. (
4
) and
Fig.
3
, each mother function consists of one or more 3D rectangle functions with
different polarities.
f
H
l
ð
x
Þg
X
N
H
ð
x
Þ
¼
p
i
R
ð
x
a
i
Þ
ð
4
Þ
i¼1
1
;
k
x
k
1
1
where polarities pi
i
¼
f
;
g
ð
Þ
¼
1
1
, R
x
denotes rectangle func-
0
;
k
x
k
1
[
1
tions and a
i
is the translation.
