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as the positives and all others are treated as negatives. Although the numbers of
positive and negative samples become highly balanced, thanks to the cascade
learning framework, we can still learn a discriminative detector for each anchor
vertebra. To reach the second goal, for each anchor vertebrae, we de
ne a set of
“
surrounding it. For example, for S1 vertebra, we use tip
of coccyx, center of sacrum and spinous process of L5, etc., as its supporting
landmarks. The strong spatial correlation between the supporting landmarks and
the anchor vertebra are exploited to achieve robust detection. Mathematically, we
employ a linear model to capture the spatial correlation between the anchor vertebra
v
T6-T8
and its supporting landmarks
supporting landmarks
”
Sð
v
T6-T8
Þ
as Eq. (
7
):
m
T6-T8
¼
C
U
ð
7
Þ
Here,
U
is a vector concatenated by coordinates of supporting landmarks and
C
denotes the linear correlation matrix. Given a set of training samples,
can be
learned by solving a least squares problem. At run-time, besides detecting anchor
vertebrae, we also detect its supporting landmarks. The learned linear correlation
matrix are then used to verify the detected anchor vertebrae. In principle, we resort
to
C
“
”
redundancy
[
24
] for highly robust anchor vertebrae detection.
4.3 Bundle Vertebrae
Different from anchor vertebrae, other vertebrae have less distinctive shapes and
appearances. These vertebrae look similar to their neighbors but different from
remote ones. On one hand, training a general detector for all bundle vertebrae is
almost infeasible due to the large variations across distal ones. On the other hand,
an attempt to learn the subtle differences between a bundle vertebra and its
neighborhoods also adversely affects the robustness. For example, T
9
and T
10
are
two neighboring vertebrae with similar appearance and shape characteristics. For
normal cases (see Fig.
5
a), the two detectors are still possible to distinguish the
subtle differences between T
9
and T
10
. However, when the appearance of T
9
becomes abnormal due to imaging artifacts or diseases, T
9
detector might have
higher responses at T
10
than T
9
(Fig.
5
b), which induces wrong/miss labeled ver-
tebrae. In fact, this problem is also observed in [
15
], where
(standard) MSL
approach may end up with detections for the most salient disks only
“
.
To avoid this problem, we propose to group neighboring vertebrae as
”
”
(Fig.
2
a). Vertebrae within the same bundle are treated as equivalent positives in the
learning algorithm. In this way, each bundle has one detector that encodes the
commonality of corresponding vertebrae and distinguishes them from other bundles.
This kind of detectors are learnable as in-bundle vertebrae often have much more
similar characteristics than cross-bundle ones. Compared to anchor vertebrae
detectors, bundle detectors are less discriminative. Bundle detectors are not able to
identify speci
“
bundles
c vertebrae, e.g., T
9
or T
10
, but speci
c vertebrae bundle, e.g., T
6
-
T
8
