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close to the mid point of the two neighboring vertebrae centers. S
2
ð
D
j
V
A
;
V
B
Þ
is
then de
ned as:
X
v
i
þ
v
i
2
2
1
þ
1
2
2
2
e
ð
1
U
ð
v
i
v
i
þ
1
Þ
D
d
i
Þ
=
k
e
kð
Þ
C
d
i
k
=
k
S
2
ð
j
V
A
;
V
B
Þ
¼
þ
ð
Þ
D
½
9
2
i
where
m
i
and
m
i
þ
1
denote the centers of the two neighboring vertebrae of disc di,
i
,
whose center and norm are D
d
i
and C
d
i
. U
ð:Þ
is the normalization operator. The
rst
and second terms of Eq. (
9
) in fact re
ects the two aforementioned spatial con-
strains between neighboring vertebrae and discs, respectively. The modeling of
spatial correlations between vertebrae and discs allow our system to propagate
information of vertebrae layer to disc layer for robust detection.
fl
6 Hierarchical Spine Detection
Based on the descriptions in Sects.
4
and
5
, we have all terms in Eq. (
3
)de
ned. At
runtime, spine detection becomes an optimization procedure of Eq. (
3
). As Eq. (
3
)
is a high-dimensional and non-linear function, we design a multi-stage algorithm to
optimize it.
Different stages target to anchor vertebrae, bundle vertebrae and inter-vertebral
discs, respectively. In each stage, we alternatively optimize the appearance terms
and spatial
cally, optimization starts from the concurrent
detection of anchor vertebrae, which is equivalent to the maximization of A
1
. Based
on the detected anchor vertebrae, S
1
is maximized to predict the positions of sub-
sequent bundle vertebrae. It determines the local regions where bundle vertebrae
detectors will be invoked to maximize A
2
. S
1
is then further maximized. In this step,
the responses of bundle vertebrae detectors are veri
terms. More speci
ed and the exact spine labels
are assigned. Subsequently, A
3
and S
2
are optimized in the same fashion.
Figure
2
a gives a schematic explanation of the optimization procedure. This
hierarchial detection scheme emulates a manual operator and achieves the robust-
ness in three aspects: (1) Anchor vertebrae are detected concurrently to provide
redundant and distributed appearance cues. With the redundant detection of sup-
porting landmarks, the detection of anchor vertebrae is very robust. Even when
some anchor vertebrae are missed due to severe local imaging artifacts, others still
provide reliable clues for spine detection. (2) Detectors of bundle vertebrae and
discs provide supporting cues. More speci
cally, instead of trying to directly derive
vertebrae labels, bundle vertebrae detectors provide a set of candidates whose labels
are mutually assigned according to relative positions to anchor vertebrae. Note that
labels assigned by different anchor vertebrae might be different, and are fused
through the maximization of S
1
. Disc detectors return a cloud of responses for disc
localization, which is robust to individual false classi
cations as well. (3) Local
articulated model propagates these appearance cues in a way robust to abnormal
spine geometry resulting from severe diseases.
