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the Point Distribution Model (PDM) [
19
] as the representation of the SSMs of the
lumbar vertebrae. The PDM was constructed from a training database consisting of
39 CT-segmentation based binary volumes of lumbar vertebrae (according to the
spine level, the distribution of these 39 binary volumes are as follows, L1 level: 3;
L2 level: 5; L3 level: 9; L4 level: 14; L5 level: 8). The PDM was constructed from
the training data based on following procedure. First, a binary volume of a L3 level
vertebra was chosen as the reference. Demon
'
s algorithm [
23
] was used to estimate
the deformation
fields between the chosen reference binary volume and the other
fl
field was then used to displace the
positions of the vertices on the reference surface model to the associated target
volume. We thus obtained a set of aligned surface models with established
correspondences.
Following the alignment, the PDM were constructed as follows. Let xi;
i
;
floating volumes. Each estimated deformation
i
¼
0
be m (m = 39) members of the aligned training surface models. Each
member is described by a vector Xi
i
with N (N = 5000) vertices:
;
1
; ...
m
1
;
x
i
¼f
x
0
;
y
0
;
z
0
;
x
1
;
y
1
;
z
1
; ...;
x
N
1
;
y
N
1
;
z
N
1
g
ð
1
Þ
A PDM was then obtained by applying principal component analysis [
24
] to the
aligned training surface models:
X
m
1
i
¼
0
ð
Þ
1
T
D
¼ðð
m
1
Þ
x
i
x
Þð
x
i
x
Þ
ð
2
Þ
p
k
¼ r
k
P
¼ð
p
0
;
p
1
; ...Þ;
D
p
k
where
x and D are the mean vector and the covariance matrix of the PDM,
respectively.
p
k
g
are the corresponding eigenvectors. The descendingly sorted eigenvalues
r
k
and the
corresponding eigenvector p
k
are the principal directions spanning a shape space
with
fr
k
g
f
are non-zero eigenvalues of the covariance matrix D, and
x representing its origin. Figure
1
shows the variability captured by the
rst
two modes of variations of the PDM.
3 Statistically Deformable 2D/3D Reconstruction
Without loss of generality, here we assume that the input image is calibrated and
image distortion is corrected. For more details about
fluoroscopic image calibration,
we refer to our previous work [
25
]. Thus, for a pixel in the input image we can always
fl
find a projection ray emitting from the focal point of the image through the pixel.
The single image based surface model reconstruction technique proposed in this
paper is based on a hybrid 2D/3D deformable registration process coupling a
landmark-based scaled rigid registration with an adapted SSM-based 2D/3D
reconstruction algorithm [
20
,
21
]. Different from the situation in our previous works
