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jth data point to the reconstruction of the ith element. The last step consists of
mapping each high-dimensional C
i
ð
u
Þ
to a low-dimensional point Yi,
i
, representing
each data point in the global coordinate system in d-space, using a cost function
which minimizes the reconstruction error:
X
M
i¼1
k
X
K
2
Uð
Y
Þ
¼
Y
i
W
ij
Y
j
k
:
ð
8
Þ
j¼1
The coordinates Yi
i
can be translated by a constant displacement without
affecting the overall cost
Uð
Y
Þ
. This degree of freedom is removed by requiring the
coordinates to be centered at the origin, such that
P
Y
i
¼
0. The optimal embed-
ding, up to a global rotation of the embedding space, is obtained from the bottom
d
M matrix enclosing the
reconstruction weights W
ij
. This helps to minimize the cost function
Uð
þ
1 eigenvectors of the sparse and symmetric M
as a
simple eigenvalue problem. The d eigenvectors form the d embedding coordinates
in
Y
Þ
M
[
54
]. Hence, a new model point can be determined in the embedded d-space
as a low-dimensional data point by
finding its optimal manifold coordinates Yi.
i
.
Given a new projection point Y
n
, an appropriately scaled model is gener-
ated from an analytical method based on nonlinear regression using a Radial Basis
Function kernel function f to perform the inverse mapping. Formally, the model is
described such that S ¼½
f
1
ð
Y
n
Þ; ...;
f
D
ð
Y
n
Þ
with S ¼
ð
s
1
;
s
2
; ...;
s
17
Þ
, where s
i
is a
3
vertebra model de
ned by s
i
¼
ð
p
1
;
p
2
; ...;
p
6
Þ
, and p
i
2<
is a 3D vertebral
landmark. Details can be found in reference [
26
].
This crude statistical 3D model is re
ned with an individual scoliotic vertebra
segmentation approach. This is achieved by extending 2D geodesic active regions
in 3D, in order to evolve prior deformable 3D surfaces by level sets optimization.
Each component model Si
i
from the atlas of vertebral triangulated meshes, repre-
sented by the triangular mesh vertices
f
v
j
j
j ¼ 1
; ...;
V
g
, are initially positioned and
oriented from their respective 6 precise landmarks s
i
composing S. The surface
evolution is then regulated by the gradient map and image intensity distributions
[
49
], where E
RAG
¼
k
E
CAG
ð
S
Þþð
1
kÞ
E
R
ð
S
Þ
is the energy function with the edge
and region-based components controlled by
k
are de
ned as:
I
X
2
1
E
CAG
¼
a
du
i
1
þjr
I
i
ð
u
i
Þj
S
i
i¼1
ZZ
X
2
E
R
¼
log
ð
p
R
ð
I
i
ð
u
i
ÞÞÞ
du
i
ð
9
Þ
P
i
ð
S
i
Þ
i
¼
1
ZZ
X
2
ð
p
Rc
ð
I
i
ð
u
i
ÞÞÞÞ
log
du
i
X
i
P
i
ð
S
i
Þ
i¼1
with
P
i
as the perspective projection parameters, while p
R
and p
Rc
are Gaussian
distributions
for bone and background regions respectively. The projected
