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4.4.2 Manifold-Driven Model Generation
The input of the method consists of calibrated coronal and sagittal X-ray images
I
i¼
f
1
;
2
g
is pre-operative spine acquired. The
personalized 3D model is achieved by means of a reconstruction method merging
statistical and image-based models [
26
]. The 3D spine centerline C
i
ð
(de
ned is space
X
i
) of the patient
'
u
Þ
is obtained
from cubic B-splines extracted from the images. The centerline is
rst embedded
onto a non-linear manifold
711) and used to
predict an initial spine. The manifold establishes the patterns of legal variations of
spine shape changes in a low-dimensional sub-space based on locally linear em-
beddings as illustrated in Fig.
10
. To map the high-dimensional 3D curve assumed
to lie on a non-linear manifold into a low-dimensional subspace, the
M
containing M scoliotic spines (M
¼
rst step
consists of selecting the K closest neighbors for each data point using the Euclidean
distance between centerlines as a closeness measure. The manifold reconstruction
weights W are then found to reconstruct point i from it
'
s K closest neighbors using
the reconstruction errors as measured by:
2
X
X
M
K
eð
W
Þ
¼
min
W
C
i
ð
u
Þ
W
ij
C
j
ð
u
Þ
ð
7
Þ
i¼
1
j¼
1
where C
i
ð
sums the squared dis-
tances between all data points and their corresponding reconstructed points. The
minimum of
eð
u
Þ
is a data B-spline described above and
eð
W
Þ
which describes the optimal weight matrix W is then determined
by solving a least-square problem. The weights W
ij
represent the importance of the
W
Þ
Fig. 10 Illustration of spine distribution embedded onto on a low-dimensional manifold. Stars
represent new sample point models which were unseen in the training set and projected onto the
manifold. Pluses are original model points from the training data which creates the manifold
distribution
