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es all the constraints is thus necessary. A simple
solution that worked well in practice was to use the one-parameter subgroup
(computed from the matrix exponential) of the rigid transformations to split a rigid
transformation into smaller equal transformations when more than two consecutive
vertebrae were missing.
To get an idea of the achievable performances of this type of method, we
selected 50 cases from a database of 3D spine models from the Sainte-Justine
hospital. We then removed a certain number of vertebrae from the models (com-
plete models include both lumbar and thoracic vertebrae, which means a total of 17
vertebrae), and reconstructed them by minimizing Eq. ( 15 ). The removed vertebrae
were equally distributed along the spine. The reconstructed vertebrae were then
compared with the original model, and the results are illustrated in Fig. 12 .
These results are a great demonstration of the power of statistical models based
on an articulated modeling of the spine. It was possible to remove eight vertebrae
from a complete model and still expect a mean absolute error of just less than 1 mm.
In this experiment, the known vertebrae were not corrupted by calibration errors.
Thus, these results should be considered as a minimum bound for real-world
experiments.
Nonetheless, these interesting results demonstrate that representing a spine using
a large number of anatomical landmarks (102 in this case) is very redundant from a
statistical point of view. A statistical model based on an articulated modeling of the
spine can harness this redundancy and reduce the amount of information needed to
recreate a valid 3D model.
A good starting point that satis
4
Mean error
Mean error +
σ
3.5
3
2.5
2
1.5
1
0.5
0
2
4
6
8
10
12
14
Number of missing vertebrae
Fig. 12 Three-dimensional spine models with missing vertebrae reconstructed using an
articulated model prior. The mean absolute error and standard deviation of the individual
landmark reconstruction errors are shown for different numbers of missing vertebrae
 
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