Information Technology Reference
In-Depth Information
4 Shape Inference
Articulated models of the spine can be used beyond their role in descriptive statistics.
They model inter-patient variability quite well and offer a compact representation of
what is and is not a valid 3D spine model. Furthermore, most common manipulations
of articulated models have closed-form solutions and are differentiable. These two
qualities make an articulated model well suited for integration with different shape
estimation algorithms. Shape estimation systems are generally large and complex
systems where image processing, system calibration, and a user interface all have to
work in synergy to produce suitable results. We will, however, focus our attention
only on the integration of the articulated statistical models to simplify our exposition,
make it more understandable, and keep it concise.
4.1 Articulated Shape Prior for 3D Reconstruction
from 2D Correspondences
A common and simple 3D reconstruction problem is the computation of the three-
dimensional coordinates of points based on image coordinates in multiple images.
This basic problem can be solved by calibrating the projective geometry of the
system and then performing triangulation on the image coordinates corresponding
to the same 3D point in multiple images. The corresponding points can be obtained
automatically using an elaborate image processing system or manually de
ned by
an expert in spinal anatomy. This general idea has been applied to 3D spine
reconstruction from multiple radiographs for many years [
33
].
The performances deteriorate quickly as the image calibration and image corre-
spondences are degraded, either by human error or simply by lower quality radio-
graphs. However, an articulated statistical shape model prior can be integrated quite
simply to mitigate some of these problems. Thus, let p
i
;
j
;
k
2D
be the image coordinates of
an anatomical landmark identi
ed in a radiograph. The index i associates a landmark
with a vertebra. The index j indicates the position of the anatomical landmark within
the set of landmarks used for the ith vertebra. Finally, k denotes the index of the
radiograph on which the coordinates were measured. In addition, let S be the
departure from the Fr
é
chet of an articulated model, which is de
ned as follows:
:
ð
with s
i
¼ T
1
i
S
¼
ð
s
1
;
s
2
; ...;
s
N
Þ
T
i
;
p
i
;
1
p
i
;
1
;
p
i
;
2
p
i
;
2
; ...;
p
i
;
M
p
i
;
M
13
Þ
A simple but effective way to combine the similarity between p
i
;
j
;
k
2D
and S with
prior knowledge of possible spine shapes is to sum the Mahalanobis distance and
the quadratic error on the anatomical landmarks. The following equation summa-
rizes this operation:
