Information Technology Reference
In-Depth Information
Using s, we scale all landmark positions on the mean model and denote them as
v
i
Mean
;
v
2
Mean
and
v
3
Mean
f
i
¼
1
;
2
;
3
;
4
g
. We then calculate the distances from
to the
v
4
Mean
and denote it as
l
2
;
1
4
and
l
3
;
1
4
v
1
Mean
line
Mean
, respectively.
Mean
v
X
ray
on the projection ray of v
X
ray
whose
Next we
find two points, point
v
X
ray
is equal to
l
2
;
1
4
v
X
ray
v
X
ray
on the projection
distance to the line
Mean
, and point
v
X
ray
is equal to
l
3
;
1
4
ray of v
X
ray
whose distance to the line
v
X
ray
Mean
. A paired-
v
i
Mean
;
v
i
X
ray
;
point matching based on
f
i
¼
1
;
2
;
3
;
4
g
and
f
i
¼
1
;
2
;
3
;
4
g
is used to
T
X
ray
Mean
calculate an updated scale s
0
and a rigid transformation
(see Fig.
4
a for
details). From now on, we assume that all information de
ned in the mean model
coordinate frame has been transformed into the
fl
fluoroscopic image coordinate
frame using s
0
and T
X
ray
Mean
. We denote the transformed mean model landmarks as
f~
v
i
Mean
;
¼
;
;
;
g
.
Iteration. The following steps are iteratively executed until convergence:
i
1
2
3
4
For a point v
i
Mean
,we
find a point on the corresponding projection ray of v
i
X
ray
which has the shortest distance to the point
v
i
X
ray
(see
Fig.
4
b). We then perform a paired-point matching using the extracted point
pairs to compute a scale
v
i
Mean
and denote it as
~
~
s and a rigid transformation update
D
T
X
ray
Mean
.
s and
D
T
X
ray
~
We update the mean model coordinate frame using
Mean
.
3.3 Statistical Shape Model-Based 2D/3D Reconstruction
The estimated scale and the rigid transformation between the mean model and the
input image are then treated as the starting values for the PDM-based 2D/3D
reconstruction scheme [
20
,
21
], which depends on an iterative image-to-model
correspondence establishing algorithm that we introduced previously [
28
]. The
image-to-model correspondence is established using a non-rigid 2D point matching
process, which iteratively uses a symmetric injective nearest-neighbor mapping
operator and 2D thin-plate splines-based deformation to
find a fraction of best
matched 2D point pairs between those contours extracted from the x-ray image as
we described above and the projections of the apparent contours extracted from the
3D model. The apparent contours of a statistically instantiated 3D model are
extracted using the approach introduced by Hertzmann and Zorin [
29
]. Previously,
we mathematically proved that the proposed non-rigid 2D point matching process
could automatically eliminate the cross-matching event [
28
], which was de
ned as
the interactions between the lines linking any matched point pair. Figure
5
a shows
the mean mode of the complete-PDM initialized with respect to the input image
using the landmark-based scaled rigid registration, and the apparent contours
extracted from the mean model. An example of building 2D/2D correspondences
between the image contours and the projections of the apparent contours of the
