Information Technology Reference
In-Depth Information
indicated by the edges of the gradient image. Given the binary gradient X-ray
image, the distance of an image point q to the projected edge structures V ¼
f
v
j
g
is
d
. However due to the poor quality of the images, a precise edge
information cannot be obtained and the gradient images may not correspond to the
edge templates. The proximity to edges can be de
ð
q
Þ
¼
min
j
j
q
v
j
j
ned by using a Gaussian
2
and weighted by the projected 2D dis-
expression controlled by the parameter
r
tance measure of the surface evolution:
X
X
2
2
p
ij
exp
ðð
q
v
ij
Þu
i
ð
x
;
y
;
z
ÞÞ
D
edges
¼
ð
11
Þ
r
2
i¼1
j
where p
ij
is the probability for pixel v
j
in image i of being an edge. The distance
measure
uð
of the surface evolution in 3D is projected on image plane i
using the rotation component
h
of the projection parameters
n
i
(Sect.
3
):
x
;
y
;
z
;
t
Þ
!
@
u
ð
x
;
y
;
z
Þ
@
cos
n
i
ðhÞþ
@
u
ð
x
;
y
;
z
Þ
sin
n
i
ðhÞ
x
@
y
u
i
ð
x
;
y
;
z
Þ
¼
uð
x
;
y
;
z
Þ
sin
:
ð
12
Þ
jruð
x
;
y
;
z
Þj
To determine the values of p
ij
, a two-dimensional proximity function can be
computed by convoluting the image with a large Gaussian kernel.
Epipolar Geometry Constraint
The calibration of the 3D radiographic viewing geometry was also used to constrain
the landmark correspondence between the biplanar images. An iterative retro-
projection method helps to re
ne landmark position, by taking the current 3D
landmark location, project it in 2D onto the coronal (PA)/sagittal (SAG) views and
measure the perpendicular distance of the projected coordinate on both views to its
corresponding epipolar line. The distance error for the L landmark points (L
¼
6
representing the 4 pedicle extremities and 2 endplates) is de
ned as:
X
L
i¼1
½Eucl
ð
w
SAG
2
;
F
T
w
PA
i
D
epipolar
¼
Þ
i
ð
13
Þ
w
PA
i
F
T
w
SAG
i
2
þ
ð^
;
^
Þ
Eucl
where Eucl
w
i
is the ana-
lytical projection of the 3D object point pi
i
obtained from standard perspective
transformation formulae. FT
T
ðÞ
denotes the Euclidean distance of a point to a line,
^
w
i
is the corresponding epipolar line on one image
based on point pi
i
from the other image, and F is the 4
^
4 fundamental matrix
integrating the geometrical parameters
n
which describes the projective 3D structure
of the scene.
