Biomedical Engineering Reference
In-Depth Information
rigid-body transformation into a single 4
4 matrix using homogeneous
coordinates:
T
rigid
x
()
1
cos
cos
cos
sin
sin
sin
cos
sin
sin
cos
sin
cos
t
x
x
y
z
cos
sin
cos
cos
sin
sin
sin
sin
cos
cos
sin
sin
t
y
sin
sin
cos
cos
cos
t
z
0
0
0
1
(3.9)
It is common to consider projection of an object defined in (
x
,
y
,
z
) space along
the
z
axis onto the
u
,
v
plane. The projection can be characterized by the
intrinsic parameters of the imaging system (
u
0
,
v
0
,
k
u
,
k
v
). For projection x-ray,
we can interpret these as follows:
u
0
and
v
0
define the ray-piercing point (the
point in the (
u
,
v
) imaging plane from which a normal vector goes through
the x-ray source), and
k
u
and
k
v
equal the pixel sizes in the horizontal (
u
) and
vertical (
v
) directions, respectively, divided by the imaging-plane-to-focal-spot
distance. These intrinsic parameters can often be determined by calibration of
the imaging system. Alternatively, they can be considered as unknowns, add-
ing four degrees of freedom to the registration algorithm. This transformation
T
projection
can be represented as a 4
3 matrix which projects the 3D object
along the
z
axis:
k
u
0
u
0
0
0
k
v
v
0
0
0010
T
projection
(3.10)
T
A 3D point in homogeneous coordinates is multiplied by
this matrix, giving a vector , and the resulting 2D point on the
projection plane is obtained by dividing the first and second ele-
ments of the vector by the third element
(
x
,
y
,
z
, 1
)
T
(
u
,
v
,
)
T
(
u
,
v
)
, which is a scaling factor.
The transformation required for rigid-body 2D-3D registration, T
2
D
-3
D
is
the composition of the projection and rigid-body transformations:
T
2
D-
3
D
T
projection
T
rigid
(3.11)
Further details about projection transformations and homogeneous
coordinates can be obtained from many topics on graphics and computer
vision, including Foley et al.
2
When we are considering structures made of bone or enclosed in bone,
this rigid-body transformation (or rigid-body transformation followed by
projection) can correctly align images of the same object in different posi-
tions. This assumption works remarkably well for images of the brain, as the
