Biomedical Engineering Reference
In-Depth Information
Because the transformation
T
maps both positions and intensities at
these positions,
(unlike the spatial mapping
T
) has to take account of the
discrete sampling.
T
3.3
Types of Transformation
The spatial mapping
T
describes the relationship between locations in one
image and corresponding locations in a second image. The images could be
two-dimensional (2D) or three-dimensional (3D), so the mapping may be from
2D space to 2D space, from 3D space to 3D space, or between 3D and 2D
space. In all cases, the object being imaged—all or part of a human subject—is
three-dimensional. There are consequently very few situations in which a
2D-2D mapping adequately aligns two images. The most common applica-
tions of image registration involve aligning pairs of 3D images. Another
important application is aligning 2D images with 3D images (2D-3D registra-
tion). In 2D-3D registration,
T
involves a 3D-3D mapping followed by pro-
jection of the 3D object onto a 2D plane.
If the images registered are of the same object that is merely in a different
position, then we can describe the required registration transformation
using just translations and rotations. This gives us a rigid-body transfor-
mation. In three dimensions, this has six degrees of freedom which can be
defined as translation in the
x
,
y
, and
z
directions, and rotations
about these three axes. From these unknowns, we can construct a
rigid
-
body
transformation matrix
T
rigid
that will map any point in one image to a
transformed point in the second. This transformation can be represented
as a rotation
R
followed by a translation
t
,
, and
T
(
t
x
,
t
y
,
t
z
)
that can be applied
to any point
x
T
in the image:
(
x
,
y
,
z
)
T
rigid
(
x
)
Rx
t
(3.7)
where the rotation matrix
R
is constructed from the rotation angles as follows:
R
cos
cos
cos
sin
sin
sin
cos
sin
sin
cos
sin
cos
(3.8)
cos
sin
cos
cos
sin
sin
sin
sin
cos
cos
sin
sin
sin
sin
cos
cos
cos
For a 2D-3D rigid-body registration, we need to consider both the rigid-body
transformation and a projection of the transformed 3D object onto a plane.
When combining rigid-body transformations with projections, it can be
useful to combine the translational and rotational components of the
