Biomedical Engineering Reference
In-Depth Information
some regions where there is missing data. For 3D-to-3D registration, as stated
above, two positions of a rigid body can always be related to one another in
terms of three translations and three rotations, giving six degrees of freedom.
The particular conditions of imaging may mean that we do not know the
pixel or voxel sizes or the fields of view, in which case the registration algo-
rithm may need to determine these. This will lead to an extra two degrees of
freedom in 2D or three degrees of freedom in 3D, equating to scaling in each
direction, also illustrated in Figure 2.3. A particular distortion in 3D images
generated from CT results from a gantry tilt, often used to reduce x-ray dose
to the eyes. Without correction this will result in a 3D volume that is skewed,
akin to a leaning stack of sliced bread (see Chapter 10, Figure 10.2). If gantry
angle is unknown, we have another degree of freedom. This is a special case
of the “affine” transformation. In the affine transformation, any straight line
in one image will transform to a straight line in the other and parallel lines
are preserved, allowing a combination of rigid body motion, scaling, and
skew about any of the three axes. The affine transformation has 12 degrees of
freedom. A mathematical definition is given in the next chapter.
In 2D-to-3D rigid-body registration, matching a perspective projection
such as an optical image or x-ray image to a volume results in up to ten
degrees of freedom. Usually four of these can be determined from a one-off
calibration of the camera or x-ray set, provided the focal length of the camera
or the distance between the x-ray set and imaging device is fixed, leaving the
six parameters of the rigid-body transformation to be determined in the reg-
istration process.
In nonrigid registration, many more degrees of freedom are required. Two
general categories of nonrigid registration occur: registration of images to an
atlas or images from another individual, so called “intersubject registration” or
registration of tissue that deforms over time, sometimes called “intrasubject
registration.” Figure 2.4 provides a 2D example of intersubject registration.
FIGURE 2.4
An example of interindividual, nonrigid registration (in 2D) of a professor (left) to a mandrill
(center), resulting in the warped image (right). The transformation was achieved by iden-
tifying a number of corresponding point landmarks and transforming with the thin-plate
spline function described in Chapter 3.
