Biomedical Engineering Reference
In-Depth Information
each point in image
A
gets transformed to a single point in image
B
, and vice
versa. There are several situations in which this does not happen. First, if the
dimensionality of the images is different such as in the registration of a
radiograph to a CT scan, a one-to-one transformation is impossible. Second,
the issues of image field of view and sampling discussed above mean that
parts of the patient sampled in one image may not be present in the second
image, even if the patient has not changed.
For various types of nonaffine registration, a one-to-one transformation
is not desirable. For example, in registration of images from different sub-
jects, or of the same subject before and after surgery, there may be struc-
tures in image
A
that are absent from image
B
, or vice versa.
3.4
Registration Algorithms
The algorithms that find
T
given two images are called registration algo-
rithms. In this section we describe algorithms that use points identified in the
images (Section 3.4.1), surfaces delineated from the images (Section 3.4.2),
and voxel intensity values (Section 3.4.3).
As stated at the start of this chapter, we are using 3D rigid-body registra-
tion as the exemplar application. All these algorithms can straightforwardly
be extended to the case of affine transformations. These approaches can also
be extended to nonaffine registration transformations, but the extension is
quite different when using points, surfaces, or voxel intensity values. These
nonaffine approaches are discussed in Chapter 13.
3.4.1
Points and the Procrustes Problem
Point-based registration involves identifying corresponding 3D points in the
images to be aligned, registering the points, and inferring the image trans-
formation from the transformation determined from the points. The 3D
points used for registration are often called
fiducial markers
or
fiducial points
.
Using the notation introduced in section 3.2, we want to find points in
image
A
and in image
B
corresponding to the set of features in the
object. The corresponding points are sometimes called homologous land-
marks, to emphasize that they should represent the same feature in the dif-
ferent images. The most common approach is then to find the least square
rigid-body or affine transformation that aligns the points. This transforma-
tion can subsequently be used to transform any arbitrary point from one
image to the other.
x
{}
x
{}
{ }
x
3.4.1.1 The Orthogonal Procrustes Problem
The orthogonal Procrustes problem draws its name from the Procrustes area
of statistics. Procrustes was a robber in Greek mythology. He would offer
travelers hospitality in his roadside house and the opportunity to stay the
