Biomedical Engineering Reference
In-Depth Information
3.1
Introduction
In the previous chapter, the concepts behind image registration were intro-
duced in a nonmathematical way. The aim of this chapter is to describe some
of the main algorithms used in image registration in greater detail, and to
compare their applicability. This requires a more mathematical approach.
This chapter uses three-dimensional (3D) rigid-body registration as an
exemplar. The algorithms described in most detail are those used for rigid-
body registration of 3D tomographic images of the same subject. Many of
the concepts and algorithms introduced here are also applicable to other
registration applications including registration of 2D and 3D images,
image-physical space registration, nonrigid intrasubject registration, and
intersubject registration.
The field of nonrigid registration for both intrasubject and intersubject
applications is an area of active current research, and is the topic of the
third part of this topic.
3.2
Notation and Terminology
In order to align two images, we need to know the transformation that relates
the
of
the corresponding feature in another image or coordinate space. We use the
symbol
position
of features in one image or coordinate space with the
position
to represent this registration transformation.
Using the language of geometry, this transformation is a spatial map-
ping. We can consider the mapping
T
from one
image to another, or from one image to the coordinate system of a treat-
ment device (image to physical registration).
T
, that transforms a position
x
T
:
x
x
⇔
T
(
x
)
x
(3.1)
B
A
B
A
T
1
It is sometimes useful to also consider the inverse mapping
that
maps
.* With image data we have to consider intensity values as
well as positions, and we refer to
x
to
x
A
B
A
(
x
) as the intensity value at the location
A
x
, and similarly for image
B
. It is important to remember that the medical
A
images
are derived from a real object, i.e., the patient. The images
have a limited field of view that does not normally cover the entire patient.
Furthermore, this field of view is likely to be different for the two images.
A
and
B
* Although rigid-body and affine transformations that form the focus of this chapter are invert-
ible, not all more complicated transformations are.
