Biomedical Engineering Reference
In-Depth Information
bone of the skull restricts the movement of the brain to less than 1 mm.
3
*
Unfortunately, repositioning patients often results in changes in the position
of objects that cannot simply be described using translations and rotations.
We can extend the transformation a little, while still maintaining a single
matrix that will transform all points in the image, if we restrict the additional
changes to stretches and skews. This gives an
affine transformation
. Unfortu-
nately, soft tissue in the body tends to deform in more complicated ways, so an
affine transformation does not add greatly to the number of registration prob-
lems that can be solved. Whereas the rigid-body transformation preserves the
distance between all points in the object transformed, an affine transformation
preserves parallel lines. Two areas where affine transformations are useful
are in correcting for scanner errors (which can be errors in scale or skew) and
in approximate alignment of brain images from different subjects. These
applications are discussed in Chapters 5 and 14, respectively.
For most organs in the body, and for accurate intersubject registration,
many more degrees of freedom are necessary to describe the tissue deforma-
tion with adequate accuracy. These nonrigid (perhaps more correctly termed
nonaffine) registration transformations are discussed further in Chapter 13.
Linear transformations:
Many authors refer to affine transformations as linear.
This is not strictly true, as a linear map is a special map
L
which satisfies:
D
L
(
x
A
x
A
)
Lx
A
()
L
(
x
A
)
x
A
,
x
A
(3.12)
The translational part of affine transformations violates this. An affine map is
more correctly described as the composition of linear transformations with
translations.
Furthermore, reflections are a linear transformation, but they are nor-
mally undesirable in medical image registration. For example, if a registra-
tion algorithm used in image-guided neurosurgery calculated a reflection
as part of the transformation, it might result in a patient having a craniot-
omy on the wrong side of his head. If there is any doubt about whether an
algorithm might have calculated a reflection, this should be checked prior
to use. Since affine transformations can be represented in matrix form, a
reflection can be detected simply from a negative value of the determinant
of this matrix.
The term
nonlinear
transformation is often used interchangeably with
non-
rigid
. Both terms are used in this topic to refer to a transformation with more
degrees of freedom than an affine transformation. As stated previously, these
types of transformations are discussed in detail in Chapters 13 through 15.
One-to-one transformations:
For intrasubject registration, the object imaged
with the same or different modalities is one patient. It would at first seem
likely that the desired transformation
T
should be one to one. This means that
* This is only valid provided the skull remains closed. In neurosurgery, for example, deformation
can be much greater.
