Biomedical Engineering Reference
In-Depth Information
In k-space, rigid-body transformations have special properties due to the
properties of the Fourier transform.
35
In particular, a spatial domain transla-
tion becomes a phase change in k-space. The modulus of k-space or power
spectrum of the image does not contain any phase information and is, there-
fore, invariant to translation. Rotations in the spatial domain are rotations by
the same angle in k-space. Rotation can be decoupled from translation by
computing the modulus of the data. By converting to a polar representation
of k-space, the rotation becomes a simple shift of angular coordinate, which
can be solved by correlation of the polar representation of the magnitude of
k-space. Once the rotation has been found, the translation can be determined
from the phase difference in the Cartesian k-space. This approach is not itera-
tive, so it can be fast. This type of approach has been applied to medical
images,
36,37
but the applicability appears to be limited by the implicit assump-
tion that the objects of interest are in the fields of view of both images being reg-
istered, i.e., that all image features are contained in the overlap domain .
Since medical images almost invariably only sample part of the patient, seg-
mentation of features of interest that lie within
T
A
,
B
T
is necessary before this
A
,
B
37
approach can be reliably used.
3.4.6
Ratio Image Uniformity (RIU)
Woods
38
This algorithm was originally introduced by for the registration of
serial PET studies, but has more recently been widely used for serial MR regis-
tration.
39
The algorithm can be thought of as working with a derived ratio image
calculated from images
A
and
B
. An iterative technique is used to find the trans-
formation
that maximizes the uniformity of this ratio image, which is quanti-
fied as the normalized standard deviation of the voxels in the ratio image. The
RIU acronym was not introduced when the algorithm was first published, and
it is also frequently referred to as the variance of intensity ratios algorithm (VIR).
The RIU algorithm is most easily thought of in terms of an intermediate ratio
image
T
T
R
comprising
N
voxels within the overlap domain
.
A
,
B
()
B
T
x
()
A
x
A
1
----
T
R
x
()
-----------------
x
A
A
,
B
,
R
R
x
()
(3.17)
T
x
A
A
,
B
1
----
2
R
x
()
(
R
)
T
x
A
A
,
B
----------------------------------------------------------------------
RIU
(3.18)
R
3.4.7
Partitioned Intensity Uniformity (PIU)
The first widely used intermodality registration algorithm that used a voxel
similarity measure was proposed by Woods for MR-PET registration soon after
he proposed his RIU algorithm.
40
Here, we refer to this intermodality algo-
rithm as partitioned intensity uniformity (PIU). This algorithm involved a
