Biomedical Engineering Reference
In-Depth Information
T
T
sets of fixed intensities within
. Since
changes with
T
, algorithms
A
,
B
A
,
B
T
that are too sensitive to changes in may be unreliable.
The difficulty caused by the different fields of views of images
A
,
B
A
and
B
is further illustrated by considering an approach to rigid-body registration
called the method of moments.
1
When applying this to images of a part of
the body, e.g., the head, that part of the body is first delineated from images
A
and
B
using a segmentation algorithm, giving the binary volumes
O
A
. The images can then be registered by first aligning the centroids
(from the first order moment) of
and
O
B
O
and
O
, and then aligning the princi-
A
B
pal axes of
(from the second order moment). This approach is,
however, unsatisfactory for most medical image registration applications
because the first and second order moments are highly sensitive to change in
image field of view. In order for this method to work accurately, the object
used for the calculations must be entirely within
O
and
O
A
B
T
, and it is frequently dif-
A
,
B
ficult to delineate structures with this property.
3.2.2
The Discrete Nature of the Images
Another important property of the medical images with which we work is
that they are discrete. That is, they sample the object at a finite number of
voxels. In general, this sampling is different for images
, and while
the sampling is commonly uniform in a given direction, it may be anisotro-
pic; that is, it varies along the different axes of the images. Discretization has
important consequences for image registration, so it is useful to build this
concept into our notational framework.
We can define our domain
A
and
B
in the following way.
˜
=:
ς
(3.6)
˜
where
is a
bounded
continuous set defining the volume of the patient
imaged, and
is an infinite discrete grid.
is our
sampling
grid, which is char-
x
,
y
,
z
acterized by the anisotropic sample spacing
ς
(
ς
ς
ς
)
.
The sampling
is normally different for the images
A
and
B
being registered, and we denote
this by introducing sampling grids
ς
A
and
ς
B
for the domains
and
.
A
B
is
likely to be the empty set, because no sample points will exactly overlap.
In order, therefore, to compare the images
For any given
T
, the intersection of the discrete domains
and
A
B
it
is necessary to interpolate between sample positions and to take account
of the differences in sample spacing and . This introduces two prob-
lems. First, fast interpolation algorithms are imperfect, introducing blur-
ring or ringing into the image. This changes the image histograms and
hence alters the isointensity sets discussed above. Second, we must be
careful when the image
A
and
B
for any estimate of
T
ς
A
ς
B
being transformed has higher resolution sam-
pling than the reference image
B
A
, or we risk aliasing when we resample
B
B
T
from
to generate
in
. In this case, we should first blur
B
with a fil-
B
A
ter of resolution
ς
A
or lower before resampling.
