Biomedical Engineering Reference
In-Depth Information
Care must be taken when truncating the interpolation kernel to ensure that
the integral of the weights of the truncated kernel is unity, or an artifactual
intensity modulation can result.
51,53
Various modifications to sinc interpolation have recently been proposed.
These fall into three categories: first, the use of sinc functions with various
radii truncated with various window functions;
54
second, approximations
to windowed sinc functions such as cubic or B-spline interpolants;
54,55
and
third, the shear transform, which involves transforming the image using a
combination of shears.
56,57
This third approach is fast, though it does result
in artifacts in the corners of the image which must be treated with caution.
An assumption implicit in the discussion above is that the original data
being interpolated are uniformly sampled. This is not always the case in
medical images. MR physics researchers are used to the problem of nonuniform
sampling in the acquisition, or k-space domain,
58,59
but this problem is less
often considered in the spatial domain. The most common circumstances
when nonuniform sampling arises are in free-hand 3D ultrasound acquisi-
tion, as discussed in Chapter 5, and in certain types of CT acquistion where
the slice spacing changes during the acquisition. The correct way of inter-
polating from nonuniformly sampled data onto a uniform grid is the
reverse of sinc interpolation. This methodology, sometimes used in k-
space regridding,
60,61
involves calculating the sinc coefficients to go from
the desired uniform sampling points to the nonuniform locations acquired,
and inverting the matrix of coefficients in order to do the correct interpo-
lation. In the cases of 3D ultrasound and CT with variable slice spacing, the
data are a long way from being band limited, so the benefits of inverse sinc
interpolation may be small in any case.
3.5.2
Interpolation during Registration
Many registration algorithms involve iteratively transforming image
B
with
respect to image
A
while optimizing a similarity measure calculated from the
voxel values. Interpolation errors can introduce modulations in the similarity
measure with
. This is most obvious for transformations involving pure
translations of datasets with equal sample spacing, where the period of the
modulation is the same as the sample spacing.
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This periodic modulation of
the similarity measure introduces local optima that can lead to the incorrect
registration solution being determined.
The computational cost of “correct” interpolation is far too high for this
approach to be used in each iteration, so lower cost interpolation tech-
niques must be used. There are several possible approaches. The first is to
use low-cost interpolation, such as trilinear or nearest neighbor, until the
transformation is close to the desired solution, then carry out the final few
iterations using more expensive interpolation. An alternative strategy is to
take advantage of the spatial-frequency dependence of interpolation
errors. Trilinear interpolation low-pass filters the data, and therefore, if the
T
