Biomedical Engineering Reference
In-Depth Information
the missing information. To provide an objective assessment of the nature of
the coverage supported by the image data, it is instructive to follow the Fou-
rier space treatment by Noll This analysis is targeted at MRI data but
is applicable more generally to other medical images. The key concept is
sampling density in the through-slice direction.
We consider the object being imaged (brain, breast, leg, etc.) and examine its
Fourier domain representation, which is a spectrum of spatial frequencies.
Since the object exists in 3D, its Fourier domain representation also has three
dimensions. For simplicity, we will look in detail at one spatial direction,
namely the through-slice direction (see Figure 4.1a). Because the object has
structure down to microscopic scales, its native spatial frequency spectrum
extends over a large range (Figure 4.1b).
We now consider the process of slice definition, for example, by a selective
excitation in MRI, the beam of x-rays used in CT, or sonic beam profile in ultra-
sound, etc. (Figure 4.1c, d). Selection of a single slice may be viewed as multi-
plication of the object by the slice sensitivity profile and summation or
projection in the through-slice direction. Slices can in principle be acquired at
any location, so to formulate the problem more generally we may consider
replacing each point on the object with the signal that would be achieved for a
slice centered at that point. The result is called a convolution of the slice profile
with the object (Figure 4.1e). Selection of any given slice is then simply a delta
function sample of the convolved distribution. Viewed from the Fourier
domain, convolution of the slice profile with the object is equivalent to multi-
plication of the object spectrum by the slice profile spectrum (Figure 4.1f). In
general, the slice spectrum has a more limited frequency content than the object
spectrum, so the process reduces available spectral content (Figure 4.1f).
Selection of a set of regularly spaced slices with separation
et al.
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can now be seen
as acquisition of a set of samples up to a frequency in the Fourier repre-
sentation (Figures 4.1g, h). The frequency content not sampled by the process
is lost and results in aliasing of information in the slice direction. The process
of reslicing resamples the data and redistributes the aliased signals along
with the correctly sampled signals to produce corrupted signal intensities
(Figure 4.1i, j).
To control the degree of aliasing requires choice of a slice profile, which
determines the bandwidth that must be sampled, and sampling with suffi-
cient density in real (object) space to cover the necessary frequency range.
This always necessitates overlapping slices, and the degree of overlap is
determined by the slice profile. An idealized top hat profile would have a
large, potentially infinite spectrum, necessitating sampling at very closely
spaced intervals. The implication is extensively overlapped slices and not the
intuitive starting point of contiguous slices. Use of softer slice profiles (e.g.,
Gaussian profiles) degrades slice definition, but requires less dense sampling.
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Many medical images are not acquired with uniform sampling; perhaps the
most striking example is freehand ultrasound. However, the concepts dis-
cussed above provide guiding principles for image acquisition with clear con-
clusions as to the results of undersampling. If data such as freehand ultrasound
