Biomedical Engineering Reference
In-Depth Information
Figure 3.3a: what X-ray absorption values for each voxel should re-
place the question marks, given the transmission measurements shown
in that Figure?
In Figure 3.3b, values of the X-ray absorption in the four voxels have
been estimated by trial and error, and the four values are consistent
with the four X-ray transmission measurements. So, have we found
the answer? Alas, no - for Figure 3.3c shows 4 different values of
X
ray absorption which fit the transmission measurements equally
well. What can one do about this ambiguity? The solution is to make
additional measurements from one or more additional angles. For
example, in Figure 3.3d a pair of measurements at 45
−
has been added
and they unequivocally imply that the solution of Figure 3.3c is the
correct one, and not that of Figure 3.3b.
In a general way we can state that one needs redundant measurements
- that is, more measurements than unknowns - typically by a factor of
°
three or so. In practice we want to resolve slices with some 512
×
512
voxels - and so need something of the order of one million measure-
ments. One can readily appreciate that this is computationally
highly demanding. The trial and error approach I used to solve the
2
×
2 problem in my head won't work in the practical situation. Never-
theless, some of the early approaches to CT reconstruction, my own
included (Goitein, 1972), used a form of guided trial and error in
which an initial guess was refined in an iterative process using, for
example, the method of least squares to fit the unknowns (the voxel
values) to the measurements (the X-ray transmission along multiple
paths). Nowadays a quite different one-pass approach is taken, using
Fourier transform methods implemented in special purpose hardware.
Also, CT scanners have evolved to, for example, make measurements
in multiple contiguous slices and with continuous rotation - sometimes
while advancing the patient support continuously (spiral scanning).
These improvements have led to much faster scanning and, in com-
bination with respiratory gating for example, allow time-variations
of anatomy to be studied.
The information content of CT
In the early days of computed tomography it was hoped that tissues
would differ sufficiently in their absorption coefficients that one
would be able to identify the histology and pathology of tissues from
from their CT values. However, this hope has not been fully rea-
lized; while many tissues show marked contrast (bone vs. muscle
and muscle vs. fat, for example), most do not. The identification of
