Biomedical Engineering Reference
In-Depth Information
A
1
A
3
in position 1 : A
ð
1
Þ¼
ð
1
Þ
A
2
A
4
in position 2 : A
ð
2
Þ¼
ð
2
Þ
A
1
A
2
in position 3 : A
ð
3
Þ¼
ð
3
Þ
A
3
A
4
in position 4 : A
ð
4
Þ¼
ð
4
Þ
In practice, only the measured absorption factors A(1), A(2), A(3), and A(4) would be
known. The problem is, therefore, to compute A
l
,A
2
,A
3
, and A
4
from the measured absorp-
tion values. The fact that the computation is possible can be seen from Eqs. (1) to (4). There
are four simultaneous equations and four unknowns, so a solution can be found. To recon-
struct a cross section containing
n
rows of blocks and
n
columns, it is necessary to make
at least
n
individual absorption measurements from at least
n
directions. For example, a dis-
play consisting of 320
320 picture elements requires a minimum of 320
320, or 102,400,
independent absorption measurements.
It was mentioned earlier that these absorption measurements are taken in the form of
profiles. Imagine a plane parallel to the x-ray beam as defining the required slice. An
absorption profile is created if the absorption of the emergent beam along a line perpendic-
ular to the x-ray beam is plotted. This profile represents the total absorption along each
of the x-ray beams. In general, the more profiles that are obtained, the better the contrast
resolution of the resulting image.
From these individual measurements, a single two-dimensional plane can be recon-
structed, and by simply (i.e., for the computer) stacking the appropriate sequence of such
planes, it is possible to reconstruct a full three-dimensional picture. Image reconstruction
of a three-dimensional object is therefore based primarily on a process of obtaining a cross
section or two-dimensional image from many one-dimensional projections. The earliest
method used to accomplish this was defined simply as “back projection,” which means that
each of the measurement profiles was projected back over the area from which it was taken.
Unfortunately, this rather simple approach was not totally successful because of blurring.
To overcome this, iterative methods were introduced that successively modified the profile
being back-projected until a satisfactory picture was obtained. Iteration is a good method,
but it is rather slow, requiring several steps to modify the original profiles into a set of
profiles that can be projected back to provide an unblurred picture of the original image.
To speed up the process, mathematic techniques involving convolution or filtering were
introduced that permit the original profile to be modified directly into the final one.
Whichever method of reconstruction is used, the final result of the computation is the
same. In each case, a file, usually known as the picture file, is created in the computer mem-
ory. The picture file contains an absorption coefficient or density reading for each element
of the final picture (e.g., 25,600 for 160
320 pixels).
The resultant absorption coefficients for each element of the image calculated in this
manner can then be displayed as gray tones or color scales on a visual display. Most CT
systems project an “image” onto the screen of a computer display terminal. Each element
or “pixel” of the picture file has a value that represents the density (or more precisely the
relative absorption coefficient) of a volume in the cross section of the body being examined.
160 pixels or over 100,000 for 320
