Digital Signal Processing Reference
In-Depth Information
with a typical energy range between 25 keV and 500 keV for medical
imaging. A conventional radiographic system contains an X-ray tube
that generates a short pulse of X-rays that travels through the human
body. X-ray photons that are not absorbed or scattered reach the large
area detector, creating an image on a film. The attenuation has a spatial
pattern. This energy- and material-dependent effect is captured by the
basic imaging equation
S
0
(
E
)
E
exp
μ
(
s
;
E
)
ds
dE
I
d
=
E
max
0
d
−
(1.1)
0
where
S
0
(
E
) is the X-ray spectrum and
μ
(
s
;
E
) is the linear attenuation
coecient along the line between the source and the detector;
s
is the
distance from the origin, and
d
is the source-to-detector distance.
The image quality is influenced by the noise stemming from the
random nature of the X-rays or their transmission. Figure 1.4 is a thorax
X-ray.
A popular imaging modality is
computed tomography (CT)
,intro-
duced by Hounsfield in 1972, that eliminates the artifacts stemming
from overlying tissues and thus hampering a correct diagnosis. In CT,
x-ray projections are collected around the patient. CT can be seen as
a series of conventional X-rays taken as the patient is rotated slightly
around an axis. The films show 2-D projections at different angles of a
3-D body. A horizontal line in a film visualizes a 1-D projection of a 2-D
axial cross section of the body. The collection of horizontal lines stem-
ming from films at the same height presents a one-axial cross section.
The 2-D cross-sectional slices of the subject are reconstructed from the
projection data based on the Radon transform [51], an integral transform
introduced by J. Radon in 1917. This transformation collects 1-D projec-
tions of a 2-D object over many angles, and the reconstruction is based
on a filtered backpropagation, which is the most frequently employed
reconstruction algorithm. The projection-slice theorem, which forms the
basis of the reconstructions, states that a 1-D Fourier transform of a
projection is a slice of the 2-D Fourier transform of the object. Figure
1.5 visualizes this.
The basic imaging equation is similar to conventional radiography,
the sole difference being that an ensemble of projections is employed in
the reconstruction of the cross-sectional images:
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