An Educational Software for Two Dimensional Computed
Tomography Image Reconstruction with Parallel and Fan X-Ray
Beam
N.ROUSSOS, E.FASOULIS, M.ZISIS, N. ARIKIDIS, P.GIAPARINOS, S.TSANTIS and
I.KANDARAKIS
1
Department of Medical Instrumentation Technology
Technological Educational Institution of Athens
Agiou Spiridonos Str. Egaleo, Athens, 12210
GREECE
Abstract: - A computer based educational software has been implemented in order to demonstrate the two
basic method of image reconstruction employed from computed tomography systems. Contemporary
computed tomography scanners are using the fan X-ray beam method while early scanners were based upon
parallel beams. The parallel beam based reconstruction is easier to be explained mathematically and is used as
an intermediate step for the explanation of the fan beam method. Image degradation and artifacts can be
explained when the mathematical basis is known. The steps for the image reconstruction process are analyzed
and parameters like linear and angular sampling are discussed. The laboratory exercise also incorporates
simulated noise filtering and identification of various artifacts.
Key – Words: - Programming Teaching Tool, Image Reconstruction, Computed Tomography
1 Introduction*
Early Computed Tomography (CT) scanners where
based on the parallel X-ray beam recording. A Xray tube produced a pencil-like beam and the
system tube-detector was shifting and rotated. The
time process was about 5 minutes. Modern CT
scanners combine the X-ray tube with many
detectors and only the rotation movement is
executed. The tube-detector system performs
rotational movement, which results in much smaller
processing time (5-10 seconds) [1].
The Computed Tomography scanners provide
medical image information not available with the
traditional X-ray systems. A CT image is the result
of the superposition of all planes normal to the
direction of propagation. Ideally, it is free of
undesirable effects caused by intervening
structures. For this reason it gives the ability of
*
Financial support for this work was provided by the
project “Upgrading of Undergraduate Curricula of
Technological Educational Institution of Athens”,
(APPS program - Τ.Ε.Ι. of Athens), financed by the
Greek Ministry of Education and the European
Union (Greek Operational Programme for Education
and Initial Vocational Training -O.P. Educationaction: 2.2.2. “Reformation of Undergraduate
Studies Programs ”).
acquiring medical information from deep inside the
human body. CT imaging has gain the confidence
of medical society due its ability to make more
prominent soft tissues compared to other
modalities. Besides that, it can derive useful
information from any imaging plane [1,3].
This exercise studies the principal mathematical
analysis of filtered back-projection algorithm,
which is used in most of the modern CT scanners.
Noisy filtering of the projections is performed in
the frequency domain. The artifacts of non-proper
linear or angular sampling at the reconstructed
image are also presented to the students. All
programs utilized in the exercise were written in
user-friendly MATLAB code.
2 Methodology
A CT image records linear attenuation coefficients.
On each coefficient value, a tone of gray scale is
attributed. So, the major problem that CT scanners
(figure 1) have to defend is the calculation of linear
attenuation coefficient’s value on each point of the
slice. CT scanners can be separated in two parts:
a) the metered part (X-ray tube, detectors,
electronic systems etc) and b) the calculated part,
which consists of the PC and its peripherals
(printer, etc) [1].
N Nyquist 
X-Ray
Tube
Ns
1

2
2S
When the field of view is x, the linear sampling N
is:
N
Detectors
The first step in CT reconstruction is to compute
the linear sampling N and the angular sampling K
as a function of the smallest anatomical structure
with diagnostic information. Under-sampling or
malfunctions may lead to artifacts that are presented
while over-sampling may extend the exposure time
and the cost of the CT scanner.
Both linear sampling and angular sampling depend
on Nyquist frequency. The line integral is the sum
of the linear attenuation coefficients along a line.
The distance S between two line integrals gives the
sampling rate Ns and defines the pixel size. The
higher spatial frequency at the reconstructed image
is the Nyquist sampling rate NNyquist:
x
S
(2)
The projection is the sum of all the line integrals for
an angle θ. The number of projections is the angular
sampling K and for 180o trajectory is:
K
Fig.1. CT scanner interior view
(1)

2
N
(3)
The image of the projections is the sinogram and
the process is called Radon transform. A typical
sinogram, derived from the Shepp-Logan phantom,
is shown at figure 2. The sinogram has no
diagnostic information however is useful for the
detection of possible malfunctions. A damaged
detector will produce a blank row, leading to
artifacts at the reconstructed image. Such artifacts
are reproduced in the laboratory to experience the
students in real situations.
Fig.2. Shepp-Logan image and its sinogram.
Then, the unknown image is transformed in the
frequency domain. The key in image reconstruction
is the Fourier Slice Theorem. According to that
theorem, the 2d Fourier transform of the unknown
image can be achieved by the 1d Fourier transform
of each projection. The frequency domain of the
projections is shown at figure 3, where the dots
correspond to the frequencies and the lines to the
projections. Low frequencies are closer to the
center of the axes [1,2].
Fig. 3. Collecting projections of an object at a
number of angles gives estimates of the Fourier
transform of the object along radial lines.
Figure 3 shows that at higher frequencies the
sampling is not the same as at lower frequencies.
Thus, high frequency information is not well
recorded and may cause image degradation. The
filters used, suspend high frequencies such that
image information is not affected. Noise filtering is
performed in the frequency domain and the filtered
projections are reconstructed with the inverse
Fourier transform (Filtered Back-Projection
algorithm) almost simultaneously with the
projection record. The noise can be simulated by
Gaussian in the laboratory and the application of
different filters is studied.
There are two types of X-Ray beams:
a) the parallel beam and b) the fan beam (figure 4).
The second type can be divided into two subtypes
according to the arraying of the rays inside the
beam: a) equiangular and b) equally spaced.
A
B
Fig. 4. (A) Parallel and (B) fan beam arrangement.
Filtered Back Projection algorithm takes place in
the following three steps for the parallel beam
arrangement and is the same with the reconstruction
from fan beam arrangement:
1) Projection Pθ(t) recording at angle θ:
P t    f t cos  s sin  , s cos  t sin  s
(4)
where
t: number of projections and s:sampling distance
2) Projection’s Fourier transform and filtering:
N 1
(5)
k 0
where:
Qθ(nτ): filtered projection (Radon transform), nτ:
filter’s size and kτ: projection’s size
A
3) Inverse Fourier Transform:
f  x, y  
s
Q n     hn  k P k 
τ: sampling interval and h(nτ-kτ): the ramp filter and
a smoothing filter

K
K
 Q x cos  y sin  
(6)
i 1
where:
f(x,y): reconstructed image and K: number of θ
angles in which projections are known
The parameters that affect the quality of the
reconstructed image are included in equations (4),
(5) and (6). It is shown to the student what artifact
is produced when those parameters are change.
Figure 5 shows the reconstruction with low linear
sampling (figure 5A) and low angular sampling
(figure 5B).
B
Fig. 5. Image reconstruction with (A) low linear sampling (32 per projection), (B) low angular sampling (9
projections).
Figure 6 shows the Shepp-Logan filter in an image
with Gaussian noise added. Only high frequencies
A
are suppressed, where the sampling in the
frequency domain is not adequate.
B
Fig. 6. (A) The Shepp-Logan filter in the frequency domain, (B) reconstructed image.
3 Conclusion
As modern technology becomes more complicated,
the biomedical engineer should be familiar with the
principal concepts of CT reconstruction. The
exercise studies the process of the image
reconstruction from CT scanners and some
parameters that affect the image quality and
produce artifacts. At the end of the exercise, the
student should be in position to recognize particular
artifacts and to optimize image quality.
4 References
[1]
[2]
[3]
I.Kandarakis, Physical and Technological
Principles of Actinodiagnostic, 3rd Edition,
2001.
Avinash C. Kak-Malcolm Slaney, Principles
of Computed Tomographic Imaging, IEEE
Press, 1999.
Edwin L. Dove, Notes on Computed
Tomography, Dove-Physics of Medical
Imaging, Oct., 2001.