X-Ray Computed Tomography
Jonathan Schock, Manuel Viermetz
March 21, 2017
X-Ray Computed Tomography (CT) is an indirect non-destructive imagingmethod. It allows to visualize the local absorption properties of a specimen.As it is an indirect method, CT requires the application of a numerical recon-struction algorithm to retrieve these absorption coefficients. In this studentlab experiment you will perform several CT scans and reconstruct tomogramsfrom them using the Filtered Backprojection algorithm. For data analysisand all the calculations MATLAB is used. You may bring along your ownsample as well for examinations. A printout of this manual is available atthe experiment!
Location: Physics department (Chair of Biomedical Physics)room: 1710 (in the basement, back left end from main entry)Technische Universitat MunchenJames-Franck-Str. 185748 Garching, Germany
Contacts: Jonathan Schock or Manuel Viermetzroom: 1.110 (open office, 1st floor)TUM Institute of Medical Engineering (IMETUM)Boltzmannstr. 11, 85748 Garching, Germanyemail: email@example.com firstname.lastname@example.org
phone: +49 (0) 89 289 10846 or +49 (0) 89 289 10861
As a non-destructive analysis technique computed tomography (CT) has been used inmedicine, materials science, biology and many other fields for a long time. CT allows tovisualize the x-ray absorption coefficient with respect to the position inside the sample.
This student lab experiment was designed to give you an insight on the backgroundof this technique. You will perform one complete CT scan of a sample to analyze someproperties of a typical reconstruction algorithm and to understand the physical theoryand the necessary steps that have to be taken to end up with a good CT reconstruction.Please bring along a suitable sample, that is non-metallic, has a mixture of differentmaterials and is about 4x4x4 cm large. As the amount of data created during theexperiment is quite large (around 10 GB), it is advisable to bring along either a largeUSB memory stick or an external hard-drive if you take your data the next day afterprocessing.
To fully understand the experimental details it is recommended to read the relevantchapters in , which is also available online , in addition to this manual. The ex-perimental control as well as the data reconstruction and analysis will be performed inMATLAB. If you are not familiar with this application, you should start reading .Section 3.5.3 gives a short list of MATLAB commands that you will most likely need.
For the interpretation of your results the Photon Cross Sections Database  and theTables of X-Ray Mass Attenuation Coefficients  may prove to be helpful.
X-ray computed tomography (CT) is not a direct imaging method, which means thatthe maintained data has to be analyzed and visualized using several complex algorithms.As a result it gives three-dimensional information about the distribution of the element-specific attenuation coefficient in the scanned volume. An x-ray producing device(e.g. an x-ray tube or a synchrotron) is used to take pictures of a sample in front ofa detector. These two-dimensional pictures are called projections. Several projectionsare taken from different angles via either rotating the sample around the vertical axisor rotating the x-ray device and detector around the sample. The obtained projectionsare then used to compute a three-dimensional representation of the spatial distributionof the attenuation coefficient inside the sample. Here, the principle of the imagereconstruction will be briefly explained (for more details see the particular references).
This part of the manual will focus mainly on the mathematics of the physical theoriesbehind CT and image reconstruction, leaving out the physical mechanisms that lead toabsorption of radiation in matter. The formulas and the approach of this section arebased on [6, (pp. 32f., 46f.)].
2.1 Attenuation Coefficient
All physical mechanisms that have to do with attenuation of radiation going through amaterial are dependent on the attenuation coefficient of the material , which is a posi-tive, material specific constant, considering absorption and scattering of radiation insidethe material. The absorption and therefore the attenuation coefficient itself depends onthe atomic number Z of the material as
and directly proportional to the mass density of the respective material. This resultsin the advantage, that by knowing the spatial distribution of the attenuation coefficient,one has a direct physical interpretation of the values. If polychromatic radiation is used,one also has to consider the dependency of the wavelength which goes like:
As mentioned in the next chapter, there is a correction for polychromaticity effects.
2.2 Lambert Beers Law
After Lambert Beers law, the intensity of radiation after traveling through a materialwith the thickness and the attenuation coefficient () is given by:
I( + ) = I() ()I() (1)
This equation can be reordered and by taking the limit one gets the differential quotient
d= ()I() (2)
Assuming that the material is homogeneous, the function () collapses to the con-stant along the entire length of the material. After simple integration and expo-nentiation one obtains
I() = I0 e (3)where I0 = I(0). As we go away from the ideal point of view and over to real materials,it is sure, that the attenuation coefficient will not be constant anymore and even dependson the energy of the radiation which leads to
I0(E) e s0 (E,) d dE (4)
where s is the travel distance of the radiation. The simplified model without energydependence is then:
I(s) = I(0) e s0 () d. (5)
As the x-ray tube employed in this experiment produces polychromatic radiation, theexact solution is provided by equation (4). However, being a quite good approximation,equation (5) can be used instead for our purposes.As mentioned above, one is especially interested in the spatial variation of the attenua-tion, therefore the projection integral is defined as:
p(s) = ln(I(s)
() d (6)
2.3 Image Reconstruction
This part covers the reconstruction of two- and three-dimensional images out of the rawdata. It follows closely the according sections in [7, (p.247 ff)] where all equations canbe found. Small changes of variables and names were introduced to preserve consistencywith the previous chapter.
2.4 Two-dimensional Reconstruction
Despite having a three-dimensional reconstruction problem to solve, we start with thetwo-dimensional problem, which can than be generalized to the three-dimensional case.The attenuation coefficient in one plane is assumed to be a well behaving function,which means that it is continuous inside the object and zero outside the object. More-over, the object has to be completely inside the detectable area, which can mathemat-ically be described as the inside of the smallest circle around the object, where thefunction of the attenuation coefficient is zero everywhere outside the circle. As we arediscussing the two-dimensional case, the distribution of the attenuation coefficient insidethe material can be described as (x, y). The easiest way to solve the problem of re-construction is transferring the whole system to cylinder coordinates, with the objectssupport fitting in the circle around the origin of the coordinate system. A ray is astraight line in cylinder coordinates and can therefore be described as:
x cos() + y sin() = r (7)
where is the view angle and r the position on the ray. The projection integral p(s)from equation (6) can then be written as:
p(, r) =
dx dy (x, y)(x cos() + y sin() r) (8)
where is Diracs delta function. The transformation from (x, y) to the projectionintegral p(, r) is called Radon transformation R. Interpreting R as an operator, one canwrite
p = R (9)
and so the problem of reconstruction is equivalent to finding the inverse operator toget the spatial distribution of the attenuation coefficient out of the measured data.Consequently this breaks down to finding R1 in
R1p = , (10)
which is not trivial.
2.5 Fourier Slice Theorem
This section is to be seen as a little mathematical insert to understand the problem offinding R1 and the essential theory behind the solution idea. The Fourier transform Fof a function f is defined via
F (u) =
f(x)e2iux dx (11)
and can be reversed via
F (u)e2iux du. (12)
The Radon transform is the result of the Fourier slice theorem, which essentially states,that the Fourier transform of the projections with respect to the distance parameter [r]is the same as the two-dimensional Fourier transform of the object expressed in radialcoordinates [(ux, uy) = (u cos , u sin )] [7, (p. 294)]. For our case this leads to thesimple identity
F (ux, uy) = F (u cos(), u sin()) = P(u) (13)
where P(u) is the Fourier transform of the projection integral, defined as
p(r) e2iur dr (14)
Here, the two-dimensional case of the projection integral in equation (6) is used whichreads:
(, ) d (15)
2.6 Simple Backprojection
Whenever enough projections are measured to have enough data points in the spectralspace, which is described via the coordinates (ux, uy), it is possible to apply the two-dimensional inverse Fourier transform to reconstruct the attenuation values. This iscalled simple backprojection, as it traces back the projection values to their origin. Theequation for simple backprojection describes the mathematical step: