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X-RAY TOMOGRAPHY ON THE BASE OF NON-COLLIMATED RADIATION
V.A. GORSHKOV Moscow Automobile-Road Construction Institute (State Technical
University), Moscow, Russia;
M. KRÖNING FhG IzfP, W-6600 Saarbrucken 11, Germany;
Y.V. ANOSOV Moscow Transport Engineers Institute (Technical University), Moscow,
Russia;
N.R. Kuzelev, V.M.Yumashev, L.I.Kosarev, FSUE-ARRI of Technical Physics and
Automation, Moscow, Russia;
V.T.Samossadny, V.Yu. Miloserdin, Moscow Physics Institute (State University), Moscow,
Russia
ABSTRACT
Issues, related to the development of reconstructed tomography based on scattered X
radiation are considered. In the most of scattering tomography based on collimated initial
and scattered radiations the distribution of the object density (linear scattering coefficient) is
reconstructed by the number of counted photons scattered in the reconstructed voxel.
Inhomogeneous of the structure along the initial path and trajectory of counted scattered
photons causes the errors of imaging. Collimation of the initial and scattered radiation allows
detecting only small part of all scattered photons. That is why the statistic of counted photons
and the resolution are very poor. The paper presents mathematical algorithms improving
images in such kind of tomographs. It is shown that the imaging on the base of scattered
collimated radiation does not allow getting the resolution comparable with transmission
tomography. One of the ways of creating scattering tomography with high resolution is
using a big non-collimated detector, which allows counting almost all scattered photons
leaves the object in direction to the detector. It gives the opportunity of numerical
reconstruction of linear scattering coefficient distribution.
Mathematical models of
measuring procedures and reconstruction algorithms are presented based both on the
collimated and non-collimated scattered radiation. The multifunction use of transmission and
scattering tomography gives an opportunity of identifying an effective atomic number and
density distribution.
PHYSICAL BASES OF SCATTERING TOMOGRAPHY
Physically backscattering tomography is based on back-scattered radiation field
characteristic dependence on density distribution and mass coefficients of photo absorption
and Compton scattering.
Photo absorption mass coefficient is characterized by a complex dependence on the
atomic number of matter. The mass coefficient of scattering is proportional to the atomic
number-to-atomic mass ratio. Since Z/A ratio for light elements (except hydrogen) is
approximately 1/2, it may mean, that this ratio is not dependent on the atomic number [1].
In case of monochromatic radiation, accompanying the process of photon propagation in
an object, induced radiation sources are produced (due to scattering) along the initial
movement trajectory (l1). Their activities (Ns ) at the distance from the source (L1) are
defined by the its activity ( N0), by the size of the scattering voxel (∆), by its Compton
scattering linear coefficient µx(E0), by linear absorption coefficient distribution µ(l1,E0) along
the initial trajectory. The intensity of back-scattered radiation (Nd) recording is in its turn
defined by the intensity of the induced source (Ns), by the energy of scattered photons (E1),
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by the distribution of linear absorption coefficient µ’(l2E1) along the reverse trajectory (l2),
and the distance from the induced source to the detector (L2):
N d = N d (∆, µ x (E 0 ), µ (l 2 , E 0 ), µ ' (l 2 , E 0 )) =
L
L


= αβN 0 1 − exp{− µ x (E0 )∆}⋅ exp − ∫ µ (l1 , E0 )dl1 − ∫ µ ' (l2 , E1 )dl2  ,
0
 0


1
2
(1)
Where α, β are coefficients, taking account of scattering angle and photo- and Compton
effect probability ratio distribution, respectively.
The given expression demonstrates the potential possibility of reconstructing linear
scattering coefficient distributions by recording the intensity of scattering radiation.
It is evident, that the intensity of back-scattered radiation is proportional to the linear
Compton scattering coefficient of the reconstructed voxel of the object.
Currently there are two basic types of scattered radiation tomography:
Collimated scattered radiation tomography;
Non-collimated scattered radiation tomography.
COLLIMATED SCATTERED RADIATION TOMOGRAPHY
Operating principle of collimated scattered radiation tomography, incorporating a
collimated source and a detector (a group of detectors), is very simple: it is based on the
visualization of scattered radiation intensity.
This principle is realized in the majority of laboratory tomographs.
Radioactive isotopes, X ray tubes and secondary-scattered radiation from special purpose
scatters are used in the back-scattered radiation tomographs as radiation sources.
The tomographs also comprise pinholes [2,3], cylindrical, cell-type and gap collimators
[4,5,6,7].
In contrast to transmission tomography, density (linear scattering coefficient) distribution
images in scattering tomography are not synthesized by calculations: they are obtained by
visualization of the recorded scattered radiation intensity distribution.
Scattered tomographs are characterized by certain drawbacks given below:
Owing to attenuation of radiation in the preceding layers the recorded
scattering intensity in such tomographs does not correspond to the actual density
distribution. The more complicated the linear attenuation factor distribution in the initial
and scattered trajectory is, the less adequate is the image obtained.
Tomographs of this category do not allow one to achieve the same resolution
as in transmission ones. The necessity for scattered radiation collimation results in a low
counting statistics. Such tomographs register only 0.1-2% of the whole amount of the
radiation scattered. Increasing resolution at the expense of reducing the size of a scattered
voxel leads to a still greater reduction of registration intensity.
Application of mathematical processing methods allows quantitatively improving the
resolution quality.
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Figure 1. Correction of radiation attenuation by initial and scattered trajectory
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Numerical estimation of linear scattering coefficients distribution
As follows from equation (1), the value of linear Compton scattering coefficient µx(∆) for
a voxel is calculated as:
µ x (∆ ) =
L
L

1 
Nd
⋅ exp ∫ µ(l1 , E 0 )dl1 + ∫ µ' (l 2 , E1 )dl 2   .
ln1 −
∆  N 0αβ
0
0

1
2
(2)
Knowing the photon energy and approximate ratio of scattering coefficient to the total
one we can reconstruct the linear scattering coefficient distribution in a layer-by-layer
manner. Since the back-scattered photons pass not only through the preceding layers
reconstructed, but partially through the current ones, a recurrent procedure of calculating the
linear coefficient of the current layer voxel scattering is introduced to the reconstruction
algorithm [10, 11, 12.].
Figure 1 shows the results of an experiment, simulation and reconstruction of
experimental data, based on equation (2) obtained by using ComScan tomograph [9].
Aluminum plates with silicon-filled holes were used as research objects. Figure 1a presents
images of the object layers, scattering photons, registered by the respective tomograph
detectors. Figure 1b presents actual distributions of linear coefficients of scattering in the
layers, whose radiation is being registered by different tomograph detectors.
Distribution of photons, registered by detectors, which was obtained in the course of
experiments and simulation, is shown in Figure 1c. Because the statistic was very poor (not
more then 250-300 photons per one reconstructed voxel) the experimental data presents
averages values of 50 measurements. Figure 1d demonstrates reconstructable values of
Compton scattering linear coefficient based on the equation 2.
It is clear, that the reconstruction has notably improved the signal adequacy to the actual
density distribution. But for that it is necessary significantly to increase the statistic of
photon registration.
Increasing resolution of reconstruction
The number of photons without regard to radiation attenuation effects S(x) in the layerby-layer object scanning will be defined by convolution detector aperture function ω(x) and
by density ρ(x):
x
s(x ) = ∫ ω(x ')ρ(x '− x )dx ' .
(3)
0
Z-transform of the convolution is presented as
S (z ) = W ( z )ρ (z ) .
(4)
From that the z-transform of the reconstructed density distribution will be
S(z )
(5)
= ρ0r + ρ1r z −1 + ρ r2 z −2 + ρ3r z −3 + K .
W (z )
Where the polynomial coefficients ριr define the reconstructed density values in layers.
ρ r (z ) =
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Figure 2. Reconstruction results
To accomplish reconstruction on the basis of z-transformation and with regard to
radiation attenuation, it is necessary to determine the quantity of photons, which would have
been registered under the absence of attenuation Sk, by actually registered quantity of backscattered Ndk photons.
The total amount of photons without regard to attenuation can be calculated as
N dk − N 0 ∑ µi ∆ω k −i +1 (1 − αi )
k −1
Sk =
i =1
1 − αk
k
+ N 0 ∑ µk −i+1∆ωi ,
(6)
i =2
where αi coefficients can be calculated as:
α0 = 0
α k = ∑ (µ ib ∆ + µ ib ∆b )
k −1
i =1
Recalculated Sk values can be used in expression (6) to obtain density distribution.
Equation (6) make it possible to directly reconstruct the linear coefficient of k-layer
scattering by the reconstructed (k-1) layers:
µk =
k −1
N − N 0 ∑ µ i ∆ω 4−i +1 (1 − α i )
k
d
i =1
∆ω 1 (1 − α 4 )
.
(7)
A three-layer object, in which each layer consisted of four razor blades, bonded by a thin
adhesive layer, was used in experimental research (Figure 2). Each layer was oriented in
three different directions. A layer thickness made up 0.45 mm.
It can be seen, that a ComScan tomograph detector registers back-scattered radiation
from 1.0-1.2 mm layer and can adequately represent only the first layer. The use of layer-bylayer reconstruction allowed one to increase resolution substantially. In contrast to ComScan
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images, cuts made by layer two and three razor blades can bee seen in the patterns
reconstructed.
The first image of layer one fragment is obtained due to the fact, that the planes of the
blades studied turned out to be non-parallel to the tomograph table surface.
Investigations have shown [13,14], that owing to the low statistics, only 2-3 times
increase of resolution is possible in collimated scattered radiation tomoghraphy, which,
naturally, prevents one from achieving the resolution, commensurable with that provided by
transmission type tomographs.
NON-COLLIMATED SCATTERED RADIATION TOMOGRAPHY
Non-collimated radiation tomography [15] (Figure 3) allows registering a substantial
portion of scattered radiation (including multiple scattered photons). Almost one hundred per
cent registration of the radiation scattered and emitted beyond the limits of an object is
possible, depending on the type of the object.
y0
Object
Detector
x0 Initial collimated
radiation
ω0
Figure 3. Non-collimated scattering tomograph
The object is represented in the form of a collection of i,j voxels, within which the linear
scattering coefficient (density) can be taken as a constant value.
The linear scattering coefficient distribution of the object µ(i,j) can be calculated from the
algebraic system of equations of albedo radial sums
∑( µ) (i, j )∆L(i, j ) = S (x
i= f x
0
, y 0 , ω 0 ),
(9)
An array of albedo radial sums S(x0,y0,ω0) along the initial trajectories can be obtained by
the array of intensity of recording scattered radiation N(x0,y0,ω0), measured at different
angles ωo and coordinates xo, yo of initial beam radiation [16, 17] as:
S (x0 , y , ω 0 ) =
1
ln(
k
N0 −
N0
k
N (x , y , ω )
α ( x0 , y 0 , ω 0 ) d 0 0 0
92
)
(10)
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where N0 is the initial number of photons, k is complete attenuation factor to Compton
scattering factor ratio, α(x0,y0,ω0) is the share of registered photons among the total amount
of photons, scattered in vector (x0,y0,ω0).
The value of α(x0,y0,ω0) depends on the position and size of detector and by density
distribution and can be determined by means of simulation during the object reconstruction
[16]. This factor (at the first stage of research) is taken as a constant value.
Thus, the problem is reduced to solving a linear system of equations with unknown linear
scattering coefficients µ(i,j).
Figure 4 presents the results of simulating the above mentioned reconstruction procedure
for a fragment of aluminum sheet with a crack and corrosion. The reconstructed picture
represents distribution of linear scattering coefficient in the object.
Object
Aluminum
sheet
Corrosion
Crack
From below
Response S(x0y0ω0)
From the right
From the top
From the left
Reconstruction of µij
By response from the top
By all responses
Figure 4. Example of reconstruction (in aluminum)
Figure 5 presents the results of simulating the honey-comb structure of composite
material.
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Figure 5. The images of the honey-comb structure with the defect (a) by one-side access (b)
and two-side access (c)
RECONSTRUCTION OF EFFECTIVE ATOMIC NUMBER
DISTRIBUTION
X-ray transmission and scattering tomography makes it possible to assess distribution of
complete linear absorption coefficient µt and linear Compton scattering coefficient µC. For
substances with relatively constant chemical composition the images produced are
practically equivalent to density distribution. However, a notable difference between the
investigated object components in terms of effective atomic number does not allow one to
clearly identify the reasons why the degree of darkening of this or that section of the
reconstructed object image varies, since this may be due both to the change of effective
atomic number and density.
In this context it may be reasonable to develop a method of density distribution
assessment (invariantly to effective atomic number distribution) and a method of effective
atomic number distribution assessment (invariantly to density distribution).
Thus, the ratio of linear coefficients is equal to the ratio of mass coefficients; assessment
of atomic number of a particular pixel can be done on the basis of certain mass coefficient
dependences on energy [17, 18].
In a number of cases assessment of the effective atomic number distribution can be
replaced by reconstruction of the Compton scattering to absorption coefficient ratio itself.
Visualization of this ratio, for instance in medicine, can serve as an additional information
(alongside with density distribution) in the diagnostics of various diseases.
Investigation into a possibility of invariant density and effective atomic number
reconstruction was conducted using a statistical model of X radiation interaction with matter.
An artificial object (Figure 6) represented a 100x100 collection of different density
voxels, their size being 1x1 mm, with different atomic numbers zef. = 1, 8, 13, 19, 20, 14. For
the sake of visualization the position of pixels with different atomic numbers is presented in
the form of periodic system element symbols, corresponding to atomic number values of
hydrogen, oxygen, aluminum, potassium, calcium (Figure 6a). The rest of pixels correspond
to the atomic number of silicon (zef=14). The pixel darkening degree is proportional to the
effective atomic number. The pixel density varied in a layer-by-layer manner within
0.5…2.0 g/cm3 range according to the distribution, shown in Figure 6b, where a deeper
darkening corresponds to lower densities.
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Atomic number
distribution Z
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Transmission
radiation detector
ρ
Scattered
radiation
detectors
Object
Initial
radiation
a)
b)
c)
Figure 6. Artificial reconstruction object
To ensure a constant character of α(x0y0, ω0) coefficient, the object scanning and
registration of scattered and transmission radiation were conducted according to a scheme,
shown in Figure 6c.
Figure 7 presents the results of reconstructing linear Compton scattering coefficient
µc(i,j), linear absorption coefficient µt(i,j), ratio of coefficients µc(i,j)/µt(i,j), effective atomic
number z and density ρ, with the quantity of initial photons per one albedo radial sums - 1
mln. Data on the object composition was used during the reconstruction as a priori
information, which means, that in determining the effective atomic number the element,
which was the closest to the reconstructed value, was selected by the ratio of scattering and
absorption coefficients.
It is evident, that both density and presence of hydrogen in the object produce an effect
on the reconstructed value of the linear scattering coefficient within the entire energy range
under investigation. This is conditioned by the fact, that the mass scattering coefficient is
proportional to the atomic number to atomic weight ratio.
For the chosen elements the most adequate reconstruction of effective atomic number
distribution is present only at 50 keV photon energy, which can be explained by the
maximum difference in the ratio of Compton scattering and complete absorption coefficients
for these elements.
The best reconstruction of the object density distribution is also observed within the same
energy range.
The values of optimal energy of initial photons are increased at higher atomic numbers.
The resolution in scattered non-collimated radiation is defined exclusively by the width
of the initial photon flux beam.
Investigations have shown, that in this case [18] generation of not less than 100 thousand
photons per a single measurement is needed for highly adequate reconstruction of density
and effective atomic number.
Conclusion
The presented results show the principle opportunity to design X-ray tomograph on the base
of non-collimated radiation with the resolution comparable with transmission tomograph.
E, KeV
µC
µt
µC/µt
Z
ρ
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20
50
100
200
300
400
500
Figure 7. Reconstruction under different values of photon energy
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