Muon tomography imaging
The principle on which muon tomography relies is multiple Coulomb scattering, which explains how the muons traversing an object are scattered with a deviation angle that has a Gaussian distribution around zero and a standard deviation calculated as follows: [2, 3]:
$$\begin{aligned} \sigma \ \approx \ \frac{13.6 \ \text{ MeV/c }}{p}\sqrt{\frac{L}{L_{0}}} \end{aligned}$$
(1)
where p is the particle momentum, L is the thickness of the traversed object, and \(L_{0}\) is the radiation length of the material the object itself is made of, which is related with the atomic number Z.
The software for the reconstruction of tomographic images that starts from the interaction points of the muons in the detectors to assess the scattering density (SD, which is proportional to the mean square deviation angle of muons, i.e., \(\sigma ^2\), passing through a unit depth of that material) has been made conforming to the algorithm developed by Schultz et al. [2, 3]. The algorithm has been modified to be geometry-independent and suitable to the detectors used in the TECNOMUSE project.
The results and images reported in this article have been obtained from tomographic reconstructions and analyses performed by using MATLAB 2017b.
In order to evaluate the imaging performance of the scanner, some Monte Carlo simulations have been performed with different geometries and materials in the scanning volume inside the middle container.
Unlike other imaging techniques (e.g., nuclear medical imaging such as PET or SPECT), for which consolidated procedures for evaluating imaging features exist (e.g., NEMA protocols [25]), there are no guidelines to assess imaging performance in muon tomography.
In this section, several simulated geometries are presented, each tailored to evaluate a different imaging performance.
Spatial resolution and voxel size
In order to evaluate the spatial resolution of the system, the object inside the scan volume has a geometry borrowed from nuclear imaging, namely the so-called “bar-phantom”.
The bar phantom, as shown in Fig. 2a, is made of a series of vertically placed square slab with 25 cm side and different thickness, namely 0.5 cm, 1 cm, 2 cm and 3 cm. The spacing between the slabs is the same as the thickness.
Tomographic reconstruction have been executed taking into account different voxel sizes in order to evaluate the best trade-off between spatial resolution and voxel size, since the latter implicates the time for the imaging reconstruction.
Images show the projection of tomographic reconstruction after an acquisition of about 4 and 8 min at sea level given the aforementioned counting efficiency, corresponding to about 12,000 tracks and 24,000 muon tracks, respectively. While the reconstruction with 5-cm voxel size does not allow to recognize (as expected) none of the section of the phantom, those with 2 cm and 1 cm allow to recognize the 2-cm bars. The 1-cm reconstruction, due to the higher level of noise, does not make the 1-cm pattern distinguishable, and it does not add any further detail to the 2-cm pattern.
Taking into account also the times needed for the reconstruction with a single, 2.60GHz, core processor, shown in Table 3, it appears that there are no advances in using 1-cm voxel size.
In a real scenario, fast reconstruction algorithms are mandatory to avoid any delay in image reconstruction, which could impact and slow down the overall inspection time.
Since the minimum recognizable thickness of the bar-phantom is in both cases 2 cm (it confirms the result in [26]), it can be inferred that this value is the spatial resolution of the system.
The following analyses have been executed with 2-cm voxel size in order to obtain the best trade-off between image quality and time for its realization.
Counting efficiency and count rate
As mentioned above, the design of the scanner with the three stacked containers is a prototypical one and also the size and geometry of the MRPC detectors have been chosen to prove the feasibility of the detector.
In fact, the detection area providing planar positional information is given by the sole intersection of the blue and green detectors (see Fig. 1), for an overall area of 176 \(\times \) 176 cm\(^{2}\) on the four planes perpendicular to the vertical axis.
For this reason, the solid angle of acceptance for this design is just 0.22sr. Taking into account the detection efficiency of the four MRPCs, this provides a count-rate of about 48 Hz (i.e., about 2900 counts per minute).
Table 3 Time needed to a single-core CPU to reconstruct an image of 12,000 muon tracks with the specified voxel size
Field-of-view
By design, the detector efficiency in term of counting is better at the center of the horizontal plane of detection since it is possible to collect more tracks with respect to the edge of the plane. In fact, muons impinging the edges of the detectors in the upper container can be scattered by the objects in the scanning volume outside the area covered by the detectors in the lower container, resulting in a lost track, which is not recorded by the system. In order to evaluate the influence of the detection geometry on the field-of-view (FoV), a grid has been simulated. The grid is made of a series of concentric square each one 5-cm-thick spaced of 5 cm with a maximum side of 170 cm. The squares are connected by a cross having the same thickness.
Figure 3 shows that the outer corners of the grid are not recognizable, and the features are clearly distinguishable up to 99 cm from the center, that is, the radius of the circle defining the use of the FoV of the scanner in this configuration. Although this size could not be enough for the scanning of a complete container, it is important to stress that the TECNOMUSE scanner is in a prototypical geometry in this phase.
Volume recognition
Two simulations have been made to evaluate the capability of the system in the identification of the volumes depending on the time and their sizes. The first has been made simulating a series of Tungsten slabs with a 10 cm \(\times \) 10 cm face and variable thicknesses from 5 to 100 mm. The reconstruction of the slabs is shown in Fig. 4a, for acquisition time from 4 to 20 min.
All the images of the slabs are visible since 4 min but those with the smaller thickness becomes more evident only after 16-min acquisition. The results show that 4-min acquisition is enough to provide, even though quite weakly, an information proving the presence of high Z materials, especially if the thickness is sufficient. Increasing the acquisition time, the square shape becomes more recognizable and the signal is less noisy. The improvement is clearer for the thicker slabs (in which the contrast increase from 62 to 82%) with respect to thinner slabs (for the 20 mm one the contrast only increase from 15 to 20%).
The other simulation is made by simulating cubes with different sides from 0.5 to 10 cm, as shown in Fig. 4b. Differently from the slabs, that are all visible since 4-min acquisition, the cubes are different since the volume of each one is not enough to produce a sufficient contrast. As the acquisition time increases, in addition to a better definition of the shapes, the number of visible cubes also increases, although the smallest cube visible is the 20 mm one. This result is in accordance with the one highlighted in Sect. 3.2. The signal from smaller cubes, since the voxel size is 2 cm, is not distinguishable from the noise after 20 min, too. It is important to stress that beyond the choice of the voxel size, the signal from cubes is proportional to the third power of the side and, for this reason, the signal from a 1-cm-side cube is just one eighth of the signal of a 2-cm-side cube. It follows that detecting very small objects is practically impossible with the aforementioned acquisition times.
It is important to stress that the detection of so small objects is beyond the scope of muon tomography; in practice, due to the limited amount of time for the scanning of the container, the goal of the technique is not the identification of very small volumes, but volumes of at least 1000 cubic centimeters [27, 28].
Elemental composition
In order to assess the capability to differentiate object made of materials of different atomic number, a simulation involving cubes of progressively increasing Z has been made. Analogously to the simulations of the slabs and cubes, in this one, a series of 10-cm-side cubes but made of different materials, namely water, aluminum (\(Z=13\)), iron (\(Z=26\)), silver (\(Z=47\)), tungsten (\(Z=74\)) and lead (\(Z=82\)) have been simulated. They have been placed along a circumference with the atomic number ascending clockwise.
The results are shown in Fig. 5. The green color indicates low atomic number and, progressively, yellow and red indicate higher atomic numbers. Until aluminum, there are no evidence of red dots (indicating high scattering density) that starts from iron and progressively increase with the atomic number. The image has been made taking into account muon interactions occurring in about 25 min at sea level.
Simulation in a real scenario
In order to assess the performance of the scanner in a real scenario, a simulation taking into account also the structure of the TECNOMUSE tomograph is reported.
In fact, the presence of the structure of the containers is expected to be an additional source of noise to the image and its contribution has to be evaluated. To evaluate the effect of the structure of the containers on the reconstructed images, additional simulations have been made.
From the comparison between Figs. 5 and 6a, it is clear the contribution due to the structure of the containers, mostly made by iron bar at their bottom and several millimeters of iron on their ceiling. The structure produces a series of scattered and spurious points that produces an homogeneous cloud of noise around the main signal.
In order to partially suppress these noisy contributions, a simple filter based on a threshold between the voxel value and the values from the first neighbors, reported in Eq. 2, allows to suppress most of those spurious counts. \(\mathrm{SD}_{x,y,z}\) is the value of the scattering density for a voxel and if the sum of its neighbors is smaller than a certain fraction (the threshold \(\tau \)) of the voxel value, then the value is set to zero. It implies that only counts in a neighborhood composing a continuous volume remain after the filter.
$$\begin{aligned} \text{ If: } \quad \sum _{i=x-1}^{i=x+1} \sum _{j=y-1}^{j=y+1} \sum _{k=z-1}^{k=z+1} \mathrm{SD}_{i,j,k} - \mathrm{SD}_{x,y,z} < \tau \cdot \mathrm{SD}_{x,y,z} \quad \text{ then: } \quad \mathrm{SD}_{x,y,z}=0 \end{aligned}$$
(2)
The result of the reconstruction after the application of the filter with threshold \(\tau =20\%\) is shown in Fig. 6b. It is clear that most of the signal is suppressed and the content of the scanning volume becomes clearer.