3.1. 3D-CT Primary Images
CT imaging can be performed by measuring the decrease in the gamma-ray intensity passing through a CT sample along a series of linear paths and various angles. The amount of decrease depends on the gamma-ray energy, the path lengths, and the material’s linear attenuation coefficients. Therefore, one can obtain a gamma-CT image for a sample using the gamma-ray transmission factor measurements of the off-resonance energy, known as off-resonance (ε
off) attenuation, which originates from the atomic effect. The procedures of the 2D gamma-CT images calculation are detailed in references [
13,
14,
21]. Since the height of the CT sample holder was 20 mm, we divided it into 20 layers and added two extra layers; one of the two extra layers was located below the bottom of the CT sample and the other above it. These layers were separated by a vertical distance of 1 mm. Therefore, the total number of scanned layers was 22. We assigned the labels L1 to L22 to the layers at the vertical distances of
y = −1 mm to
y = 21 mm, respectively. Consequently, we obtained one 2D gamma-CT image (25 × 25) with a resolution of 1 mm/pixel for each row of rods in the
y direction of the CT sample scan. The pixel size was determined by the scanning step in the
x direction. In total, 22 reconstructed 2D gamma-CT images were obtained, which we named
to
. Since Layer L1 was located beneath the CT sample holder and Layer L2 was located at the bottom edge of the holder, the LCS gamma-ray beam traveled underneath the rods. Therefore, the reconstructed images for the first two layers
and
did not show any characterization of the rods within the CT sample.
Figure 3a shows the cross-sectional slices of the CT sample for the horizontal layers from L3 to L8. The total height of these layers was 6 mm, which was identical to the height of each rod within these layers. Two rods of enriched lead isotopes (
208Pb and
206Pb) were inserted into two of the holes in the CT sample, and an Fe rod was inserted into the third hole.
Figure 3b shows the reconstructed CT images from
to
due to the atomic attenuation measured by the LaBr
3(Ce) detector. Because the lead isotope (
206Pb and
208Pb) rods had the same atomic attenuation, the two high-attenuation areas in the white-colored area caused by the atomic process corresponding to the locations of these lead rods were plainly evident in all reconstructed images. However, it was difficult to discriminate between the two isotopes. Furthermore, the atomic attenuation caused by the Fe rod was smaller than that caused by the lead rods, making the Fe rod appears as a faint shadow.
Figure 4a shows the cross-sectional slices of the CT sample for the horizontal layers from L9 to L14. The total height of these layers was also 6 mm. Rods of
208Pb, Fe, and Al were inserted into the first, second, and third holes, respectively. The reconstructed CT images from
to
are shown in
Figure 4b. A high-attenuation area caused by the atomic process corresponding to the
208Pb rod is clearly visible in the reconstructed images. The atomic attenuation originating from the Fe rods was less than that caused by the
208Pb rod, making the Fe rod discernible but not as apparent. Since the CT sample holder was Al, the Al rod was not visible.
The results for the horizontal layers from L15 to L20 are shown in
Figure 5. Two lead isotope rods (
208Pb and
206Pb) were inserted into the two holes, while the third hole remained unfilled as shown in
Figure 5a.
Figure 5b shows the reconstructed images from
to
. The two high-attenuation areas corresponding to the
206Pb and
208Pb rods can be clearly observed but they cannot be distinguished. The low-intensity area at the third hole corresponds to the vacant space.
Since Layers L21 and L22 were located above the inserted rods within the CT sample, the LCS gamma-ray beam did not pass through these rods. Therefore, there was no characterization for the rods in the two reconstructed images,
and
. The 3D-CT image was, in general, primarily made up of the 2D-CT images taken at various positions. Thus, the 22 measured 2D gamma-CT images were visualized together to create one 3D gamma-CT image. We used the MicroAVS data visualization tool [
22] to create the 3D gamma-CT image with a resolution of 1 mm/pixel.
Movie S1 shows the visualized 3D gamma-CT image.
Figure 6 shows a shot of the 3D gamma-CT image, which was captured from the visualized 3D movie (see
Movie S1 in the Supplementary Materials). We adjust a value of approximately 60% of the maximum intensity of the 2D gamma-CT images as a threshold limit for visualizing the 3D surface. Since the Fe or Al rods induced less atomic attenuation than the lead rods, this threshold limit ensured that only the lead rods appeared on the 3D surface. Intensity values greater (less) than the limit appeared (disappeared) on the 3D image. The five high-attenuation areas caused by the atomic process, which correspond to the positions of the five enriched lead isotope (
208Pb and
206Pb) rods within the CT sample, are clearly visible. The gamma-CT imaging technique is capable of clearly identifying the presence of the materials to be examined. In addition, we obtained the 3D gamma-CT image with a higher resolution and shorter acquisition time than the NRF-CT imaging technique. However, it lacked the isotope-selective capability needed to distinguish between different isotopes of the same element. Therefore, we utilized the 3D gamma-CT image as an additional information source to improve the quality of the 3D isotope-selective CT imaging based on the NRF transmission method.
For the primary 3D NRF-CT image, we chose the layers at the vertical distances of
z = 3, 11, and 17 mm, measured from the holder’s bottom, to reconstruct the 2D NRF-CT images of
,
, and
, respectively. One 2D NRF-CT (7 × 7) image was obtained for each horizontal row of rods (see
Figure 1a). In our previous study [
14], we reported the visualization of the reconstructed 3D images with a resolution of 4 or 8 mm/pixel for the horizontal or vertical plane, respectively, which were based on the atomic attenuation and the pure NRF attenuation. The measurement procedure, numerical equations, and 2D NRF-CT images were reported in [
14]. If a 3D NRF-CT image were measured under the current gamma-ray flux and detection efficiency available at the BL1U in the UVSOR-III facility, the total required time for obtaining a 3D NRF-CT with a spatial resolution of 1 mm is expected to be 2640 h. Thus, we need an alternative method to improve the resolution of the 3D NRF-CT image without needing a long measurement time. We used an FV numerical treatment to obtain a higher-quality image with a resolution of 1 mm/pixel with a measuring time shorter than that required to get the desired resolution without this treatment. In the following sub-section, we detail the FV approach.
3.2. Post-Multiply FV Method
One of the FV numerical approaches is the so-called “post-multiply FV method”, which analyzes a 3D gamma-CT image with a resolution of 1 mm/pixel as an additional data source to improve the resolution of the 3D NRF-CT image [
14]. The procedure of the post-multiply FV method is as follows: (i) sinogram adjustment, (ii) layer alignment (
x,
θ), (iii) 2D gamma-CT image segmentation, (iv) 2D-CT image overlapping, and (v) image visualization in 3D.
- (I)
Sinogram Adjustment
The transmission factors of the off-resonance attenuation (ε
off) and the on-resonance attenuation (ε
NRF) [
14] were gathered into a 2D matrix (sinogram). The size of the two-dimensional sonogram was determined by the scanning steps in the
x direction and the
θ angle. We obtained the 2D gamma-CT images by scanning the CT sample in the
x direction with a 1 mm step size across the range from −12 to +12 (25 positions). Furthermore, we scanned the CT sample in the
θ direction with an angle step of 30° across the range from 0° to 150° (six angles). Therefore, we obtained sinograms of the ε
off transmission factor
in a matrix with a size of 25 × 6. Moreover, the reconstructed 2D gamma-CT images had a matrix size of 25 × 25. For the 2D NRF-CT images, we scanned the CT sample in the
x direction with a 4 mm step size across the range from −12 to +12 (seven positions). We scanned the CT sample in the
θ direction with an angle step of 30° across the range from 0° to 150° (six angles). Therefore, we obtained sinograms of the Ԑ
NRF transmission factor
with a size of 7 × 6 [
14]. The reconstructed 2D NRF-CT images had a matrix size of 7 × 7. For overlapping the two 2D-CT images resulting from the sinograms of the reconstructed transmission factors Ԑ
NRF and Ԑ
off, the images needed to have the same size in both directions (
x and
z). In order to ensure the two images had the same size in both directions, we introduced a sinogram adjustment process to numerically divide each Ԑ
NRF sinogram from
with a dimension of 7 × 6 to
with a dimension of 25 × 6 as follows: we divided each value in the
x direction equally into 25 values to create a sinogram (
x,
θ) with a dimension of 175, 6. We then combined each seven consecutive values together into one value to create a new sinogram in a dimension of 25 × 6. Therefore, the newly reconstructed images
had a size of 25 × 25.
- (II)
Layer Alignment (x, θ)
We measured the 22 layers of the gamma-CT measurement (
to
) in addition to the three layers of the NRF-CT measurement (
), and we aligned every
image to a group with six 2D gamma-CT images measured for six consecutive layers.
Figure 7 illustrates the aligned layers in the vertical direction for both kinds of images. We aligned the 2D gamma-CT images from
to
, which included the rods of
208Pb,
206Pb and Fe, with the
image of
. We also aligned the 2D gamma-CT images from
to
, which included the rods of
208Pb, Fe, and Al, with the
image of
. The 2D gamma-CT images from
to
, including the
206Pb and
208Pb rods in addition to the vacant area, were aligned with the
image of
.
- (III)
2D Gamma-CT Image Segmentation
Image segmentation is a process of partitioning an image into numerous segments or a process of placing pixels such that they have non-overlapping areas [
23]. This segmentation procedure is the first stage of image analysis [
24], object representation, visualization, and other image processing strategies used in a variety of fields [
25]. The main goal of image segmentation is to simplify and/or transform an image into one that can be readily analyzed [
24]. In the case of segmentation of a 2D gamma-CT image, we assumed that the atomic absorption intensities varied in some regions of the CT sample and that the intensity at each pixel in a region where a rod existed would be almost constant. The first step of 2D gamma-CT image segmentation was primary image scaling. We scaled the primary gamma-CT images to a value between 0 and 255. The range of 0 to 255 was chosen to provide for an 8-bit representation of each pixel. The black, gray, and white colors were represented by values of 0, 128, and 255, respectively. One of the most commonly used approaches used for image segmentation is the threshold method [
24,
26]. Thresholding plays a crucial role in various algorithms used for image analysis, object representation, and visualization [
27]. In this method, we chose a threshold value (T
r) for transforming an image from a grayscale into a binary image, in which each pixel has a value of 0 or 1, to distinguish the foreground and background of the image [
28]. A pixel with an intensity greater (lower) than T
r is shown by a white-colored (black-colored) area. Threshold methods are grouped into two types: local thresholding and global thresholding. The local thresholding methods apply different threshold values to different regions of the image. Each T
r value is determined by the neighborhood of the pixel to which the threshold is applied [
26,
29]. On the other hand, in global thresholding, a single T
r value is used to separate the foreground and the background for all pixels in an image [
24,
30]. In the present analysis, we performed global thresholding with a common threshold T
r value of 128 as the second step of gamma-CT image segmentation. The black and white 2D segmented gamma-CT images of the scanned layers are shown in
Figure 8, with the layers labeled from
to
. The positions of the inserted rod turned entirely white, while the residual areas turned black. The image segmentation procedure resulted in a set of segments that collectively covered the whole image or a set of contours as the object’s edge. Each pixel in the inserted rods’ positions had a common property such as color or intensity. Furthermore, the neighboring areas differed substantially in terms of the same characteristic. When 2D segmented gamma-CT images are created for the employment of the FV technique with the NRF-CT images after the gamma-CT image segmentation, the resulting contours can be used to precisely preserve the
208Pb rod locations within the gamma-ray images of the CT sample and cut the surrounding noise and distortions.
- (IV)
2D-CT Image Overlapping
We overlapped the primary 2D-CT images to obtain a 2D FV NRF-CT image as follows: the
images of Layers
,
, and
were overlapped with the 2D segmented gamma-CT images from
to
,
to
, and
to
, respectively, according to the following equation:
We labeled the resulting 2D FV NRF-CT images as
to
. While the
208Pb rod is clearly visible in the fused images from
to
, as shown in
Figure 9a, the Fe and
206Pb rods are almost invisible.
Figure 9b shows the fused images from
to
. Only the
208Pb rod can be seen clearly, but neither the Fe nor the Al rods are visible. Moreover, we can clearly pinpoint the location of the
208Pb rod in the reconstructed images from
to
, as shown in
Figure 9c. The
206Pb rod and the vacant area disappeared. Obviously, the noise in all 2D FV NRF-CT image backgrounds caused by using the post-multiply FV method for the primary data sources was totally eliminated from the reconstructed image background. Since the overlapping process of the post-multiply FV method was performed by multiplying the intensity value at each pixel in the
by a value of 1 within the
208Pb locations or by a value of 0 in the surrounding regions, the method preserved the locations of the isotopes of interest and completely eliminated the surrounding distortion and background noise. Furthermore, it kept the intensity value at each pixel constant, making this approach useful for isotope quantification.
- (V)
Image Visualization in 3D
The 22 2D FV NRF-CT images were combined together using the MicroAVS data visualization tool to create the 3D FV NRF-CT image. We also adjusted a value within the intensity range of the 2D FV NRF-CT images as a threshold limit, so that only the intensity values greater than the threshold limit (
208Pb rods) appeared on the 3D surface.
Movie S2 shows a visualization of the fused CT-image of the NRF attenuation caused by the
208Pb isotope rods within the CT sample (pure NRF) in three dimensions.
Figure 10 shows a shot of the 3D FV NRF-CT image, which was captured from the visual-ized 3D movie (see
Movie S2 in the Supplementary Materials). The visualization clearly shows the locations of the enriched lead isotope (
208Pb) rods. In contrast, the rods of
206Pb, Fe, and Al and the empty areas are not visible.
We obtained an isotope-selective 3D FV NRF-CT image with a resolution of 1 mm/pixel for the distribution of an enriched lead isotope (208Pb) inserted with rods of different materials within a cylindrical sample 20 mm in height and 25 mm in diameter. Since the fused image has the same pixel resolution as the primary gamma-CT image, the image distortion and the background noise vanished, and the locations of the 208Pb rods were clearly visible. In contrast, the other rods completely disappeared, since their intensity values were less than the selected threshold limit of the 3D surface. Therefore, the combination of primary 3D NRF-CT images with high-resolution 3D gamma-CT images obtained by the FV technique provided a beneficial improvement in the image resolution and the isotope-selective capability in three dimensions.
The numerical treatment of the FV technique can be used in different approaches for primary 3D-CT images. Most of these approaches are similar to the post-multiply FV method, except for a few minor differences. One of the approaches is to directly overlap a 2D NRF-CT image with a 2D gamma-CT image without the 2D gamma-CT image segmentation process according to the following equation:
The 2D FV NRF-CT image obtained by this approach in the current study had the required quality. However, it could not be used for isotope quantification because the intensity at each pixel of the NRF-CT image was not conserved. Furthermore, even if the background noise were significantly reduced, it would be impossible to eliminate all noise. Another approach is the post-sum FV method, for which the overlapping is as follows:
The fused image obtained using the post-sum FV method shows the change in the intensity at each pixel but retains some of the background noise, so this method is useless for isotope quantification.
Figure 11 shows a comparison between the 2D FV NRF-CT images for the layer L4 obtained by (a) the post-multiply FV method, (b) the post-multiply FV method without 2D gamma-CT image segmentation, and (c) the post-sum FV method.
These results show that the alternative approaches may change the intensity of each pixel, so that they are unable to maintain the quantity of the isotope of interest. Furthermore, the noise cannot be completely removed even if it can be greatly decreased. Therefore, we recommend the post-multiply FV method as the most beneficial method for NRF-CT imaging that is likely to be effectively applicable in a variety of fields, including nuclear engineering and nuclear safety.