## 3 Results and Discussion
### 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 (\(x,\theta\)), (iii) 2D gamma-CT image segmentation, (iv) 2D-CT image overlapping, and (v) image visualization in 3D. The transmission factors of the off-resonance attenuation (\(\varepsilon_{\mathit{off}}\)) and the on-resonance attenuation (\(\varepsilon_{\mathit{NRF}}\)) [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 \(\theta\) 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 \(\theta\) direction with an angle step of 30\({}^{\circ}\) across the range from 0\({}^{\circ}\) to 150\({}^{\circ}\) (six angles). Therefore, we obtained sinograms of the \(\varepsilon_{\mathit{off}}\) transmission factor M\({}_{\varepsilon_{\mathit{eff}}}\) (\(x_{i},\theta_{i}\)) in a matrix with a size of 25 \(\times\) 6. Moreover, the reconstructed 2D gamma-CT images had a matrix size of 25 \(\times\) 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 \(\theta\) direction with an angle step of 30\({}^{\circ}\) across the range from 0\({}^{\circ}\) to 150\({}^{\circ}\) (six angles). Therefore, we obtained sinograms of the \(\varepsilon_{\mathit{NRF}}\) transmission factor M\({}_{\varepsilon_{\mathit{NRF}}}\) (\(x_{i},\theta_{i}\)) with a size of 7 \(\times\) 6 [14]. The reconstructed 2D NRF-CT images had a matrix size of 7 \(\times\) 7. For overlapping the two 2D-CT images resulting from the sinograms of the reconstructed transmission factors \(\varepsilon_{\mathit{NRF}}\) and \(\varepsilon_{\mathit{off}}\), 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 \(\varepsilon_{\mathit{NRF}}\) sinogram from M\({}_{\varepsilon_{\mathit{NRF}}}\) (\(x_{i},\theta_{i}\)) with a dimension of 7 \(\times\) 6 to M\({}^{\circ}_{\varepsilon_{\mathit{NRF}}}\) (\(x_{i},\theta_{i}\)) with a dimension of 25 \(\times\) 6 as follows: we divided each value in the \(x\) direction equally into 25 values to create a sinogram (\(x\), \(\theta\)) 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 \(\times\) 6. Therefore, the newly reconstructed images (2D NRF\({}^{\sim}\) - CT) had a size of 25 \(\times\) 25. Layer Alignment (\(x\), \(\theta\))  
We measured the 22 layers of the gamma-CT measurement (L\({}_{\mathit{off}}\)1 to L\({}_{\mathit{off}}\)22) in addition to the three layers of the NRF-CT measurement (L\({}_{\mathit{NRF}}\)4, L\({}_{\mathit{NRF}}\)12, L\({}_{\mathit{NRF}}\)18), and we aligned every 2D NRF\({}^{\sim}\) - CT 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 L\({}_{\mathit{off}}\) 3 to L\({}_{\mathit{off}}\) 8, which included the rods of \({}^{208}\)Pb, \({}^{206}\)Pb and Fe, with the 2D NRF\({}^{\sim}\) - CT image of L\({}_{\mathit{NRF}}\)4. We also aligned the 2D gamma-CT images from L\({}_{\mathit{off}}\) 9 to L\({}_{\mathit{off}}\) 14, which included the rods of \({}^{208}\)Pb, Fe, and Al, with the 2D NRF\({}^{\sim}\) - CT image of L\({}_{\mathit{NRF}}\)12.