# Universal Behavior of the Image Resolution for Different Scanning Trajectories
## 3 Results and Discussion
### Image Reconstruction  
The image was then reconstructed using all of the investigated scanning trajectories [31]. MPI is one of the primary uses of the aforementioned scanning methods. As a result, additional analysis will be carried out utilizing image reconstruction in MPI. System matrix method was used to reconstruct the image. This method is preferred because it may be used for any type of trajectory. The phantom was designed to be 3.2 cm wide and 1.6 cm tall, with the supposed word \"NU\" filled with super paramagnetic iron oxide nanoparticles (SPION) (see Figure 5a). In order to guarantee a high fill factor for all trajectories, the Nyquist-Shannon sampling theorem was used, with a pixel size set at half of \(H_{max}\). Table 5 contains some of the factors that were used to perform the image reconstruction using the system matrix method. Figure 5: (a) Phantom image used for the reconstruction and (b) the outcomes of the reconstructed images using bidirectional Cartesian mesh (i.e., 0.5 \(H_{max}\) value of BC was used as the pixel size) for various scanning patterns and \(N_{P}\). As previously stated, the number of pixels in each side length of the phantom was determined by dividing the phantom's width and height by 0.5 \(H_{max}\). Thus, as the \(H_{max}\) decreases, the number of pixels represents an image increase. We chose to perform image reconstruction for all four scanning trajectories using the \(H_{max}\) value of the BC trajectory because this value ensures that all trajectories have a fill factor of at least 90%. Using different mesh sizes that reduce the fill factor for some trajectory to less than 90% may produce bias in the reconstruction as well as the quality for a fair comparison. Figure 5b depicts the reconstructed image obtained by utilizing the \(H_{max}\) value for the BC scanning trajectory. The reconstructed image using the \(H_{max}\) value of the SL scanning pattern is shown in Figure A3 in Appendix A. To assess the similarity between the reconstructed and phantom images, the peak signal-to-noise ratio (PSNR) is used [32]:  
\[PSNR=10 log_{10}\left(\frac{1}{n}\sum_{i=1}^{n}\left(x_{r}-x_{p}\right)^{2} \right)^{-1} \tag{15}\]  
Here, \(x_{r}\) represents the intensity unit of the reconstructed image, while \(x_{p}\) represents the intensity unit of the phantom. The unit of PSNR is in decibels (dB). A high PSNR score indicates that the reconstructed image is comparable to the phantom image. Figure 6a depicts the estimation of PSNR for image reconstruction of all scanning trajectories using the BC mesh. Another metric for quantifying the degree of accuracy of the reconstructed image is the structural similarity index measure (SSIM), which measures the similarity between two images. The SSIM is defined as follows [33; 34; 35; 36]:  
\[SSIM=\frac{\left(2\overline{xy}+c_{1}\right)\left(2\sigma_{xy}+c_{2}\right)}{ \left(\overline{x}^{2}+\overline{y}^{2}+c_{1}\right)\left(\sigma_{x}^{2}+ \sigma_{y}^{2}+c_{2}\right)} \tag{16}\]  
where \(\overline{x}\) and \(\overline{y}\) are the mean over a window in Image \(X\) and Image \(Y\), respectively. \(\sigma_{x}\) and \(\sigma_{y}\) are the standard deviation over a window in Image \(X\) and Image \(Y\). \(\sigma_{xy}\) is the co-variance over a window between Image \(X\) and Image \(Y\) and \(c_{1}\) and \(c_{2}\) are constants. Figure 6b illustrates the values of SSIM for all scanning trajectories. In order to quantify the quality of the reconstructed image, it is also important to quantify the error. The formula for calculating the Root Mean Square Error (RMSE) is as follows [37; 38; 39]:  
\[RMSE=\sqrt{mean(\left(x_{r}-x_{p}\right)^{2})} \tag{17}\]  
Figure 6c shows the RMSE of various scanning methods calculated using (16) above. The Normalized Total Error (NTE) is another approach for quantifying the error, and it is defined as [40]:  
\[NTE=\frac{Sum\left(\left|x_{p}-x_{r}\right|\right)}{Sum\left(x_{p}\right)} \times 100 \tag{18}\]