# Real-Time 3D PET Image with Pseudoinverse Reconstruction
## 4 Discussion  
In this work, we showed that pseudoinverse reconstruction of 3D PET data using PINV\({}_{\text{z}}\) + 2D-PINV provided better (Tables 1 and 2) and faster reconstructions than SSRB + FBP (Tables 3 and 4). It is worth noting that in FBP, it is further needed to estimate the missing data (filling in the gaps in between detectors) [22], and this time was left out from Table 4. On the other hand, pseudoinverse reconstruction only requires knowing the position of the missing data in order to skip those, further speeding the process. Pseudoinverse axial rebinning was much faster than FORE, with just a moderately worse performance in terms of noise/resolution. As has been noted in previous works [3], there is a 2D-PINV matrix that would be fully equivalent to FBP. However, in the construction of the 2D-PINV, it is also possible to add geometrical information and physics effects, which introduce blurring to the TORs. This is the main reason for the improved resolution of in-slice PINV images compared to FBP ones. The matrices for PINV rebinning and slice reconstruction can be precomputed and stored once and for all for a given scanner, making it easy for the user to produce their own reconstructions. Although it would be desirable to apply PINV methods in 3D altogether instead of the division into axial and transaxial components employed in this work, building a 3D PINV matrix currently seems impractical as 3D PINV arrays for modern complex scanners will not fit in the memory of common computers. Furthermore, as we have presented here, the noise exhibited by PINV rebinning methods is slightly larger than that of FORE. Better regularization techniques [50] or improvements in the SRM could remedy this. Figure 12: YZ plane reconstructed with PINV\({}_{\text{z}}\) + 2D-PINV of a 13 g mouse injected with 120 \(\upmu\)Ci of \({}^{18}\)FDG. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{3}{c}{**In-Plane**} & \multicolumn{1}{c}{**3D Image**} \\ \cline{2-5}  & \(\mathbf{XY}\) & \(\mathbf{XZ}\) & \(\mathbf{YZ}\) & **Reconstruction** \\ \hline Time (s) & 0.08 & 0.11 & 0.11 & 1.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Total (wall clock) time required for direct reconstruction of direct 2D projections using the 2D PINV + PINV\({}_{\text{z}}\) code with eight threads of an Intel E5-2640 v4 @ 2.40 GHz processor (multithreaded enabled, up to 20 threads out of 10 physical cores in this processor). Data were arranged into 175 x 128 x 1185 sinograms. New regularization techniques could involve filtering of the reconstructed image or even the data before reconstruction. Any linear filter (\(F\)) can be applied during PINV reconstruction from its matrix representation:  
\[X\cong A^{+}\cdot Y\rightarrow\widetilde{X}=F\cdot X=F\cdot A^{+}\cdot Y=\Big{(}F \cdot A^{+}\Big{)}\cdot Y, \tag{14}\]  
Therefore, applying the filter in the pseudoinverse before the reconstruction would result in the same image as using the filter over the reconstructed image. The same applies to any linear filter applied in the data domain:  
\[X\cong A^{+}\cdot\widetilde{Y}=A^{+}\cdot F\cdot Y=\Big{(}A^{+}\cdot F\Big{)} \cdot Y, \tag{15}\]  
Matrices to represent the SRM in this work were obtained analytically for convenience, but they can also be obtained from Monte Carlo simulations, with likely improvement in image quality without affecting the speed of the method. Reconstruction times presented in Section 3.4 corresponded to only one CPU core. Multi-CPU or Graphics Processing Unit (GPU) strategies can be used to further accelerate the reconstructions. For instance, we have verified a very straightforward method to use multicore 2D by assigning each slice to a CPU-thread, reducing the total time almost linearly within the number of threads employed. In Tables 3 and 4, reconstruction times for smaller sinograms were also presented. Although some resolution was lost with these smaller sinograms, their very short reconstruction times make them suitable for real time imaging.