# Real-Time 3D PET Image with Pseudoinverse Reconstruction
## 3 Results
### Preclinical Data  
Results of a PINV reconstruction (PINV\({}_{\mathrm{z}}\) + 2D-PINV) were compared with the FORE + FBP images. We also studied the isolated 2D reconstruction with FBP and PINV from the same FORE rebinned sinogram for the IQ-NEMA phantom. These results are presented in Figure 7 and Table 1.  
In Table 1, we compared the recovery coefficients for the IQ-NEMA phantom at approximately the same level of uniformity for every rebinning + 2D reconstruction combination. The 2D-PINV slice reconstruction systematically outperformed (around 5-10% better for the smallest capillary and up to 15% in the 2 mm capillary). As expected, FORE rebinning provided the best recovery, improving marginally on the pseudoinverse rebinning results.  
Furthermore, in Figure 8, we present the results for a Derenzo filled with 250 uCi of \({}^{18}\)FDG measured with a well-counter with an accuracy of \(\pm\)5% [11]. It can be seen as an improvement in resolution for 2D-PINV reconstruction when compared to FBP.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Method** & **Rod** & **1 mm** & **2 mm** & **3 mm** & **4 mm** & **5 mm** & **\% Uniformity** \\ \hline FORE + FBP & \% RC & 23.3 & 55.2 & 85.0 & 92.2 & 101.2 & 12.2 \\ FORE + 2 DPINV & \% RC & 24.4 & 62.0 & 96.9 & 104.0 & 105.3 & 11.7 \\ PINV\({}_{\mathrm{z}}\) + FBP & \% RC & 20.8 & 57.1 & 85.9 & 96.8 & 101.7 & 12.2 \\ PINV\({}_{\mathrm{z}}\) + 2 DPINV & \% RC & 22.8 & 62.7 & 95.7 & 105.6 & 101.4 & 12.9 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Recovery coefficients and uniformity according to NEMA 4-2008.  
Figure 6: Axial profile of a simulated thin-slice (XX mm slices thickness) Defrise phantom for the SuperArgus scanner reconstructed from the three rebinning methods (**top**) and a profile along the axis of the scanner (**bottom**), that is, along the yellow line in the top panel. PINVz rebinning employed 20 iterations.  
Figure 9 shows the cardiac region of a 15 g mouse injected with 170 \(\upmu\)Ci of \({}^{18}\)FDG, with slices reconstructed with FBP and 2D-PINV from a PINV rebinned sinogram. Improvement in spatial resolution (a relative 15%) for 2D-PINV reconstructions can be clearly seen.  
Figure 8: Five centimeter diameter Derenzo phantom. Rods were 1.2, 1.6, 2.4 3.2, 4, and 4.8 mm in diameter. (**Left**) Reconstructed with 2D-PINV and FBP, respectively, from same sinogram rebinned with FORE. We show the 10 axial slices summed up. (**Right**) Profile along the yellow line showing the 1.6 and 3.2 mm rods.  
Figure 7: Fourier Rebinning (FORE) and axial Pseudoinverse (PINV\({}_{x}\)) rebinned data, coupled with different 2D reconstruction (Filtered Back-Projection (FBP) and 2D Pseudoinverse (2D-PINV)) for the Image Quality National Electrical Manufacturers Association (IQ NEMA) phantom.  
Figure 10 shows the PINV\({}_{\rm z}\) + 2D-PINV versus SSRB + FBP reconstructions of a 200 g rat injected with 300 uCi of \({}^{18}\)FDG where the improvement using PINV was evident. Images from both 30 min (bottom) and 5 s (top) acquisitions are shown, the second demonstrating the capabilities for near real time imaging.