# Reconstruction of Preclinical PET Images via Chebyshev Polynomial Approximation of the Sinogram
## 4 Results
As shown in Figure 5, Chebyshev reconstructions exhibited similar \(RC\) values with FBP for the hot rods with diameters 1 to 3 mm, whereas for the hot rods with diameters 4 and 5 mm exhibited higher \(RC\) values. The \(RC\) values decrease as the rod size decreases for both algorithms [41, 42]. In Figure 6, we present the spill-over ratio (\(SOR\)) measurements for the water-filled and the air-filled chambers in the cold region of the simulated IQ phantom, for all noise levels, and for Chebyshev and FBP reconstructions. We observe that the impact of %\(STD\) in \(SOR\) is minimal. It is worth mentioning that in both Figures 5 and 6, the small standard error of the mean (SEM) bars indicate that the variation of the mean values is negligible in the metrics evaluated, namely \(RC\) and \(SOR\). Furthermore, in Table 2 we present the \(CNR\) measurements, resulting from the combination of the corresponding ones of \(RC\) and \(\%STD\). As shown in Table 2, Chebyshev reconstructions exhibited consistently higher \(CNR\) values than FBP, for all hot rods and for all noise levels investigated. Figure 7 illustrates the \(RC\) versus \(\%STD\) plots, for each of the hot fillable rods (diameters 1 to 5 mm) of the simulated IQ phantom, for Chebyshev and FBP reconstructions, corresponding to all sigma values, namely from right to left: NL3 (no smoothing), 0.5\(\times\), 1\(\times\), and 1.5\(\times\) sigma, respectively. We observe that, for all rod sizes, Chebyshev-based reconstructions exhibit higher \(RC\) values for \(\%STD\) between 3% to 6%, whereas, for \(\%STD\) between 1% to 2%, FBP reconstructions have higher \(RC\) values than the corresponding ones of Chebyshev. As expected, more Gaussian smoothing (larger sigma) leads to lower \(\%STD\) as well as \(RC\) values. Finally, in Figure 8 we have plotted \(SOR\) versus \(\%STD\), for the water-filled and air-filled chambers in the cold region of the simulated IQ phantom, for Chebyshev and FBP reconstructions, corresponding to all sigma values, namely from right to left: NL3 (no smoothing), 0.5\(\times\), 1\(\times\), and 1.5\(\times\) sigma, respectively. We observe that the calculated \(SOR\) values for both analytic reconstruction algorithms are similar. As expected, for both FBP and Chebyshev reconstructions, the \(SOR\) value increases as the \(\%STD\) increases. \begin{table}
\begin{tabular}{c c c c} \hline
**Rod** & **Noise Level** & **Chebyshev** & **FBP** \\ \hline \multirow{3}{*}{1 mm} & NL1 & 6.62 & 5.94 \\  & NL2 & 4.68 & 4.15 \\  & NL3 & 3.00 & 2.47 \\ \hline \multirow{3}{*}{2 mm} & NL1 & 17.90 & 15.74 \\  & NL2 & 12.91 & 10.93 \\  & NL3 & 8.03 & 6.84 \\ \hline \multirow{3}{*}{3 mm} & NL1 & 32.42 & 24.65 \\  & NL2 & 23.34 & 17.21 \\  & NL3 & 14.41 & 10.66 \\ \hline \multirow{3}{*}{4 mm} & NL1 & 42.75 & 29.52 \\  & NL2 & 30.62 & 20.42 \\  & NL3 & 18.85 & 12.86 \\ \hline \multirow{3}{*}{5 mm} & NL1 & 46.84 & 28.73 \\  & NL2 & 33.76 & 20.06 \\ \cline{1-1}  & NL3 & 20.84 & 12.39 \\ \hline \end{tabular}
\end{table} 
Table 2: Contrast-to-noise-ratio (\(CNR\)) measurements for the hot region of the simulated IQ phantom for Chebyshev and FBP reconstructions, for all hot fillable rods, for all noise levels.