# Reconstruction of Preclinical PET Images via Chebyshev Polynomial Approximation of the Sinogram
## 4 Results  
For the simulated NEMA IQ phantom, reconstructed images using Chebyshev and FBP for all noise levels are presented in Figure 2. The figure was created in the free software tool for multimodality medical image analysis AMIDE [40]. The reconstructions depicted in this figure are representative reconstructions of one out of ten Poisson noise realizations at all noise levels, namely NL1, NL2 and NL3, since no visual differences were apparent between the realizations simulated. In Figure 2, scalar values are represented accordingly in the colorbar, namely zero values are represented by the all-black color, whereas the maximum values of the simulated IQ phantom by the all-white color. By visual comparison, the FBP reconstructed images appear of higher noise than the Chebyshev reconstructed images. Furthermore, in Figure 3 we present the line profiles of the IQ phantom, in its uniform and cold regions, respectively, for all noise levels. In Table 1, we present the percentage standard deviation (\(\%STD\)) measurements in all noise levels, for both Chebyshev and FBP reconstructions. The images reconstructed via Chebyshev exhibited considerably lower noise, compared to the ones reconstructed via FBP, in all noise levels. We note that the \(\%STD\) measurements in NL3 for FBP agree with the corresponding measurements in [18]. In this direction, the FBP level of noise in our \(\%STD\) measurements in NL3 is consistent with the one corresponding to FBP for acquisition duration of 1200 s, recommended by the NEMA protocol [23]. Figure 4 illustrates the recovery coefficient, \(RC\), calculated for each of the five hot rods in the hot region of the simulated IQ phantom, namely for the rods with diameters of 1 mm, 2 mm, 3 mm, 4 mm, and 5 mm, respectively. The \(RC\) values are plotted for all three noise levels, and for Chebyshev and FBP reconstructions. We observe that the FBP reconstructions present slightly higher \(RC\) values than the Chebyshev ones in the expense of higher noise. Furthermore, we observe that the \(RC\) is independent of the noise level, as expected. It is worth noting that in the Chebyshev-based reconstructions, the recovery coefficient values for the hot rods with diameters 4 and 5 mm are larger than 1. This overestimation is not uncommon in PET, and is usually accepted by several commercial preclinical systems, including the Mediso nanoScan(r) PC small-animal PET/CT scanner used in [18; 37]. \(RC\) values greater than 1 are due to few-pixel measurements from the average image, generated by averaging larger axial regions, especially in cases were small, pointlike sources are located inside regions with no background activity [41]. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **NL1** & **NL2** & **NL3** \\ \hline Chebyshev & 2.72 & 3.65 & 5.90 \\ FBP & 3.58 & 5.17 & 8.31 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Percentage standard deviation (\(\%STD\)) measurements for the uniform region of the simulated IQ phantom for Chebyshev and FBP reconstructions, for all noise levels. Figure 2: Reconstructions of the simulated IQ phantom via Chebyshev and FBP, for NL1, NL2, and NL3. Figure 4: Recovery coefficient (_RC_) measurements for each of the hot fillable rods (diameters 1 to 5 mm) of the simulated IQ phantom, over the percentage standard deviation (\(\%STD\)), for Chebyshev and FBP reconstructions, corresponding to all noise levels. Note that the leftmost data point in each curve corresponds to NL1 (less noise), whereas the rightmost data point corresponds to the NL3 case. Figure 3: Line profiles of the IQ phantom, in its uniform (**left**) and cold (**right**) regions, for all noise levels. In Figure 5, we illustrate the \(RC\) values as a function of the five rod diameters of the simulated IQ phantom, namely 1 mm, 2 mm, 3 mm, 4 mm and 5 mm, for both reconstruction methods investigated. This specific plot may be referred to as the _recovery coefficient curve_.