# Reconstruction of Preclinical PET Images via Chebyshev Polynomial Approximation of the Sinogram
## 5 Discussion  
In this work we have presented a new analytic reconstruction technique based on Chebyshev polynomials of the first kind, and have evaluated it, in comparison to the analytic industry-standard method of FBP, in a simulated IQ phantom according to the NEMA NU 4-2008 protocol. For the evaluation of the performance of our new method,  
Figure 8: Spill-over ratio (_SOR_) measurements for the water-filled and air-filled chambers in the cold region of the simulated IQ phantom, as functions of percentage standard deviation (\(\%STD\)), 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. Figure 7: Recovery coefficient (_RC_) measurements for each of the hot fillable rods (diameters 1 to 5 mm) of the simulated IQ phantom, as functions of percentage standard deviation (\(\%STD\)), 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 employed several image metrics, including the percentage standard deviation for the uniform region of the phantom, the recovery coefficient for the hot region of the phantom and the spill-over ratio for the cold region of the phantom. Our results suggest that our Chebyshev-based reconstruction algorithm is capable of producing accurate analytic reconstructions for the simulated IQ phantom. It is important to note that the main hypotheses involved in our current work are supported by our findings. Firstly, the hypothesis that Chebyshev polynomials of the first kind operate exclusively in the physical space, without incorporating the Fourier transform, which is expected to benefit the contrast of the reconstructed images. Secondly, the fact that Chebyshev polynomials operate globally, instead of locally, when interpolating the sinogram, which proved to be beneficial in terms of noise in the reconstructed images. Furthermore, we note that Chebyshev polynomials for PET reconstruction have similar computational efficiency to other analytical reconstruction formulations, such as SRT (based on splines), as in [18], where a thorough comparison of the spline technique with FBP has been performed following the same NEMA protocol. Chebyshev had similar computation complexity to other polynomial-based analytical transformations [16], however simpler transformations like FBP are less computationally expensive. In particular, for the uniform region of the simulated IQ phantom, our new analytic method provided images with percentage standard deviation (\(\%STD\)) considerably lower than FBP in all noise levels investigated. In the uniform region, Chebyshev reconstructions were consistently improved over FBP, in terms of percentage standard deviation (up to 29% lower \(\%STD\) values, depending on the noise level). For the hot regions of the simulated IQ phantom, our Chebyshev-based method provided images with recovery coefficients similar to FBP in the rods with diameter 1, 2 and 3 mm, whereas higher \(RC\) values in the rods with diameter 4 and 5 mm. When post-smoothing was applied, in order to compare at similar noise levels, Chebyshev demonstrated \(RC\) values up to 14% and 42% higher, for rods 1-3 mm and 4-5 mm, respectively, compared to FBP, depending on the smoothing sigma values. Especially in the cases of the 4 and 5 mm rods, \(RC\) values were overestimated and above 1. However, the measured contrast-to-noise-ratio indicated an improvement of Chebyshev versus FBP of up to 68%, depending on the noise level. Our \(CNR\) measurements provide a better indication of the advantages of Chebyshev reconstructions. Furthermore, at equal noise levels, namely between 3% to 6%, higher \(RC\) values were achieved via Chebyshev, whereas at lower noise levels FBP seems to have an advantage.