# Reconstruction of Preclinical PET Images via Chebyshev Polynomial Approximation of the Sinogram

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## Abstract

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## 1. Introduction

## 2. Mathematical Formulation

#### 2.1. Inversion of the Radon Transform via Chebyshev Polynomials

#### 2.2. Numerical Implementation of the Inversion of the Radon Transform via Chebyshev Polynomials

Algorithm 1: Computational steps of the proposed Chebyshev-based reconstruction algorithm |

Reconstruction algorithm: Chebyshev approximation of the sinogram |

Input: $\widehat{f}(\rho ,\theta )$ given at ${\rho}_{\lambda},\phantom{\rule{3.33333pt}{0ex}}\forall \phantom{\rule{3.33333pt}{0ex}}\lambda \u2a7d\mathsf{\Lambda}$ and ${\theta}_{k},\phantom{\rule{3.33333pt}{0ex}}\forall \phantom{\rule{3.33333pt}{0ex}}k\u2a7dK$ |

Compute: ${t}_{r}={\displaystyle \frac{1}{(1-{\rho}^{2})}}$ for all ${x}_{1}$, ${x}_{2}$ and ${\theta}_{k}$, via Equation (2),${t}_{\ell}=ln\left({\displaystyle \frac{1-\rho}{1+\rho}}\right)$ for all ${x}_{1}$, ${x}_{2}$ and ${\theta}_{k}$, via Equation (2), ${T}_{n}\left(\rho \right)$ for all n, via Equation (7), ${T}_{n}^{\prime}\left(\rho \right)$ for all n, via Equation (12), ${c}_{n}\left(\theta \right)$ for all n, via Equation (6) as in [28], ${t}_{1}=-{t}_{r}{c}_{0}\left(\theta \right)$ for all $\rho $ and ${\theta}_{k}$, ${t}_{2}=-2{t}_{r}\sum _{n=1}^{N}{c}_{n}\left(\theta \right){T}_{n}\left(\rho \right)$ for all $\rho $ and ${\theta}_{k}$, ${t}_{3}={t}_{\ell}{\displaystyle \sum _{n=1}^{N}{c}_{n}\left(\theta \right){T}_{n}^{\prime}\left(\rho \right)}$ for all $\rho $ and ${\theta}_{k}$, $t}_{4}=4\sum _{n=1}^{N}{c}_{n}\left(\theta \right)\underset{k=1}{\overset{\u230a\frac{n+1}{2}\u230b}{\sum \prime}}{\displaystyle \frac{{T}_{n-2k+1}^{\prime}\left(\rho \right)}{2k-1}$ for all ${x}_{1}$, ${x}_{2}$ and ${\theta}_{k}$, $\displaystyle \frac{\partial H(\rho ,\theta )}{\partial \rho}}=\sum _{i=1}^{4}{t}_{i$, $-{\displaystyle \frac{1}{4{\pi}^{2}}}\underset{0}{\overset{2\pi}{\int}}{\displaystyle \frac{\partial H(\rho ,\theta )}{\partial \rho}}\mathrm{d}\theta$, via numerical integration |

Output: $f({x}_{1},{x}_{2})$ |

## 3. Materials and Methods

#### 3.1. Phantom Study: Simulated Micro-PET IQ Phantom via the NEMA NU 4-2008 Protocol

#### 3.2. Sinograms

^{®}R2019b (The Mathworks Inc., Natick, MA, USA). In particular, all sinograms involved had dimensions of 119 × 180 pixels, corresponding to 119 detectors and 180 angles (180 views over 180 degrees), respectively, with a bin size of 1.17 mm and 161 image slices. For the purposes of our Chebyshev and FBP-based simulations we created two types of sinograms: sinograms with equally spaced $\rho $ for FBP reconstructions, and sinograms evaluated in $\rho $ locations corresponding to the roots of the Chebyshev polynomials, as in Equation (15). The simulated sinograms were acquired for three noise levels (NL), taking into account 100% (NL1), 50% (NL2) and 20% (NL3) of the total counts, respectively. The noisy sinograms were generated by adding Poisson noise in ten realizations ($R=10$) at each of the three noise levels.

#### 3.3. Reconstructions

#### 3.4. Image Metrics

#### 3.4.1. Percentage Standard Deviation

#### 3.4.2. Recovery Coefficient

#### 3.4.3. Spill-Over Ratio

#### 3.4.4. Image Analysis

## 4. Results

^{®}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].

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Line profiles of the IQ phantom, in its uniform (

**left**) and cold (

**right**) regions, for all noise levels.

**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 5.**Recovery coefficient ($RC$) for all hot fillable rods in the hot region of the simulated IQ phantom as a function of rod diameter (in mm), for Chebyshev and FBP reconstructions in NL2.

**Figure 6.**Spill-over ratio ($SOR$) measurements for the two water-filled and air-filled chambers in the cold region 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 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×, 1×, and 1.5× sigma, respectively.

**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×, 1×, and 1.5× sigma, respectively.

**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.

NL1 | NL2 | NL3 | |
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Chebyshev | 2.72 | 3.65 | 5.90 |

FBP | 3.58 | 5.17 | 8.31 |

**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.

Rod | Noise Level | Chebyshev | FBP |
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1 mm | NL1 | 6.62 | 5.94 |

NL2 | 4.68 | 4.15 | |

NL3 | 3.00 | 2.47 | |

2 mm | NL1 | 17.90 | 15.74 |

NL2 | 12.91 | 10.93 | |

NL3 | 8.03 | 6.84 | |

3 mm | NL1 | 32.42 | 24.65 |

NL2 | 23.34 | 17.21 | |

NL3 | 14.41 | 10.66 | |

4 mm | NL1 | 42.75 | 29.52 |

NL2 | 30.62 | 20.42 | |

NL3 | 18.85 | 12.86 | |

5 mm | NL1 | 46.84 | 28.73 |

NL2 | 33.76 | 20.06 | |

NL3 | 20.84 | 12.39 |

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**MDPI and ACS Style**

Protonotarios, N.E.; Fokas, A.S.; Vrachliotis, A.; Marinakis, V.; Dikaios, N.; Kastis, G.A.
Reconstruction of Preclinical PET Images via Chebyshev Polynomial Approximation of the Sinogram. *Appl. Sci.* **2022**, *12*, 3335.
https://doi.org/10.3390/app12073335

**AMA Style**

Protonotarios NE, Fokas AS, Vrachliotis A, Marinakis V, Dikaios N, Kastis GA.
Reconstruction of Preclinical PET Images via Chebyshev Polynomial Approximation of the Sinogram. *Applied Sciences*. 2022; 12(7):3335.
https://doi.org/10.3390/app12073335

**Chicago/Turabian Style**

Protonotarios, Nicholas E., Athanassios S. Fokas, Alexandros Vrachliotis, Vangelis Marinakis, Nikolaos Dikaios, and George A. Kastis.
2022. "Reconstruction of Preclinical PET Images via Chebyshev Polynomial Approximation of the Sinogram" *Applied Sciences* 12, no. 7: 3335.
https://doi.org/10.3390/app12073335