# Reconstruction of Preclinical PET Images via Chebyshev Polynomial Approximation of the Sinogram
## 2 Mathematical Formulation  
### Inversion of the Radon Transform via Chebyshev Polynomials  
For analytic PET image reconstruction, one has to \"reconstruct\" the image \(f\) from its corresponding sinogram, denoted by \(\widehat{f}\). The sinogram involves the Radon transform of the image, denoted by \(\mathcal{R}f\), and is given by  
\[\widehat{f}(\rho,\theta)=\mathcal{R}\{f(x_{1},x_{2})\}=\int\limits_{L}f(x_{1}( s,\rho,\theta),x_{2}(s,\rho,\theta))\mathrm{d}s, \tag{1}\]  
where \(\rho\) represents the signed distance from the origin, i.e.,  
\[\rho:=\rho(x_{1},x_{2},\theta)=x_{2}\cos\theta-x_{1}\sin\theta, \tag{2}\]  
ds denotes an arc length differential along parallel lines \(L\) for all angles \(\theta\), \(s\) represents a parameter along the line \(L\), and \(x_{1}\), \(x_{2}\) are the usual Cartesian coordinates, as illustrated in Figure 1. We assume that the integral on the right-hand-side of Equation (1) has finite support for \(|\rho|\leqslant 1\), as commonly considered in medical applications [24, 25], i.e.,  
\[\widehat{f}(\rho,\theta)=0,\quad\text{for}\quad|\rho|>1,\quad\text{and}\quad \forall\;\theta. \tag{3}\]  
As Equation (1) suggests, analytic PET reconstructions entail the inversion of the Radon transform. In this direction, it has been established in [15, 16] that the inverse Radon transform \((\mathcal{R}^{-1})\) is given by the following expression:  
\[f(x_{1},x_{2})=\mathcal{R}^{-1}\Big{\{}\widehat{f}(\rho,\theta)\Big{\}}=- \frac{1}{4\pi^{2}}\int\limits_{0}^{2\pi}\frac{\partial H(\rho,\theta)}{ \partial\rho}\Bigg{|}_{\rho=x_{2}\cos\theta-x_{1}\sin\theta}\mathrm{d}\theta, \tag{4}\]  
where \(H(\rho,\theta)\) denotes the Hilbert transform (\(\mathcal{H}\)) of the sinogram, namely  
\[H(\rho,\theta):=\mathcal{H}\Big{\{}\widehat{f}(\rho,\theta)\Big{\}}=\oint \limits_{-\infty}^{\infty}\frac{\widehat{f}(\rho^{\prime},\theta)}{\rho^{ \prime}-\rho}\mathrm{d}\rho^{\prime}=\oint\limits_{-1}^{1}\frac{\widehat{f}( \rho^{\prime},\theta)}{\rho^{\prime}-\rho}\mathrm{d}\rho^{\prime}, \tag{5}\]  
and \(\oint\) denotes principal value integral. For the numerical implementation of the inversion, we employ Chebyshev polynomials of the first kind [21]. To this end, we approximate the sinogram function \(\widehat{f}(\rho,\theta)\) by  
\[\widehat{f}(\rho,\theta)\simeq\frac{1}{2}c_{0}(\theta)+\sum\limits_{n=1}^{N} c_{n}(\theta)T_{n}(\rho), \tag{6}\]  
where \(T_{n}(\rho)\) denotes Chebyshev polynomials of the first kind [26], i.e.,  
Figure 1: Cartesian geometry of the Radon transform. \[T_{n}(\rho)=\cos\left(n\arccos\rho\right),\quad\rho\in[-1,1],\quad n\in\mathbb{N}, \tag{7}\]  
and \(N\) denotes the maximum degree of the Chebyshev polynomial \(T_{n}(\rho)\) taken into account in the sum on the right-hand-side of Equation (6). Taking into account Equation (6), Equation (5) may be rewritten as  
\[H(\rho,\theta)=\frac{c_{0}(\theta)}{2}\ln\left(\frac{1-\rho}{1+\rho}\right)+ \sum_{n=1}^{N}c_{n}(\theta)\oint\limits_{-1}^{1}\frac{T_{n}(\rho^{\prime})}{ \rho^{\prime}-\rho}\mathrm{d}\rho^{\prime}. \tag{8}\]  
Proposition 2.3(i) and Equation (2.17) of [27] suggest that:  
\[\rho:=\rho(x_{1},x_{2},\theta)=x_{2}\cos\theta-x_{1}\sin\theta, \tag{2}\]0  
where \(\left\lfloor a\right\rfloor\) denotes the integer part of a real number \(a\), while the primed sum implies that the last term of the sum must be halved when \(n\) is odd.