## 2 Theory
### Objects Convergence
In this model, by approximating a finite forward projection along an LOR with the integral, we have \(\sum_{\xi}c_{i,\xi}\Big{(}\lambda_{\xi}^{k}-\lambda_{\xi}^{*}\Big{)}\approx \gamma d\left(\lambda_{j}^{k}-\lambda_{j}^{*}\right)\). Therefore, Equation (6) becomes  
\[\lambda_{j}^{k+1}-\lambda_{j}^{*}\approx\Big{(}\lambda_{j}^{k}-\lambda_{j}^{*} \Big{)}\times\Bigg{[}1-\frac{\gamma}{\eta_{j}}\sum_{i}\frac{c_{i,j}\lambda_{j} ^{*}d}{\bar{y_{i}}^{*}}\Bigg{]} \tag{8}\]where all terms are now independently computed for each pixel. We can estimate a convergence rate \(\alpha_{j}\) by dividing both side of the equation by \(\lambda_{j}^{k}-\lambda_{j}^{*}\) and obtaining  
\[\alpha_{j}\approx 1-\frac{\gamma}{\eta_{j}}\sum_{i}c_{ij}\frac{c_{ij}\lambda_{j}^ {*}d}{\sum_{\xi}c_{i\xi}\lambda_{\xi}^{*}} \tag{9}\]  
In this equation, it can be seen that the signal convergence depends on its spatial extension and on the fraction of counts due to the signal over all the LORs. This equation holds for a generic signal in a generic background. For an exemplary setup, the forward projections can be analytically computed further simplifying this expression. Let us assume an image composed of a background circle of diameter \(D\) with a smaller concentric circle of diameter \(d\) and activity ratio \(\beta\), as shown in Figure 1. In the central pixel \(j\), \(c_{ij}\lambda_{j}^{*}=\phi\beta\) for every LOR \(i\) crossing pixel \(j\), with \(\phi\) a constant accounting for the direct relation between a pixel activity and the expected counts. Since \(\sum_{\xi}c_{i\xi}\lambda_{\xi}^{*}=\phi[D+(\beta-1)]d\), we can further simplify Equation (9) as  
\[\lambda_{j}^{k+1}=\frac{\lambda_{j}^{k}}{\eta_{j}}\sum_{i}c_{i,j}\frac{y_{i} }{\bar{y}_{i}} \tag{2}\]0  
With TOF, the same equation holds with \(D_{eff}\) in place of \(D\), if \(D_{eff}<D\). A few things can be noticed:  
1. With increasing the signal to background contrast \(\beta\), the convergence is faster. 2. With increasing object size \(d\), the convergence speed also increases. 3. If an object is totally cold, i.e., \(\beta=0\), the convergence rate approaches 1. This makes cold object convergence extremely slow. 4. Without TOF, the convergence speed decreases with background diameter \(D\). With TOF, instead, convergence does not depend on \(D\) (for \(D>D_{eff}\)). 5. With TOF, convergence depends on \(D_{eff}\); if \(d\approx D_{eff}\) convergence is almost instantaneous. 6. For the full reconstruction of a circle, not only low frequency components need to converge but also high frequency components. Each frequency converges according to the same equation by replacing \(d\) with \(\frac{1}{f}\). Equation (10) with \(\beta=1\) and \(d=\frac{1}{f}\) can also describe the convergence of background noise frequencies \(f\). Without TOF, the larger the background diameter \(D\), the slower the noise converges. This implies that, without TOF, a fixed stopping rule produces comparable noise levels on reconstructed images, even if the noise at convergence is higher for larger objects. For higher noise frequencies, \(\alpha\to 1\). Since for a number \(\varepsilon\) close to 1  
\[\lambda_{j}^{k+1}=\frac{\lambda_{j}^{k}}{\eta_{j}}\sum_{i}c_{i,j}\frac{y_{i} }{\bar{y}_{i}} \tag{2}\]1 The exemplar object for which we will perform the calculation. (**a**) 2D view of the object. Dotted blue line represent a projection direction, passing by the central pixel. (**b**) Profile of the projection line represented in blue in the left panel. \((1+\varepsilon)^{k}\approx 1+k\times\varepsilon\), it can be said that high frequency noise increases almost linearly with iterations in early stopped reconstructions.