## 2 Theory  
### Objects Convergence  
We start by analysing the MLEM convergence proprieties. In PET reconstruction, the tomographic problem is the minimization of the negative likelihood  
\[\lambda^{*}=\arg\min_{\lambda}\sum_{i}y_{i}-y_{i}\ln\bar{y}_{i} \tag{1}\]  
with \(\bar{y}_{i}=\sum_{j}c_{i,j}\lambda_{j}\) as the expected number of counts in each sinogram bin, given the current image estimate \(\lambda\). The update equation is  
\[\lambda_{j}^{k+1}=\frac{\lambda_{j}^{k}}{\eta_{j}}\sum_{i}c_{i,j}\frac{y_{i} }{\bar{y}_{i}} \tag{2}\]It has been shown that MLEM can be also written as a gradient descent algorithm, which is expected to converge linearly with a unitary step size and a diagonal preconditioner equal to the current image estimate divided by the normalization matrix [10]. If we look at Equation (2) as \(\lambda_{j}^{k+1}=f_{j}\Big{(}\lambda^{k}\Big{)}\), we can compute the Taylor expansion around the solution \(\lambda^{*}\), \(\lambda_{j}^{k+1}\approx\lambda_{j}^{*}+\sum_{\xi}\frac{\partial\lambda_{j}^{k +1}}{\partial\lambda_{\xi}}\Big{(}\lambda_{\xi}^{k}-\lambda_{\xi}^{*}\Big{)}\) The derivative of \(\lambda_{j}^{k+1}\) against \(\lambda_{\xi}^{k}\), considering that \(\frac{\partial y_{i}}{\partial\lambda_{\xi}}=c_{i,\xi}\), is  
\[\frac{\partial\lambda_{j}^{k+1}}{\partial\lambda_{\xi}^{k}}=\frac{\delta_{j, \xi}}{\eta_{j}}\sum_{i}c_{i,j}\frac{y_{i}}{\bar{y}_{i}^{k}}-\frac{\lambda_{j}^ {k}}{\eta_{j}}\sum_{i}c_{i,j}\frac{y_{i}}{\big{(}\bar{y}_{i}^{k}\big{)}}c_{i, \xi} \tag{3}\]  
By substituting in the Taylor expansion around the solution \(\lambda^{*}\), we obtain  
\[\lambda_{j}^{k+1}-\lambda_{j}^{*}\approx\sum_{\xi}\Bigg{[}\delta_{j,\xi} \frac{1}{\eta_{j}}\sum_{i}c_{i,j}\frac{y_{i}}{\bar{y}_{i}^{*}}-\frac{\lambda_ {j}^{*}}{\eta_{j}}\sum_{i}\frac{c_{i,j}}{\big{(}\bar{y}_{i}^{*}\big{)}^{2}}c_{ i,\xi}\Bigg{]}\Big{(}\lambda_{\xi}^{k}-\lambda_{\xi}^{*}\Big{)} \tag{4}\]  
This equation closely mimics that of an algorithm linearly converging. We will prove that we are able to estimate a linear convergence rate \(\alpha_{j}=\frac{\left\|\lambda_{j}^{k+1}-\lambda_{j}^{*}\right\|}{\left\| \lambda_{j}^{k}-\lambda_{j}^{*}\right\|}\) for each pixel for different kinds of signals. A propriety of MLEM is that, after the first iteration [2],  
\[\sum_{i}y_{i}=\sum_{i}\bar{y_{i}}^{k} \tag{5}\]  
We therefore approximate \(\sum_{i}c_{i,j}\bar{y_{i}}^{*}\approx\eta_{j}\) and \(\sum_{i}c_{i,j}\frac{y_{i}}{\big{(}\bar{y_{i}}^{*}\big{)}}c_{i,\xi}\approx\sum _{i}\frac{c_{i,j}}{\bar{y_{i}}^{*}}c_{i,\xi}\). These two approximations modify Equation (4) to  
\[\lambda_{j}^{k+1}-\lambda_{j}^{*}\approx\sum_{\xi}\Bigg{[}\delta_{j,\xi}- \frac{\lambda_{j}^{*}}{\eta_{j}}\sum_{i}\frac{c_{i,j}}{\bar{y_{i}}^{*}}c_{i, \xi}\Bigg{]}\Big{(}\lambda_{\xi}^{k}-\lambda_{\xi}^{*}\Big{)}= \tag{6}\]  
where \(\lambda^{k}\) is the free parameter of the Taylor expansion. To simplify this expression, we need to compute \(\sum_{\xi}c_{i,\xi}\Big{(}\lambda_{\xi}^{k}-\lambda_{\xi}^{*}\Big{)}\). Exploiting again the propriety mentioned in Equation (5), we can state that, in uniform large regions, which generally make up the biological background, \(\sum_{\xi}c_{i,\xi}\Big{(}\lambda_{\xi}^{k}-\lambda_{\xi}^{*}\Big{)}\approx 0\). If, instead, centered around a pixel \(j\) there is a signal characterized by a \"dimension\" \(d\) to be recovered, we can model:  
\[\lambda_{\xi}^{k}-\lambda_{\xi}^{*}=\Big{(}\lambda_{j}^{k}-\lambda_{j}^{*} \Big{)}g\Big{(}\frac{x}{d}\Big{)}; \tag{7}\]  
where \(g\) is a function of the distance \(x\) from pixel \(j\) to pixel \(\xi\), to which it is only required to have a finite integral \(\int_{-r}^{+r}g(\
icefrac{{x}}{{d}})dx=\gamma d\), with \(\gamma\) as the scale-independent integral value. In general \(\gamma<1\) (e.g., only for a step function \(\gamma=1\). A smooth function like half a cosine period has \(\gamma=\
icefrac{{2}}{{\gamma}}\)).