## 2 Methods
### Derivation of the potential function  
The space invariant linear system \(\mathbf{g}=\mathbf{R}\mathbf{f}+\mathbf{e}\) with additive and zero-mean Gaussian noise assumed uncorrelated between pixels is considered hereafter. The task is to restore an image in the form of a probability distribution \(\mathbf{f}=\{p_{n}\}\), where \(p_{n}=f_{n}/U\), \(f_{n}\) is the number of quanta in the \(n\)-th pixel, \(U\) is the total number of quanta in the image, given the measured \"blurred\" data \(g_{m},\,m=1,2,...,M\) affected by the errors \(e_{m},\,m=1,2,...,M\), and which complies with a finite set of constraints:  
1. Conservation of the total number of photons in the measured image, \(\mathbf{g}\), and the model image \(\mathbf{f}\): \[\sum_{n=1}^{N}p_{n}=1\] (42)
2. Linear transform between the model space and the data space: \[g_{m}=\sum_{n=1}^{N}U\cdot R_{mn}\cdot p_{n}+e_{m},\,m=1,2,...,M\] (43)
3. The errors \(e_{m},\,m=1,2,...,M\), are normally distributed with zero mean, \(\overline{e}=0\), and variances \(\sigma_{m}^{2},\,m=1,2,...,M\): \[\sum_{m=1}^{M}\frac{e_{m}^{2}}{\sigma_{m}^{2}}=\Omega\] (44) where \(\Omega\) denotes the expected value of the statistical goodness-of-fit \(\chi^{2}\). Generally, the \(\chi^{2}\) statistic associated with \(e_{m},\,m=1,2,...,M\) independent Gaussian random variables:  
\[\chi^{2}=\sum_{m=1}^{M}\frac{\left(e_{m}-\overline{e}\right)^{2}}{\sigma_{m}^ {2}} \tag{45}\]  
has the expectation \(M\pm\sqrt{M}\). Since the mean is fitted by setting \(\overline{e}=\overline{\sum_{m=1}^{M}e_{m}}=0\) inferred by the conservation of the number of quanta spread over the pixels, one degree of freedom has already been used up. Consequently, the expectation of \(\chi^{2}\) is reduced by 1 and set \(\Omega=M-1\). Though this distinction is commonly ignored, the difference is essential in cases where the number of free parameters is similar to the number of data points like interpolation [29]. In all cases, the potential function, \(Z\), is useful if reliable estimates of the standard errors \(\sigma_{m}^{2},\,m=1,2,...,M\) that determine the severity of blurring are _a priori_ selected on some theoretical and/or experimental base. Noise degrades the overall quality of an image and makes it difficult to reveal certain details of interest. The major source of noise in X-ray imaging is the random distribution of photons over the surface of the image. The standard deviation of the photon distribution is the best quantitative estimator of the noise in an image. As such, the standard deviation \(\sigma_{m}\) of the photon distribution in each pixel \(m=1,2,...,M\) was set in agreement with [PERSON]'s law as proportional with the square root of the recorded quanta \(g_{m}\), such as:  
\[\sigma_{m}=\alpha\cdot\sqrt{g_{m}},\,m=1,2,...,M \tag{46}\]The model parameter \(\alpha\) sets down the proper scattering range of the signal-to-noise ratio (SNR) in each pixel around some common level and ensures convergence of the algorithm after a reasonable number of iterations. Assuming that the errors \(e_{m},\)\(m=1,2,...,M\) are normally distributed with zero mean and variances \(\sigma_{m}^{2},\)\(m=1,2,...,M\), then the expected value of their \(\chi^{2}\) distribution should lie within the interval \(\left[M-\sqrt{M},M+\sqrt{M}\right]\). In a similar approach, [PERSON] and [PERSON] assumed as convenient but not essential, that the covariance matrix of the errors being proportional to the unit matrix such as \(\mathbf{C}=\sigma^{2}\mathbf{I}\)[30]. Among all admissible probability distributions of the model image, \(\mathbf{f}=\{p_{n}\}\), which satisfy the imposed constraints, eqs. (42), (43), and (44), the one that complies with the ME principle is the most unbiased choice to estimate the true image. Accordingly, maximizing the image entropy:  
\[S(\mathbf{f})=-\sum_{n=1}^{N}f_{n}\ln\left(\frac{f_{n}}{U}\right)=-U\sum_{n=1}^{N} p_{n}\ln p_{n} \tag{47}\]  
leads to the smoothest and uniform distribution, \(\mathbf{f}\), among the admissible set of images.