## 2 Methods
### Bayesian image restoration
If \(\mathbf{g}=\mathbf{R}\mathbf{f}+k\mathbf{\sigma}\) and the noise standard error is the same for all pixels, \(\sigma_{m}=\sigma,\,m=1,2,...,M\), the likelihood of data \(P\left(\mathbf{g}|\mathbf{f},\mathbf{C},H\right)\) can alternatively be written as:  
\[P\left(\mathbf{g}|\mathbf{f},\beta,H\right)=\frac{1}{Z_{b}(\beta)}\cdot\exp\left[- \sum_{m=1}^{M}\beta E_{b}\left(\mathbf{g}|\mathbf{f},H\right)\right] \tag{31}\]where \(\beta=1/\sigma^{2}\) is a measure of the error (noise) in each pixel, \(Z_{b}(\beta)=\left(2\pi/\beta\right)^{\frac{M}{2}}\) is the noise partition function, and the error function is:  
\[E_{b}\left(\mathbf{g}|\mathbf{f},H\right)=\frac{1}{2}\cdot\frac{\mathbf{e}^{T}\mathbf{e }}{\sigma^{2}}=\frac{1}{2}\sum_{m=1}^{M}\frac{\left(g_{m}-\sum_{n=1}^{N}R_{mn} f_{n}\right)^{2}}{\sigma^{2}}=\frac{1}{2}\chi^{2}(\mathbf{f}) \tag{32}\]  
Prior probability  
Applying the ME principle amounts to assigning a distribution \(\{p_{i}\}_{i=1,2,\ldots}\) on some hypothesis space \(\{H_{i}\}_{i=1,2,\ldots}\) by the criterion that it shall maximize some form of entropy subject to constraints that express the desired properties of the distribution, but are not sufficient to determine it. The ME methods require specifying in advance a definite hypothesis space which sets down the possibilities to be taken into consideration. The result is a probability distribution, rather than a probability. The ME probability of a single hypothesis \(H\) that is not embedded in a space of alternative hypotheses does not make any sense. The ME approaches do not need numerical values of any probabilities on that space as input, rather they assign numerical values to available information as expressed by the choice of the hypothesis space and constraints. By all means, any set of constraints on \(\mathbf{f}\) is affecting the amount by which the image restoration is offset from reality. In the ergodic case, where reasons are to believe that all _a priori_ probabilities of the microscopic configurations are the same, the Bayesian prior probability \(P\left(\mathbf{f}|\alpha,H\right)\) for a macroscopic state \(\mathbf{f}\) with entropy \(S(\mathbf{f})\) is postulated [27] as given by a general potential function \(\Phi\left(\mathbf{f}\right)\), such as:  
\[P\left(\mathbf{f}|\alpha,H\right)=\frac{1}{Z(\alpha)}\,\exp\left[-\alpha\,\Phi \left(\mathbf{f}\right)\right] \tag{33}\]  
where \(\alpha\) is a positive parameter and \(Z(\alpha)\) is a normalizing factor. When partial information on a random process is available only, the entropic prior is assumed to follow the ME law that complies with prior information. The entropic prior in the discrete case corresponds to potential functions like:  
\[\Phi\left(\mathbf{f}\right)=-S(\mathbf{f})=\sum_{n=1}^{N}f_{n}\ln\left(\frac{f_{n}}{ U}\right) \tag{34}\]  
resulting in:  
\[P\left(\mathbf{f}|\alpha,H\right)=\frac{1}{Z(\alpha)}\,\exp\left[-\alpha\sum_{n=1 }^{N}f_{n}\ln\left(\frac{f_{n}}{U}\right)\right] \tag{35}\]  
The best suited prior probability distribution, \(P\left(\mathbf{f}|\alpha,H\right)\), should incorporate as much as possible of the available statistical characteristics of the ensemble of images to which the original image, \(\mathbf{f}\), is assumed to belong. Apart from the entropic one [28], there are more different choices to be considered and evaluated for prior probability distribution of an image. The full joint posterior probability \(P\left(\mathbf{f},\theta|\mathbf{g},H\right)\) of an image \(\mathbf{f}\) and the unknown PSF parameters denoted generically by \(\theta\) need to be evaluated. Then the required inference about the posterior probability \(P\left(\mathbf{f}|\mathbf{g},H\right)\) is obtained as a marginal integral of this joint posterior over the PSF parameters:  
\[P(k)=(2\pi)^{-\frac{1}{2}}\exp\left(-\frac{k^{2}}{2}\right) \tag{27}\]0  
In this respect, [PERSON] theorem applied for the parameters \(\theta\) gives:  
\[P(k)=(2\pi)^{-\frac{1}{2}}\exp\left(-\frac{k^{2}}{2}\right) \tag{27}\]1  
which substituted in eq.