## 3 Methods
### Bayesian Posterior Expectation for the Restoration  
An overview of the posterior expectation is presented in Figure 1. The posterior expectation of the latent intensity, \(\tilde{u}=E(u|v)\), is the minimum mean squared error. The Bayesian expansion for \(p_{(u|v)}(u|v)\) produces  
\[\tilde{u}=E(u|v)=\iint_{0}^{\infty}p_{(u|v)}(u|v)udu=\frac{\iint_{0}^{\infty}p _{(v|u)}(v|u)p_{u}(u)udu}{\iint_{0}^{\infty}p_{(v|u)}(v|u)p_{u}(u)du}. \tag{6}\]The likelihood \(p_{(v|u)}(v|u)\) in Equation (6) is the non-stationary distortion as given in Section 3.1 for the joint intensity statistics. This expression expanded in Equation (6) also involves the prior distribution \(p_{u}(u)\) of latent image \(u\). This is estimated in Equation (5) with the deconvolution of the distortion \(p_{b}(v-u|u)\) from the intensity co-occurrence distribution \(p_{v}(v)\), (\(C_{v}\)), to provide the non-parametric prior \(\hat{p}_{u}(u)=\bar{C}_{u}\) of \(u\). These two factors are substituted into Equation (6) to produce  
\[\hat{u}=E(u|v)=\frac{\iint_{0}^{\infty}p_{b}(v-u;0,(\sigma_{b}u)^{2})\cdot\hat {p}_{u}(u)\cdot udu}{\iint_{0}^{\infty}p_{b}(v-u;0,(\sigma_{b}u)^{2})\cdot\hat{ p}_{u}(u)du}. \tag{7}\]  
The magnitude of \(p_{b}(v-u;(\sigma_{b}u)^{2})\) is significant for neighborhood \(\Delta u\in\mathcal{N}_{u}\) in co-occurrence space and tends to zero for intensities \(u\) far from \(v\). The image domain is discrete and the necessary discretization of \(\hat{u}=E(u|v)\) in Equation (7) becomes  
\[\hat{u}=E(u|v)=\frac{\sum_{u\in\mathcal{N}_{u}}p_{b}(\Delta u;0,(\sigma_{b}u)^ {2})\cdot\bar{C}_{u}(u+\Delta u)\cdot(u+\Delta u)}{\sum_{u\in\mathcal{N}_{u}}p _{b}(\Delta u;0,(\sigma_{b}u)^{2})\cdot\bar{C}_{u}(u+\Delta u)}. \tag{8}\]  
The extent of \(P_{b}\) and \(\mathcal{N}_{u}\) increases linearly with intensity \(u\). General Equation (8) for the auto-co-occurrence statistics offers the posterior expectation of the intensities in polar coordinates \((f_{i},\hat{\phi}_{i})^{T}=E((r_{i},\phi_{i})^{T}|(r_{i}^{\prime},\phi_{i}^{ \prime})^{T})\) in auto-co-occurrence neighborhoods \(\Delta r_{i}\in\mathcal{N}_{r_{i}}\) and \(\Delta\phi_{i}\in\mathcal{N}_{\phi_{i}}\) as  
\[\left( \begin{array}{c}\hat{r}_{i}\\ \hat{\phi}_{i}\end{array} \right)=\frac{\sum_{\mathcal{N}_{i}}\sum_{\mathcal{N} _{\phi_{i}}}P_{b}\left(\left( \begin{array}{c}\Delta r_{i}\\ \phi_{i}\end{array} \right);\left( \begin{array}{c}(\sigma_{b_{r}r_{i}})^{2}\\ \sigma_{b,\phi_{i}}^{2}\end{array} \right)\right)\cdot\bar{C}_{u_{i}u_{i}}\left( \begin{array}{c}r_{i}+\Delta r_{i}\\ \phi_{i}+\Delta\phi_{i}\end{array} \right)}{\sum_{\mathcal{N}_{r_{i}}}\sum_{ \mathcal{N}_{\phi_{i}}}P_{b}\left(\left( \begin{array}{c}\Delta r_{i}\\ \phi_{i}\end{array} \right);\left( \begin{array}{c}(\sigma_{b_{r}r_{i}})^{2} \\ \sigma_{b,\phi_{i}}^{2}\end{array} \right)\right)\cdot\bar{C}_{u_{i}u_{i}}\left( \begin{array}{c}r_{i}+\Delta r_{i}\\ \phi_{i}+\Delta\phi_{i}\end{array} \right)}, \tag{9}\]  
where \(P_{b}(\Delta r_{i},\phi_{i})\) is as used in the RHS of Equation (3). Equation (8) for the joint-co-occurrence statistics provides the posterior expectation of the intensities in Cartesian coordi  
Figure 1: Overview of the non-parametric Bayesian coring derivation of the posterior conditional expectation. The conditional expectation is further expanded with Bayes’ rule. The expressions for the prior and the likelihood are then substituted. It provides the intermediate vector field \(\hat{u}=E(u|v,x)\) for image restoration. nates \((\eta_{0},\eta_{1})^{T}=E((\eta_{0},\eta_{1})^{T}|(\eta_{0}^{\prime},\eta_{1}^{ \prime})^{T})\) in joint-co-occurrence neighborhoods \(\Delta\eta_{0}\in\mathcal{N}_{\eta_{0}}\) and \(\Delta\eta_{1}\in\mathcal{N}_{\eta_{1}}\) with  
\[\left( \begin{array}{c}\eta_{0}\\ \eta_{1}\end{array} \right)=\frac{\sum_{\mathcal{N}_{\eta_{0}}}\sum_{\mathcal{N }_{\eta_{1}}}P_{b}\left(\left( \begin{array}{c}\Delta\eta_{0}\\ \Delta\eta_{1}\end{array} \right);\left( \begin{array}{c}(\sigma_{b,\eta_{0}} \eta_{0})^{2}\\ (\sigma_{b,\eta_{1}}\eta_{1})^{2}\end{array}