## 3 Methods
### Spatial Image Restoration
#### 3.4.2 Anisotropic Smoothing of the Restoration Field  
The nonuniformities of individual coils are smooth and, in this case of multiple coils, they are piecewise smooth due to the differences between the sensitivities of different coils. In particular, the junctions between the coils are on axial planes \((x,z)\) of the 3D data. Hence, the nonuniformities on axial planes \((x,z)\) are unaffected and remain smooth. This is imposed to the initial rough estimates \(\vec{b^{-1}}(\mathbf{x})\) with a smooth 2D anisotropic axial spatial Gaussian filter \(G(x,z;\sigma_{s,i}^{2})\). This filtering does not affect the piecewise nonuniformities in the axial direction resulting from different coils. It produces the restoration fields at this phase as  
\[W^{\prime}_{i,t}(\mathbf{x})=\vec{b^{-1}}_{i,t}(\mathbf{x})\ast G(x,z;\sigma_{ s,i}^{2}),i=0,1. \tag{13}\]  
The application of the restoration fields to the images produces \(I_{v_{1}}(\mathbf{x})\gets I_{v_{1}}(\mathbf{x})W^{\prime}_{i,t}(\mathbf{x})\). The piecewise nonuniformity along the axial direction, \(y\), is accounted for with spatially varying inhomogeneous smoothing. The method used is based on the 2D Minimum Description Length (MDL) principle method that has been developed for piecewise smoothness of image regions [41, 42]. In the 2D rectangular image grid, the 8-connected neighborhood in used. The 2D method considers the anisotropy of the grid. The method for 2D piecewise smoothness is extended in his work for 3D piecewise smoothness. In 3D, the 26-connected neighborhood is used. The considerations for the anisotropy artifact of the 8-connected neighborhood in the 2D grid are extended to account for the spatial anisotropy of the 26-connected neighborhood of the 3D grid used in this work. It considers the resolution of the 3D grid of the image within the neighborhood of the anisotropic filter. The 3D anisotropic smoothing with MDL is applied to \(W_{i,t}(\mathbf{x})\) to offer the final restoration field \(W\). The MDL principle can be formulated equivalently in terms of the Bayesian Maximum a Posteriori (MAP) estimate. This uses the Bayesian formulation:  
\[P(W_{i}|W_{i}^{\prime})=\frac{P(W_{i}^{\prime}|W_{i})P(W_{i})}{P(W_{i}^{\prime} )},i=0,1, \tag{14}\]  
where \(W_{i}^{\prime}\) offers initial rough estimates of the restoration and \(W_{i}\) are the final restoration fields. In fact, in this case, it is the minimum rather than the maximum of the Bayesian posterior that is of interest. The MAP strategy is to choose the \(W_{i}\) that minimizes \(P(W_{i}|W_{i}^{\prime})\). That is, the objective is to obtain the MAP estimate that is given by  
\[\hat{W}_{i,MAP} = \arg\min_{W_{i}}P(W_{i}|W_{i}^{\prime}) \tag{15}\] \[\stackrel{{(\ref{eq:1})}}{{=}} \arg\min_{W_{i}}\frac{P(W_{i}^{\prime}|W_{i})P(W_{i})}{P(W_{i}^{ \prime})}\] \[\stackrel{{(\ref{eq:2})}}{{=}} \arg\min_{W_{i}}P(W_{i}^{\prime}|W_{i})P(W_{i}).\]  
Equality (1) in Equation (15) holds from the Bayesian formulation in Equation (14). Equality (2) in Equation (15) holds from the assumption that the marginal likelihood \(P(W_{i}^{\prime})\) is constant. The probability of the likelihood \(P(W_{i}^{\prime}|W_{i})\) is a Gaussian distribution of the noise \(P(W_{i}^{\prime}|W_{i})=\sum_{\mathbf{x}}(W_{i}^{\prime}-W_{i})\) with a zero mean and variance \(\sigma_{W}^{2}\) in the spatial nonuniformity. In the piecewise constant case, the probability of prior for a piecewise constant field \(W_{i}\) is a Gaussian normal distribution of the length of the boundary between the different regions. The length of the boundary is \(P(W_{i})=\sum_{\mathbf{x}}\sum_{\mathbf{x}^{\prime}}(1-\delta(W_{i}(\mathbf{ x})-W_{i}(\mathbf{x}^{\prime})))\), where \(\mathbf{x}\) and \(\mathbf{x}^{\prime}\) are neighboring spatial locations.