## 4 Experimental Results
### Implementation  
#### 4.2.1 Parameters of the Method and Their Values  
The size of the spherical window for the computation of the co-occurrences is \(\rho=24\) mm. Within a sphere of this size, the nonuniformity field is assumed constant or piecewise constant. To expedite computation, a subsampling step set to \(\rho_{step}=4\) mm is used. As described in Section 3.2.1, the standard deviation of the deconvolution filter for the co-occurrence statistics is \(\sigma_{r_{i}}\propto r_{i}\), where \(r_{i}\) is the magnitude of the co-occurrences. The standard deviation of the deconvolution filter for the joint-co-occurrence statistics is \(\sigma_{\eta_{i}}\propto\eta_{i}\), where \(\eta_{i}\) is the magnitude of the joint-co-occurrences. The maximum values of the standard deviations of the deconvolution filters, \(\sigma_{r_{i}}\) and \(\sigma_{\eta_{i}}\), at the maxima of the dynamic ranges are set to 6% of these dynamic ranges extents. The standard deviations of the deconvolution filers are accumulated along the iterations, \(t_{max}=10\), to cumulatively remove the total standard deviations of the distortions present in the statistics.  
The Gaussian filtering for the rough nonuniformity estimates is implemented with separable Gaussian filtering along the two axes of the transverse plane of the image. There is no Gaussian smoothing in the axial direction \(y\). The rough estimate of the nonuniformity after the preliminary smoothing with the Gaussian is processed with the MDL principle 3D anisotropic smoothing to separate the field into piecewise linear regions. Higher orders for the MDL smoothing are possible, and in some cases would be more representable. However, they would significantly increase the computational complexity of the MDL smoothing. The size of the windows for MDL smoothing is \(3\times 3\times 3\) with associated weights for each location inversely proportional to the distance of the voxels in space from the center considering anisotropy. The extent of the incremental smoothing both for the case of anisotropic Gaussian smoothing and anisotropic MDL smoothing is 33% of the extent for the respective cumulative smoothing. That is, \(\sigma_{s,incr}=\frac{1}{3}\sigma_{s,cum}\) and \(K_{incr}=\frac{1}{3}K_{cum}\). The cumulative smoothing parameter values are \(\sigma_{s,cum}=\)107 mm and \(K_{cum}=15\). A lower smoothing for the incremental nonuniformity is used to avoid discontinuities in the incremental nonuniformity field that is also assumed to be smooth. A larger smoothing for the cumulative nonuniformity fields represents the actual coil field nonuniformities.  
The size of the images is very large. To expedite processing, the images are subsampled by a certain factor along each of their axes. In the experiments, this factor is \(\alpha=0.5\). That is, the number of pixels in the subsampled image is only 12.5% of the number of pixels in the original image. The values of several of the program parameters, namely \(\rho,\rho_{step},\sigma_{s,cum}\), decrease proportionately to the image axes subsampling factor, \(\alpha\). That is, the parametersbecome \(\rho\mapsto\alpha\rho\), \(\rho_{step}\mapsto\alpha\rho_{step}\), and \(\sigma_{s,cum}\mapsto\alpha\sigma_{s,cum}\). The value of the parameter \(\alpha\) is a tradeoff between computational complexity and accuracy of the method.