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Gradient of the Perturbed Field

The exact analytical gradient of the perturbed field $ B^{(1)}$ can be calculated by simply replacing the convolution kernel, $ H$, with its gradient and neglecting the first term, since it is proportional to $ \chi_1
B^{(0)}$ and has zero gradient at the centre of the voxel. This can be also be seen from equation 24 which gives

$\displaystyle \frac{\partial B^{(1)}}{\partial q} = \sum_{\mathbf{x}'} \chi_1(\...
..., \left. \frac{\partial H}{\partial q} \right\vert _{(\mathbf{x}- \mathbf{x}')}$ (26)

where $ q$ stands for $ x, y$ or $ z$.