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Combining Voxels

Due to the linearity of equation 18 the single voxel solutions derived in the previous section can be added together to give the total field for the object as

$\displaystyle B_z^{(1)}(\mathbf{x}) = \sum_{\mathbf{x}'} \chi_1(\mathbf{x}') H(\mathbf{x}- \mathbf{x}').$ (24)

This takes the form of a discrete convolution of the discrete susceptibility map, $ \chi_1$, and the single voxel solution, $ H$. Therefore the calculation can be efficiently implemented by using the 3D Fast Fourier Transform (FFT) as

$\displaystyle B_z = {\cal F}^{-1} \left( {\cal F}(Z(\chi_1)) {\cal F}(H) \right)$ (25)

where $ {\cal F}(\cdots)$ is the FFT and $ Z(\cdots)$ is a zero-padding function, used to ensure that there is no period wrap-around in the convolution.