Fig. 1.

Schematic illustration of the proposed multi-scale dipole inversion (MSDI) method. Each row illustrates the application of Eqs. (1), (2), (3), (4)) across spatial scales. The routine is initialised with SMV filtering with a small kernel radius of 2 mm. The first-scale deconvolution operation uses magnitude priors to ensure accurate depiction of the vasculature and other focal susceptibility gradients if they are co-localised with rapid magnitude variations. Gradually increasing the background-filtering kernel radius in subsequent scales (without using the magnitude prior) gradually recovers sparse susceptibility distributions from increasingly larger-scale fields. In MSDI, to control for the impact of data inconsistencies, a weighting matrix, W_l, is applied to compensate for phase-noise non-uniformities in a scale-dependent manner. In addition, the masking rule imposed by Q_l increasingly lowers the threshold for exclusion of noisy phase-neighbourhoods from the data fidelity term.