Simulation scheme.

<p>A gradient echo measurement is used to estimate a <math><msubsup><mi>T</mi><mn>2</mn><mo>*</mo></msubsup></math> map. Using the <math><msubsup><mi>T</mi><mn>2</mn><mo>*</mo></msubsup></math> anatomy and rescaling the values to an appropriate range, a susceptibility distribution is generated. Random susceptibility structures are added and dipole convolution generates the susceptibility-induced field map and a local reference. External disturbances are introduced as randomised spherical harmonic functions and noise is added. After BFR correction the results are compared to the reference field.</p

Measurement series at 3T.

<p>From top to bottom the field map, DIPF, MUBAFIRE and MUBAFIRE Local correction for five samples of the 3T study are shown. The bottom row illustrates histograms of the field distribution for the correction algorithms in each sample.</p

Measurement c): <i>in vivo</i> BFR results at 9.4T.

<p>(a) shows orthogonal sample slices, from left to right: raw, GF, SPHINX, DIPF, MUBAFIRE, MUBAFIRE Local; (b) histogram overview; (c) shows position of the line plot in the MUBAFIRE Local correction and (d) the according line plot comparing MUBAFIRE and MUBAFIRE Local in magnified view.</p

Results of BFR on Monte Carlo Simulation.

<p>Units are Hz. The rows show mean and standard deviation of the</p

Parameter Optimisation.

<p>From left to right, the resulting error for standalone SPHINX, DIPF and MUBAFIRE are shown for a range of parameterisations. The rightmost graph illustrates the required computing time for MUBAFIRE.</p

Intra-voxel field gradients.

<p>A simulated numerical phantom of a solid sphere with high susceptibility (top left) is resampled in complex domain by 1/4, 1/8 and 1/16 of the native resolution. The field is determined from the phase of the resampled complex signal by unwrapping. Differences higher than 5% between true and calculated field are indicated red. Especially near objects with an extent of only few voxels, but of high susceptibility contrast, phase gradients are strong and lead to incorrect field estimates (right column).</p

Spatial view of BFR on 9.4T <i>in vivo</i> measurement.

<p>Actual field map (left) and MUBAFIRE Local corrected field (right) recorded and computed from the 9.4T <i>in vivo</i> measurement. In the middle, long-range distortions (MUBAFIRE) and local distortions (MUBAFIRE Local) are visualised.</p

Workflow of the MUBAFIRE algorithm.

<p>The raw data are corrected with respect to constant offsets and linear gradients. Then, SPHINX and finally the DIPF are applied. The lower row shows the optional local distortion correction featured by MUBAFIRE Local: After thresholding, a local erosion filter is applied, followed by a final DIPF correcting for local distortions (from [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0138325#pone.0138325.ref024" target="_blank">24</a>]).</p

Results of BFR on 9.4T measurements.

<p>Units are Hz. The mean field and standard deviation of the corrected field from the measurements are shown.</p><p>Results of BFR on 9.4T measurements.</p

Background Filter Performance.

<p>Field shifts are calculated based on the numerical susceptibility phantoms (first column) by dipole convolution and are falsified by artificial harmonic background inhomogeneities (second column). The top row shows a homogeneous phantom, the bottom row includes strong susceptibility spikes. SPHINX, DIPF and MUBAFIRE (Local) are applied to the cases. Red arrows indicate artefacts.</p