  Quantitative susceptibility mapping (QSM) aims to visualize the three
dimensional susceptibility distribution by solving the field-to-source inverse
problem using the phase data in magnetic resonance signal. However, the inverse
problem is ill-posed since the Fourier transform of integral kernel has zeroes
in the frequency domain. Although numerous regularization based models have
been proposed to overcome this problem, the incompatibility in the field data
has not received enough attention, which leads to deterioration of the
recovery. In this paper, we show that the data acquisition process of QSM
inherently generates a harmonic incompatibility in the measured local field.
Based on such discovery, we propose a novel regularization based susceptibility
reconstruction model with an additional sparsity based regularization term on
the harmonic incompatibility. Numerical experiments show that the proposed
method achieves better performance than the existing approaches.
