Two Dimensional Magnetic Current Imaging Via L1-Curl Regularized Divergence Free Wavelet Reconstruction

The reconstruction of current distributions from samples of their induced magnetic field is a challenging problem due to multiple factors. First, the problem of reconstructing general three dimensional current distributions is ill-posed. Second, the current-to-field operator performs a low-pass filter that dampens high-spatial frequency information, so that even in situations where the inversion is formally possible, attempting to employ the formal inverse will result in solutions with unacceptable noise. Most contemporary methods for reconstructing current distributions in two dimensions are based on Fourier techniques and apply a low pass filter to the $B$-field data, which prevents excessive noise amplification during reconstruction at the cost of admitting blurring in the reconstructed solution. In this report, we present a method of current recovery based on penalizing the $L1$ norm of the curl of the current distribution. The utility of this method is based on the observation that in microelectronics settings, the conductivity is piecewise constant. We also reconstruct the current fields using a divergence-free wavelet basis. This has the advantage of automatically enforcing current continuity and halving the number of unknowns that must be solved for. Additionally, the curl operator can be computed exactly and analytically in this wavelet expansion, which simplifies the application of the $L1-\textrm{curl}$ regularizer. We demonstrate improved reconstruction quality relative to Fourier-based techniques on both simulated and laboratory-acquired magnetic field data.