  Quantum Diamond Microscopy using Nitrogen-Vacancy (NV) defects in diamond
crystals has enabled the magnetic field imaging of a wide variety of nanoscale
current profiles. Intimately linked with the imaging process is the problem of
reconstructing the current density, which provides critical insight into the
structure under study. This manifests as a non-trivial inverse problem of
current reconstruction from noisy data, typically conducted via Fourier-based
approaches. Learning algorithms and Bayesian methods have been proposed as
novel alternatives for inference-based reconstructions. We study the
applicability of Fourier-based and Bayesian methods for reconstructing
two-dimensional current density maps from magnetic field images obtained from
NV imaging. We discuss extensive numerical simulations to elucidate the
performance of the reconstruction algorithms in various parameter regimes, and
further validate our analysis via performing reconstructions on experimental
data. Finally, we examine parameter regimes that favor specific reconstruction
algorithms and provide an empirical approach for selecting regularization in
Bayesian methods.
