PurposeThe presence of dipole‐inconsistent data due to substantial noise or artifacts causes streaking artifacts in quantitative susceptibility mapping (QSM) reconstructions. Often used Bayesian approaches rely on regularizers, which in turn yield reduced sharpness. To overcome this problem, we present a novel L1‐norm data fidelity approach that is robust with respect to outliers, and therefore prevents streaking artifacts.MethodsQSM functionals are solved with linear and nonlinear L1‐norm data fidelity terms using functional augmentation, and are compared with equivalent L2‐norm methods. Algorithms were tested on synthetic data, with phase inconsistencies added to mimic lesions, QSM Challenge 2.0 data, and in vivo brain images with hemorrhages.ResultsThe nonlinear L1‐norm‐based approach achieved the best overall error metric scores and better streaking artifact suppression. Notably, L1‐norm methods could reconstruct QSM images without using a brain mask, with similar regularization weights for different data fidelity weighting or masking setups.ConclusionThe proposed L1‐approach provides a robust method to prevent streaking artifacts generated by dipole‐inconsistent data, renders brain mask calculation unessential, and opens novel challenging clinical applications such asassessing brain hemorrhages and cortical layers.