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Magn Reson Med. 2019 Feb;81(2):1399-1411. doi: 10.1002/mrm.27483. Epub 2018 Sep 28.

Weak-harmonic regularization for quantitative susceptibility mapping.

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Department of Electrical Engineering, Pontificia Universidad Catolica de Chile, Santiago, Chile.
Biomedical Imaging Center, Pontificia Universidad Catolica de Chile, Santiago, Chile.
Martinos Center for Biomedical Imaging, Harvard Medical School, Boston, Massachusetts.
Department of Neurology, Medical University of Graz, Graz, Austria.
Wellcome Centre for Human Neuroimaging, UCL Institute of Neurology, University College London, London, United Kingdom.



Background-field removal is a crucial preprocessing step for quantitative susceptibility mapping (QSM). Remnants from this step often contaminate the estimated local field, which in turn leads to erroneous tissue-susceptibility reconstructions. The present work aimed to mitigate this undesirable behavior with the development of a new approach that simultaneously decouples background contributions and local susceptibility sources on QSM inversion.


Input phase data for QSM can be seen as a composite scalar field of local effects and residual background components. We developed a new weak-harmonic regularizer to constrain the latter and to separate the 2 components. The resulting optimization problem was solved with the alternating directions of multipliers method framework to achieve fast convergence. In addition, for convenience, a new alternating directions of multipliers method-based preconditioned nonlinear projection onto dipole fields solver was developed to enable initializations with wrapped-phase distributions. Weak-harmonic QSM, with and without nonlinear projection onto dipole fields preconditioning, was compared with the original (alternating directions of multipliers method-based) total variation QSM algorithm in phantom and in vivo experiments.


Weak-harmonic QSM returned improved reconstructions regardless of the method used for background-field removal, although the proposed nonlinear projection onto dipole fields method often obtained better results. Streaking and shadowing artifacts were substantially suppressed, and residual background components were effectively removed.


Weak-harmonic QSM with field preconditioning is a robust dipole inversion technique and has the potential to be extended as a single-step formulation for initialization with uncombined multi-echo data.


alternating direction method of multipliers (ADMM); augmented lagrangian; background field removal; nonlinear inversion; total variation


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