Fast quantitative susceptibility mapping with L1-regularization and automatic parameter selection

Magn Reson Med. 2014 Nov;72(5):1444-59. doi: 10.1002/mrm.25029. Epub 2013 Nov 20.


Purpose: To enable fast reconstruction of quantitative susceptibility maps with total variation penalty and automatic regularization parameter selection.

Methods: ℓ(1) -Regularized susceptibility mapping is accelerated by variable splitting, which allows closed-form evaluation of each iteration of the algorithm by soft thresholding and fast Fourier transforms. This fast algorithm also renders automatic regularization parameter estimation practical. A weighting mask derived from the magnitude signal can be incorporated to allow edge-aware regularization.

Results: Compared with the nonlinear conjugate gradient (CG) solver, the proposed method is 20 times faster. A complete pipeline including Laplacian phase unwrapping, background phase removal with SHARP filtering, and ℓ(1) -regularized dipole inversion at 0.6 mm isotropic resolution is completed in 1.2 min using MATLAB on a standard workstation compared with 22 min using the CG solver. This fast reconstruction allows estimation of regularization parameters with the L-curve method in 13 min, which would have taken 4 h with the CG algorithm. The proposed method also permits magnitude-weighted regularization, which prevents smoothing across edges identified on the magnitude signal. This more complicated optimization problem is solved 5 times faster than the nonlinear CG approach. Utility of the proposed method is also demonstrated in functional blood oxygen level-dependent susceptibility mapping, where processing of the massive time series dataset would otherwise be prohibitive with the CG solver.

Conclusion: Online reconstruction of regularized susceptibility maps may become feasible with the proposed dipole inversion.

Keywords: L-curve; Quantitative susceptibility mapping; Regularization; Total variation.

Publication types

  • Research Support, N.I.H., Extramural

MeSH terms

  • Adult
  • Algorithms*
  • Brain Mapping / methods*
  • Computer Simulation
  • Echo-Planar Imaging
  • Fourier Analysis
  • Healthy Volunteers
  • Humans
  • Image Processing, Computer-Assisted / methods*
  • Imaging, Three-Dimensional / methods
  • Magnetic Resonance Imaging / methods*
  • Male
  • Phantoms, Imaging