Rapid reconstruction of quantitative susceptibility mapping via improved ℓ₀ norm approximation

Quantitative susceptibility mapping (QSM) reconstruction is a well-known ill-posed problem. Various regularization techniques have been proposed for solving this problem. In this paper, a rapid method is proposed that uses ℓ₀ norm minimization in a gradient domain. Because ℓ₀ minimization is an NP-hard problem, a special alternating optimization strategy is employed to simplify the reconstruction algorithm. The proposed algorithm uses only simple point-wise multiplications and thresholding operations, and significantly speeds up the calculation. Both numerical simulations and in vivo experiments demonstrate that the proposed method can reconstruct susceptibility fast and accurately. Because morphology information weighted methods have achieved considerable success in QSM, we performed a quantitative comparison with some typical weighted methods, such as MEDI (morphology enabled dipole inversion), iLSQR (improved least squares algorithm), and wℓ₁ (weighted ℓ₁ norm minimization). The reconstructed results show that the proposed method can provide accurate results with a satisfactory speed.