Module ilsqr 

iLSQR: Iterative LSQR for QSM with streaking artifact removal

Reference: Li, W., Wang, N., Yu, F., Han, H., Cao, W., Romero, R., Tantiwongkosi, B., Duong, T.Q., Liu, C. (2015). “A method for estimating and removing streaking artifacts in quantitative susceptibility mapping.” NeuroImage, 108:111-122. https://doi.org/10.1016/j.neuroimage.2014.12.043

Reference implementation: https://github.com/kamesy/QSM.m

The algorithm consists of 4 steps:

1. Initial LSQR solution with Laplacian-based weights
2. FastQSM estimate using sign(D) approximation
3. Streaking artifact estimation using LSMR
4. Artifact subtraction

Structs

IlsqrParams

Parameters for the iLSQR algorithm.

Functions

axpy 🔒

axpy_complex 🔒

Complex axpy: y += a * x

compute_laplacian 🔒

Compute Laplacian of a 3D field using mask-adaptive finite differences

dipole_kspace_weights_ilsqr 🔒

K-space weights for FastQSM (Equation 10)

fastqsm_step 🔒

Step 2: FastQSM estimate

fgrad_masked 🔒

Mask-adaptive forward gradient (matching MATLAB’s gradfm_mex)

gradient_weights_component 🔒

gradient_weights_ilsqr 🔒

Gradient weights for streaking artifact estimation (Equation 15)

ilsqr

iLSQR: A method for estimating and removing streaking artifacts in QSM

lap1_axis 🔒

Compute second derivative along one axis using mask-adaptive stencil.

lap1_backward 🔒

Backward one-sided second derivative (matching MATLAB’s bd/bf functions)

lap1_forward 🔒

Forward one-sided second derivative (matching MATLAB’s fd/ff functions)

laplacian_weights_ilsqr 🔒

Laplacian weights for iLSQR (Equation 7)

lsmr

LSMR iterative solver for Ax = b

lsqr

LSQR iterative solver for Ax = b

lsqr_complex

LSQR iterative solver for Ax = b (complex version)

lsqr_step 🔒

Step 1: Initial LSQR solution with Laplacian weights

multiply_elementwise 🔒

norm 🔒

norm_complex 🔒

Complex norm

prctile 🔒

Percentile with linear interpolation (matching MATLAB’s prctile)

scale_complex_inplace 🔒

Complex scale in place

scale_inplace 🔒

sign_array 🔒

susceptibility_artifacts_step 🔒

Step 3: Estimate streaking artifacts using LSMR