Simultaneous auto‐calibration and gradient delays estimation (SAGE) in non‐Cartesian parallel MRI using low‐rank constraints

Purpose To correct gradient timing delays in non‐Cartesian MRI while simultaneously recovering corruption‐free auto‐calibration data for parallel imaging, without additional calibration scans. Methods The calibration matrix constructed from multi‐channel k‐space data should be inherently low‐rank. This property is used to construct reconstruction kernels or sensitivity maps. Delays between the gradient hardware across different axes and RF receive chain, which are relatively benign in Cartesian MRI (excluding EPI), lead to trajectory deviations and hence data inconsistencies for non‐Cartesian trajectories. These in turn lead to higher rank and corrupted calibration information which hampers the reconstruction. Here, a method named Simultaneous Auto‐calibration and Gradient delays Estimation (SAGE) is proposed that estimates the actual k‐space trajectory while simultaneously recovering the uncorrupted auto‐calibration data. This is done by estimating the gradient delays that result in the lowest rank of the calibration matrix. The Gauss‐Newton method is used to solve the non‐linear problem. The method is validated in simulations using center‐out radial, projection reconstruction and spiral trajectories. Feasibility is demonstrated on phantom and in vivo scans with center‐out radial and projection reconstruction trajectories. Results SAGE is able to estimate gradient timing delays with high accuracy at a signal to noise ratio level as low as 5. The method is able to effectively remove artifacts resulting from gradient timing delays and restore image quality in center‐out radial, projection reconstruction, and spiral trajectories. Conclusion The low‐rank based method introduced simultaneously estimates gradient timing delays and provides accurate auto‐calibration data for improved image quality, without any additional calibration scans.