Gradient timing delay errors in radial, spiral and EPI trajectories often produce spurious geometric image distortions. Moreover, misaligned k-space center data results in auto-calibration errors for parallel imaging methods. Recently, a number of promising works ^1,2,3 have shown that promoting self-consistency that is inherent in neighboring multi-channel k-space measurements can correct for trajectory errors, without separate trajectory calibration scans. However, GRAPPA based methods^1,2 cannot be extended to arbitrary non-Cartesian trajectory and SPIRiT based method³ still requires prior calibration data to obtain accurate SPIRiT kernels.

In this work, we propose a method that estimates the gradient delays and auto-calibration data simultaneously. Our method is applicable to any non-Cartesian trajectory, and does not require prior GRAPPA⁸/SPIRiT⁹ calibration. We make use of the fact that uncorrupted multi-channel data is inherently low dimensional, ^4,5,6,7 and data corruption induces inconsistencies that violate this property. Hence, we solve directly for the gradient delays corrections that result in consistent low-dimensional calibration data. We formulate the joint estimation as a low-rank minimization problem, implemented using a Gauss-Newton method.

We exploit the low-rank property of calibration matrices that are used in GRAPPA⁸, SPIRiT⁹ and ESPIRiT ⁶ methods as constraints to the following optimization problem:

$$ \underset{\Delta(\vec t),x}{\text{minimize}} \| D(\vec t+\Delta(\vec t))x - y \| _2^2 \\ \text{subject to } rank(H(x)) \leq k $$

where D is gridding operator, which is parameterized by gradient delay in time $$$\Delta(\vec t)$$$ , x is a Cartesian auto-calibration data, y is the non-Cartesian k-space measurements, H is the Hankel structured operator, which reformats the calibration data into the usual calibration matrix and k is the expected calibration matrix rank.⁶ Since solving for delay is non-convex, we use a Gauss-Newton method to obtain a local minimum. In each step we linearize the problem about a point and solve the sub-problem using alternating minimization between data consistency and low-rank projection. Figure 1 illustrates the proposed method.

The proposed method was verified on simulated 2D phantom data (256x256 matrix, 8 coils), with a ramp-sampled radial trajectory (kx/ky), and interleaved time-optimal spiral trajectory. Various gradient delay amounts, ranging from -2 to 2 samples for X and Y gradient respectively, were added to the data. Experimentally, we synthetically added a range of gradient delays from -2 to 2 samples to a phantom scan with a 2D slice-selective RF-spoiled sequence of a ramp-sampled, center-out radial trajectory (256x256 matrix, 32 coils, 256 readout points, 1.02­­ ms readout, 754 spokes, 24 cm FOV and 1 mm in-plane resolution), on a 7T scanner (GE Healthcare). For computation purposes, the delay estimation was performed on the beginning 56 readout points and 150 spokes.Figure 2 shows a simulation instance with gradient delay = [1 2], which is 1 sample delay in X gradient and 2 samples delay on Y gradient. As the middle row of Figure 2 shows, gradient delay induces both image artifacts and corrupted sensitivity maps. Our proposed method estimates the delay as [0.9989 2.0075] and yields the correct sensitivity maps within 50 iterations. Simulations on 2D radial and spiral trajectories with various delays result in an RMS delay estimation error of 0.0020/0.0028 (unit: sample) respectively. Experimentally, the proposed method effectively corrects for gradient timing delays, removing the delay-induced artifacts, and providing a corrected calibration region for the following parallel imaging processing. Figure 3 shows the case of gradient delay = [-1 2], with the estimated delay of [-1.21 1.92]. The errors are consistent across different datasets, regardless of delay amount. It might be related to eddy currents induced errors, which our approach does not model.

We have presented a low-rank minimization method that effectively corrects for the gradient delays and also estimates consistent auto-calibration data. It can be easily implemented for 2D radial, spiral, EPI ⁴ and other non-Cartesian trajectories, as well as for 3D non-Cartesian trajectories.

Even though the proposed model only estimates the gradient timing delay to satisfy low rank constraints, short-term eddy currents can be approximated as delay, and our preliminary results indicate that the proposed model can also compensate for these terms as well. To accurately model both eddy currents and timing delay, the proposed method can be extended to a filter design problem to generalize the trajectory errors for non-Cartesian MRI.