Simultaneous Estimation of Auto-calibration Data and Gradient Delays in non-Cartesian Parallel MRI using Low-rank Constraints

Wenwen Jiang^{1}, Peder E.Z Larson^{2}, and Michael Lustig^{3}

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
method^{3} still requires prior calibration data to obtain accurate SPIRiT
kernels.

We exploit
the low-rank property of calibration matrices that are used in GRAPPA^{8},
SPIRiT^{9} 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.

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.

Iterations of alternating minimization
between a low-rank constraint and data consistency: yielding both the timing
delay of individual gradient axes and a
a
consistent k-space calibration region

Simulation for gradient timing delay effects for auto-calibration: (top)
original image without any delay; (middle) radial trajectory suffering 1 sample
delay at X gradient and 2 samples delay at Y gradient, denoted as [1 2]; (bottom)
corrected image and estimated delay is [0.9989 2.0014] samples on X, Y axis respectively

7T phantom data with a synthetically enhanced delay (-1 sample on X
gradient and 2 samples on Y gradient from delays calculated at installation)
with (a) no compensation (b) corrected based on calibrated system delays (c)
corrected based on estimated delays using the proposed method

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

0939