Rodrigo A Lobos^{1}, Ahsan Javed^{1}, Krishna S Nayak^{1}, W Scott Hoge^{2,3}, and Justin P Haldar^{1}

The presence of ghost artifacts is a recurrent problem in EPI images, which has been recently addressed using structured low-rank matrix (SLM) methods. In this work we propose a new SLM ghost correction method called Robust Autocalibrated LORAKS (RAC-LORAKS). RAC-LORAKS considers autocalibrated k-space constraints (similar to GRAPPA) to deal with the ill-posedness of existing SLM EPI ghost correction methods. RAC-LORAKS additionally adapts these constraints to enable robustness to possible imperfections in the autocalibration data. We illustrate the capabilities of RAC-LORAKS in two challenging scenarios: highly accelerated EPI of the brain, and cardiac EPI with double-oblique slice orientation.

The
principle behind SLM methods is that, because k-space data is often linearly
predictable (due to support, phase, parallel imaging, and sparsity
constraints), it can be embedded into structured Toeplitz/Hankel matrices which
will have low-rank characteristics. If
the data is undersampled, then information can be recovered by applying
low-rank matrix recovery to these matrices^{2-8,10-12}. It has been
shown in earlier work^{4,5} that SLM EPI ghost correction can be
challenging from a theoretical perspective unless prior information is used,
and that nonconvex formulations have substantial advantages over convex
formulations. Reference 5 proposed a
nonconvex formulation based on the LORAKS framework that incorporates prior
information in the form of autocalibrated (AC) k-space constraints^{8}. This “AC-LORAKS” approach was shown to be
particularly powerful when compared against other approaches.

However, the good performance of the previous AC-LORAKS for EPI ghost correction relies on having high-quality autocalibration (ACS) data. This requirement is nonideal, because ACS data can often suffer from artifacts due to effects such as respiration, motion, and concomitant fields, and ACS data acquired at the beginning of a long experiment is not always consistent with EPI data measured at different timepoints. In this work, we propose a generalization of this AC-LORAKS approach called Robust Autocalibrated LORAKS (RAC-LORAKS) which is designed to be robust against ACS data imperfections. The main idea is that we do not totally trust the ACS data, and use a formulation that balances the information learned from the ACS data with information from the measured data being reconstructed. Using an alternating minimization approach, RAC-LORAKS solves the following constrained optimization problem subject to data consistency:

$$\{\hat{\mathbf{k}},\hat{\mathbf{N}}\}=\arg\min_{\mathbf{k},\mathbf{N}}\|\mathbf{N}\mathbf{C}(\mathbf{k})\|_2^2+ \lambda_c\|\mathbf{N}\mathbf{C}(\mathbf{k_{acs}})\|_2^2+\lambda_sJ(\mathbf{S}(\mathbf{k})),$$

where $$$\mathbf{C}(\mathbf{k})$$$ and $$$\mathbf{S}(\mathbf{k})$$$ are C-LORAKS (which encourages support and parallel
imaging constraints) and S-LORAKS (which encourages support, parallel imaging,
and phase constraints) matrices^{6,7} formed from the multi-channel
k-space data $$$\mathbf{k}$$$; $$$\mathbf{C}(\mathbf{k}_{acs})$$$ is the C-LORAKS
matrix of the ACS data $$$\mathbf{k_{acs}}$$$; $$$\mathbf{N}$$$ is an approximate
nullspace that is shared between $$$\mathbf{C}(\mathbf{k})$$$ and $$$\mathbf{C}(\mathbf{k}_{acs})$$$; $$$J$$$ is a nonconvex function that encourages
low-rank^{8}; and $$$\lambda_c$$$ and $$$\lambda_s$$$ are regularization
parameters. The previous AC-LORAKS approach for EPI ghost correction^{5}
can be obtained in the limit as $$$\lambda_c\rightarrow \infty$$$, in which case
the approximate nullspace $$$\mathbf{N}$$$ is a fixed matrix that is influenced
only by the ACS data. The extent to
which the ACS data is trusted is controlled by the value of $$$\lambda_c$$$.

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[5] Lobos RA, Kim TH, Hoge WS, Haldar JP. "Navigator-free EPI ghost correction with structured low-rank matrix models: New theory and methods. " arXiv: 1708.05095, 2017.

[6] Haldar JP. "Low-rank modeling of
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(top) Magnitude and (bottom) phase images
corresponding to the (ﬁrst two columns) ACS data and (remaining columns) image
reconstruction results obtained from multi-channel EPI data prospectively
undersampled by a factor or 6. Considerable ghost artifacts (which are
particularly visible in the phase images) are observed in the ACS data. The red
arrow indicates the phase encoding direction.

Magnitude images corresponding to the (ﬁrst
column) positive gradient polarity image of the ACS data and (remaining
columns) image reconstruction results obtained from unaccelerated double-oblique
cardiac EPI data. In the ﬁrst six columns, the colorscale has been adjusted to
highlight ghost artifacts. The last column shows a standard intensity window
for reference, using the same RAC-LORAKS image as in the sixth column. The red
arrow indicates the phase encoding direction.
The yellow arrows are used to indicate undesired ghost artifacts.