Rodrigo A. Lobos^{1}, Tae Hyung Kim^{1}, W. Scott Hoge^{2,3}, and Justin P. Haldar^{1}

While the formation of ghost-free images from EPI data can be a difficult problem, recent low-rank matrix modeling methods have demonstrated promising results. In this abstract, we provide new theoretical insight into these approaches, and show that the low-rank ghost correction optimization problem has infinitely many solutions without using additional constraints. However, we also show that SENSE-like or GRAPPA-like constraints can be successfully used to make the problem well-posed, even for single-channel data. Additionally, we show that substantial performance gains can be achieved over previous low-rank ghost correction implementations by using nonconvex low-rank regularization instead of previous convex approaches.

EPI collects multiple k-space lines after a single excitation, employing alternating positive and negative readout gradients to quickly traverse k-space. Due to hardware miscalibration, eddy currents, and several other factors, the RO+ and RO- datasets are not directly compatible. While images reconstructed from different polarities have similar magnitudes, they typically have different phase profiles, leading to ghost artifacts if these differences are not corrected before combining the RO+ and RO- data.

Existing
LORAKS-based approaches^{6,7} observe that the RO+ and RO- data can be
treated as different channels from a parallel imaging acquisition, similar to
dual-polarity GRAPPA^{11}. This
is useful because earlier work^{10,12} showed that subsampled parallel
imaging k-space data can be embedded into structured low-rank matrices, and
then reconstructed using low-rank matrix recovery. A typical approach is to minimize
$$$J(\mathbf{C}(\mathbf{k}))$$$ subject to independent channel-by-channel data
consistency constraints, where $$$\mathbf{C}(\mathbf{k})$$$ is a structured
matrix formed from the multi-channel k-space data $$$\mathbf{k}$$$, and
$$$J(\cdot)$$$ is a regularization functional that encourages
$$$\mathbf{C}(\mathbf{k})$$$ to have low-rank.

While
LORAKS-based ghost correction has shown promise^{6,7}, we can
mathematically prove that the
problem with channel-by-channel data consistency constraints is ill-posed. Specifically, there are infinitely many
optimal solutions to this problem, and including many undesirable solutions. We
omit the details due to space constraints, but the proof relies on the
fact that it is possible to spatially-shift the image without violating
channel-by-channel k-space data consistency, and without changing the singular
values, rank, or nuclear norm (a regularization functional used to encourage
low rank^{6,7}) of $$$\mathbf{C}(\mathbf{k})$$$. An illustration is shown in Fig. 1.

This
theory implies that additional prior information is necessary for robust
LORAKS-based ghost correction. Previous
ghost correction work^{6,7} did not emphasize the importance of such
prior information, although one reference^{7} implicitly incorporated
information by using sensitivity maps within the SENSE framework^{13,14}.
Using SENSE-like information^{7,13}, LORAKS-based ghost correction is
reformulated as
$$\hat{\mathbf{k}}=\arg\min_\mathbf{k}\|\mathbf{E}\mathbf{p}-\mathbf{d}\|_2^2+\lambda J(\mathbf{C}(\mathbf{k})),$$
where $$$\mathbf{E}$$$ is the SENSE encoding matrix incorporating sensitivity
map information^{14}, $$$\mathbf{d}$$$ is the acquired data,
$$$\mathbf{p}$$$ is the SENSE image to be estimated, and $$$\mathbf{k}$$$ is
the Fourier transform of $$$\mathbf{p}$$$ after sensitivity weighting.

In this
work, we make the novel observation that it is also possible to perform
LORAKS-based ghost correction using GRAPPA-like prior information^{15}. The formulation in this case is
$$$\|\mathbf{k}-\mathbf{d}\|_2^2+\|\mathbf{N}\mathbf{C}(\mathbf{k})\|_2^2+\lambda J(\mathbf{C}(\mathbf{k}))$$$,
where $$$\mathbf{N}$$$ is an estimate of the nullspace of
$$$\mathbf{C}(\mathbf{k})$$$ obtained from autocalibration data^{16}. This GRAPPA-like approach is potentially more
powerful than the SENSE-like approach, because it does not require explicit
estimates of the coil sensitivity---which can be difficult to obtain---and can
be used for single-channel EPI data (which features two-channel data when
separated by readout polarity).

Finally,
we investigated the choice of the regularization functional $$$J(\cdot)$$$,
comparing the nuclear norm functional used by recent ghost correction work^{6,7}
against the nonconvex functional used in earlier LORAKS work^{9,10,13}.

Twelve-channel EPI data was acquired using
a temporal encoding scheme (PLACE)^{5} that enables construction of
ghost-free fully-sampled RO+ and RO- images (unlike the original PLACE^{5},
we do not subsequently combine polarities).
This data was then retrospectively undersampled to simulate EPI datasets
with interleaved RO+ and RO- encoding at different acceleration factors.

Fig. 2 shows LORAKS-based ghost correction
with and without SENSE-based prior information, confirming the power of the
LORAKS-based approach and illustrating the importance of prior
information. Fig. 3 shows similar
results for an *in-vivo* brain dataset.
Fig. 4 shows that LORAKS-based ghost correction of single-channel data is
feasible using GRAPPA-based prior information. Finally, Fig. 5 shows the
advantages of using nonconvex regularization over nuclear norm regularization.

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Fig. 1. Demonstration that shifting the image does not
change the LORAKS matrix rank. (Top Left)
The original simulated RO+ and RO- images have similar magnitude images but
different phase characteristics. In
unaccelerated EPI, we would measure half of the RO+ k-space lines and half of
the RO- lines, meaning that both the (top left) original images and the (bottom
left) images shifted by half the FOV will be perfectly data consistent. (Right) The LORAKS matrices for these two
cases have exactly the same singular values.
Further derivations show that LORAKS-based ghost correction has
infinitely many solutions.

Fig. 2. Comparison of navigator-free ghost correction
methods for different acceleration factors R, showing only RO+ due to space
constraints. (a) Ghost-free PLACE image
of a phantom. (b) Independent SENSE reconstruction of each polarity (GESTE^{2})
provides one approach for navigator-free ghost correction. However, the problem becomes ill-posed for
highly-accelerated scans, since each polarity is undersampled by 2 relative to
the combined k-space data. (c) LORAKS-based joint reconstruction of both
polarities is highly ill-posed, and the reconstruction results are often quite
poor. (d) However, using LORAKS to jointly reconstruct both polarities with
SENSE-like prior information is much more successful.

Fig. 3. Comparison of navigator-free ghost correction
methods for different acceleration factors R.
(a) Ghost-free PLACE image of an *in-vivo*
brain. (b) Independent SENSE reconstruction of each gradient polarity
(GESTE). (c) LORAKS-based joint reconstruction
of both polarities reconstruction without additional constraints. (d) LORAKS-based
joint reconstruction of both polarities combined with SENSE-like prior
information. The results are consistent with those described in Fig. 2.

Fig. 4. Single-channel
EPI ghost correction of a phantom image using GRAPPA-like prior information.
(a) EPI image without ghost correction shows substantial ghost artifacts.
K-space data was fully sampled, but since each polarity is undersampled by 2
relative to the full data, this leads to obvious aliasing that is not corrected
when RO+ and RO- are combined without phase correction. (b) Ghost-free PLACE
images for each gradient polarity. (c) LORAKS-based joint reconstruction of
both polarities with GRAPPA-like prior information.

Fig. 5. A comparison of LORAKS-based reconstruction
using SENSE-like prior information with convex and nonconvex
regularization. The gold standard image
is obtained from PLACE, leading to fully-sampled RO+ and RO- datasets. The LORAKS-based reconstructions are obtained
for an acceleration factor of $$$R=3$$$.
Results suggest that the convex nuclear norm approach^{6,7} is
substantially outperformed by the use of nonconvex regularization^{9,10,13}.