Bo Zhao^{1,2}, Borjan Gagoski^{2,3}, Justin P. Haldar^{4}, Elfar Adalsteinsson^{5}, Ellen Grant^{3,6}, and Lawrence L. Wald^{1,2}

HAlf-fourier
Single-shot Turbo spin Echo (HASTE) acquisition is widely used in fetal MR
imaging due to its T_{2} contrast and motion robustness, but speed and
T_{2}-blurring remain a problem for fully sampled acquisitions. In the
work, we describe a new reconstruction approach based on low-rank and subspace
modeling of local k-space neighborhood to accelerate HASTE acquisition. The
proposed approach decreases the echo-train length with improved image quality
and noise robustness compared to conventional reconstruction. It is compatible
with the vendor-provided acquisition. The effectiveness and utility of the
proposed approach is evaluated with both retrospectively and prospectively undersampled
fetal imaging data.

Early work has shown that an
approximately low-rank matrix can be constructed by collecting patches of k-space
data, if the underlying image has a slow-varying image phase^{5}. Here
we form such a data matrix, denoted as $$$\mathbf{S}\in\mathbb{C}^{M\times N}$$$. Moreover, to account for correlation
between multi-coil data^{6}, we further form the matrix $$$\mathbf{S}^P = [\mathbf{S^1}, \mathbf{S}^2,\cdots, \mathbf{S}^L]\in\mathbb{C}^{M\times LN}$$$. Here, by enforcing the
low-rank property, i.e., $$$\mathrm{rank}(\mathbf{S}^P) \leq r$$$, we can formulate a matrix completion problem that enables reconstruction from sub-Nyquist data. Although
directly solving the matrix completion problem often provides good accuracy, it
often results in an expensive computational problem. To enhance the
computational efficiency, we further introduce a null space constraint^{7},
which pre-estimates the null space of $$$\mathbf{S}^P$$$ from the full-sampled
calibration/training data. Figure 1 shows the singular
value decays respectively from $$$\mathbf{S}^P$$$ and the calibration data, which follow a very similar
trend. More specifically, let the
column of $$$\mathbf{V}_s$$$ span the null space of $$$\mathbf{S}^P$$$. Enforcing both the low-rank constraint and the subspace constraint, we can
formulate the following image reconstruction problem: $$\hat{\mathbf{z}} =\mathrm{arg}\underset{\mathbf{z} }{\mathrm{~min}}\parallel\mathcal{P}_s(\mathcal{T}(\mathbf{d})+\mathcal{T}^{c}(\mathbf{z}))\mathbf{V}_s\parallel^2_2$$ where $$$\mathcal{T}$$$ is the linear operator that
maps the sampled k-space data into the complete k-space data vector, while
filling unsampled k-space locations with zeros; $$$\mathcal{T}^c$$$ is the linear operator
performing the complementary operation; and $$$\mathcal{P}_s$$$ maps the complete k-space
data to the data matrix $$$\mathbf{S}^P$$$. Note that with the subspace
constraint, the reconstruction problem reduces to a sparse linear least-squares
problem, which can be efficiently solved by a number of numerical algorithms
(e.g., the iterative LSQR algorithm).

[1] A. Gholipour, J. A. Estroff, C. E. Barnewolt, R. L. Robertson, P. E. Grant, B. Gagoski, S. K. Warfield, O. Afacan, S. A. Connolly, J. J. Neil, A. Wolfberg, and R. V. Mulkern. “Fetal MRI: A technical update with educational aspirations,” Concepts Magn. Reson. Part A, vol. 43, pp. 237-266, 2014.

[2] D. C. Noll, D. G. Nishmura, and A. Macovski, “Homodyne detection in magnetic resonance imaging,” IEEE Trans. Med. Imag. vol. 10, pp. 154-163, 1991.

[3] M. A. Griswold, P. M. Jakob, R. M. Heidemann, M. Nittka, V. Jellus, J. Wang, B. Kiefer, and A. Haase. “Generalized autocalibrating partially parallel acquisitions (GRAPPA),” Magn. Reson. Med., vol. 47, pp. 1202-1210, 2002.

[4] B. Zhao, Borjan Gagoski, Elfar Adalsteinsson, P. Ellen Grant, and Lawrence L. Wald, “Accelerated HASTE-Based Fetal MRI with Low-Rank Modeling”, In: Proc. Int. Soc. Magn. Reson. Med. 2016; p. 4820.

[5] J. P. Haldar, “Low-rank modeling of local k-space neighborhoods (LORAKS),” IEEE Trans. Med. Imag., vol. 33, pp. 668-681, 2014.

[6] J. P. Haldar and J. Zhuo, “P-LORAKS: Low-rank modeling of local k-space neighborhoods with parallel imaging data,” Magn. Reson. Med. vol.75, pp. 1499-1513, 2016.

[7] J. P. Haldar, “Autocalibrated LORAKS for fast constrained MRI reconstruction”, in Proc. IEEE Int. Symp. Biomed. Imaging, pp. 910-913, 2015.

Figure 1: Singular values for the S^{p} matrix and the training matrix for estimating the null space.

Figure 2: K-space sampling scheme
for accelerated HASTE imaging. Here slightly more than half of k-space region
is sampled, within which the central k-space is fully-sampled, while the other
region is undersampled.

Figure 3: Reconstruction of the
retrospectively-undersampled data
at the net AF = 3.0. (a) Reference (Fully-sampled).
(b) GRAPPA and Half-Fourier
reconstruction. (c) Proposed method with the low-rank and subspace constraint. It is clear that the proposed
method well preserves the anatomical structure, while effectively overcoming
the noise contamination (as shown in the GRAPPA and Half-Fourier reconstruction).

Figure 4: Three
imaging slices reconstructed using the proposed method for the prospectively-undersampled data
at
the net AF = 2.8.