Evan Levine^{1,2}, Kathryn Stevens^{2}, and Brian Hargreaves^{2}

3D multispectral imaging (MSI) corrects most distortion in MRI near metallic implants at the cost of prolonged scan time by phase encoding to resolve slice distortions. However, existing methods to accelerate 3D MSI do not exploit the redundancy of slice-phase encoding for the dominant on-resonance signal. A novel compact representation of 3D-MSI images based on a decomposition of on- and off-resonance via robust principal component analysis (RPCA) is introduced to exploit this redundancy in a calibration and model-free reconstruction and push the current limits of accelerated 3D MSI. A complementary randomized sampling strategy is used to vary undersampling in different spectral bins to enable the separation. Experiments with retrospective and prospective undersampling show comparable image quality between standard MSI images and 2.6-3.4-fold accelerated RPCA and improvement over bin-by-bin compressed sensing reconstruction.

The signal in spectral bin
$$$b$$$ and voxel $$$(x,y,z)$$$ is $$$s(x,y,z,b)=RF_0(\frac{\gamma}{2\pi} G_zz−f_b)s_0(x,y,z)+e(x,y,z,b)$$$,
where $$$RF_0$$$ is the on-resonance bin RF profile, which weights the
on-resonance magnetization $$$s_0$$$, and $$$e$$$ is the off-resonance
component. At a fixed $$$z$$$, on-resonance bin profiles at any $$$(x,y)$$$ are
spanned by $$$RF_0$$$, and thus, a matrix of bin profiles from different $$$(x,y)$$$
is rank-one plus a spatially sparse error $$$e$$$ due to off-resonance. Figure
1 illustrates this property, showing that the first principal component of this
matrix represents on-resonance (93% of the energy) and the residual,
off-resonance (Figure 1). Correspondingly, on- and off-resonance components, $$$\hat{L}$$$
and $$$\hat{S}$$$, can be reconstructed from undersampled k-space data, $$$y$$$
by solving
$$\hat{L},\hat{S}=\arg\min_{L,S}\|y–\mathcal{F}_u(L+S)\|_2^2+\lambda_1\|TS\|_1+J_L(L)
+J_C(L+S)$$ (1),
where $$$F_u$$$ is an
undersampled Fourier operator, $$$T$$$ is a wavelet transform applied along spatial
dimensions, $$$\|\cdot\|_1$$$ is an entry-wise 1 norm, $$$J_L$$$ is a rank-one
or low-rank-inducing regularizer, and $$$J_C$$$ is a regularizer previously introduced
to extend image-domain calibration-free parallel imaging (CLEAR^{9}) for
partial Fourier parallel imaging.^{10} (1) can be minimized using the
alternating direction method of multipliers,^{11} which can be easily extended
with other constraints. We refer to this reconstruction as RPCA. To enable joint
reconstruction of bins, a calibrationless variable-density complementary
Poisson-disc sampling pattern was used to vary the undersampling in different
bins.^{12}

Initial experiments with
retrospective undersampling were used to evaluate reconstruction accuracy. Additional
experiments were performed with prospective undersampling. PD-weighted hip images
in 7 patients were acquired using standard MSI (2×2
autocalibrating parallel imaging and partial Fourier) and prospectively
undersampled MSI also with partial Fourier. Scan parameters were 3T; Coronal;
TE 6.5ms; TR 4s; FOV 40×40cm^{2}; matrix
384×256; 20-40 sections; 24
bins; radial echo-train view ordering; slice thickness 4mm; ETL 20; half-Fourier;
cut k-space corners. Acquisition times were 6-8.3min for standard MSI and 2-3.5
min for accelerated, a **2.6-3.4-fold
reduction, 18-24-fold overall**. Images from standard and accelerated 3D MSI were evaluated by an
experienced MSK radiologist using a 5-point scale (nondiagnostic; limited; diagnostic; good; excellent) in three categories: 1) image
quality, 2) blockiness, 3) ripple or pipe-up artifact near metal.

Figure 1: 3D MSI bin images can
be analyzed at each $$$z$$$ location by constructing matrices with bin profiles
mapping to columns. The first principal component corresponds to on-resonance
signal (93% of the energy) spanned by the bin profile $$$RF_0$$$, and the
residual off-resonance signal exhibits sparsity.

Figure 2: The enhanced sparsity
due to the separation of on- and off-resonance in RPCA greatly improves reconstruction
accuracy.

Figure 3: Standard
and prospectively accelerated MSI with RPCA and bin-by-bin CS reconstructions are
shown. Left (top row) and right (bottom row) hips from the same subject are shown.
Bin-by-bin CS shows blocky artifacts along the arrows due to sparsity-based
regularization of the entire image, while RPCA reconstructions assume only
sparsity of the off-resonance.

Figure 4: Standard
and prospectively undersampled MSI images reconstructed with RPCA show
comparable image quality. $$$L$$$ and $$$S$$$ components show the expected separation into
on- and off-resonance.

Table 1: Scores from
an experienced MSK radiologist comparing standard MSI and accelerated
MSI with bin-by-bin CS and RPCA reconstructions are shown. Of the image quality
scores for RPCA, Exam 1, shown in Figure 4, had the worst.