Andres Saucedo^{1}, Manoj K. Sarma^{1}, and M. Albert Thomas^{1}

Compressed sensing (CS) combined with non-uniform
undersampling, such as the low-rank Hankel matrix completion method, have accelerated the acquisition time of 2D magnetic resonance spectroscopy (MRS). This
technique relies on reconstructing the vector of all t_{1} points separately for each F_{2} point. We introduce a CS-based method that implements
joint Hankel low rank regularization, which enforces the low-rankness of
all Hankel matrices formed from the entire F_{2}-t_{1} data simultaneously. We compare this method with
group sparsity CS reconstruction of retrospectively undersampled localized correlated spectroscopy (COSY) acquisitions in a brain phantom
and calf muscle.

The reconstruction of the F_{2}-t_{1}
data *x* using JHLR regularization is
posed as the following minimization problem: $$\operatorname*{min}_{x \in \mathbb{C}^{N_{2} \times N_{1}}} \frac{1}{2}\| y - Ax\|^{2}_{2} + \tau \sum_{n = 1}^{N_2} \|H_{n} x\|_{S_{1}}$$ where y is the under-sampled F_{2}-t_{1}
data, A is the undersampling operator, N_{1} the number of F_{1}
points, N_{2} the number of F_{2} points,$$$\| \cdot \|_{S_{1}} $$$ denotes the Schatten 1-norm^{9} , $$$ \tau $$$ is a regularization
parameter, and H_{n} is the operator that forms a Hankel matrix from
all t_{1} measurements corresponding to the n^{th} F_{2}
point of *x*. The regularization parameters
for each algorithm were chosen empirically as those minimizing the normalized
root mean square error (nRMSE) for selected diagonal and cross peaks. Both the JHLR and GS algorithms are based on the alternating direction method of multipliers (ADMM)^{10,11}

A COSY spectrum of a brain phantom composed of several metabolites at physiological concentrations was acquired with the following
parameters: VOI = 3x3x3 cm^{3}, TR=2 s, TE=30 ms, 1024 t_{2} points, 128 t_{1}
points, BW_{2}=2000 Hz, BW_{1}=1250 Hz, and 12 averages. A COSY spectrum of the soleus calf muscle from a healthy volunteer was acquired with: VOI = 2.5x2.5x2.5 cm^{3}, TR=1.5 s, TE=30 ms, 1024 t_{2} points,
96 t_{1} points, BW_{2}=2000 Hz, BW_{1}=1250 Hz, and 8 averages. These data sets were retrospectively undersampled at reduction factors
(RF) of 2, 2.5, 3, 3.5, 4, and 5 using NUS masks generated with a skewed sine bell squared sampling density function.

To assess reconstruction and
quantitation accuracy, the peak integrals of the fully-sampled,
GS-reconstructed, and JHLR-reconstructed spectra were computed, as well and the
nRMSE’s of selected diagonal and cross peaks. For the brain phantom the diagonal
peaks are: N-acetylaspartate (NAA), Creatine (Cr-3.0), Choline (Ch-3.2), myo-Inositol
(mI-3.2), and Cr-3.9; the cross peaks are: Alanine (Ala), Lactate (Lac), Threonine (Thr), glutamine/glutamate (Glx), N-acetylaspartate (NAA), Aspartate (Asp), and mI-Ch. For the
calf muscle, the diagonal peaks are the (CH_{2})_{N} lipids,
lipid methylene peaks, Cr-3.0, Ch-3.2, Cr-3.9, and the olefinic (CH=CH) peaks;
the cross peaks are the extra- and intra-myocellular lipids (EMCL_{1}/EMCL_{2} and IMCL_{1}/IMCL_{2}) and the triglyceride backbone fatty (TGBF) acids.

[1] Thomas, M.A., et al. Localized two-dimensional shift correlated MR spectroscopy of human brain. Magnetic Resonance in Medicine, 2001; 46(1): 58-67.

[2] Thomas, M.A., et al. Evaluation of two-dimensional L-COSY and JPRESS using a 3T MRI scanner: from phantoms to human brain in vivo. NMR in Biomedicine, 2003; 16(5):245-251.

[3] Furuyama, J.K., et al. Application of Compressed Sensing to Multi-dimensional Spectroscopic Imaging in human prostate. Magnetic Resonance in Medicine, 2012; 67(6):1499-1505.

[4] Wilson, N.E., et al. Accelerated Five-dimensional Echo Planar J-resolved Spectroscopic Imaging: Implementation and pilot validation in human brain. Magnetic Resonance in Medicine, 2016; 75(1):42-51.

[5] Burns, B.L., et al. Group Sparse Reconstruction of Multi-dimensional Spectroscopic Imaging in Human Brain in vivo. Algorithms, 2014; 7(3):276-294.

[6] Burns, B.L., et al. Non-Uniformly under-sampled Multi-dimensional Spectroscopic Imaging in vivo: maximum entropy versus compressed sensing reconstruction. NMR in Biomedicine, 2014; 27(2):191-201.

[7] Qu, X., et al. Accelerated NMR Spectroscopy with Low-Rank Reconstruction. Angewandte Chemie International Edition, 2015; 54(3):852-854.

[8] Guo, D., et al. A Fast Low Rank Hankel Matrix Factorization Reconstruction Method for Non-Uniformly Sampled Magnetic Resonance Spectroscopy. IEEE Access, 2017; 5:16033-16039.

[9] Lefkimmiatis, S., et al. Hessian Schatten-norm Regularization for Linear Inverse Problems. IEEE Transactions on Image Processing, 2013; 22(5):1873-1888.

[10] Goldstein, T., Osher, S. The Split-Bregman Method for L1-regularized problems. SIAM Journal on Imaging Sciences, 2009; 2(2):323-343.

[11] Afonso, M.V., et al. An Augmented Lagrangian Approach to the Constrained Optimization Formulation of Imaging Inverse Problems. IEEE Transactions on Image Processing, 2011; 20(3):681-695.

Figure 1: (A) NUS mask (RF = 3). (B)
Fully-sampled COSY spectrum of brain phantom. (C) localization of brain
phantom. (D) Zero-filled reconstruction (RF = 3). (E) Group sparse (GS)
reconstruction. (F) Joint Hankel low-rank (JHLR) reconstruction. (G) absolute
difference map (relative to the fully-sampled spectrum) of the zero-filled
reconstruction. (H) absolute difference map of the GS reconstruction. (I) absolute
difference map of the JHLR reconstruction. Note the lower intensity of the absolute
difference map of JHLR. The absolute difference images are windowed and leveled identically.

Figure
2: (A) NUS mask (RF = 3.5). (B) Fully-sampled COSY spectrum of calf
muscle. (C) localization of soleus calf muscle. (D) Zero-filled reconstruction
(RF = 3.5). (E) Group sparse (GS) reconstruction. (F) Joint Hankel low-rank (JHLR) reconstruction. (G)
absolute difference map (relative to the fully-sampled spectrum) of the zero-filled
reconstruction. (H) absolute difference map of the GS reconstruction. (I) absolute
difference map of the JHLR reconstruction. Note the lower intensity of the absolute
difference map of JHLR. The absolute difference images are windowed and leveled identically.

Table 1: Normalized root mean square error (nRMSE) values for brain phantom
and calf muscle COSY reconstructions at various reduction factors (RF) for selected
diagonal peaks and cross peaks. Note the lower nRMSE values of JHLR
reconstruction compared to GS.

Table 2: Peak integral values for
fully-sampled and reconstructed brain phantom and calf muscle COSY spectra at
all reduction factors (RF) for selected diagonal and cross peaks.
Note the general trend in lower percent errors of JHLR compared to GS. The peak integral values for the fully-sampled COSY are shown in the leftmost column.