Yunsong Liu^{1}, Congyu Liao^{2}, Kawin Setsompop^{2}, and Justin P. Haldar^{1}

^{1}Signal and Image Processing Institute, University of Southern California, Los Angeles, CA, United States, ^{2}Martinos Center for Biomedical Imaging, Charlestown, MA, United States

gSlider is a diffusion MRI method that achieves fast high-resolution data acquisition using a novel slab-selective RF-encoding strategy. Recent work has proposed subsampling of the multidimensional gSlider encoding space (diffusion-encoding/RF-encoding) for further improved scan efficiency. Two different q-space regularization approaches (i.e., Laplace-Beltrami smoothness and spherical ridgelet sparsity) have been proposed to compensate for missing data, but there have been no systematic comparisons between the two. We compare and evaluate the potential synergies of these regularization approaches. Results suggest that there can be small advantages to combining both regularization strategies together, although Laplace-Beltrami regularization alone is simpler and not much worse.

Formulation To evaluate the different regularization strategies, we consider a gSlider formulation that combines both penalties together:

$$ \hat{\mathbf{x}} = \arg\min_{\mathbf{x}} \frac{1}{2} \| \mathbf{b} - \mathbf{A}\mathbf{x} \|_2^2 + \beta R(\mathbf{x}) + \lambda L(\mathbf{x}).$$

Here, $$$\mathbf{x}$$$ is the vector of high-resolution image values (unknown and to be estimated) for all of the different diffusion encodings, $$$\mathbf{b}$$$ is the vector of measured thick-slab data for all of the different RF encodings and diffusion encodings, $$$\mathbf{A}$$$ is the operator that models gSlider data acquisition with interlaced subsampling, $$$R(\cdot)$$$ is an $\ell_1$-norm regularization penalty that promotes sparsity of the spherical ridgelet coefficients of the estimated data

This optimization problem was solved using the FISTA algorithm

Using this formulation, simulated gSlider data was reconstructed with systematic variation of the $$$(\lambda, \beta)$$$ parameters to elucidate the characteristics of the different regularization methods. The case with $\lambda=0$ yields Laplace-Beltrami regularization by itself, and the case with $$$\beta=0$$$ yields spherical ridgelet regularization by itself. Both regularization penalties are active when both $$$\lambda$$$ and $$$\beta$$$ are nonzero.

Four-average gSlider data with 860 $$$\mu m$$$ isotropic resolution were acquired and combined to get a high quality reference dataset. For each average, the scan time is about 20 min. Interlaced subsampling was simulated as illustrated in Figure 1 where 3 or 4 RF encodings were acquired at each position in q-space out of 5 nominal RF encodings (the average was 3.5 RF encodings per q-space location, which corresponds to an acceleration factor of 1.4).

We computed normalized root mean square error (NRMSE) metrics for the recovered diffusion weighted images (DWIs) and for the quantitative fractional anisotropy (FA) estimate obtained from diffusion tensor fitting. We also calculated NRMSE metrics for two different orientation distribution function estimation methods: the Funk-Radon Transform (FRT)

Interestingly, different error metrics were associated with different optimal regularization parameters. Further, the optimal regularization parameters were also substantially different for the low-anisotropy and high-anisotropy voxels. This suggests that when such regularization is used in practical diffusion MRI applications, regularization parameters may need to be chosen carefully based on the objectives of the specific study.

Notably, Laplace-Beltrami regularization by itself leads to a simple linear least-squares problem that has an analytical closed-form solution that is easy to implement, while spherical ridgelet regularization or a combination of Laplace-Beltrami with spherical ridgelet regularization is nonlinear and requires iterative optimization. As a result, Laplace-Beltrami regularization by itself may be preferred if computational complexity is a concern.

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Figure 1 Illustration of
interlaced q-space subsampling for gSlider with 5 RF encodings. At each q-space
location, either three or four of the five RF encodings are sampled.
Subsampling is depicted by using different colors (blue, green, white, yellow
and red) to represent each of the different RF encodings.

Figure 2 Optimal regularization parameters $$$\lambda^*$$$
and $$$\beta^*$$$ corresponding optimal NRMSE values for different error metrics.

Figure 3 Visualization of selected performance metrics as
a function of the regularization parameters $$$\beta$$$ (spherical ridgelet
regularization) and $$$\lambda$$$ (Laplace-Beltrami regularization).
High-Anisotropy (FA > 0.3) case.

Figure 4 Visualization of selected performance metrics as
a function of the regularization parameters $$$\beta$$$ (spherical ridgelet
regularization) and $$$\lambda$$$ (Laplace-Beltrami regularization).
Low-Anisotropy (FA < 0.3) case.