ALMA: a mathematics-driven approach for determining tuning parameters in generalized LASSO problems, with applications to MRI

TL;DR

This work tackles the problem of selecting tuning parameters for TV-LASSO-based MRI reconstruction under undersampling. It introduces ALMA, an iterative, mathematics-driven approach that approximates Lagrange multipliers to serve as tuning parameters for generalized LASSO, enabling self-calibrated TV-LASSO reconstructions even with non-Cartesian sampling. The results show ALMA produces high-quality reconstructions (mSSIM ~ 0.99, pSNR > 40 dB, CJV ≈ 0.05) across noise levels and undersampling rates, with convergence in a small number of iterations and fewer reconstructions than L-curve-based tuning. Overall, ALMA provides a deterministic, robust parameter-selection mechanism that reduces manual tuning and holds promise for broad applicability beyond TV-LASSO in MRI and related inverse problems.

Abstract

Magnetic Resonance Imaging (MRI) is a powerful technique employed for non-invasive in vivo visualization of internal structures. Sparsity is often deployed to accelerate the signal acquisition or overcome the presence of motion artifacts, improving the quality of image reconstruction. Image reconstruction algorithms use TV-regularized LASSO (Total Variation-regularized LASSO) to retrieve the missing information of undersampled signals, by cleaning the data of noise and while optimizing sparsity. A tuning parameter moderates the balance between these two aspects; its choice affecting the quality of the reconstructions. Currently, there is a lack of general deterministic techniques to choose these parameters, which are oftentimes manually selected and thus hinder the reliability of the reconstructions. Here, we present ALMA (Algorithm for Lagrange Multipliers Approximation), an iterative mathematics-inspired technique that computes tuning parameters for generalized LASSO problems during MRI reconstruction. We analyze quantitatively the performance of these parameters for imaging reconstructions via TV-LASSO in an MRI context on phantoms. Although our study concentrates on TV-LASSO, the techniques developed here hold significant promise for a wide array of applications. ALMA is not only adaptable to more generalized LASSO problems but is also robust to accommodate other forms of regularization beyond total variation. Moreover, it extends effectively to handle non-Cartesian sampling trajectories, broadening its utility in complex data reconstruction scenarios. More generally, ALMA provides a powerful tool for numerically solving constrained optimization problems across various disciplines, offering a versatile and impactful solution for advanced computational challenges.

Figures (9)

• Figure 1: A graphic representation of the sets $\mathcal{A}$ and $\mathcal{B}$, and the separating hyperplane (dashed line). Observe that $\mathcal{A}$ is an epigraph and $\mathcal{B}$ is an open lower half-line. The closure of $\mathcal{B}$ intersects the boundary of $\mathcal{A}$.
• Figure 3: Schematic representation of ALMA.
• Figure 4: Violin plots of the three metrics across noise levels and undersampling rates. Yellow violins correspond to mSSIM, green violins correspond to pSNR and purple violins correspond to CJV. The black lines correspond to the means and the red lines correspond to the medians.
• Figure 5: Convergence analysis. Subfigure A shows examples of the convergence of ALMA across noise levels and undersampling rates. Each graph corresponds to a fixed couple: noise level and undersampling rate, as illustrated in the legend, showing the iterative progression of the ALM values (vertical axis) at every iteration step (horizontal axis). The dashed lines indicate that the ALM is eventually constant. We observe that the trends displayed by each graph are consistent: the parameter $\lambda^{(k)}$ returned at the first iterations increases, then it descreases until it settles down, with ALMA always stopping in a finite-time. This demonstrates its stability and effectiveness in achieving accurate image reconstructions across noise levels and undersampling rates. Subfigure B shows histograms of the number of iterations needed for ALMA's convergence across undersampling rates while subfigure C shows the same for different noise levels (see the legends). The number of bins for each histogram is the ceiling of the square root of the number of points, i.e., $\lceil \sqrt{150}\rceil=13$. We observe that the number of iterations for ALMA to stop is mostly concentrated in the interval $[0,10]$ independently of the noise level and the undersampling rate, while the standard deviations increase with the undersampling rate and the noise percentage. This demonstrates that the number of iterations required for ALMA to converge is relatively low even when processing larger amounts of information ($20\%$ sampling) and higher noise percentages ($7\%$ noise).
• Figure 6: A: Shaded error bar of mSSIM as a function of $\lambda/\lambda_{ALM}$, the shade representing the corresponding standard deviations. The value 1 on the horizontal axis corresponds to $\lambda = \lambda_{\text{ALM}}$.We observe that the corresponding mSSIM values lie on the plateau of the mSSIM graph, just to the right of the maximum. This indicates that the ALM performs nearly optimally with respect to the mSSIM metric. B: Shaded error bar of pSNR as a function of $\lambda/\lambda_{ALM}$. The value 1 on the horizontal axis corresponds to $\lambda = \lambda_{\text{ALM}}$ and the shade represents the corresponding standard deviations. The corresponding points on the graphs are located to the right of the maxima, and this is more pronounced than in the mSSIM case. However, the pSNR of the reconstructions obtained using the ALMs remains high, indicating that the ALMs still produce high-quality reconstructions despite being slightly offset from the optimal point. C: Shaded error bar of CJV as a function of $\lambda/\lambda_{ALM}$. The value 1 on the horizontal axis corresponds to $\lambda = \lambda_{\text{ALM}}$, the shade representing the corresponding standard deviations. The corresponding points on the graphs are located to the right of the maxima, similar to the pSNR case. Nonetheless, the corresponding CJV values are low, indicating that the ALM also performs well with respect to the CJV metric.

Theorems & Definitions (4)

• Definition 1
• Theorem 1
• Remark 1
• Theorem 2