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

2024-06-27

arXiv.org

AI Summary

In MRI reconstruction, the tuning parameter (Lagrange multiplier) of the TV-LASSO model lacks a universal, deterministic selection criterion, leading to reliance on manual empirical tuning and poor reproducibility. To address this, we propose the Adaptive Lagrange Multiplier Algorithm (ALMA), the first method to establish an analytical relationship between the LASSO penalty parameter and the duality gap. ALMA employs a dual optimization framework grounded in the Karush–Kuhn–Tucker (KKT) conditions and equipped with adaptive step sizing to automatically determine the optimal multiplier. The method is compatible with non-Cartesian sampling schemes and generalizable to diverse regularizers, ensuring both theoretical rigor and cross-modal applicability. Validation on simulated and phantom data demonstrates that ALMA’s automatic parameter selection yields an average 3.2 dB PSNR improvement over manual tuning, significantly enhancing structural fidelity and noise suppression.

Technology Category

• Machine Learning: Optimization
• Search and Optimization: Algorithm Configuration
• Computer Vision: Learning & Optimization for CV

Application Category

• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Responsible Web: Machine-in-the-loop, human agency and autonomy

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.

Problem

• Determining tuning parameters in generalized LASSO problems for MRI reconstruction
• Lack of general deterministic techniques for selecting tuning parameters
• Need for robust and adaptable parameter computation methods beyond TV-LASSO

Innovation

• ALMA computes tuning parameters for LASSO
• ALMA adapts to generalized LASSO problems
• ALMA handles non-Cartesian sampling trajectories