Accelerating Convergence of Proximal Methods for Compressed Sensing using Polynomials with Application to MRI

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   This work aims to accelerate the convergence of proximal gradient algorithms (like FISTA) by designing a preconditioner using polynomials that targets the eigenvalue spectrum of the forward linear model to enable faster convergence. The preconditioner does not assume any explicit structure and can be applied to various imaging applications. The efficacy of the preconditioner is validated on four varied imaging applications, where it seen to achieve 2-4x faster convergence.

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   We propose a k-space preconditioning formulation for accelerating the convergence of iterative Magnetic Resonance Imaging (MRI) reconstructions from non-uniformly sampled k-space data. Existing methods either use sampling density compensations which sacrifice reconstruction accuracy, or circulant preconditioners which increase per-iteration computation. Our approach overcomes both shortcomings. Concretely, we show that viewing the reconstruction problem in the dual formulation allows us to precondition in k-space using density-compensation-like operations. Using the primal-dual hybrid gradient method, the proposed preconditioning method does not have inner loops and are competitive in accele…

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   Résumé scientifique

   Model-based methods are widely used for reconstruction in compressed sensing (CS) magnetic resonance imaging (MRI), using regularizers to describe the images of interest. The reconstruction process is equivalent to solving a composite optimization problem. Accelerated proximal methods (APMs) are very popular approaches for such problems. This paper proposes a complex quasi-Newton proximal method (CQNPM) for the wavelet and total variation based CS MRI reconstruction. Compared with APMs, CQNPM requires fewer iterations to converge but needs to compute a more challenging proximal mapping called weighted proximal mapping (WPM). To make CQNPM more practical, we propose efficient methods to solve…

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   Résumé scientifique

   Model-based methods play a key role in the reconstruction of compressed sensing (CS) MRI. Finding an effective prior to describe the statistical distribution of the image family of interest is crucial for model-based methods. Plug-and-play (PnP) is a general framework that uses denoising algorithms as the prior or regularizer. Recent work showed that PnP methods with denoisers based on pretrained convolutional neural networks outperform other classical regularizers in CS MRI reconstruction. However, the numerical solvers for PnP can be slow for CS MRI reconstruction. This paper proposes a preconditioned PnP P2nP method to accelerate the convergence speed. Moreover, we provide proofs of the f…

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   Résumé scientifique

   Imaging inverse problems are commonly addressed by minimizing measurement consistency and signal prior terms. While huge attention has been paid to developing highperformance priors, even the most advanced signal prior may lose its effectiveness when paired with an ill-conditioned sensing matrix that hinders convergence and degrades reconstruction quality. In optimization theory, preconditioners allow improving the algorithm’s convergence by transforming the gradient update. Traditional linear preconditioning techniques enhance convergence, but their performance remains limited due to their dependence on the structure of the sensing matrix. Learning-based linear preconditioners have been pro…

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   Conventional non‐Cartesian compressed sensing requires multiple nonuniform Fourier transforms every iteration, which is computationally expensive. Accordingly, time‐consuming reconstructions have slowed the adoption of undersampled 3D non‐Cartesian acquisitions into clinical protocols. In this work we investigate several approaches to minimize reconstruction times without sacrificing accuracy.

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   HighlightsA generalized form for the analysis‐based sparsity reconstruction, which contains the total variation (TV) regularization.A novel method to accelerate the optimization for different structured sparsity patterns using the proposed preconditioning.An entirely new algorithm is proposed to solve the TV and joint TV based reconstruction, by the proposed preconditioner and incomplete LU decomposition.Extensive comparisons with 14 state‐of‐the‐art algorithms. Graphical abstract Figure. No caption available. ABSTRACT In this paper, we propose a novel algorithm for analysis‐based sparsity reconstruction. It can solve the generalized problem by structured sparsity regularization with an orth…

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   Over the past quarter-century, model-based Fourier image reconstruction methods have typically modeled the continuous image as a linear combination of shifted voxel basis functions. Despite its popularity, this model suffers from long-standing practical issues (e.g., expensive computations and a propensity for artifacts). While substantial effort has been spent to alleviate these well-known problems, we observe in this work that this model also suffers from more fundamental issues that do not appear to be previously known. This includes an implicit periodicity assumption (which can limit modeling accuracy) and a systematic tendency to produce unrealistic cross-like structures in k-space. Imp…

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