A Conditional Normalizing Flow for Accelerated Multi-Coil MR Imaging.

Accelerated magnetic resonance (MR) imaging attempts to reduce acquisition time by collecting data below the Nyquist rate. As an ill-posed inverse problem, many plausible solutions exist, yet the majority of deep learning approaches generate only a single solution. We instead focus on sampling from the posterior distribution, which provides more comprehensive information for downstream inference tasks. To do this, we design a novel conditional normalizing flow (CNF) that infers the signal component in the measurement operator's nullspace, which is later combined with measured data to form complete images. Using fastMRI brain and knee data, we demonstrate fast inference and accuracy that surpasses recent posterior sampling techniques for MRI. Code is available at https://github.com/jwen307/mri_cnf.

EXPECTATION CONSISTENT PLUG-AND-PLAY FOR MRI.

For image recovery problems, plug-and-play (PnP) methods have been developed that replace the proximal step in an optimization algorithm with a call to an application-specific denoiser, often implemented using a deep neural network. Although such methods have been successful, they can be improved. For example, the denoiser is often trained using white Gaussian noise, while PnP's denoiser input error is often far from white and Gaussian, with statistics that are difficult to predict from iteration to iteration. PnP methods based on approximate message passing (AMP) are an exception, but only when the forward operator behaves like a large random matrix. In this work, we design a PnP method using the expectation consistent (EC) approximation algorithm, a generalization of AMP, that offers predictable error statistics at each iteration, from which a deep-net denoiser can be effectively trained.

MRI RECOVERY WITH A SELF-CALIBRATED DENOISER.

Plug-and-play (PnP) methods that employ application-specific denoisers have been proposed to solve inverse problems, including MRI reconstruction. However, training application-specific denoisers is not feasible for many applications due to the lack of training data. In this work, we propose a PnP-inspired recovery method that does not require data beyond the single, incomplete set of measurements. The proposed self-supervised method, called recovery with a self-calibrated denoiser (ReSiDe), trains the denoiser from the patches of the image being recovered. The denoiser training and a call to the denoising subroutine are performed in each iteration of a PnP algorithm, leading to a progressive refinement of the reconstructed image. For validation, we compare ReSiDe with a compressed sensing-based method and a PnP method with BM3D denoising using single-coil MRI brain data.

Denoising Generalized Expectation-Consistent Approximation for MR Image Recovery.

To solve inverse problems, plug-and-play (PnP) methods replace the proximal step in a convex optimization algorithm with a call to an application-specific denoiser, often implemented using a deep neural network (DNN). Although such methods yield accurate solutions, they can be improved. For example, denoisers are usually designed/trained to remove white Gaussian noise, but the denoiser input error in PnP algorithms is usually far from white or Gaussian. Approximate message passing (AMP) methods provide white and Gaussian denoiser input error, but only when the forward operator is sufficiently random. In this work, for Fourier-based forward operators, we propose a PnP algorithm based on generalized expectation-consistent (GEC) approximation-a close cousin of AMP-that offers predictable error statistics at each iteration, as well as a new DNN denoiser that leverages those statistics. We apply our approach to magnetic resonance (MR) image recovery and demonstrate its advantages over existing PnP and AMP methods.

Plug-and-Play Methods for Magnetic Resonance Imaging: Using Denoisers for Image Recovery.

Magnetic Resonance Imaging (MRI) is a non-invasive diagnostic tool that provides excellent soft-tissue contrast without the use of ionizing radiation. Compared to other clinical imaging modalities (e.g., CT or ultrasound), however, the data acquisition process for MRI is inherently slow, which motivates undersampling and thus drives the need for accurate, efficient reconstruction methods from undersampled datasets. In this article, we describe the use of "plug-and-play" (PnP) algorithms for MRI image recovery. We first describe the linearly approximated inverse problem encountered in MRI. Then we review several PnP methods, where the unifying commonality is to iteratively call a denoising subroutine as one step of a larger optimization-inspired algorithm. Next, we describe how the result of the PnP method can be interpreted as a solution to an equilibrium equation, allowing convergence analysis from the equilibrium perspective. Finally, we present illustrative examples of PnP methods applied to MRI image recovery.

FREE-BREATHING CARDIOVASCULAR MRI USING A PLUG-AND-PLAY METHOD WITH LEARNED DENOISER.

Cardiac magnetic resonance imaging (CMR) is a noninvasive imaging modality that provides a comprehensive evaluation of the cardiovascular system. The clinical utility of CMR is hampered by long acquisition times, however. In this work, we propose and validate a plug-and-play (PnP) method for CMR reconstruction from undersampled multicoil data. To fully exploit the rich image structure inherent in CMR, we pair the PnP framework with a deep learning (DL)-based denoiser that is trained using spatiotemporal patches from high-quality, breath-held cardiac cine images. The resulting "PnP-DL" method iterates over data consistency and denoising subroutines. We compare the reconstruction performance of PnP-DL to that of compressed sensing (CS) using eight breath-held and ten real-time (RT) free-breathing cardiac cine datasets. We find that, for breath-held datasets, PnP-DL offers more than one dB advantage over commonly used CS methods. For RT free-breathing datasets, where ground truth is not available, PnP-DL receives higher scores in qualitative evaluation. The results highlight the potential of PnP-DL to accelerate RT CMR.

Regularization by Denoising: Clarifications and New Interpretations.

Regularization by Denoising (RED), as recently proposed by Romano, Elad, and Milanfar, is powerful image-recovery framework that aims to minimize an explicit regularization objective constructed from a plug-in image-denoising function. Experimental evidence suggests that the RED algorithms are state-of-the-art. We claim, however, that explicit regularization does not explain the RED algorithms. In particular, we show that many of the expressions in the paper by Romano et al. hold only when the denoiser has a symmetric Jacobian, and we demonstrate that such symmetry does not occur with practical denoisers such as non-local means, BM3D, TNRD, and DnCNN. To explain the RED algorithms, we propose a new framework called Score-Matching by Denoising (SMD), which aims to match a "score" (i.e., the gradient of a log-prior). We then show tight connections between SMD, kernel density estimation, and constrained minimum mean-squared error denoising. Furthermore, we interpret the RED algorithms from Romano et al. and propose new algorithms with acceleration and convergence guarantees. Finally, we show that the RED algorithms seek a consensus equilibrium solution, which facilitates a comparison to plug-and-play ADMM.

Sparsity adaptive reconstruction for highly accelerated cardiac MRI.

PURPOSE: To enable parameter-free, accelerated cardiovascular magnetic resonance (CMR). METHODS: Regularized reconstruction methods, such as compressed sensing (CS), can significantly accelerate MRI data acquisition but require tuning of regularization weights. In this work, a technique, called Sparsity adaptive Composite Recovery (SCoRe) that exploits sparsity in multiple, disparate sparsifying transforms is presented. A data-driven adjustment of the relative contributions of different transforms yields a parameter-free CS recovery process. SCoRe is validated in a dynamic digital phantom as well as in retrospectively and prospectively undersampled cine CMR data. RESULTS: The results from simulation and 6 retrospectively undersampled datasets indicate that SCoRe with auto-tuned regularization weights yields lower root-mean-square error (RMSE) and higher structural similarity index (SSIM) compared to state-of-the-art CS methods. In 45 prospectively undersampled datasets acquired from 15 volunteers, the image quality was scored by 2 expert reviewers, with SCoRe receiving a higher average score (p  <  0.01) compared to other CS methods. CONCLUSIONS: SCoRe enables accelerated cine CMR from highly undersampled data. In contrast to other acceleration techniques, SCoRe adapts regularization weights based on noise power and level of sparsity in each transform, yielding superior performance without admitting any free parameters.

Phase diagram of matrix compressed sensing.

In the problem of matrix compressed sensing, we aim to recover a low-rank matrix from a few noisy linear measurements. In this contribution, we analyze the asymptotic performance of a Bayes-optimal inference procedure for a model where the matrix to be recovered is a product of random matrices. The results that we obtain using the replica method describe the state evolution of the Parametric Bilinear Generalized Approximate Message Passing (P-BiG-AMP) algorithm, recently introduced in J. T. Parker and P. Schniter [IEEE J. Select. Top. Signal Process. 10, 795 (2016)1932-455310.1109/JSTSP.2016.2539123]. We show the existence of two different types of phase transition and their implications for the solvability of the problem, and we compare the results of our theoretical analysis to the numerical performance reached by P-BiG-AMP. Remarkably, the asymptotic replica equations for matrix compressed sensing are the same as those for a related but formally different problem of matrix factorization.