Parallel reconstruction using null operations

Knowledge-driven deep learning for fast MR imaging: Undersampled MR image reconstruction from supervised to un-supervised learning

Deep learning (DL) has emerged as a leading approach in accelerating MRI. It employs deep neural networks to extract knowledge from available datasets and then applies the trained networks to reconstruct accurate images from limited measurements. Unlike natural image restoration problems, MRI involves physics-based imaging processes, unique data properties, and diverse imaging tasks. This domain knowledge needs to be integrated with data-driven approaches. Our review will introduce the significant challenges faced by such knowledge-driven DL approaches in the context of fast MRI along with several notable solutions, which include learning neural networks and addressing different imaging application scenarios. The traits and trends of these techniques have also been given which have shifted from supervised learning to semi-supervised learning, and finally, to unsupervised learning methods. In addition, MR vendors' choices of DL reconstruction have been provided along with some discussions on open questions and future directions, which are critical for the reliable imaging systems.

Annihilation-Net: Learned annihilation relation for dynamic MR imaging

BACKGROUND

Deep learning methods driven by the low-rank regularization have achieved attractive performance in dynamic magnetic resonance (MR) imaging. The effectiveness of existing methods lies mainly in their ability to capture interframe relationships using network modules, which are lack interpretability.

PURPOSE

This study aims to design an interpretable methodology for modeling interframe relationships using convolutiona networks, namely Annihilation-Net and use it for accelerating dynamic MRI.

METHODS

Based on the equivalence between Hankel matrix product and convolution, we utilize convolutional networks to learn the null space transform for characterizing low-rankness. We employ low-rankness to represent interframe correlations in dynamic MR imaging, while combining with sparse constraints in the compressed sensing framework. The corresponding optimization problem is solved in an iterative form with the semi-quadratic splitting method (HQS). The iterative steps are unrolled into a network, dubbed Annihilation-Net. All the regularization parameters and null space transforms are set as learnable in the Annihilation-Net.

RESULTS

Experiments on the cardiac cine dataset show that the proposed model outperforms other competing methods both quantitatively and qualitatively. The training set and test set have 800 and 118 images, respectively.

CONCLUSIONS

The proposed Annihilation-Net improves the reconstruction quality of accelerated dynamic MRI with better interpretability.

New Theory and Faster Computations for Subspace-Based Sensitivity Map Estimation in Multichannel MRI

Sensitivity map estimation is important in many multichannel MRI applications. Subspace-based sensitivity map estimation methods like ESPIRiT are popular and perform well, though can be computationally expensive and their theoretical principles can be nontrivial to understand. In the first part of this work, we present a novel theoretical derivation of subspace-based sensitivity map estimation based on a linear-predictability/structured low-rank modeling perspective. This results in an estimation approach that is equivalent to ESPIRiT, but with distinct theory that may be more intuitive for some readers. In the second part of this work, we propose and evaluate a set of computational acceleration approaches (collectively known as PISCO) that can enable substantial improvements in computation time (up to  ∼ 100× in the examples we show) and memory for subspace-based sensitivity map estimation.

Accelerated submillimeter wave-encoded magnetic resonance imaging via deep untrained neural network

BACKGROUND

Wave gradient encoding can adequately utilize coil sensitivity profiles to facilitate higher accelerations in parallel magnetic resonance imaging (pMRI). However, there are limitations in mainstream pMRI and a few deep learning (DL) methods for recovering missing data under wave encoding framework: the former is prone to introduce errors from the auto-calibration signals (ACS) signal acquisition and is time-consuming, while the latter requires a large amount of training data.

PURPOSE

To tackle the above issues, an untrained neural network (UNN) model incorporating wave-encoded physical properties and deep generative model, named WDGM, was proposed with additional ACS- and training data-free.

METHODS

Generally, the proposed method can provide powerful missing data interpolation capability using the wave physical encoding framework and designed UNN to characterize the MR image (k-space data) priors. Specifically, the MRI reconstruction combining physical wave encoding and elaborate UNN is modeled as a generalized minimization problem. The designation of UNN is driven by the coil sensitivity maps (CSM) smoothness and k-space linear predictability. And then, the iterative paradigm to recover the full k-space signal is determined by the projected gradient descent, and the complex computation is unrolled to the network with optimized parameters by the optimizer. Simulated wave encoding and in vivo experiments are exploited to demonstrate the feasibility of the proposed method. The best quantitative metrics RMSE/SSIM/PSNR of 0.0413, 0.9514, and 37.4862 gave competitive results in all experiments with at least six-fold acceleration, respectively.

RESULTS

In vivo experiments of human brains and knees showed that the proposed method can achieve comparable reconstruction quality and even has superiority relative to the comparison, especially at a high resolution of 0.67 mm and fewer ACS. In addition, the proposed method has a higher computational efficiency achieving a computation time of 9.6 s/per slice.

CONCLUSIONS

The model proposed in this work addresses two limitations of MRI reconstruction in the wave encoding framework. The first is to eliminate the need for ACS signal acquisition to perform the time-consuming calibration process and to avoid errors such as motion during the acquisition procedure. Furthermore, the proposed method has clinical application friendly without the need to prepare large training datasets, which is difficult in the clinical. All results of the proposed method demonstrate more confidence in both quantitative and qualitative metrics. In addition, the proposed method can achieve higher computational efficiency.

One-Shot Generative Prior in Hankel-k-Space for Parallel Imaging Reconstruction

Magnetic resonance imaging serves as an essential tool for clinical diagnosis. However, it suffers from a long acquisition time. The utilization of deep learning, especially the deep generative models, offers aggressive acceleration and better reconstruction in magnetic resonance imaging. Nevertheless, learning the data distribution as prior knowledge and reconstructing the image from limited data remains challenging. In this work, we propose a novel Hankel-k-space generative model (HKGM), which can generate samples from a training set of as little as one k-space. At the prior learning stage, we first construct a large Hankel matrix from k-space data, then extract multiple structured k-space patches from the Hankel matrix to capture the internal distribution among different patches. Extracting patches from a Hankel matrix enables the generative model to be learned from the redundant and low-rank data space. At the iterative reconstruction stage, the desired solution obeys the learned prior knowledge. The intermediate reconstruction solution is updated by taking it as the input of the generative model. The updated result is then alternatively operated by imposing low-rank penalty on its Hankel matrix and data consistency constraint on the measurement data. Experimental results confirmed that the internal statistics of patches within single k-space data carry enough information for learning a powerful generative model and providing state-of-the-art reconstruction.

WKGM: weighted k-space generative model for parallel imaging reconstruction

Deep learning based parallel imaging (PI) has made great progress in recent years to accelerate MRI. Nevertheless, it still has some limitations: for example, the robustness and flexibility of existing methods are greatly deficient. In this work, we propose a method to explore the k-space domain learning via robust generative modeling for flexible calibrationless PI reconstruction, coined the weighted k-space generative model (WKGM). Specifically, WKGM is a generalized k-space domain model, where the k-space weighting technology and high-dimensional space augmentation design are efficiently incorporated for score-based generative model training, resulting in good and robust reconstructions. In addition, WKGM is flexible and thus can be synergistically combined with various traditional k-space PI models, which can make full use of the correlation between multi-coil data and realize calibrationless PI. Even though our model was trained on only 500 images, experimental results with varying sampling patterns and acceleration factors demonstrate that WKGM can attain state-of-the-art reconstruction results with the well learned k-space generative prior.

Multi-band multi-shot diffusion MRI reconstruction with joint usage of structured low-rank constraints and explicit phase mapping

PURPOSE

To develop a joint reconstruction method for multi-band multi-shot diffusion MRI.

THEORY AND METHODS

Multi-band multi-shot EPI acquisition is an effective approach for high-resolution diffusion MRI, but requires specific algorithms to correct the inter-shot phase variations. The phase correction can be done by first estimating the explicit phase map and then feeding it into the k-space signal formulation model. Alternatively, the phase information can be used indirectly as structured low-rank constraints in k-space. The 2 methods differ in reconstruction accuracy and efficiency. We aim to combine the 2 different approaches for improved image quality and reconstruction efficiency simultaneously, termed "joint usage of structured low-rank constraints and explicit phase mapping" (JULEP). The proposed JULEP reconstruction is tested on both single-band and multi-band, multi-shot diffusion data, with different resolutions and b values. The results of JULEP are compared with conventional methods with explicit phase mapping (i.e., multiplexed sensitivity-encoding [MUSE]) and structured low-rank constraints (i.e., MUSSELS), and another joint reconstruction method (i.e., network estimated artifacts for tempered reconstruction [NEATR]).

RESULTS

JULEP improves the quality of the navigator and subsequently facilitates the reconstruction of final diffusion images. Compared with all 3 other methods (MUSE, MUSSELS, and NEATR), JULEP mitigates residual structural bias and improves temporal SNRs in the final diffusion image, particularly at high multi-band factors. Compared with MUSSELS, JULEP also improves computational efficiency.

CONCLUSION

The proposed JULEP method significantly improves the image quality and reconstruction efficiency of multi-band multi-shot diffusion MRI, which can promote a broader application of high-resolution diffusion MRI.

One-Dimensional Deep Low-Rank and Sparse Network for Accelerated MRI

Deep learning has shown astonishing performance in accelerated magnetic resonance imaging (MRI). Most state-of-the-art deep learning reconstructions adopt the powerful convolutional neural network and perform 2D convolution since many magnetic resonance images or their corresponding k-space are in 2D. In this work, we present a new approach that explores the 1D convolution, making the deep network much easier to be trained and generalized. We further integrate the 1D convolution into the proposed deep network, named as One-dimensional Deep Low-rank and Sparse network (ODLS), which unrolls the iteration procedure of a low-rank and sparse reconstruction model. Extensive results on in vivo knee and brain datasets demonstrate that, the proposed ODLS is very suitable for the case of limited training subjects and provides improved reconstruction performance than state-of-the-art methods both visually and quantitatively. Additionally, ODLS also shows nice robustness to different undersampling scenarios and some mismatches between the training and test data. In summary, our work demonstrates that the 1D deep learning scheme is memory-efficient and robust in fast MRI.

A 2D-GRAPPA Algorithm with a Boomerang Kernel for 3D MRI Data Accelerated along Two Phase-Encoding Directions

For the reconstruction of 3D MRI data that are accelerated along the two phase-encoding directions, the 2D-generalized autocalibrating partially parallel acquisitions (GRAPPA) algorithm can be used to estimate the missing data in the k-space. We propose a new boomerang-shaped kernel based on theoretic and systemic analyses of the shape and dimensions of the kernel. The reconstruction efficiency of the 2D-GRAPPA algorithm with the proposed boomerang-shaped kernel (i.e., boomerang kernel (BK)-2D-GRAPPA) was compared with other 2D-GRAPPA algorithms that utilize different types of kernels (i.e., EX-2D-GRAPPA and SK-2D-GRAPPA) based on computer simulation, phantom and in vivo experiments. The proposed method was validated for different sets of ACS lines with acceleration factors from four to eight and various sizes of the kernels. A quantitative analysis was also performed by comparing the normalized root mean squared error (nRMSE) in the images and the undersampled edges. Computer simulation, in vivo and phantom experiments, and the quantitative analysis, showed that the proposed method could reduce aliasing artifacts without reducing the SNRs of the reconstructed images.

Generalized self-calibrating simultaneous multi-slice MR image reconstruction from 3D Fourier encoding perspective

This work introduces a novel, k-space based one-step solution for simultaneous multi-slice MR image reconstruction from 3D Fourier encoding perspective. With undersampled SMS imaging, image reconstruction suffers from both inter-slice leakages and in-plane aliasing artifacts. Aliasing separation becomes further challenging in the presence of discrepancies between calibration and imaging. To address them, in this work a measured SMS 3D k-space with additional calibrating signals is decomposed into SMS imaging and self-calibrating data sets. Extended controlled aliasing is performed by upsampling the measured data in the k_z-direction. A slice-specific null space operator is then learned using extended self-calibration exploiting target slices and additional in-plane-shifted images. Inter-slice leakages and in-plane aliasing artifacts are jointly resolved in a single step by solving a constrained optimization problem in which null space reconstruction consistency is balanced with a Hankel-structured low rank prior while data fidelity in 3D Fourier space is enforced. Retrospective and prospective studies are performed to validate the effectiveness of the proposed method in various regions including knee and L-spine.

High-Quality MR Fingerprinting Reconstruction Using Structured Low-Rank Matrix Completion and Subspace Projection

Due to the capability of fast multiparametric quantitative imaging, magnetic resonance fingerprinting (MRF) is becoming a promising quantitative magnetic resonance imaging approach. However, the artifacts caused by the highly undersampled data acquisition lead to inaccurate estimation of the tissue parameter maps. Based on the assumption that the 3-D MRF data can be modeled as a piecewise smooth signal, with the discontinuities localized to the zero sets of a bandlimited function, we exploit the low-rank property of the structured Toeplitz matrix constructed from the Fourier measurements. In addition, we adopt the subspace projection scheme to improve the accuracy of parameter estimation. In order to efficiently solve the regularized problem, we propose an iterative two-stage algorithm, which alternately updates the k -space data and projects the space-time matrix into the dictionary space. Numerical experiments demonstrate that the proposed algorithm shows significant improvement in MRF time-series images reconstruction and can provide more accurate parameter maps over the state-of-the-art algorithms.

A review and experimental evaluation of deep learning methods for MRI reconstruction

Following the success of deep learning in a wide range of applications, neural network-based machine-learning techniques have received significant interest for accelerating magnetic resonance imaging (MRI) acquisition and reconstruction strategies. A number of ideas inspired by deep learning techniques for computer vision and image processing have been successfully applied to nonlinear image reconstruction in the spirit of compressed sensing for accelerated MRI. Given the rapidly growing nature of the field, it is imperative to consolidate and summarize the large number of deep learning methods that have been reported in the literature, to obtain a better understanding of the field in general. This article provides an overview of the recent developments in neural-network based approaches that have been proposed specifically for improving parallel imaging. A general background and introduction to parallel MRI is also given from a classical view of k-space based reconstruction methods. Image domain based techniques that introduce improved regularizers are covered along with k-space based methods which focus on better interpolation strategies using neural networks. While the field is rapidly evolving with plenty of papers published each year, in this review, we attempt to cover broad categories of methods that have shown good performance on publicly available data sets. Limitations and open problems are also discussed and recent efforts for producing open data sets and benchmarks for the community are examined.

Automated Parameter Selection for Accelerated MRI Reconstruction via Low-Rank Modeling of Local k-Space Neighborhoods

PURPOSE

Image quality in accelerated MRI rests on careful selection of various reconstruction parameters. A common yet tedious and error-prone practice is to hand-tune each parameter to attain visually appealing reconstructions. Here, we propose a parameter tuning strategy to automate hybrid parallel imaging (PI) - compressed sensing (CS) reconstructions via low-rank modeling of local k-space neighborhoods (LORAKS) supplemented with sparsity regularization in wavelet and total variation (TV) domains.

METHODS

For low-rank regularization, we leverage a soft-thresholding operation based on singular values for matrix rank selection in LORAKS. For sparsity regularization, we employ Stein's unbiased risk estimate criterion to select the wavelet regularization parameter and local standard deviation of reconstructions to select the TV regularization parameter. Comprehensive demonstrations are presented on a numerical brain phantom and in vivo brain and knee acquisitions. Quantitative assessments are performed via PSNR, SSIM and NMSE metrics.

RESULTS

The proposed hybrid PI-CS method improves reconstruction quality compared to PI-only techniques, and it achieves on par image quality to reconstructions with brute-force optimization of reconstruction parameters. These results are prominent across several different datasets and the range of examined acceleration rates.

CONCLUSION

A data-driven parameter tuning strategy to automate hybrid PI-CS reconstructions is presented. The proposed method achieves reliable reconstructions of accelerated multi-coil MRI datasets without the need for exhaustive hand-tuning of reconstruction parameters.

Autoregression and Structured Low-Rank Modeling of Sinogram Neighborhoods

Sinograms are commonly used to represent the raw data from tomographic imaging experiments. Although it is already well-known that sinograms posess some amount of redundancy, in this work, we present novel theory suggesting that sinograms will often possess substantial additional redundancies that have not been explicitly exploited by previous methods. Specifically, we derive that sinograms will often satisfy multiple simple data-dependent autoregression relationships. This kind of autoregressive structure enables missing/degraded sinogram samples to be linearly predicted using a simple shift-invariant linear combination of neighboring samples. Our theory also further implies that if sinogram samples are assembled into a structured Hankel/Toeplitz matrix, then the matrix will be expected to have low-rank characteristics. As a result, sinogram restoration problems can be formulated as structured low-rank matrix recovery problems. Illustrations of this approach are provided using several different (real and simulated) X-ray imaging datasets, including comparisons against a state-of-the-art deep learning approach. Results suggest that structured low-rank matrix methods for sinogram recovery can have comparable performance to state-of-the-art approaches. Although our evaluation focuses on competitive comparisons against other approaches, we believe that autoregressive constraints are actually complementary to existing approaches with strong potential synergies.

High-dimensional fast convolutional framework (HICU) for calibrationless MRI

PURPOSE

To present a computational procedure for accelerated, calibrationless magnetic resonance image (Cl-MRI) reconstruction that is fast, memory efficient, and scales to high-dimensional imaging.

THEORY AND METHODS

Cl-MRI methods can enable high acceleration rates and flexible sampling patterns, but their clinical application is limited by computational complexity and large memory footprint. The proposed computational procedure, HIgh-dimensional fast convolutional framework (HICU), provides fast, memory-efficient recovery of unsampled k-space points. For demonstration, HICU is applied to 6 2D T2-weighted brain, 7 2D cardiac cine, 5 3D knee, and 1 multi-shot diffusion weighted imaging (MSDWI) datasets.

RESULTS

The 2D imaging results show that HICU can offer 1-2 orders of magnitude computation speedup compared to other Cl-MRI methods without sacrificing imaging quality. The 2D cine and 3D imaging results show that the computational acceleration techniques included in HICU yield computing time on par with SENSE-based compressed sensing methods with up to 3 dB improvement in signal-to-error ratio and better perceptual quality. The MSDWI results demonstrate the feasibility of HICU for a challenging multi-shot echo-planar imaging application.

CONCLUSIONS

The presented method, HICU, offers efficient computation and scalability as well as extendibility to a wide variety of MRI applications.

Robust autocalibrated structured low-rank EPI ghost correction

PURPOSE

We propose and evaluate a new structured low-rank method for echo-planar imaging (EPI) ghost correction called Robust Autocalibrated LORAKS (RAC-LORAKS). The method can be used to suppress EPI ghosts arising from the differences between different readout gradient polarities and/or the differences between different shots. It does not require conventional EPI navigator signals, and is robust to imperfect autocalibration data.

METHODS

Autocalibrated LORAKS is a previous structured low-rank method for EPI ghost correction that uses GRAPPA-type autocalibration data to enable high-quality ghost correction. This method works well when the autocalibration data are pristine, but performance degrades substantially when the autocalibration information is imperfect. RAC-LORAKS generalizes Autocalibrated LORAKS in two ways. First, it does not completely trust the information from autocalibration data, and instead considers the autocalibration and EPI data simultaneously when estimating low-rank matrix structure. Second, it uses complementary information from the autocalibration data to improve EPI reconstruction in a multi-contrast joint reconstruction framework. RAC-LORAKS is evaluated using simulations and in vivo data, including comparisons to state-of-the-art methods.

RESULTS

RAC-LORAKS is demonstrated to have good ghost elimination performance compared to state-of-the-art methods in several complicated EPI acquisition scenarios (including gradient-echo brain imaging, diffusion-encoded brain imaging, and cardiac imaging).

CONCLUSIONS

RAC-LORAKS provides effective suppression of EPI ghosts and is robust to imperfect autocalibration data.

CONVOLUTIONAL FRAMEWORK FOR ACCELERATED MAGNETIC RESONANCE IMAGING

Magnetic Resonance Imaging (MRI) is a noninvasive imaging technique that provides exquisite soft-tissue contrast without using ionizing radiation. The clinical application of MRI may be limited by long data acquisition times; therefore, MR image reconstruction from highly undersampled k-space data has been an active area of research. Many works exploit rank deficiency in a Hankel data matrix to recover unobserved k-space samples; the resulting problem is non-convex, so the choice of numerical algorithm can significantly affect performance, computation, and memory. We present a simple, scalable approach called Convolutional Framework (CF). We demonstrate the feasibility and versatility of CF using measured data from 2D, 3D, and dynamic applications.

Structured Low-Rank Algorithms: Theory, Magnetic Resonance Applications, and Links to Machine Learning

In this survey, we provide a detailed review of recent advances in the recovery of continuous domain multidimensional signals from their few non-uniform (multichannel) measurements using structured low-rank matrix completion formulation. This framework is centered on the fundamental duality between the compactness (e.g., sparsity) of the continuous signal and the rank of a structured matrix, whose entries are functions of the signal. This property enables the reformulation of the signal recovery as a low-rank structured matrix completion, which comes with performance guarantees. We will also review fast algorithms that are comparable in complexity to current compressed sensing methods, which enables the application of the framework to large-scale magnetic resonance (MR) recovery problems. The remarkable flexibility of the formulation can be used to exploit signal properties that are difficult to capture by current sparse and low-rank optimization strategies. We demonstrate the utility of the framework in a wide range of MR imaging (MRI) applications, including highly accelerated imaging, calibration-free acquisition, MR artifact correction, and ungated dynamic MRI.

Navigator-Free EPI Ghost Correction With Structured Low-Rank Matrix Models: New Theory and Methods

Structured low-rank matrix models have previously been introduced to enable calibrationless MR image reconstruction from sub-Nyquist data, and such ideas have recently been extended to enable navigator-free echo-planar imaging (EPI) ghost correction. This paper presents a novel theoretical analysis which shows that, because of uniform subsampling, the structured low-rank matrix optimization problems for EPI data will always have either undesirable or non-unique solutions in the absence of additional constraints. This theory leads us to recommend and investigate problem formulations for navigator-free EPI that incorporate side information from either image-domain or k-space domain parallel imaging methods. The importance of using nonconvex low-rank matrix regularization is also identified. We demonstrate using phantom and in vivo data that the proposed methods are able to eliminate ghost artifacts for several navigator-free EPI acquisition schemes, obtaining better performance in comparison with the state-of-the-art methods across a range of different scenarios. Results are shown for both single-channel acquisition and highly accelerated multi-channel acquisition.

Wave-LORAKS: Combining wave encoding with structured low-rank matrix modeling for more highly accelerated 3D imaging

PURPOSE

Wave-CAIPI is a novel acquisition approach that enables highly accelerated 3D imaging. This paper investigates the combination of Wave-CAIPI with LORAKS-based reconstruction (Wave-LORAKS) to enable even further acceleration.

METHODS

LORAKS is a constrained image reconstruction framework that can impose spatial support, smooth phase, sparsity, and/or parallel imaging constraints. LORAKS requires minimal prior information, and instead uses the low-rank subspace structure of the raw data to automatically learn which constraints to impose and how to impose them. Previous LORAKS implementations addressed 2D image reconstruction problems. In this work, several recent advances in structured low-rank matrix recovery were combined to enable large-scale 3D Wave-LORAKS reconstruction with improved quality and computational efficiency. Wave-LORAKS was investigated by retrospective subsampling of two fully sampled Wave-encoded 3D MPRAGE datasets, and comparisons were made against existing Wave reconstruction approaches.

RESULTS

Results show that Wave-LORAKS can yield higher reconstruction quality with 16×-accelerated data than is obtained by traditional Wave-CAIPI with 9×-accerated data.

CONCLUSIONS

There are strong synergies between Wave encoding and LORAKS, which enables Wave-LORAKS to achieve higher acceleration and more flexible sampling compared to Wave-CAIPI.

ROBUST AUTOCALIBRATED LORAKS FOR EPI GHOST CORRECTION

Nyquist ghosts are a longstanding problem in a variety of fast MRI experiments that use echo-planar imaging (EPI). Recently, several structured low-rank matrix modeling approaches have been proposed that achieve state-of-the-art ghost-elimination, although the performance of these approaches is still inadequate in several important scenarios. We present a new structured low-rank matrix recovery ghost correction method that we call Robust Autocalibrated LORAKS (RAC-LORAKS). RAC-LORAKS incorporates constraints from autocalibration data to avoid ill-posedness, but allows adaptation of these constraints to gain robustness against possible autocalibration imperfections. RAC-LORAKS is tested in two challenging scenarios: highly-undersampled multi-channel EPI of the brain, and cardiac EPI with a double-oblique slice orientation. Results show that RAC-LORAKS can provide substantial improvements over existing ghost correction methods, and potentially enables new imaging applications that were previously confounded by ghost artifacts.

A Fast Algorithm for Convolutional Structured Low-rank Matrix Recovery

Fourier domain structured low-rank matrix priors are emerging as powerful alternatives to traditional image recovery methods such as total variation (TV) and wavelet regularization. These priors specify that a convolutional structured matrix, i.e., Toeplitz, Hankel, or their multi-level generalizations, built from Fourier data of the image should be low-rank. The main challenge in applying these schemes to large-scale problems is the computational complexity and memory demand resulting from a lifting the image data to a large scale matrix. We introduce a fast and memory efficient approach called the Generic Iterative Reweighted Annihilation Filter (GIRAF) algorithm that exploits the convolutional structure of the lifted matrix to work in the original un-lifted domain, thus considerably reducing the complexity. Our experiments on the recovery of images from undersampled Fourier measurements show that the resulting algorithm is considerably faster than previously proposed algorithms, and can accommodate much larger problem sizes than previously studied.

Multi-shot sensitivity-encoded diffusion data recovery using structured low-rank matrix completion (MUSSELS)

PURPOSE

To introduce a novel method for the recovery of multi-shot diffusion weighted (MS-DW) images from echo-planar imaging (EPI) acquisitions.

METHODS

Current EPI-based MS-DW reconstruction methods rely on the explicit estimation of the motion-induced phase maps to recover artifact-free images. In the new formulation, the k-space data of the artifact-free DWI is recovered using a structured low-rank matrix completion scheme, which does not require explicit estimation of the phase maps. The structured matrix is obtained as the lifting of the multi-shot data. The smooth phase-modulations between shots manifest as null-space vectors of this matrix, which implies that the structured matrix is low-rank. The missing entries of the structured matrix are filled in using a nuclear-norm minimization algorithm subject to the data-consistency. The formulation enables the natural introduction of smoothness regularization, thus enabling implicit motion-compensated recovery of the MS-DW data.

RESULTS

Our experiments on in-vivo data show effective removal of artifacts arising from inter-shot motion using the proposed method. The method is shown to achieve better reconstruction than the conventional phase-based methods.

CONCLUSION

We demonstrate the utility of the proposed method to effectively recover artifact-free images from Cartesian fully/under-sampled and partial Fourier acquired data without the use of explicit phase estimates. Magn Reson Med 78:494-507, 2017. © 2016 International Society for Magnetic Resonance in Medicine.

Trimmed autocalibrating k-space estimation based on structured matrix completion

PURPOSE

Parallel imaging allows the reconstruction of undersampled data from multiple coils. This provides a means to reject and regenerate corrupt data (e.g. from motion artefact). The purpose of this work is to approach this problem using the SAKE parallel imaging method.

THEORY AND METHODS

Parallel imaging methods typically require calibration by fully sampling the center of k-space. This is a challenge in the presence of corrupted data, since the calibration data may be corrupted which leads to an errors-in-variables problem that cannot be solved by least squares or even iteratively reweighted least squares. The SAKE method, based on matrix completion and structured low rank approximation, was modified to detect and trim these errors from the data.

RESULTS

Simulated and actual corrupted datasets were reconstructed with SAKE, the proposed approach and a more standard reconstruction method (based on solving a linear equation) with a data rejection criterion. The proposed approach was found to reduce artefacts considerably in comparison to the other two methods.

CONCLUSION

SAKE with data trimming improves on previous methods for reconstructing images from grossly corrupted data.

SMS-HSL: Simultaneous multislice aliasing separation exploiting hankel subspace learning

PURPOSE

To develop a novel, simultaneous multislice reconstruction method that exploits Hankel subspace learning (SMS-HSL) for aliasing separation in the slice direction.

METHODS

An SMS signal model with the Hankel-structured matrix was proposed. To efficiently suppress interslice leakage artifacts from a signal subspace perspective, a null space was learned from the reference data combined over all slices other than a slice of interest using singular value decomposition. Given the fact that the Hankel-structured matrix is rank-deficient while the magnitude between the reference and its estimate is similar in k-space, the SMS-HSL was reformulated as a constrained optimization problem with both low-rank and magnitude priors. SMS signals were projected onto a slice-specific subspace while undesired signals were eliminated using the null space operator. The simulations and experiments were performed with increasing multiband factors up to 6 using the SMS-HSL and the split slice-GRAPPA.

RESULTS

Compared with the split slice-GRAPPA, the SMS-HSL shows superior performance in suppressing aliasing artifacts and noises at high multiband factors even with: insufficient reference signals, a small number of coils, and a short distance between aliasing voxels.

CONCLUSION

We successfully demonstrated the effectiveness of the SMS-HSL over the split slice-GRAPPA for aliasing separation in the slice direction. Magn Reson Med 78:1392-1404, 2017. © 2016 International Society for Magnetic Resonance in Medicine.

LORAKS makes better SENSE: Phase-constrained partial fourier SENSE reconstruction without phase calibration

PURPOSE

Parallel imaging and partial Fourier acquisition are two classical approaches for accelerated MRI. Methods that combine these approaches often rely on prior knowledge of the image phase, but the need to obtain this prior information can place practical restrictions on the data acquisition strategy. In this work, we propose and evaluate SENSE-LORAKS, which enables combined parallel imaging and partial Fourier reconstruction without requiring prior phase information.

THEORY AND METHODS

The proposed formulation is based on combining the classical SENSE model for parallel imaging data with the more recent LORAKS framework for MR image reconstruction using low-rank matrix modeling. Previous LORAKS-based methods have successfully enabled calibrationless partial Fourier parallel MRI reconstruction, but have been most successful with nonuniform sampling strategies that may be hard to implement for certain applications. By combining LORAKS with SENSE, we enable highly accelerated partial Fourier MRI reconstruction for a broader range of sampling trajectories, including widely used calibrationless uniformly undersampled trajectories.

RESULTS

Our empirical results with retrospectively undersampled datasets indicate that when SENSE-LORAKS reconstruction is combined with an appropriate k-space sampling trajectory, it can provide substantially better image quality at high-acceleration rates relative to existing state-of-the-art reconstruction approaches.

CONCLUSION

The SENSE-LORAKS framework provides promising new opportunities for highly accelerated MRI. Magn Reson Med 77:1021-1035, 2017. © 2016 International Society for Magnetic Resonance in Medicine.

A subspace-based coil combination method for phased-array magnetic resonance imaging

PURPOSE

Coil-by-coil reconstruction methods are followed by coil combination to obtain a single image representing a spin density map. Typical coil combination methods, such as square-root sum-of-squares and adaptive coil combining, yield images that exhibit spatially varying modulation of image intensity. Existing practice is to first combine coils according to a signal-to-noise criterion, then postprocess to correct intensity inhomogeneity. If inhomogeneity is severe, however, intensity correction methods can yield poor results. The purpose of this article is to present an alternative optimality criterion for coil combination; the resulting procedure yields reduced intensity inhomogeneity while preserving contrast.

THEORY AND METHODS

A minimum mean squared error criterion is adopted for combining coils via a subspace decomposition. Techniques are compared using both simulated and in vivo data.

RESULTS

Experimental results for simulated and in vivo data demonstrate lower bias, higher signal-to-noise ratio (about 7×) and contrast-to-noise ratio (about 2×), compared to existing coil combination techniques.

CONCLUSION

The proposed coil combination method is noniterative and does not require estimation of coil sensitivity maps or image mask; the method is particularly suited to cases where intensity inhomogeneity is too severe for existing approaches.

Efficient Compressed Sensing SENSE pMRI Reconstruction With Joint Sparsity Promotion

The theory and techniques of compressed sensing (CS) have shown their potential as a breakthrough in accelerating k-space data acquisition for parallel magnetic resonance imaging (pMRI). However, the performance of CS reconstruction models in pMRI has not been fully maximized, and CS recovery guarantees for pMRI are largely absent. To improve reconstruction accuracy from parsimonious amounts of k-space data while maintaining flexibility, a new CS SENSitivity Encoding (SENSE) pMRI reconstruction framework promoting joint sparsity (JS) across channels (JS CS SENSE) is proposed in this paper. The recovery guarantee derived for the proposed JS CS SENSE model is demonstrated to be better than that of the conventional CS SENSE model and similar to that of the coil-by-coil CS model. The flexibility of the new model is better than the coil-by-coil CS model and the same as that of CS SENSE. For fast image reconstruction and fair comparisons, all the introduced CS-based constrained optimization problems are solved with split Bregman, variable splitting, and combined-variable splitting techniques. For the JS CS SENSE model in particular, these techniques lead to an efficient algorithm. Numerical experiments show that the reconstruction accuracy is significantly improved by JS CS SENSE compared with the conventional CS SENSE. In addition, an accurate residual-JS regularized sensitivity estimation model is also proposed and extended to calibration-less (CaL) JS CS SENSE. Numerical results show that CaL JS CS SENSE outperforms other state-of-the-art CS-based calibration-less methods in particular for reconstructing non-piecewise constant images.

ESPIRiT--an eigenvalue approach to autocalibrating parallel MRI: where SENSE meets GRAPPA

PURPOSE

Parallel imaging allows the reconstruction of images from undersampled multicoil data. The two main approaches are: SENSE, which explicitly uses coil sensitivities, and GRAPPA, which makes use of learned correlations in k-space. The purpose of this work is to clarify their relationship and to develop and evaluate an improved algorithm.

THEORY AND METHODS

A theoretical analysis shows: (1) The correlations in k-space are encoded in the null space of a calibration matrix. (2) Both approaches restrict the solution to a subspace spanned by the sensitivities. (3) The sensitivities appear as the main eigenvector of a reconstruction operator computed from the null space. The basic assumptions and the quality of the sensitivity maps are evaluated in experimental examples. The appearance of additional eigenvectors motivates an extended SENSE reconstruction with multiple maps, which is compared to existing methods.

RESULTS

The existence of a null space and the high quality of the extracted sensitivities are confirmed. The extended reconstruction combines all advantages of SENSE with robustness to certain errors similar to GRAPPA.

CONCLUSION

In this article the gap between both approaches is finally bridged. A new autocalibration technique combines the benefits of both.

P-LORAKS: Low-rank modeling of local k-space neighborhoods with parallel imaging data

PURPOSE

To propose and evaluate P-LORAKS a new calibrationless parallel imaging reconstruction framework.

THEORY AND METHODS

LORAKS is a flexible and powerful framework that was recently proposed for constrained MRI reconstruction. LORAKS was based on the observation that certain matrices constructed from fully sampled k-space data should have low rank whenever the image has limited support or smooth phase, and made it possible to accurately reconstruct images from undersampled or noisy data using low-rank regularization. This paper introduces P-LORAKS, which extends LORAKS to the context of parallel imaging. This is achieved by combining the LORAKS matrices from different channels to yield a larger but more parsimonious low-rank matrix model of parallel imaging data. This new model can be used to regularize the reconstruction of undersampled parallel imaging data, and implicitly imposes phase, support, and parallel imaging constraints without needing to calibrate phase, support, or sensitivity profiles.

RESULTS

The capabilities of P-LORAKS are evaluated with retrospectively undersampled data and compared against existing parallel MRI reconstruction methods. Results show that P-LORAKS can improve parallel imaging reconstruction quality, and can enable the use of new k-space trajectories that are not compatible with existing reconstruction methods.

CONCLUSION

The P-LORAKS framewok provides a new and effective way to regularize parallel imaging reconstruction.

Edge Sharpness Assessment by Parametric Modeling: Application to Magnetic Resonance Imaging

In biomedical imaging, edge sharpness is an important yet often overlooked image quality metric. In this work, a semi-automatic method to quantify edge sharpness in the presence of significant noise is presented with application to magnetic resonance imaging (MRI). The method is based on parametric modeling of image edges. First, an edge map is automatically generated and one or more edges-of-interest (EOI) are manually selected using graphical user interface. Multiple exclusion criteria are then enforced to eliminate edge pixels that are potentially not suitable for sharpness assessment. Second, at each pixel of the EOI, an image intensity profile is read along a small line segment that runs locally normal to the EOI. Third, the profiles corresponding to all EOI pixels are individually fitted with a sigmoid function characterized by four parameters, including one that represents edge sharpness. Last, the distribution of the sharpness parameter is used to quantify edge sharpness. For validation, the method is applied to simulated data as well as MRI data from both phantom imaging and cine imaging experiments. This method allows for fast, quantitative evaluation of edge sharpness even in images with poor signal-to-noise ratio. Although the utility of this method is demonstrated for MRI, it can be adapted for other medical imaging applications.

STEP: Self-supporting tailored k-space estimation for parallel imaging reconstruction

PURPOSE

A new subspace-based iterative reconstruction method, termed Self-supporting Tailored k-space Estimation for Parallel imaging reconstruction (STEP), is presented and evaluated in comparison to the existing autocalibrating method SPIRiT and calibrationless method SAKE.

THEORY AND METHODS

In STEP, two tailored schemes including k-space partition and basis selection are proposed to promote spatially variant signal subspace and incorporated into a self-supporting structured low rank model to enforce properties of locality, sparsity, and rank deficiency, which can be formulated into a constrained optimization problem and solved by an iterative algorithm. Simulated and in vivo datasets were used to investigate the performance of STEP in terms of overall image quality and detail structure preservation.

RESULTS

The advantage of STEP on image quality is demonstrated by retrospectively undersampled multichannel Cartesian data with various patterns. Compared with SPIRiT and SAKE, STEP can provide more accurate reconstruction images with less residual aliasing artifacts and reduced noise amplification in simulation and in vivo experiments. In addition, STEP has the capability of combining compressed sensing with arbitrary sampling trajectory.

CONCLUSION

Using k-space partition and basis selection can further improve the performance of parallel imaging reconstruction with or without calibration signals.

GRAPPA reconstruction with spatially varying calibration of self-constraint

PURPOSE

To develop and evaluate a novel method of generalized auto-calibrating partially parallel acquisition (GRAPPA) with spatially varying calibration of self-constraint for parallel magnetic resonance imaging (MRI) reconstruction.

THEORY AND METHODS

The conventional GRAPPA independently estimates each missing sample with adjacent acquired data over multiple coils, thereby ignoring correlations inside missing data. Self-constrained methods can exploit correlations inside missing data by imposing linear dependence within full neighborhood kernels and showing improved reconstruction compared with GRAPPA. However, self-constraint kernels are currently calibrated by using auto-calibration signals. Thus, they may be suboptimal for reconstructing outer k-space because of spatially varying correlations. This study proposes a novel GRAPPA method with separate self-constraints (SSC-GRAPPA). In this method, the spatially varying self-constraint coefficients are adaptively calibrated by separately exploiting correlations inside missing and acquired data in the outer k-space. Both phantom and in vivo imaging experiments were conducted with retrospective undersampling to evaluate the performance of the proposed method.

RESULTS

Compared with GRAPPA and self-constrained GRAPPA, the proposed SSC-GRAPPA generates images with reduced artifacts and noise.

CONCLUSION

The proposed method provides an effective and efficient approach to improve parallel MRI reconstruction, and has potential to benefit routine clinical practice in the future.

Low-rank modeling of local k-space neighborhoods (LORAKS) for constrained MRI

Recent theoretical results on low-rank matrix reconstruction have inspired significant interest in low-rank modeling of MRI images. Existing approaches have focused on higher-dimensional scenarios with data available from multiple channels, timepoints, or image contrasts. The present work demonstrates that single-channel, single-contrast, single-timepoint k-space data can also be mapped to low-rank matrices when the image has limited spatial support or slowly varying phase. Based on this, we develop a novel and flexible framework for constrained image reconstruction that uses low-rank matrix modeling of local k-space neighborhoods (LORAKS). A new regularization penalty and corresponding algorithm for promoting low-rank are also introduced. The potential of LORAKS is demonstrated with simulated and experimental data for a range of denoising and sparse-sampling applications. LORAKS is also compared against state-of-the-art methods like homodyne reconstruction, l1-norm minimization, and total variation minimization, and is demonstrated to have distinct features and advantages. In addition, while calibration-based support and phase constraints are commonly used in existing methods, the LORAKS framework enables calibrationless use of these constraints.

Calibrationless parallel imaging reconstruction based on structured low-rank matrix completion

PURPOSE

A calibrationless parallel imaging reconstruction method, termed simultaneous autocalibrating and k-space estimation (SAKE), is presented. It is a data-driven, coil-by-coil reconstruction method that does not require a separate calibration step for estimating coil sensitivity information.

METHODS

In SAKE, an undersampled, multichannel dataset is structured into a single data matrix. The reconstruction is then formulated as a structured low-rank matrix completion problem. An iterative solution that implements a projection-onto-sets algorithm with singular value thresholding is described.

RESULTS

Reconstruction results are demonstrated for retrospectively and prospectively undersampled, multichannel Cartesian data having no calibration signals. Additionally, non-Cartesian data reconstruction is presented. Finally, improved image quality is demonstrated by combining SAKE with wavelet-based compressed sensing.

CONCLUSION

Because estimation of coil sensitivity information is not needed, the proposed method could potentially benefit MR applications where acquiring accurate calibration data is limiting or not possible at all.

Nonlinear coil sensitivity estimation for parallel magnetic resonance imaging using data-adaptive steering kernel regression method

The parallel magnetic resonance imaging (parallel imaging) technique reduces the MR data acquisition time by using multiple receiver coils. Coil sensitivity estimation is critical for the performance of parallel imaging reconstruction. Currently, most coil sensitivity estimation methods are based on linear interpolation techniques. Such methods may result in Gibbs-ringing artifact or resolution loss, when the resolution of coil sensitivity data is limited. To solve the problem, we proposed a nonlinear coil sensitivity estimation method based on steering kernel regression, which performs a local gradient guided interpolation to the coil sensitivity. The in vivo experimental results demonstrate that this method can effectively suppress Gibbs ringing artifact in coil sensitivity and reduces both noise and residual aliasing artifact level in SENSE reconstruction.

SC-GRAPPA: Self-constraint noniterative GRAPPA reconstruction with closed-form solution

PURPOSE

Parallel MRI (pMRI) reconstruction techniques are commonly used to reduce scan time by undersampling the k-space data. GRAPPA, a k-space based pMRI technique, is widely used clinically because of its robustness. In GRAPPA, the missing k-space data are estimated by solving a set of linear equations; however, this set of equations does not take advantage of the correlations within the missing k-space data. All k-space data in a neighborhood acquired from a phased-array coil are correlated. The correlation can be estimated easily as a self-constraint condition, and formulated as an extra set of linear equations to improve the performance of GRAPPA. The authors propose a modified k-space based pMRI technique called self-constraint GRAPPA (SC-GRAPPA) which combines the linear equations of GRAPPA with these extra equations to solve for the missing k-space data. Since SC-GRAPPA utilizes a least-squares solution of the linear equations, it has a closed-form solution that does not require an iterative solver.

METHODS

The SC-GRAPPA equation was derived by incorporating GRAPPA as a prior estimate. SC-GRAPPA was tested in a uniform phantom and two normal volunteers. MR real-time cardiac cine images with acceleration rate 5 and 6 were reconstructed using GRAPPA and SC-GRAPPA.

RESULTS

SC-GRAPPA showed a significantly lower artifact level, and a greater than 10% overall signal-to-noise ratio (SNR) gain over GRAPPA, with more significant SNR gain observed in low-SNR regions of the images.

CONCLUSIONS

SC-GRAPPA offers improved pMRI reconstruction, and is expected to benefit clinical imaging applications in the future.