Parallel Magnetic Resonance Imaging Research Articles

An $\mathcal O(1/{k})$ Convergence Rate for the Variable Stepsize Bregman Operator Splitting Algorithm

An earlier paper proved the convergence of a variable stepsize Bregman operator splitting algorithm (BOSVS) for minimizing $\phi(Bu)+H(u)$, where $H$ and $\phi$ are convex functions, and $\phi$ is possibly nonsmooth. The algorithm was shown to be relatively efficient when applied to partially parallel magnetic resonance image reconstruction problems. In this paper, the convergence rate of BOSVS is analyzed. When $H(u) = \|Au-f\|^2$, where $A$ is a matrix, it is shown that for an ergodic approximation ${\bf u}_k$ obtained by averaging $k$ BOSVS iterates, the error in the objective value $\phi (B{\bf u}_k) + H({\bf u}_k)$ is ${\cal O}(1/k)$. When the optimization problem has a unique solution $u^*$, we obtain the estimate $\|{\bf u}_k - u^*\| = {\cal O}(1/\sqrt{k})$. The theoretical analysis is compared to observed convergence rates for partially parallel magnetic resonance image reconstruction problems where $A$ is a large dense ill-conditioned matrix.

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.

High-resolution MRI of spinal cords by compressive sensing parallel imaging.

Spinal Cord Injury (SCI) is a common injury due to diseases or accidents. Noninvasive imaging methods play a critical role in diagnosing SCI and monitoring the response to therapy. Magnetic Resonance Imaging (MRI), by the virtue of providing excellent soft tissue contrast, is the most promising imaging method for this application. However, spinal cord has a very small cross-section, which needs high-resolution images for better visualization and diagnosis. Acquiring high-resolution spinal cord MRI images requires long acquisition time due to the physical and physiological constraints. Moreover, long acquisition time makes MRI more susceptible to motion artifacts. In this paper, we studied the application of compressive sensing (CS) and parallel imaging to achieve high-resolution imaging from sparsely sampled and reduced k-space data acquired by parallel receive arrays. In particular, the studies are limited to the effects of 2D Cartesian sampling with different subsampling schemes and reduction factors. The results show that compressive sensing parallel MRI has the potential to provide high-resolution images of the spinal cord in 1/3 of the acquisition time required by the conventional methods.

Partial fourier and parallel MR image reconstruction with integrated gradient nonlinearity correction.

To describe how integrated gradient nonlinearity (GNL) correction can be used within noniterative partial Fourier (homodyne) and parallel (SENSE and GRAPPA) MR image reconstruction strategies, and demonstrate that performing GNL correction during, rather than after, these routines mitigates the image blurring and resolution loss caused by postreconstruction image domain based GNL correction. Starting from partial Fourier and parallel magnetic resonance imaging signal models that explicitly account for GNL, noniterative image reconstruction strategies for each accelerated acquisition technique are derived under the same core mathematical assumptions as their standard counterparts. A series of phantom and in vivo experiments on retrospectively undersampled data were performed to investigate the spatial resolution benefit of integrated GNL correction over conventional postreconstruction correction. Phantom and in vivo results demonstrate that the integrated GNL correction reduces the image blurring introduced by the conventional GNL correction, while still correcting GNL-induced coarse-scale geometrical distortion. Images generated from undersampled data using the proposed integrated GNL strategies offer superior depiction of fine image detail, for example, phantom resolution inserts and anatomical tissue boundaries. Noniterative partial Fourier and parallel imaging reconstruction methods with integrated GNL correction reduce the resolution loss that occurs during conventional postreconstruction GNL correction while preserving the computational efficiency of standard reconstruction techniques. Magn Reson Med 75:2534-2544, 2016. © 2015 Wiley Periodicals, Inc.

An Alternating Direction Approximate Newton Algorithm for Ill-Conditioned Inverse Problems with Application to Parallel MRI

An alternating direction approximate Newton (ADAN) method is developed for solving inverse problems of the form \(\min \{\phi (Bu) +(1/2)\Vert Au-f\Vert _{2}^{2}\},\) where \(\phi \) is convex and possibly nonsmooth, and A and B are matrices. Problems of this form arise in image reconstruction where A is the matrix describing the imaging device, f is the measured data, \(\phi \) is a regularization term, and B is a derivative operator. The proposed algorithm is designed to handle applications where A is a large dense, ill-conditioned matrix. The algorithm is based on the alternating direction method of multipliers (ADMM) and an approximation to Newton’s method in which a term in Newton’s Hessian is replaced by a Barzilai–Borwein (BB) approximation. It is shown that ADAN converges to a solution of the inverse problem. Numerical results are provided using test problems from parallel magnetic resonance imaging. ADAN was faster than a proximal ADMM scheme that does not employ a BB Hessian approximation, while it was more stable and much simpler than the related Bregman operator splitting algorithm with variable stepsize algorithm which also employs a BB-based Hessian approximation.

Parallel magnetic resonance imaging as approximation in a reproducing kernel Hilbert space

In magnetic resonance imaging data samples are collected in the spatial frequency domain (k-space), typically by time-consuming line-by-line scanning on a Cartesian grid. Scans can be accelerated by simultaneous acquisition of data using multiple receivers (parallel imaging), and by using more efficient non-Cartesian sampling schemes. To understand and design k-space sampling patterns, a theoretical framework is needed to analyze how well arbitrary sampling patterns reconstruct unsampled k-space using receive coil information. As shown here, reconstruction from samples at arbitrary locations can be understood as approximation of vector-valued functions from the acquired samples and formulated using a reproducing kernel Hilbert space with a matrix-valued kernel defined by the spatial sensitivities of the receive coils. This establishes a formal connection between approximation theory and parallel imaging. Theoretical tools from approximation theory can then be used to understand reconstruction in k-space and to extend the analysis of the effects of samples selection beyond the traditional image-domain g-factor noise analysis to both noise amplification and approximation errors in k-space. This is demonstrated with numerical examples.

Highly undersampled peripheral Time-of-Flight magnetic resonance angiography: optimized data acquisition and iterative image reconstruction.

The aim of this study was to investigate the acceleration of peripheral Time-of-Flight magnetic resonance angiography using Compressed Sensing and parallel magnetic resonance imaging (MRI) while preserving image quality and vascular contrast. An analytical sampling pattern is proposed that combines aspects of parallel MRI and Compressed Sensing. It is used in combination with a dedicated Split Bregman algorithm. This approach is compared with current state-of-the-art patterns and reconstruction algorithms. The acquisition time was reduced from 30 to 2.5min in a study using ten volunteer data sets, while showing improved sharpness, better contrast and higher accuracy compared to state-of-the-art techniques. This study showed the benefits of the proposed dedicated analytical sampling pattern and Split Bregman algorithm for optimizing the Compressed Sensing reconstruction of highly accelerated peripheral Time-of-Flight data.

Analysis of the noise correlation in MRI coil arrays loaded with metamaterial magnetoinductive lenses.

A numerical method is shown for calculating the noise correlation coefficient in arrays of magnetic resonance imaging (MRI) coils loaded with capacitively-loaded ring metamaterial lenses, and in the presence of a conducting half-space resembling a sample. This numerical method is validated by comparison with experimental results obtained in two different experimental procedures for double check: noise resistance measurements with a network analyzer and noise correlation measurements in an MRI system. It is found that, for practical array configurations such as overlapping coils or capacitively-decoupled coils, the noise correlation coefficient turns negative for coils loaded with metamaterial lenses. In particular, the analysis is carried out with metamaterial structures known as magnetoinductive lenses, which have been demonstrated in previous works to improve the signal-to-noise ratio of MRI coils. Results are also shown to demonstrate that negative noise correlations have as an effect the improvement of the g-factor in coil arrays for parallel MRI.

GRAPPA reconstruction with spatially varying calibration of self-constraint.

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

Denoising Multi-Channel Images in Parallel MRI by Low Rank Matrix Decomposition

Parallel magnetic resonance imaging (pMRI) techniques can speed up MRI scan through a multi-channel coil array receiving signal simultaneously. Nevertheless, noise amplification and aliasing artifacts are serious in pMRI reconstructed images at high accelerations. This study presents a patch-wise denoising method for pMRI by exploiting the rank deficiency of multi-channel coil images and sparsity of artifacts. For each processed patch, similar patches are searched in spatial domain and throughout all coil elements, and arranged in appropriate matrix forms. Then, noise and aliasing artifacts are removed from the structured matrix by applying sparse and low rank matrix decomposition method. The proposed method has been validated using both phantom and in vivo brain data sets, producing encouraging results. Specifically, the method can effectively remove both noise and residual aliasing artifact from pMRI reconstructed noisy images, and produce higher peak signal noise rate (PSNR) and structural similarity index matrix (SSIM) than other state-of-the-art denoising methods.

Metamaterial magnetoinductive lens performance as a function of field strength

Metamaterials are artificial composites that exhibit exotic electromagnetic properties, as the ability of metamaterial slabs to behave like lenses with sub-wavelength resolution for the electric or the magnetic field. In previous works, the authors investigated magnetic resonance imaging (MRI) applications of metamaterial slabs that behave like lenses for the radiofrequency magnetic field. In particular, the authors investigated the ability of MRI metamaterial lenses to increase the signal-to-noise ratio (SNR) of surface coils, and to localize the field of view (FOV) of the coils, which is of interest for parallel MRI (pMRI) applications. A metamaterial lens placed between a surface coil and the tissue enhances the sensitivity of the coil. Although the metamaterial lens introduces losses which add to the losses of the tissue, the enhancement of the sensitivity can compensate these additional losses and the SNR of the coil is increased. In a previous work, an optimization procedure was followed to find a metamaterial structure with minimum losses that will maximize the SNR. This structure was termed magnetoinductive (MI) lens by the authors. The properties of surface coils in the presence of MI lenses were investigated in previous works at the proton frequency of 1.5T systems. The different frequency dependence of the losses in both the MI lenses and the tissue encouraged us to investigate the performance of MI lenses at different frequencies. Thus, in the present work, the SNR and the pMRI ability of MI lenses are investigated as a function of field strength. A numerical analysis is carried out with an algorithm developed by the authors to predict the SNR behavior of a surface coil loaded with a MI lens at the proton frequencies of 0.5T, 1.5T and 3T systems. The results show that, at 0.5T, there is a gain in the SNR for short distances, but the SNR is highly degraded at deeper distances. However, at 1.5T and 3T, the MI lenses provide a gain in the SNR up to a certain penetration depth, which is deeper at 3T, and do not degrade the SNR at deeper distances. These numerical results are checked by means of an experiment. Moreover, a second experiment developed with two-channel arrays of surface coils loaded with MI lenses shows that the pMRI ability of the lenses also improves from 1.5T to 3T. This improvement was quantified by means of the calculation of the GRAPPA g-factor.

Chaotic Compressed Sensing and Its Application to Magnetic Resonance Imaging

Fast image acquisition in magnetic resonance imaging (MRI) is important, due to the need to find ways that help relieve patient’s stress during MRI scans. Methods for fast MRI have been proposed, most notably among them are pMRI (parallel MRI), SWIFT (SWeep Imaging with Fourier Transformation), and compressed sensing (CS) based MRI. Although it promises to significantly reduce acquisition time, applying CS to MRI leads to difficulties with hardware design because of the randomness nature of the measurement matrix used by the conventional CS methods. In this paper, we propose a novel method that combines the above-mentioned three approaches for fast MRI by designing a compound measurement matrix from a series of single measurement matrices corresponding to pMRI, SWIFT, and CS. In our method, the CS measurement matrix is designed to be deterministic via chaotic systems. This chaotic compressed sensing (CCS) measurement matrix, while retaining most features of the random CS matrix, is simpler to realize in hardware. Several compound measurement matrices have been constructed and examined in this work, including CCS-MRI, CCS-pMRI, CCS-SWIFT, and CCS-pSWIFT. Simulation results showed that the proposed method allows an increase in the speed of the MRI acquisition process while not compromising the quality of the acquired MR images.

Spatio-temporal wavelet regularization for parallel MRI reconstruction: application to functional MRI

Parallel magnetic resonance imaging (MRI) is a fast imaging technique that helps acquiring highly resolved images in space/time. Its performance depends on the reconstruction algorithm, which can proceed either in the k-space or in the image domain. To improve the performance of the widely used SENSE algorithm, 2D regularization in the wavelet domain has been investigated. In this paper, we first extend this approach to 3D-wavelet representations and the 3D sparsity-promoting regularization term, in order to address reconstruction artifacts that propagate across adjacent slices. The resulting optimality criterion is convex but nonsmooth, and we resort to the parallel proximal algorithm to minimize it. Second, to account for temporal correlation between successive scans in functional MRI (fMRI), we extend our first contribution to 3D + t acquisition schemes by incorporating a prior along the time axis into the objective function. Our first method (3D-UWR-SENSE) is validated on T1-MRI anatomical data for gray/white matter segmentation. The second method (4D-UWR-SENSE) is validated for detecting evoked activity during a fast event-related functional MRI protocol. We show that our algorithm outperforms the SENSE reconstruction at the subject and group levels (15 subjects) for different contrasts of interest (motor or computation tasks) and two parallel acceleration factors (R = 2 and R = 4) on 2 × 2 × 3 MM(3) echo planar imaging (EPI) images.

MR Imaging Artifacts and Parallel Imaging Techniques with Calibration Scanning: A New Twist on Old Problems

The application of parallel magnetic resonance (MR) imaging is increasing as clinicians continue to strive for improved spatial and temporal resolution, benefits that arise from the use of fewer phase encodings during imaging. To reconstruct images, extra information is needed to map the spatial sensitivity of each coil element, which may be accomplished by acquiring a calibration image in one common implementation of parallel MR imaging. Although obtaining a quick calibration image is an efficient method for gathering this information, corruption of the image or disharmony with subsequent images may lead to errors in reconstruction. Although conventional MR imaging sequences may be employed with parallel MR imaging, the altered image reconstruction introduces several new artifacts and changes the appearance of conventional artifacts. The altered appearance of traditional artifacts may obscure the source of the problem, and, in some cases, the severity of artifacts associated with parallel MR imaging may be exacerbated, hindering image interpretation. Several artifacts arise in the context of parallel MR imaging, including both traditional artifacts and those associated with parallel MR imaging.

Improve GRAPPA with Cross-sampled ACS Lines and Nonlinear Kernel Model

Among the methods of parallel magnetic resonance imaging (PMRI), Generalized Auto-calibrating Partially Parallel Acquisitions (GRAPPA) technique reconstructs the missing k-space data by a set of weights, which are derived from auto-calibration signal (ACS) lines acquired in parallel to the reduced lines. In this paper, a novel hybrid method is proposed to reconstruct by cross sampling the ACS lines orthogonal to the reduced lines and estimating weights with a second-order nonlinear model. The proposed method can mix the benefits of cross sampling and the nonlinear kernel model. The in vivo experiments demonstrate this method, named as cross-sampled nonlinear (CSNL) GRAPPA, can effectively reduce the aliasing artifacts and noises when high acceleration is desired.

Calibrationless Parallel Magnetic Resonance Imaging: A Joint Sparsity Model

State-of-the-art parallel MRI techniques either explicitly or implicitly require certain parameters to be estimated, e.g., the sensitivity map for SENSE, SMASH and interpolation weights for GRAPPA, SPIRiT. Thus all these techniques are sensitive to the calibration (parameter estimation) stage. In this work, we have proposed a parallel MRI technique that does not require any calibration but yields reconstruction results that are at par with (or even better than) state-of-the-art methods in parallel MRI. Our proposed method required solving non-convex analysis and synthesis prior joint-sparsity problems. This work also derives the algorithms for solving them. Experimental validation was carried out on two datasets—eight channel brain and eight channel Shepp-Logan phantom. Two sampling methods were used—Variable Density Random sampling and non-Cartesian Radial sampling. For the brain data, acceleration factor of 4 was used and for the other an acceleration factor of 6 was used. The reconstruction results were quantitatively evaluated based on the Normalised Mean Squared Error between the reconstructed image and the originals. The qualitative evaluation was based on the actual reconstructed images. We compared our work with four state-of-the-art parallel imaging techniques; two calibrated methods—CS SENSE and l1SPIRiT and two calibration free techniques—Distributed CS and SAKE. Our method yields better reconstruction results than all of them.

Quantifying the Statistical Impact of GRAPPA in fcMRI Data With a Real-Valued Isomorphism

The interpolation of missing spatial frequencies through the generalized auto-calibrating partially parallel acquisitions (GRAPPA) parallel magnetic resonance imaging (MRI) model implies a correlation is induced between the acquired and reconstructed frequency measurements. As the parallel image reconstruction algorithms in many medical MRI scanners are based on the GRAPPA model, this study aims to quantify the statistical implications that the GRAPPA model has in functional connectivity studies. The linear mathematical framework derived in the work of Rowe , 2007, is adapted to represent the complex-valued GRAPPA image reconstruction operation in terms of a real-valued isomorphism, and a statistical analysis is performed on the effects that the GRAPPA operation has on reconstructed voxel means and correlations. The interpolation of missing spatial frequencies with the GRAPPA model is shown to result in an artificial correlation induced between voxels in the reconstructed images, and these artificial correlations are shown to reside in the low temporal frequency spectrum commonly associated with functional connectivity. Through a real-valued isomorphism, such as the one outlined in this manuscript, the exact artificial correlations induced by the GRAPPA model are not simply estimated, as they would be with simulations, but are precisely quantified. If these correlations are unaccounted for, they can incur an increase in false positives in functional connectivity studies.

Accelerated magnetic resonance imaging using the sparsity of multi-channel coil images

Joint estimation of coil sensitivities and output image (JSENSE) is a promising approach that improves the reconstruction of parallel magnetic resonance imaging (pMRI). However, when acceleration factor increases, the signal to noise ratio (SNR) of JSENSE reconstruction decreases as quickly as that of the conventional pMRI. Although sparse constraints have been used to improve the JSENSE reconstruction in recent years, these constraints only use the sparsity of the output image, which cannot fully exploit the prior information of pMRI. In this paper, we use the sparsity of coil images, instead of the output image, to exploit more prior information for JSENSE. Numerical simulation, phantom and in vivo experiments demonstrate that the proposed method has better performance than the SparseSENSE method and the constrained JSENSE method using the sparsity of the output image only.

A Majorize-Minimize Memory Gradient method for complex-valued inverse problems

Complex-valued data are encountered in many application areas of signal and image processing. In the context of the optimization of functions of real variables, subspace algorithms have recently attracted much interest, owing to their efficiency for solving large-size problems while simultaneously offering theoretical convergence guarantees. The goal of this paper is to show how some of these methods can be successfully extended to the complex case. More precisely, we investigate the properties of the proposed complex-valued Majorize-Minimize Memory Gradient (3MG) algorithm. Important practical applications of these results arise in inverse problems. Here, we focus on image reconstruction in Parallel Magnetic Resonance Imaging (PMRI). The linear operator involved in the observation model then includes a subsampling operator over the k-space (2D Fourier domain) the choice of which is analyzed through our numerical results. In addition, sensitivity matrices associated with the multiple channel coils come into play. Comparisons with existing optimization methods confirm the better performance of the proposed algorithm.

Contrast echocardiography to assess left ventricular volume and function in Beagle dogs: Comparison with 3-Tesla dual source parallel cardiac magnetic resonance imaging

This study was performed to evaluate the effect and feasibility of contrast echocardiography (CE) compared with unenhanced echocardiography (UE) and cardiac magnetic resonance imaging (CMRI) to assess left ventricular (LV) volume and function, including end-diastolic volume (EDV), end-systolic volume (ESV), stroke volume (SV), and ejection fraction (EF) in six healthy Beagles.When the dogs were conscious, LV measurements using CE were significantly higher than those obtained using UE, except for EF, and were similar to the values obtained using CMRI. Additionally, EDV, SV, and EF obtained using UE from anesthetized dogs were significantly lower than those obtained using CE or CMRI. Measurements of EDV, SV and EF using CE were not significantly different from the corresponding measurements obtained using CMRI (31.13±2.18 vs. 32.88±1.17mL, 18.41±1.25 vs. 17.92±0.96mL, 59.29±2.29% vs. 53.33±1.69%, respectively). Inter-observer agreements for UE (0.74±0.05) were lower than those for CE (0.80±0.04) and CMRI (0.92±0.03). In conclusion, LV function was assessed reproducibly using CE, and the measurements obtained were consistent with reference standard measurements obtained using CMRI. Measurements made using CE agreed more closely with CMRI than those made using UE.