Reconstruction of undersampled non-Cartesian data sets using pseudo-Cartesian GRAPPA in conjunction with GROG.

Most k-space-based parallel imaging reconstruction techniques, such as Generalized Autocalibrating Partially Parallel Acquisitions (GRAPPA), necessitate the acquisition of regularly sampled Cartesian k-space data to reconstruct a nonaliased image efficiently. However, non-Cartesian sampling schemes offer some inherent advantages to the user due to their better coverage of the center of k-space and faster acquisition times. On the other hand, these sampling schemes have the disadvantage that the points acquired generally do not lie on a grid and have complex k-space sampling patterns. Thus, the extension of Cartesian GRAPPA to non-Cartesian sequences is nontrivial. This study introduces a simple, novel method for performing Cartesian GRAPPA reconstructions on undersampled non-Cartesian k-space data gridded using GROG (GRAPPA Operator Gridding) to arrive at a nonaliased image. Because the undersampled non-Cartesian data cannot be reconstructed using a single GRAPPA kernel, several Cartesian patterns are selected for the reconstruction. This flexibility in terms of both the appearance and number of patterns allows this pseudo-Cartesian GRAPPA to be used with undersampled data sets acquired with any non-Cartesian trajectory. The successful implementation of the reconstruction algorithm using several different trajectories, including radial, rosette, spiral, one-dimensional non-Cartesian, and zig-zag trajectories, is demonstrated.     

Generalized autocalibrating partially parallel acquisitions (GRAPPA).

In this study, a novel partially parallel acquisition (PPA) method is presented which can be used to accelerate image acquisition using an RF coil array for spatial encoding. This technique, GeneRalized Autocalibrating Partially Parallel Acquisitions (GRAPPA) is an extension of both the PILS and VD-AUTO-SMASH reconstruction techniques. As in those previous methods, a detailed, highly accurate RF field map is not needed prior to reconstruction in GRAPPA. This information is obtained from several k-space lines which are acquired in addition to the normal image acquisition. As in PILS, the GRAPPA reconstruction algorithm provides unaliased images from each component coil prior to image combination. This results in even higher SNR and better image quality since the steps of image reconstruction and image combination are performed in separate steps. After introducing the GRAPPA technique, primary focus is given to issues related to the practical implementation of GRAPPA, including the reconstruction algorithm as well as analysis of SNR in the resulting images. Finally, in vivo GRAPPA images are shown which demonstrate the utility of the technique.     

Self-calibration method for radial GRAPPA/k-t GRAPPA.

Generalized autocalibrating partially parallel acquisitions (GRAPPA), an important parallel imaging technique, can be easily applied to radial k-space data by segmenting the k-space. The previously reported radial GRAPPA method requires extra calibration data to determine the relative shift operators. In this work it is shown that pseudo-full k-space data can be generated from the partially acquired radial data by filtering in image space followed by inverse gridding. The relative shift operators can then be approximated from the pseudo-full k-space data. The self-calibration method using pseudo-full k-space data can be applied in both k and k-t space. This technique avoids the prescans and hence improves the applicability of radial GRAPPA to image static tissue, and makes k-t GRAPPA applicable to radial trajectory. Experiments show that radial GRAPPA calibrated with pseudo-full calibration data generates results similar to radial GRAPPA calibrated with the true full k-space data for that image. If motion occurs during acquisition, self-calibrated radial GRAPPA protects structural information better than externally calibrated GRAPPA. However, radial GRAPPA calibrated with pseudo-full calibration data suffers from residual streaking artifacts when the reduction factor is high. Radial k-t GRAPPA calibrated with pseudo-full calibration data generates reduced errors compared to the sliding-window method and temporal GRAPPA (TGRAPPA).     

Artifact reduction in moving-table acquisitions using parallel imaging and multiple averages.

Two-dimensional (2D) axial continuously-moving-table imaging has to deal with artifacts due to gradient nonlinearity and breathing motion, and has to provide the highest scan efficiency. Parallel imaging techniques (e.g., generalized autocalibrating partially parallel acquisition GRAPPA)) are used to reduce such artifacts and avoid ghosting artifacts. The latter occur in T(2)-weighted multi-spin-echo (SE) acquisitions that omit an additional excitation prior to imaging scans for presaturation purposes. Multiple images are reconstructed from subdivisions of a fully sampled k-space data set, each of which is acquired in a single SE train. These images are then averaged. GRAPPA coil weights are estimated without additional measurements. Compared to conventional image reconstruction, inconsistencies between different subsets of k-space induce less artifacts when each k-space part is reconstructed separately and the multiple images are averaged afterwards. These inconsistencies may lead to inaccurate GRAPPA coil weights using the proposed intrinsic GRAPPA calibration. It is shown that aliasing artifacts in single images are canceled out after averaging. Phantom and in vivo studies demonstrate the benefit of the proposed reconstruction scheme for free-breathing axial continuously-moving-table imaging using fast multi-SE sequences.     

Artifact and noise suppression in GRAPPA imaging using improved k-space coil calibration and variable density sampling.

A parallel imaging technique, GRAPPA (GeneRalized Auto-calibrating Partially Parallel Acquisitions), has been used to improve temporal or spatial resolution. Coil calibration in GRAPPA is performed in central k-space by fitting a target signal using its adjacent signals. Missing signals in outer k-space are reconstructed. However, coil calibration operates with signals that exhibit large amplitude variation while reconstruction is performed using signals with small amplitude variation. Different signal variations in coil calibration and reconstruction may result in residual image artifact and noise. The purpose of this work was to improve GRAPPA coil calibration and variable density (VD) sampling for suppressing residual artifact and noise. The proposed coil calibration was performed in local k-space along both the phase and frequency encoding directions. Outer k-space was acquired with two different reduction factors. Phantom data were reconstructed by both the conventional GRAPPA and the improved technique for comparison at an acceleration of two. Under the same acceleration, optimal sampling and calibration parameters were determined. An in vivo image was reconstructed in the same way using the predetermined optimal parameters. The performance of GRAPPA was improved by the localized coil calibration and VD sampling scheme.     

Iterative GRAPPA (iGRAPPA) for improved parallel imaging reconstruction.

In this work an iterative reconstruction method based on generalized autocalibrating partially parallel acquisitions (GRAPPA) reconstruction is introduced. In the new method the reconstructed lines are used to reestimate and refine the weights from all the acquired data by applying the GRAPPA procedure iteratively with regularization. Both phantom and in vivo MRI experiments demonstrated that, compared to GRAPPA, the iterative approach reduces parallel imaging artifacts and permits high-quality image reconstruction with a relatively small number of calibration lines and slight changes of GRAPPA weights.     

k-t GRAPPA: a k-space implementation for dynamic MRI with high reduction factor.

A novel technique called "k-t GRAPPA" is introduced for the acceleration of dynamic magnetic resonance imaging. Dynamic magnetic resonance images have significant signal correlations in k-space and time dimension. Hence, it is feasible to acquire only a reduced amount of data and recover the missing portion afterward. Generalized autocalibrating partially parallel acquisitions (GRAPPA), as an important parallel imaging technique, linearly interpolates the missing data in k-space. In this work, it is shown that the idea of GRAPPA can also be applied in k-t space to take advantage of the correlations and interpolate the missing data in k-t space. For this method, no training data, filters, additional parameters, or sensitivity maps are necessary, and it is applicable for either single or multiple receiver coils. The signal correlation is locally derived from the acquired data. In this work, the k-t GRAPPA technique is compared with our implementation of GRAPPA, TGRAPPA, and sliding window reconstructions, as described in Methods. The experimental results manifest that k-t GRAPPA generates high spatial resolution reconstruction without significant loss of temporal resolution when the reduction factor is as high as 4. When the reduction factor becomes higher, there might be a noticeable loss of temporal resolution since k-t GRAPPA uses temporal interpolation. Images reconstructed using k-t GRAPPA have less residue/folding artifacts than those reconstructed by sliding window, much less noise than those reconstructed by GRAPPA, and wider temporal bandwidth than those reconstructed by GRAPPA with residual k-space. k-t GRAPPA is applicable to a wide range of dynamic imaging applications and is not limited to imaging parts with quasi-periodic motion. Since only local information is used for reconstruction, k-t GRAPPA is also preferred for applications requiring real time reconstruction, such as monitoring interventional MRI.     

Nonlinear GRAPPA: a kernel approach to parallel MRI reconstruction

GRAPPA linearly combines the undersampled k-space signals to estimate the missing k-space signals where the coefficients are obtained by fitting to some auto-calibration signals (ACS) sampled with Nyquist rate based on the shift-invariant property. At high acceleration factors, GRAPPA reconstruction can suffer from a high level of noise even with a large number of auto-calibration signals. In this work, we propose a nonlinear method to improve GRAPPA. The method is based on the so-called kernel method which is widely used in machine learning. Specifically, the undersampled k-space signals are mapped through a nonlinear transform to a high-dimensional feature space, and then linearly combined to reconstruct the missing k-space data. The linear combination coefficients are also obtained through fitting to the ACS data but in the new feature space. The procedure is equivalent to adding many virtual channels in reconstruction. A polynomial kernel with explicit mapping functions is investigated in this work. Experimental results using phantom and in vivo data demonstrate that the proposed nonlinear GRAPPA method can significantly improve the reconstruction quality over GRAPPA and its state-of-the-art derivatives.     

Fast four-dimensional coronary MR angiography with k-t GRAPPA

PURPOSE: To investigate the effectiveness of k-t GRAPPA for accelerating four-dimensional (4D) coronary MRA in comparison with GRAPPA and the feasibility of combining variable density undersampling with conventional k-t GRAPPA (k-t(2) GRAPPA) to alleviate the overhead of acquiring autocalibration signals. MATERIALS AND METHODS: The right coronary artery of nine healthy volunteers was scanned at 1.5 Tesla. The 4D k-space datasets were fully acquired and subsequently undersampled to simulate partially parallel acquisitions, namely, GRAPPA, k-t GRAPPA, and k-t(2) GRAPPA. Comparisons were made between the images reconstructed from full k-space datasets and those reconstructed from undersampled k-space datasets. RESULTS: k-t GRAPPA significantly reduced artifacts compared with GRAPPA and high acceleration factors were achieved with only minimal sacrifices in vessel depiction. k-t(2) GRAPPA could further increase imaging speed without significant losses in image quality. CONCLUSION: By exploiting high-degree spatiotemporal correlations during the rest period of a cardiac cycle, k-t GRAPPA and k-t(2) GRAPPA can greatly increase data acquisition efficiency and, therefore, are promising solutions for fast 4D coronary MRA.     

Iterative Next-Neighbor Regridding (INNG): improved reconstruction from nonuniformly sampled k-space data using rescaled matrices.

The reconstruction of MR images from nonrectilinearly sampled data is complicated by the fact that the inverse 2D Fourier transform (FT) cannot be performed directly on the acquired k-space data set. k-Space gridding is commonly used because it is an efficient reconstruction method. However, conventional gridding requires optimized density compensation functions (DCFs) to avoid profile distortions. Oftentimes, the calculation of optimized DCFs presents an additional challenge in obtaining an accurately gridded reconstruction. Another type of gridding algorithm, the block uniform resampling (BURS) algorithm, often requires singular value decomposition (SVD) regularization to avoid amplification of data imperfections, and under some conditions it is difficult to adjust the regularization parameters. In this work, new reconstruction algorithms for nonuniformly sampled k-space data are presented. In the newly proposed algorithms, high-quality reconstructed images are obtained from an iterative reconstruction that is performed using matrices scaled to sizes greater than that of the target image matrix. A second version partitions the sampled k-space region into several blocks to avoid limitations that could result from performing multiple 2D-FFTs on large data matrices. The newly proposed algorithms are a simple alternative approach to previously proposed optimized gridding algorithms.     

RT‐GROG: parallelized self‐calibrating GROG for real‐time MRI

A real‐time implementation of self‐calibrating Generalized Autocalibrating Partially Parallel Acquisitions (GRAPPA) operator gridding for radial acquisitions is presented. Self‐calibrating GRAPPA operator gridding is a parallel‐imaging‐based, parameter‐free gridding algorithm, where coil sensitivity profiles are used to calculate gridding weights. Self‐calibrating GRAPPA operator gridding's weight‐set calculation and image reconstruction steps are decoupled into two distinct processes, implemented in C++ and parallelized. This decoupling allows the weights to be updated adaptively in the background while image reconstruction threads use the most recent gridding weights to grid and reconstruct images. All possible combinations of two‐dimensional gridding weights GG are evaluated for m,n = {?0.5, ?0.4, …, 0, 0.1, …, 0.5} and stored in a look‐up table. Consequently, the per‐sample two‐dimensional weights calculation during gridding is eliminated from the reconstruction process and replaced by a simple look‐up table access. In practice, up to 34× faster reconstruction than conventional (parallelized) self‐calibrating GRAPPA operator gridding is achieved. On a 32‐coil dataset of size 128 × 64, reconstruction performance is 14.5 frames per second (fps), while the data acquisition is 6.6 fps. Magn Reson Med 64:306–312, 2010. © 2010 Wiley‐Liss, Inc.     

Parallel imaging reconstruction for arbitrary trajectories using k-space sparse matrices (kSPA).

Although the concept of receiving MR signal using multiple coils simultaneously has been known for over two decades, the technique has only recently become clinically available as a result of the development of several effective parallel imaging reconstruction algorithms. Despite the success of these algorithms, it remains a challenge in many applications to rapidly and reliably reconstruct an image from partially-acquired general non-Cartesian k-space data. Such applications include, for example, three-dimensional (3D) imaging, functional MRI (fMRI), perfusion-weighted imaging, and diffusion tensor imaging (DTI), in which a large number of images have to be reconstructed. In this work, a systematic k-space-based reconstruction algorithm based on k-space sparse matrices (kSPA) is introduced. This algorithm formulates the image reconstruction problem as a system of sparse linear equations in k-space. The inversion of this system of equations is achieved by computing a sparse approximate inverse matrix. The algorithm is demonstrated using both simulated and in vivo data, and the resulting image quality is comparable to that of the iterative sensitivity encoding (SENSE) algorithm. The kSPA algorithm is noniterative and the computed sparse approximate inverse can be applied repetitively to reconstruct all subsequent images. This algorithm, therefore, is particularly suitable for the aforementioned applications.     

Cross-validation-based kernel support selection for improved GRAPPA reconstruction.

The extended version of the generalized autocalibrating partially parallel acquisition (GRAPPA) technique incorporates multiple lines and multiple columns of measured k-space data to estimate missing data. For a given accelerated dataset, the selection of the measured data points for fitting a missing datum (i.e., the kernel support) that provides optimal reconstruction depends on coil array configuration, noise level in the acquired data, imaging configuration, and number and position of autocalibrating signal lines. In this work, cross-validation is used to select the kernel support that best balances the conflicting demands of fit accuracy and stability in GRAPPA reconstruction. The result is an optimized tradeoff between artifacts and noise. As demonstrated with experimental data, the method improves image reconstruction with GRAPPA. Because the method is simple and applied in postprocessing, it can be used with GRAPPA routinely.     

Active catheter tracking using parallel MRI and real-time image reconstruction.

In this work active MR catheter tracking with automatic slice alignment was combined with an autocalibrated parallel imaging technique. Using an optimized generalized autocalibrating partially parallel acquisitions (GRAPPA) algorithm with an acceleration factor of 2, we were able to reduce the acquisition time per image by 34%. To accelerate real-time GRAPPA image reconstruction, the coil sensitivities were updated only after slice reorientation. For a 2D trueFISP acquisition (160 x 256 matrix, 80% phase matrix, half Fourier acquisition, TR = 3.7 ms, GRAPPA factor = 2) real-time image reconstruction was achieved with up to six imaging coils. In a single animal experiment the method was used to steer a catheter from the vena cava through the beating heart into the pulmonary vasculature at an image update rate of about five images per second. Under all slice orientations, parallel image reconstruction was accomplished with only minor image artifacts, and the increased temporal resolution provided a sharp delineation of intracardial structures, such as the papillary muscle.     

A fast Edge‐preserving Bayesian reconstruction method for Parallel Imaging applications in cardiac MRI

Among recent parallel MR imaging reconstruction advances, a Bayesian method called Edge‐preserving Parallel Imaging reconstructions with GRAph cuts Minimization (EPIGRAM) has been demonstrated to significantly improve signal‐to‐noise ratio when compared with conventional regularized sensitivity encoding method. However, EPIGRAM requires a large number of iterations in proportion to the number of intensity labels in the image, making it computationally expensive for high dynamic range images. The objective of this study is to develop a Fast EPIGRAM reconstruction based on the efficient binary jump move algorithm that provides a logarithmic reduction in reconstruction time while maintaining image quality. Preliminary in vivo validation of the proposed algorithm is presented for two‐dimensional cardiac cine MR imaging and three‐dimensional coronary MR angiography at acceleration factors of 2–4. Fast EPIGRAM was found to provide similar image quality to EPIGRAM and maintain the previously reported signal‐to‐noise ratio improvement over regularized sensitivity encoding method, while reducing EPIGRAM reconstruction time by 25–50 times. Magn Reson Med, 2010. © 2010 Wiley‐Liss, Inc.     

Parallel magnetic resonance imaging using the GRAPPA operator formalism.

In this article it is shown that GRAPPA reconstruction can be reformulated as a matrix operator, similar to ladder or propagator operators used in quantum mechanics, that shifts data in k-space. Using this formalism, it is shown that there exists an infinitesimal GRAPPA operator that shifts data in k-space by arbitrarily small amounts. Other desired k-space shifts can then be accomplished through repeated applications of this infinitesimal GRAPPA operator. Implications of these ideas are described.     

Parallel MR imaging

Parallel imaging is a robust method for accelerating the acquisition of magnetic resonance imaging (MRI) data, and has made possible many new applications of MR imaging. Parallel imaging works by acquiring a reduced amount of k-space data with an array of receiver coils. These undersampled data can be acquired more quickly, but the undersampling leads to aliased images. One of several parallel imaging algorithms can then be used to reconstruct artifact-free images from either the aliased images (SENSE-type reconstruction) or from the undersampled data (GRAPPA-type reconstruction). The advantages of parallel imaging in a clinical setting include faster image acquisition, which can be used, for instance, to shorten breath-hold times resulting in fewer motion-corrupted examinations. In this article the basic concepts behind parallel imaging are introduced. The relationship between undersampling and aliasing is discussed and two commonly used parallel imaging methods, SENSE and GRAPPA, are explained in detail. Examples of artifacts arising from parallel imaging are shown and ways to detect and mitigate these artifacts are described. Finally, several current applications of parallel imaging are presented and recent advancements and promising research in parallel imaging are briefly reviewed.     

Exploiting sparsity to accelerate noncontrast MR angiography in the context of parallel imaging

Noncontrast techniques for peripheral MR angiography are receiving renewed interest because of safety concerns about the use of gadolinium in patients with renal insufficiency. One class of techniques involves subtraction of dark-blood images acquired during fast systolic flow from bright-blood images obtained during slow diastolic flow. The goal of this work was to determine whether the inherent sparsity of the difference images could be exploited to achieve greater acceleration without loss of image quality in the context of generalized autocalibrating partially parallel acquisition (GRAPPA). It is shown that noise amplification at high acceleration factors can be reduced by performing subtraction on the raw data, before calculation of the GRAPPA weights, rather than on the final magnitude images. Use of the difference data to calculate the GRAPPA weights decreases the geometry factor (g-factor), because the difference data represent a sparse image set. This demonstrates an inherent property of GRAPPA and does not require the use of compressed sensing. Application of this approach to highly accelerated data from healthy volunteers resulted in similar depiction of large arteries to that obtained with low acceleration and standard reconstruction. However, visualization of very small vessels and arterial branches was compromised.     

2D-GRAPPA-operator for faster 3D parallel MRI.

When using parallel MRI (pMRI) methods in combination with three-dimensional (3D) imaging, it is beneficial to subsample the k-space along both phase-encoding directions because one can then take advantage of coil sensitivity variations along two spatial dimensions. This results in an improved reconstruction quality and therefore allows greater scan time reductions as compared to subsampling along one dimension. In this work we present a new approach based on the generalized autocalibrating partially parallel acquisitions (GRAPPA) technique that allows Fourier-domain reconstructions of data sets that are subsampled along two dimensions. The method works by splitting the 2D reconstruction process into two separate 1D reconstructions. This approach is compared with an extension of the conventional GRAPPA method that directly regenerates missing data points of a 2D subsampled k-space by performing a linear combination of acquired data points. In this paper we describe the theoretical background and present computer simulations and in vivo experiments.     

Improved parallel MR imaging using a coefficient penalized regularization for GRAPPA reconstruction

A novel coefficient penalized regularization method for generalized autocalibrating partially parallel acquisitions (GRAPPA) reconstruction is developed for improving MR image quality. In this method, the fitting coefficients of the source data are weighted with different penalty factors, which are highly dependent upon the relative displacements from the source data to the target data in k‐space. The imaging data from both phantom testing and in vivo MRI experiments demonstrate that the coefficient penalized regularization method in GRAPPA reconstruction is able to reduce noise amplification to a greater degree. Therefore, the method enhances the quality of images significantly when compared to the previous least squares and Tikhonov regularization methods. Magn Reson Med 69:1109–1114, 2013. © 2012 Wiley Periodicals, Inc.