Varying kernel-extent gridding reconstruction for undersampled variable-density spirals.

Nonuniform, non-Cartesian k-space trajectories enable fast scanning with reduced motion and flow artifacts. In such cases, the data are usually convolved with a kernel and resampled onto a Cartesian grid before reconstruction. For trajectories such as undersampled variable-density spirals, the mainlobe width of the kernel for undersampled high spatial frequencies has to be larger to limit the amount of aliasing energy. Continuously varying the kernel extent is time consuming. By dividing k-space into several annuli and using appropriate mainlobe widths for each, the aliasing energy and noise can be reduced at the expense of lower resolution towards the edge of the field of view (FOV). Resolution can instead be preserved at the center of the FOV, which is expected to be free of artifacts, without any artifact reduction. The image reconstructed from each annulus can be deapodized separately. The method can be applied to most k-space trajectories used in MRI.     

Self-calibrating GRAPPA operator gridding for radial and spiral trajectories.

Self-calibrating GRAPPA operator gridding (GROG) is a method by which non-Cartesian MRI data can be gridded using spatial information from a multichannel coil array without the need for an additional calibration dataset. Using self-calibrating GROG, the non-Cartesian datapoints are shifted to nearby k-space locations using parallel imaging weight sets determined from the datapoints themselves. GROG employs the GRAPPA Operator, a special formulation of the general reconstruction method GRAPPA, to perform these shifts. Although GROG can be used to grid undersampled datasets, it is important to note that this method uses parallel imaging only for gridding, and not to reconstruct artifact-free images from undersampled data. The innovation introduced here, namely, self-calibrating GROG, allows the shift operators to be calculated directly out of the non-Cartesian data themselves. This eliminates the need for an additional calibration dataset, which reduces the imaging time and also makes the GROG reconstruction more robust by removing possible inconsistencies between the calibration and non-Cartesian datasets. Simulated and in vivo examples of radial and spiral datasets gridded using self-calibrating GROG are compared to images gridded using the standard method of convolution gridding.     

Applying the uniform resampling (URS) algorithm to a lissajous trajectory: fast image reconstruction with optimal gridding.

Various kinds of nonrectilinear Cartesian k-space trajectories have been studied, such as spiral, circular, and rosette trajectories. Although the nonrectilinear Cartesian sampling techniques generally have the advantage of fast data acquisition, the gridding process prior to 2D-FFT image reconstruction usually requires a number of additional calculations, thus necessitating an increase in the computation time. Further, the reconstructed image often exhibits artifacts resulting from both the k-space sampling pattern and the gridding procedure. To date, it has been demonstrated in only a few studies that the special geometric sampling patterns of certain specific trajectories facilitate fast image reconstruction. In other words, the inherent link among the trajectory, the sampling scheme, and the associated complexity of the regridding/reconstruction process has been investigated to only a limited extent. In this study, it is demonstrated that a Lissajous trajectory has the special geometric characteristics necessary for rapid reconstruction of nonrectilinear Cartesian k-space trajectories with constant sampling time intervals. Because of the applicability of a uniform resampling (URS) algorithm, a high-quality reconstructed image is obtained in a short reconstruction time when compared to other gridding algorithms.     

Fast method for 1D non-cartesian parallel imaging using GRAPPA.

MRI with non-Cartesian sampling schemes can offer inherent advantages. Radial acquisitions are known to be very robust, even in the case of vast undersampling. This is also true for 1D non-Cartesian MRI, in which the center of k-space is oversampled or at least sampled at the Nyquist rate. There are two main reasons for the more relaxed foldover artifact behavior: First, due to the oversampling of the center, high-energy foldover artifacts originating from the center of k-space are avoided. Second, due to the non-equidistant sampling of k-space, the corresponding field of view (FOV) is no longer well defined. As a result, foldover artifacts are blurred over a broad range and appear less severe. The more relaxed foldover artifact behavior and the densely sampled central k-space make trajectories of this type an ideal complement to autocalibrated parallel MRI (pMRI) techniques, such as generalized autocalibrating partially parallel acquisitions (GRAPPA). Although pMRI can benefit from non-Cartesian trajectories, this combination has not yet entered routine clinical use. One of the main reasons for this is the need for long reconstruction times due to the complex calculations necessary for non-Cartesian pMRI. In this work it is shown that one can significantly reduce the complexity of the calculations by exploiting a few specific properties of k-space-based pMRI.     

Non-Cartesian data reconstruction using GRAPPA operator gridding (GROG).

A novel approach that uses the concepts of parallel imaging to grid data sampled along a non-Cartesian trajectory using GRAPPA operator gridding (GROG) is described. GROG shifts any acquired data point to its nearest Cartesian location, thereby converting non-Cartesian to Cartesian data. Unlike other parallel imaging methods, GROG synthesizes the net weight for a shift in any direction from a single basis set of weights along the logical k-space directions. Given the vastly reduced size of the basis set, GROG calibration and reconstruction requires fewer operations and less calibration data than other parallel imaging methods for gridding. Instead of calculating and applying a density compensation function (DCF), GROG requires only local averaging, as the reconstructed points fall upon the Cartesian grid. Simulations are performed to demonstrate that the root mean square error (RMSE) values of images gridded with GROG are similar to those for images gridded using the gold-standard convolution gridding. Finally, GROG is compared to the convolution gridding technique using data sampled along radial, spiral, rosette, and BLADE (a.k.a. periodically rotated overlapping parallel lines with enhanced reconstruction [PROPELLER]) trajectories.     

Parallel magnetic resonance imaging with adaptive radius in k-space (PARS): constrained image reconstruction using k-space locality in radiofrequency coil encoded data.

A parallel image reconstruction algorithm is presented that exploits the k-space locality in radiofrequency (RF) coil encoded data. In RF coil encoding, information relevant to reconstructing an omitted datum rapidly diminishes as a function of k-space separation between the omitted datum and the acquired signal data. The proposed method, parallel magnetic resonance imaging with adaptive radius in k-space (PARS), harnesses this physical property of RF coil encoding via a sliding-kernel approach. Unlike generalized parallel imaging approaches that might typically involve inverting a prohibitively large matrix for arbitrary sampling trajectories, the PARS sliding-kernel approach creates manageable and distributable independent matrices to be inverted, achieving both computational efficiency and numerical stability. An empirical method designed to measure total error power is described, and the total error power of PARS reconstructions is studied over a range of k-space radii and accelerations, revealing "minimal-error" conditions at comparatively modest k-space radii. PARS reconstructions of undersampled in vivo Cartesian and non-Cartesian data sets are shown and are compared selectively with traditional SENSE reconstructions. Various characteristics of the PARS k-space locality constraint (such as the tradeoff between signal-to-noise ratio and artifact power and the relationship with iterative parallel conjugate gradient approaches or nonparallel gridding approaches) are discussed.     

Sampling density compensation in MRI: rationale and an iterative numerical solution.

Data collection of MRI which is sampled nonuniformly in k-space is often interpolated onto a Cartesian grid for fast reconstruction. The collected data must be properly weighted before interpolation, for accurate reconstruction. We propose a criterion for choosing the weighting function necessary to compensate for nonuniform sampling density. A numerical iterative method to find a weighting function that meets that criterion is also given. This method uses only the coordinates of the sampled data; unlike previous methods, it does not require knowledge of the trajectories and can easily handle trajectories that "cross" in k-space. Moreover, the method can handle sampling patterns that are undersampled in some regions of k-space and does not require a post-gridding density correction. Weighting functions for various data collection strategies are shown. Synthesized and collected in vivo data also illustrate aspects of this method.     

Non‐Cartesian parallel imaging reconstruction

Non‐Cartesian parallel imaging has played an important role in reducing data acquisition time in MRI. The use of non‐Cartesian trajectories can enable more efficient coverage of k‐space, which can be leveraged to reduce scan times. These trajectories can be undersampled to achieve even faster scan times, but the resulting images may contain aliasing artifacts. Just as Cartesian parallel imaging can be used to reconstruct images from undersampled Cartesian data, non‐Cartesian parallel imaging methods can mitigate aliasing artifacts by using additional spatial encoding information in the form of the nonhomogeneous sensitivities of multi‐coil phased arrays. This review will begin with an overview of non‐Cartesian k‐space trajectories and their sampling properties, followed by an in‐depth discussion of several selected non‐Cartesian parallel imaging algorithms. Three representative non‐Cartesian parallel imaging methods will be described, including Conjugate Gradient SENSE (CG SENSE), non‐Cartesian generalized autocalibrating partially parallel acquisition (GRAPPA), and Iterative Self‐Consistent Parallel Imaging Reconstruction (SPIRiT). After a discussion of these three techniques, several potential promising clinical applications of non‐Cartesian parallel imaging will be covered. J. Magn. Reson. Imaging 2014;40:1022–1040 . © 2014 Wiley Periodicals, Inc.     

Parallel spectroscopic imaging reconstruction with arbitrary trajectories using k‐space sparse matrices

Parallel imaging reconstruction has been successfully applied to magnetic resonance spectroscopic imaging (MRSI) to reduce scan times. For undersampled k‐space data on a Cartesian grid, the reconstruction can be achieved in image domain using a sensitivity encoding (SENSE) algorithm for each spectral data point. Alternative methods for reconstruction with undersampled Cartesian k‐space data are the SMASH and GRAPPA algorithms that do the reconstruction in the k‐space domain. To reconstruct undersampled MRSI data with arbitrary k‐space trajectories, image‐domain‐based iterative SENSE algorithm has been applied at the cost of long computing times. In this paper, a new k‐space domain‐based parallel spectroscopic imaging reconstruction with arbitrary k‐space trajectories using k‐space sparse matrices is applied to MRSI with spiral k‐space trajectories. The algorithm achieves MRSI reconstruction with reduced memory requirements and computing times. The results are demonstrated in both phantom and in vivo studies. Spectroscopic images very similar to that reconstructed with fully sampled spiral k‐space data are obtained at different reduction factors. Magn Reson Med 61:267–272, 2009. © 2009 Wiley‐Liss, Inc.     

Using the GRAPPA operator and the generalized sampling theorem to reconstruct undersampled non‐Cartesian data

As expected from the generalized sampling theorem of Papoulis, the use of a bunched sampling acquisition scheme in conjunction with a conjugate gradient (CG) reconstruction algorithm can decrease scan time by reducing the number of phase‐encoding lines needed to generate an unaliased image at a given resolution. However, the acquisition of such bunched data requires both modified pulse sequences and high gradient performance. A novel method of generating the “bunched” data using self‐calibrating GRAPPA operator gridding (GROG), a parallel imaging method that shifts data points by small distances in k‐space (with Δk usually less than 1.0, depending on the receiver coil) using the GRAPPA operator, is presented here. With the CG reconstruction method, these additional “bunched” points can then be used to reconstruct an image with reduced artifacts from undersampled data. This method is referred to as GROG‐facilitated bunched phase encoding (BPE), or GROG‐BPE. To better understand how the patterns of bunched points, maximal blip size, and number of bunched points affect the reconstruction quality, a number of simulations were performed using the GROG‐BPE approach. Finally, to demonstrate that this method can be combined with a variety of trajectories, examples of images with reduced artifacts reconstructed from undersampled in vivo radial, spiral, and rosette data are shown. Magn Reson Med, 2009. © 2009 Wiley‐Liss, Inc.     

MRI using a concentric rings trajectory.

The concentric rings two-dimensional (2D) k-space trajectory provides an alternative way to sample polar data. By collecting 2D k-space data in a series of rings, many unique properties are observed. The concentric rings are inherently centric-ordered, provide a smooth weighting in k-space, and enable shorter total scan times. Due to these properties, the concentric rings are well-suited as a readout trajectory for magnetization-prepared studies. When non-Cartesian trajectories are used for MRI, off-resonance effects can cause blurring and degrade the image quality. For the concentric rings, off-resonance blur can be corrected by retracing rings near the center of k-space to obtain a field map with no extra excitations, and then employing multifrequency reconstruction. Simulations show that the concentric rings exhibit minimal effects due to T(2) (*) modulation, enable shorter scan times for a Nyquist-sampled dataset than projection-reconstruction imaging or Cartesian 2D Fourier transform (2DFT) imaging, and have more spatially distributed flow and motion properties than Cartesian sampling. Experimental results show that off-resonance blurring can be successfully corrected to obtain high-resolution images. Results also show that concentric rings effectively capture the intended contrast in a magnetization-prepared sequence.     

Direct parallel image reconstructions for spiral trajectories using GRAPPA.

The use of spiral trajectories is an efficient way to cover a desired k-space partition in magnetic resonance imaging (MRI). Compared to conventional Cartesian k-space sampling, it allows faster acquisitions and results in a slight reduction of the high gradient demand in fast dynamic scans, such as in functional MRI (fMRI). However, spiral images are more susceptible to off-resonance effects that cause blurring artifacts and distortions of the point-spread function (PSF), and thereby degrade the image quality. Since off-resonance effects scale with the readout duration, the respective artifacts can be reduced by shortening the readout trajectory. Multishot experiments represent one approach to reduce these artifacts in spiral imaging, but result in longer scan times and potentially increased flow and motion artifacts. Parallel imaging methods are another promising approach to improve image quality through an increase in the acquisition speed. However, non-Cartesian parallel image reconstructions are known to be computationally time-consuming, which is prohibitive for clinical applications. In this study a new and fast approach for parallel image reconstructions for spiral imaging based on the generalized autocalibrating partially parallel acquisitions (GRAPPA) methodology is presented. With this approach the computational burden is reduced such that it becomes comparable to that needed in accelerated Cartesian procedures. The respective spiral images with two- to eightfold acceleration clearly benefit from the advantages of parallel imaging, such as enabling parallel MRI single-shot spiral imaging with the off-resonance behavior of multishot acquisitions.     

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.     

Time-resolved undersampled projection reconstruction magnetic resonance imaging of the peripheral vessels using multi-echo acquisition.

The hybrid projection reconstruction (PR) imaging provides high temporal resolution through an undersampled PR acquisition for the in-plane dimensions and Cartesian slice encoding for the through-plane dimension. The undersampling of projection data introduces streak artifact, which may severely compromise image quality. This study reports on a combination of multi-echo acquisition with time-resolved undersampled PR imaging and its application to peripheral magnetic resonance angiography. Multi-echo acquisition improved imaging speed effectively, thereby reducing the undersampling streak artifact and improving the temporal resolution. The gradient distortion was reduced through gradient calibration and accurate k-space trajectory measurement.     

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.     

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.     

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.     

k-t BLAST reconstruction from non-Cartesian k-t space sampling.

Current implementations of k-t Broad-use Linear Acqusition Speed-up Technique (BLAST) require the sampling in k-t space to conform to a lattice. To permit the use of k-t BLAST with non-Cartesian sampling, an iterative reconstruction approach is proposed in this work. This method, which is based on the conjugate gradient (CG) method and gridding reconstruction principles, can efficiently handle data that are sampled along non-Cartesian trajectories in k-t space. The approach is demonstrated on prospectively gated radial and retrospectively gated Cartesian imaging. Compared to a sliding window (SW) reconstruction, the resulting image series exhibit lower artifact levels and improved temporal fidelity. The proposed approach thus allows investigators to combine the specific advantages of non-Cartesian imaging or retrospective gating with the acceleration provided by k-t BLAST.     

Decomposed direct matrix inversion for fast non-cartesian SENSE reconstructions.

A new k-space direct matrix inversion (DMI) method is proposed here to accelerate non-Cartesian SENSE reconstructions. In this method a global k-space matrix equation is established on basic MRI principles, and the inverse of the global encoding matrix is found from a set of local matrix equations by taking advantage of the small extension of k-space coil maps. The DMI algorithm's efficiency is achieved by reloading the precalculated global inverse when the coil maps and trajectories remain unchanged, such as in dynamic studies. Phantom and human subject experiments were performed on a 1.5T scanner with a standard four-channel phased-array cardiac coil. Interleaved spiral trajectories were used to collect fully sampled and undersampled 3D raw data. The equivalence of the global k-space matrix equation to its image-space version, was verified via conjugate gradient (CG) iterative algorithms on a 2x undersampled phantom and numerical-model data sets. When applied to the 2x undersampled phantom and human-subject raw data, the decomposed DMI method produced images with small errors (< or = 3.9%) relative to the reference images obtained from the fully-sampled data, at a rate of 2 s per slice (excluding 4 min for precalculating the global inverse at an image size of 256 x 256). The DMI method may be useful for noise evaluations in parallel coil designs, dynamic MRI, and 3D sodium MRI with fixed coils and trajectories.