Deconvolution-interpolation gridding (DING): accurate reconstruction for arbitrary k-space trajectories.

A simple iterative algorithm, termed deconvolution-interpolation gridding (DING), is presented to address the problem of reconstructing images from arbitrarily-sampled k-space. The new algorithm solves a sparse system of linear equations that is equivalent to a deconvolution of the k-space with a small window. The deconvolution operation results in increased reconstruction accuracy without grid subsampling, at some cost to computational load. By avoiding grid oversampling, the new solution saves memory, which is critical for 3D trajectories. The DING algorithm does not require the calculation of a sampling density compensation function, which is often problematic. DING's sparse linear system is inverted efficiently using the conjugate gradient (CG) method. The reconstruction of the gridding system matrix is simple and fast, and no regularization is needed. This feature renders DING suitable for situations where the k-space trajectory is changed often or is not known a priori, such as when patient motion occurs during the scan. DING was compared with conventional gridding and an iterative reconstruction method in computer simulations and in vivo spiral MRI experiments. The results demonstrate a stable performance and reduced root mean square (RMS) error for DING in different k-space trajectories.     

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.     

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.     

Polar sampling in k-space: Reconstruction effects

Magnetic resonance images are most commonly computed by taking the inverse Fourier transform of the k-space data. This transformation can potentially create artifacts in the im age, depending on the reconstruction algorithm used. For equally spaced radial and azimuthal k-space polar sampling, both gridding and convolution backprojection are applicable. However, these algorithms potentially can yield different res olution, signal-to-noise ratio, and aliasing characteristics in the reconstructed image. Here, these effects are analyzed and their tradeoffs are discussed. It is shown that, provided the modulation transfer function and the signal-to-noise ratio are considered together, these algorithms perform similarly. In contrast, their aliasing behavior is different, since their re spective point spread functions (PSF) differ. In gridding, the PSF is composed of the mainlobe and ringlobes that lead to aliasing. Conversely, there are no ringlobes in the convolution backprojection PSF, thus radial aliasing effects are mini mized. Also, a hybrid gridding and convolution backprojection reconstruction is presented for radially nonequidistant k-space polar sampling.     

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.     

On optimality of parallel MRI reconstruction in k-space.

Parallel MRI reconstruction in k-space has several advantages, such as tolerance to calibration data errors and efficient non-Cartesian data processing. These benefits largely accrue from the approximation that a given unsampled k-space datum can be synthesized from only a few local samples. In this study, several aspects of parallel MRI reconstruction in k-space are studied: the design of optimized reconstruction kernels, the effect of regularization on image error, and the accuracy of different k-space-based parallel MRI methods. Reconstruction of parallel MRI data in k-space is posed as the problem of approximating the pseudoinverse with a sparse matrix. The error of the approximation is used as an optimization criterion to find reconstruction kernels optimized for the given coil setup. An efficient algorithm for automatic selection of reconstruction kernels is described. Additionally, a total error metric is introduced for validation of the reconstruction kernel and choice of regularization parameters. The new methods yield reduced reconstruction and noise errors in both simulated and real data studies when compared with existing methods. The new methods may be useful for reduction of image errors, faster data processing, and validation of parallel MRI reconstruction design for a given coil system and k-space trajectory.     

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.     

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.     

New approach to gridding using regularization and estimation theory.

When sampling under time-varying gradients, data is acquired over a non-equally spaced grid in k-space. The most computationally efficient method of reconstruction is first to interpolate the data onto a Cartesian grid, enabling the subsequent use of the inverse fast Fourier transform (IFFT). The most commonly used interpolation technique is called gridding, and is comprised of four steps: precompensation, convolution with a Kaiser-Bessel window, IFFT, and postcompensation. Recently, the author introduced a new gridding method called Block Uniform ReSampling (BURS), which is both optimal and efficient. The interpolation coefficients are computed by solving a set of linear equations using singular value decomposition (SVD). BURS requires neither the pre- nor the postcompensation steps, and resamples onto an n x n grid rather than the 2n x 2n matrix required by conventional gridding. This significantly decreases the computational complexity. Several authors have reported that although the BURS algorithm is very accurate, it is also sensitive to noise. As a consequence, even in the presence of a low level of measurement noise, the resulting image is often highly contaminated with noise. In this work, the origin of the noise sensitivity is traced back to the potentially ill-posed matrix inversion performed by BURS. Two approaches to the solution are presented. The first uses regularization theory to stabilize the inversion process. The second formulates the interpolation as an estimation problem, and employs estimation theory for the solution. The new algorithm, called rBURS, contains a regularization parameter, which is used to trade off the accuracy of the result against the signal-to-noise ratio (SNR). The results of the new method are compared with those obtained using conventional gridding via simulations. For the SNR performance of conventional gridding, it is shown that the rBURS algorithm exhibits equal or better accuracy. This is achieved at a decreased computational cost compared to conventional gridding.     

Modified block uniform resampling (BURS) algorithm using truncated singular value decomposition: fast accurate gridding with noise and artifact reduction.

The block uniform resampling (BURS) algorithm is a newly proposed regridding technique for nonuniformly-sampled k-space MRI. Even though it is a relatively computationally intensive algorithm, since it uses singular value decomposition (SVD), its procedure is simple because it requires neither a pre- nor a postcompensation step. Furthermore, the reconstructed image is generally of high quality since it provides accurate gridded values when the local k-space data SNR is high. However, the BURS algorithm is sensitive to noise. Specifically, inaccurate interpolated data values are often generated in the BURS algorithm if the original k-space data are corrupted by noise, which is virtually guaranteed to occur to some extent in MRI. As a result, the reconstructed image quality is degraded despite excellent performance under ideal conditions. In this article, a method is presented which avoids inaccurate interpolated k-space data values from noisy sampled data with the BURS algorithm. The newly proposed technique simply truncates a series of singular values after the SVD is performed. This reduces the computational demand when compared with the BURS algorithm, avoids amplification of noise resulting from small singular values, and leads to image SNR improvements over the original BURS algorithm.     

Robust GRAPPA reconstruction and its evaluation with the perceptual difference model

PURPOSE: To develop and optimize a new modification of GRAPPA (generalized autocalibrating partially parallel acquisitions) MR reconstruction algorithm named "Robust GRAPPA." MATERIALS AND METHODS: In Robust GRAPPA, k-space data points were weighted before the reconstruction. Small or zero weights were assigned to "outliers" in k-space. We implemented a Slow Robust GRAPPA method, which iteratively reweighted the k-space data. It was compared to an ad hoc Fast Robust GRAPPA method, which eliminated (assigned zero weights to) a fixed percentage of k-space "outliers" following an initial estimation procedure. In comprehensive experiments the new algorithms were evaluated using the perceptual difference model (PDM), whereby image quality was quantitatively compared to the reference image. Independent variables included algorithm type, total reduction factor, outlier ratio, center filling options, and noise across multiple image datasets, providing 10,800 test images for evaluation. RESULTS: The Fast Robust GRAPPA method gave results very similar to Slow Robust GRAPPA, and showed significant improvements as compared to regular GRAPPA. Fast Robust GRAPPA added little computation time compared with regular GRAPPA. CONCLUSION: Robust GRAPPA was proposed and proved useful for improving the reconstructed image quality. PDM was helpful in designing and optimizing the MR reconstruction algorithms.     

Iterative image reconstruction for PROPELLER‐MRI using the nonuniform fast fourier transform

Purpose:

To investigate an iterative image reconstruction algorithm using the nonuniform fast Fourier transform (NUFFT) for PROPELLER (Periodically Rotated Overlapping ParallEL Lines with Enhanced Reconstruction) MRI.

Materials and Methods:

Numerical simulations, as well as experiments on a phantom and a healthy human subject were used to evaluate the performance of the iterative image reconstruction algorithm for PROPELLER, and compare it with that of conventional gridding. The trade‐off between spatial resolution, signal to noise ratio, and image artifacts, was investigated for different values of the regularization parameter. The performance of the iterative image reconstruction algorithm in the presence of motion was also evaluated.

Results:

It was demonstrated that, for a certain range of values of the regularization parameter, iterative reconstruction produced images with significantly increased signal to noise ratio, reduced artifacts, for similar spatial resolution, compared with gridding. Furthermore, the ability to reduce the effects of motion in PROPELLER‐MRI was maintained when using the iterative reconstruction approach.

Conclusion:

An iterative image reconstruction technique based on the NUFFT was investigated for PROPELLER MRI. For a certain range of values of the regularization parameter, the new reconstruction technique may provide PROPELLER images with improved image quality compared with conventional gridding. J. Magn. Reson. Imaging 2010;32:211–217. © 2010 Wiley‐Liss, Inc.     

Advances in sensitivity encoding with arbitrary k-space trajectories.

New, efficient reconstruction procedures are proposed for sensitivity encoding (SENSE) with arbitrary k-space trajectories. The presented methods combine gridding principles with so-called conjugate-gradient iteration. In this fashion, the bulk of the work of reconstruction can be performed by fast Fourier transform (FFT), reducing the complexity of data processing to the same order of magnitude as in conventional gridding reconstruction. Using the proposed method, SENSE becomes practical with nonstandard k-space trajectories, enabling considerable scan time reduction with respect to mere gradient encoding. This is illustrated by imaging simulations with spiral, radial, and random k-space patterns. Simulations were also used for investigating the convergence behavior of the proposed algorithm and its dependence on the factor by which gradient encoding is reduced. The in vivo feasibility of non-Cartesian SENSE imaging with iterative reconstruction is demonstrated by examples of brain and cardiac imaging using spiral trajectories. In brain imaging with six receiver coils, the number of spiral interleaves was reduced by factors ranging from 2 to 6. In cardiac real-time imaging with four coils, spiral SENSE permitted reducing the scan time per image from 112 ms to 56 ms, thus doubling the frame-rate.     

Parallel imaging reconstruction using automatic regularization.

Increased spatiotemporal resolution in MRI can be achieved by the use of parallel acquisition strategies, which simultaneously sample reduced k-space data using the information from multiple receivers to reconstruct full-FOV images. The price for the increased spatiotemporal resolution in parallel MRI is the degradation of the signal-to-noise ratio (SNR) in the final reconstructed images. Part of the SNR reduction results when the spatially correlated nature of the information from the multiple receivers destabilizes the matrix inversion used in the reconstruction of the full-FOV image. In this work, a reconstruction algorithm based on Tikhonov regularization is presented that reduces the SNR loss due to geometric correlations in the spatial information from the array coil elements. Reference scans are utilized as a priori information about the final reconstructed image to provide regularized estimates for the reconstruction using the L-curve technique. This automatic regularization method reduces the average g-factors in phantom images from a two-channel array from 1.47 to 0.80 in twofold sensitivity encoding (SENSE) acceleration. In vivo anatomical images from an eight-channel system show an averaged g-factor reduction of 1.22 to 0.84 in 2.67-fold acceleration.     

High-resolution diffusion-weighted imaging with interleaved variable-density spiral acquisitions

PURPOSE: To develop a multishot magnetic resonance imaging (MRI) pulse sequence and reconstruction algorithm for diffusion-weighted imaging (DWI) in the brain with submillimeter in-plane resolution. MATERIALS AND METHODS: A self-navigated multishot acquisition technique based on variable-density spiral k-space trajectory design was implemented on clinical MRI scanners. The image reconstruction algorithm takes advantage of the oversampling of the center k-space and uses the densely sampled central portion of the k-space data for both imaging reconstruction and motion correction. The developed DWI technique was tested in an agar gel phantom and three healthy volunteers. RESULTS: Motions result in phase and k-space shifts in the DWI data acquired using multishot spiral acquisitions. With the two-dimensional self-navigator correction, diffusion-weighted images with a resolution of 0.9 x 0.9 x 3 mm3 were successfully obtained using different interleaves ranging from 8-32. The measured apparent diffusion coefficient (ADC) in the homogenous gel phantom was (1.66 +/- 0.09) x 10(-3) mm2/second, which was the same as measured with single-shot methods. The intersubject average ADC from the brain parenchyma of normal adults was (0.91 +/- 0.01) x 10(-3) mm2/second, which was in a good agreement with the reported literature values. CONCLUSION: The self-navigated multishot variable-density spiral acquisition provides a time-efficient approach to acquire high-resolution diffusion-weighted images on a clinical scanner. The reconstruction algorithm based on motion correction in the k-space data is robust, and measured ADC values are accurate and reproducible.     

Accelerating MR diffusion tensor imaging via filtered reduced-encoding projection-reconstruction.

MR diffusion tensor imaging (DTI) is a promising tool for characterizing the microstructure of ordered tissues. However, its practical applications are hampered by relatively low signal-to-noise-ratio and spatial and temporal resolution. Reduced-encoding imaging (REI) via k-space sharing with constrained reconstruction has previously been shown to be effective for accelerating DTI, although the implementation was based on rectilinear k-space sampling. Due to the intrinsic oversampling of central k-space and allowance for isotropic downsampling, projection-reconstruction (PR) imaging may be better suited for REI. In this study, regularization procedures, including radial filtering and baseline signal correction to adequately reconstruct reduced encoded PR imaging data, are investigated. The proposed filtered reduced-encoding projection-reconstruction (FREPR) technique is applied to DTI tissue fiber orientation and fractional anisotropy (FA) measurements. Results show that FREPR offers improved reconstructions of the reduced encoded images and on an equal total scan-time basis provides more accurate fiber orientation and FA measurements compared to rectilinear k-space sampling-based REI methods or a control experiment consisting of only fully encoded images. These findings suggest a potentially significant role of FREPR in accelerating repeated imaging and improving the data acquisition-time efficiency of DTI experiments.     

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.     

Variable resolution reconstruction for Cartesian data acquired with nonconstant sampling density in phase-encoding direction.

The variable-kernel extent technique is applied for providing local high-resolution images from k-space data sampled on a Cartesian sampling grid with gradually decreasing sampling density in the phase-encoding direction. The approach is based on a variable spatial resolution reconstruction technique providing gradually decreasing resolution in the phase-encoding direction with increasing distance to the image center, while preserving full spatial resolution in a narrow slab centered in spatial domain. Reconstruction is performed by a variable convolution kernel gridding technique. The convolution kernel width is chosen proportional to the k-space sampling spacing to utilize the respective apodization in the image for reduction of the aliasing artifacts. Application of this technique to carotid artery wall imaging shows the potential of the technique for a significant reduction of image acquisition time without sacrificing image quality in the region of the carotid arteries.     

Removal of lipid artifacts in 1H spectroscopic imaging by data extrapolation

Proton MR spectroscopic imaging (MRSI) of human cerebral cortex is complicated by the presence of an intense signal from subcutaneous lipids, which, if not suppressed before Fourier reconstruction, causes ringing and signal contamination throughout the metabolite images as a result of limited k-space sampling. In this article, an improved reconstruction of the lipid region is obtained using the Papoulis-Gerchberg algorithm. This procedure makes use of the narrow-band-limited nature of the subcutaneous lipid signal to extrapolate to higher k-space values without alteration of the metabolite signal region. Using computer simulations and in vivo experimental studies, the implementation and performance of this algorithm were examined. This method was found to permit MRSI brain spectra to be obtained without applying any lipid suppression during data acquisition, at echo times of 50 ms and longer. When applied together with optimized acquisition methods, this provides an effective procedure for imaging metabolite distributions in cerebral cortical surface regions.     

Spiral reconstruction by regridding to a large rectilinear matrix: a practical solution for routine systems.

Spiral trajectories offer a number of attractive features for fast imaging. A practical problem for the implementation on routine magnetic resonance scanners is the lack of appropriate and efficient reconstruction algorithms in the available scanner software. In this paper, a simple way to implement a spiral reconstruction algorithm is described that avoids the data interpolation required by gridding approaches commonly used. Using the optimized fast Fourier transform built into each scanner, it offers image reconstruction times of less than 1 second and thus allows the introduction of spiral imaging to routine scanners.