Michael Lustig

Sparse MRI: The Application of Compressed Sensing for Rapid MR Imaging

The sparsity which is implicit in MR images is exploited to significantly undersample k‐space. Some MR images such as angiograms are already sparse in the pixel representation; other, more complicated images have a sparse representation in some transform domain–for example, in terms of spatial finite‐differences or their wavelet coefficients. According to the recently developed mathematical theory of compressed‐sensing, images with a sparse representation can be recovered from randomly undersampled k‐space data, provided an appropriate nonlinear recovery scheme is used. Intuitively, artifacts due to random undersampling add as noise‐like interference. In the sparse transform domain the significant coefficients stand out above the interference. A nonlinear thresholding scheme can recover the sparse coefficients, effectively recovering the image itself. In this article, practical incoherent undersampling schemes are developed and analyzed by means of their aliasing interference. Incoherence is introduced by pseudo‐random variable‐density undersampling of phase‐encodes. The reconstruction is performed by minimizing the ℓ1 norm of a transformed image, subject to data fidelity constraints. Examples demonstrate improved spatial resolution and accelerated acquisition for multislice fast spin‐echo brain imaging and 3D contrast enhanced angiography. Magn Reson Med, 2007.

An Interior-Point Method for Large-Scale

Recently, a lot of attention has been paid to regularization based methods for sparse signal reconstruction (e.g., basis pursuit denoising and compressed sensing) and feature selection (e.g., the Lasso algorithm) in signal processing, statistics, and related fields. These problems can be cast as -regularized least-squares programs (LSPs), which can be reformulated as convex quadratic programs, and then solved by several standard methods such as interior-point methods, at least for small and medium size problems. In this paper, we describe a specialized interior-point method for solving large-scale -regularized LSPs that uses the preconditioned conjugate gradients algorithm to compute the search direction. The interior-point method can solve large sparse problems, with a million variables and observations, in a few tens of minutes on a PC. It can efficiently solve large dense problems, that arise in sparse signal recovery with orthogonal transforms, by exploiting fast algorithms for these transforms. The method is illustrated on a magnetic resonance imaging data set.

\ell_1

This article reviews the requirements for successful compressed sensing (CS), describes their natural fit to MRI, and gives examples of four interesting applications of CS in MRI. The authors emphasize on an intuitive understanding of CS by describing the CS reconstruction as a process of interference cancellation. There is also an emphasis on the understanding of the driving factors in applications, including limitations imposed by MRI hardware, by the characteristics of different types of images, and by clinical concerns.

-Regularized Least Squares

A new approach to autocalibrating, coil‐by‐coil parallel imaging reconstruction, is presented. It is a generalized reconstruction framework based on self‐consistency. The reconstruction problem is formulated as an optimization that yields the most consistent solution with the calibration and acquisition data. The approach is general and can accurately reconstruct images from arbitrary k‐space sampling patterns. The formulation can flexibly incorporate additional image priors such as off‐resonance correction and regularization terms that appear in compressed sensing. Several iterative strategies to solve the posed reconstruction problem in both image and k‐space domain are presented. These are based on a projection over convex sets and conjugate gradient algorithms. Phantom and in vivo studies demonstrate efficient reconstructions from undersampled Cartesian and spiral trajectories. Reconstructions that include off‐resonance correction and nonlinear ℓ1‐wavelet regularization are also demonstrated. Magn Reson Med, 2010.

Compressed Sensing MRI

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

SPIRiT: Iterative self-consistent parallel imaging reconstruction from arbitrary k-space

High polarization of nuclear spins in liquid state through dynamic nuclear polarization has enabled the direct monitoring of 13C metabolites in vivo at very high signal-to-noise, allowing for rapid assessment of tissue metabolism. The abundant SNR afforded by this hyperpolarization technique makes high-resolution 13C 3D-MRSI feasible. However, the number of phase encodes that can be fit into the short acquisition time for hyperpolarized imaging limits spatial coverage and resolution. To take advantage of the high SNR available from hyperpolarization, we have applied compressed sensing to achieve a factor of 2 enhancement in spatial resolution without increasing acquisition time or decreasing coverage. In this paper, the design and testing of compressed sensing suited for a flyback 13C 3D-MRSI sequence are presented. The key to this design was the undersampling of spectral k-space using a novel blipped scheme, thus taking advantage of the considerable sparsity in typical hyperpolarized 13C spectra. Phantom tests validated the accuracy of the compressed sensing approach and initial mouse experiments demonstrated in vivo feasibility.

ESPIRiT — An Eigenvalue Approach to Autocalibrating Parallel MRI: Where SENSE meets GRAPPA

We present l1 -SPIRiT, a simple algorithm for auto calibrating parallel imaging (acPI) and compressed sensing (CS) that permits an efficient implementation with clinically-feasible runtimes. We propose a CS objective function that minimizes cross-channel joint sparsity in the wavelet domain. Our reconstruction minimizes this objective via iterative soft-thresholding, and integrates naturally with iterative self-consistent parallel imaging (SPIRiT). Like many iterative magnetic resonance imaging reconstructions, l1-SPIRiTs image quality comes at a high computational cost. Excessively long runtimes are a barrier to the clinical use of any reconstruction approach, and thus we discuss our approach to efficiently parallelizing l1 -SPIRiT and to achieving clinically-feasible runtimes. We present parallelizations of l1 -SPIRiT for both multi-GPU systems and multi-core CPUs, and discuss the software optimization and parallelization decisions made in our implementation. The performance of these alternatives depends on the processor architecture, the size of the image matrix, and the number of parallel imaging channels. Fundamentally, achieving fast runtime requires the correct trade-off between cache usage and parallelization overheads. We demonstrate image quality via a case from our clinical experimentation, using a custom 3DFT spoiled gradient echo (SPGR) sequence with up to 8× acceleration via Poisson-disc undersampling in the two phase-encoded directions.

Compressed Sensing for Resolution Enhancement of Hyperpolarized 13C Flyback 3D-MRSI

PURPOSE To develop a method that combines parallel imaging and compressed sensing to enable faster and/or higher spatial resolution magnetic resonance (MR) imaging and show its feasibility in a pediatric clinical setting. MATERIALS AND METHODS Institutional review board approval was obtained for this HIPAA-compliant study, and informed consent or assent was given by subjects. A pseudorandom k-space undersampling pattern was incorporated into a three-dimensional (3D) gradient-echo sequence; aliasing then has an incoherent noiselike pattern rather than the usual coherent fold-over wrapping pattern. This k-space-sampling pattern was combined with a compressed sensing nonlinear reconstruction method that exploits the assumption of sparsity of medical images to permit reconstruction from undersampled k-space data and remove the noiselike aliasing. Thirty-four patients (15 female and 19 male patients; mean age, 8.1 years; range, 0-17 years) referred for cardiovascular, abdominal, and knee MR imaging were scanned with this 3D gradient-echo sequence at high acceleration factors. Obtained k-space data were reconstructed with both a traditional parallel imaging algorithm and the nonlinear method. Both sets of images were rated for image quality, radiologist preference, and delineation of specific structures by two radiologists. Wilcoxon and symmetry tests were performed to test the hypothesis that there was no significant difference in ratings for image quality, preference, and delineation of specific structures. RESULTS Compressed sensing images were preferred more often, had significantly higher image quality ratings, and greater delineation of anatomic structures (P < .001) than did images obtained with the traditional parallel reconstruction method. CONCLUSION A combination of parallel imaging and compressed sensing is feasible in a clinical setting and may provide higher resolution and/or faster imaging, addressing the challenge of delineating anatomic structures in pediatric MR imaging.

Fast

High polarization of nuclear spins in liquid state through hyperpolarized technology utilizing dynamic nuclear polarization has enabled the direct monitoring of 13C metabolites in vivo at a high signal‐to‐noise ratio. Acquisition time limitations due to T1 decay of the hyperpolarized signal require accelerated imaging methods, such as compressed sensing, for optimal speed and spatial coverage. In this paper, the design and testing of a new echo‐planar 13C three‐dimensional magnetic resonance spectroscopic imaging (MRSI) compressed sensing sequence is presented. The sequence provides up to a factor of 7.53 in acceleration with minimal reconstruction artifacts. The key to the design is employing x and y gradient blips during a fly‐back readout to pseudorandomly undersample kf‐kx‐ky space. The design was validated in simulations and phantom experiments where the limits of undersampling and the effects of noise on the compressed sensing nonlinear reconstruction were tested. Finally, this new pulse sequence was applied in vivo in preclinical studies involving transgenic prostate cancer and transgenic liver cancer murine models to obtain much higher spatial and temporal resolution than possible with conventional echo‐planar spectroscopic imaging methods. Magn Reson Med, 2010.

\ell_1

In this paper, we present a multi-scale dictionary learning paradigm for sparse and redundant signal representations. The appeal of such a dictionary is obvious-in many cases data naturally comes at different scales. A multi-scale dictionary should be able to combine the advantages of generic multi-scale representations (such as Wavelets), with the power of learned dictionaries, in capturing the intrinsic characteristics of a family of signals. Using such a dictionary would allow representing the data in a more efficient, i.e., sparse, manner, allowing applications to take a more global look at the signal. In this paper, we aim to achieve this goal without incurring the costs of an explicit dictionary with large atoms. The K-SVD using Wavelets approach presented here applies dictionary learning in the analysis domain of a fixed multi-scale operator. This way, sub-dictionaries at different data scales, consisting of small atoms, are trained. These dictionaries can then be efficiently used in sparse coding for various image processing applications, potentially outperforming both single-scale trained dictionaries and multi-scale analytic ones. In this paper, we demonstrate this construction and discuss its potential through several experiments performed on fingerprint and coastal scenery images.