Following the success of deep learning in a wide range of applications, neural network-based machine-learning techniques have received interest as a means of accelerating magnetic resonance imaging (MRI). A number of ideas inspired by deep-learning techniques for computer vision and image processing have been successfully applied to nonlinear image reconstruction in the spirit of compressed sensing for both low-dose computed tomography and accelerated MRI. The additional integration of multicoil information to recover missing k-space lines in the MRI reconstruction process is studied less frequently, even though it is the de facto standard for the currently used accelerated MR acquisitions. This article provides an overview of the recent machine-learning approaches that have been proposed specifically for improving parallel imaging. A general background introduction to parallel MRI is given and structured around the classical view of image- and k-space-based methods. Linear and nonlinear methods are covered, followed by a discussion of the recent efforts to further improve parallel imaging using machine learning and, specifically, artificial neural networks. Image domain-based techniques that introduce improved regularizers are covered as well as k-space-based methods, where the focus is on better interpolation strategies using neural networks. Issues and open problems are discussed and recent efforts for producing open data sets and benchmarks for the community are examined.

Functional magnetic resonance imaging (fMRI) has become an indispensable tool for investigating the human brain. However, the inherently poor signal-to-noise-ratio (SNR) of the fMRI measurement represents a major barrier to expanding its spatiotemporal scale as well as its utility and ultimate impact. Here we introduce a denoising technique that selectively suppresses the thermal noise contribution to the fMRI experiment. Using 7-Tesla, high-resolution human brain data, we demonstrate improvements in key metrics of functional mapping (temporal-SNR, the detection and reproducibility of stimulus-induced signal changes, and accuracy of functional maps) while leaving the amplitude of the stimulus-induced signal changes, spatial precision, and functional point-spread-function unaltered. We demonstrate that the method enables the acquisition of ultrahigh resolution (0.5 mm isotropic) functional maps but is also equally beneficial for a large variety of fMRI applications, including supra-millimeter resolution 3- and 7-Tesla data obtained over different cortical regions with different stimulation/task paradigms and acquisition strategies.

In this paper, we study the number of measurements required to recover a sparse signal in C^M with L nonzero coefficients from compressed samples in the presence of noise. We consider a number of different recovery criteria, including the exact recovery of the support of the signal, which was previously considered in the literature, as well as new criteria for the recovery of a large fraction of the support of the signal, and the recovery of a large fraction of the energy of the signal. For these recovery criteria, we prove that O(L) (an asymptotically linear multiple of L) measurements are necessary and sufficient for signal recovery, whenever L grows linearly as a function of M. This improves on the existing literature that is mostly focused on variants of a specific recovery algorithm based on convex programming, for which O(L log(M - L)) measurements are required. In contrast, the implementation of our proof method would have a higher complexity. We also show that O(L log(M - L)) measurements are required in the sublinear regime (L - o(M)). For our sufficiency proofs, we introduce a Shannon-theoretic decoder based on joint typicality, which allows error events to be defined in terms of a single random variable in contrast to previous information-theoretic work, where comparison of random variables are required. We also prove concentration results for our error bounds implying that a randomly selected Gaussian matrix will suffice with high probability. For our necessity proofs, we rely on results from channel coding and rate-distortion theory.

This paper develops a novel linear-time algorithm, termed SPARse Truncated Amplitude flow (SPARTA), to reconstruct a sparse signal from a small number of magnitude-only measurements. It deals with what is also known as sparse phase retrieval (PR), which is NP-hard in general and emerges in many science and engineering applications. Upon formulating sparse PR as an amplitude-based nonconvex optimization task, SPARTA works iteratively in two stages: In stage one, the support of the underlying sparse signal is recovered using an analytically well-justified rule, and subsequently a sparse orthogonality-promoting initialization is obtained via power iterations restricted on the support; and, in stage two, the initialization is successively refined by means of hard thresholding based truncated gradient iterations. SPARTA is a simple yet effective, scalable, and fast sparse PR solver. On the theoretical side, for any $n$-dimensional $k$-sparse ($k\ll n$) signal $\bm{x}$ with minimum (in modulus) nonzero entries on the order of $(1/\sqrt{k})\|\bm{x}\|_2$, SPARTA recovers the signal exactly (up to a global unimodular constant) from about $k^2\log n$ random Gaussian measurements with high probability. Furthermore, SPARTA incurs computational complexity on the order of $k^2n\log n$ with total runtime proportional to the time required to read the data, which improves upon the state-of-the-art by at least a factor of $k$. Finally, SPARTA is robust against additive noise of bounded support. Extensive numerical tests corroborate markedly improved recovery performance and speedups of SPARTA relative to existing alternatives.