Sparsity and Parallel Acquisition: Optimal Uniform and Nonuniform Recovery Guarantees

The problem of multiple sensors simultaneously acquiring measurements of a single object can be found in many applications. In this paper, we present the optimal recovery guarantees for the recovery of compressible signals from multi-sensor measurements using compressed sensing. In the first half of the paper, we present both uniform and nonuniform recovery guarantees for the conventional sparse signal model in a so-called distinct sensing scenario. In the second half, using the so-called sparse and distributed signal model, we present nonuniform recovery guarantees which effectively broaden the class of sensing scenarios for which optimal recovery is possible, including to the so-called identical sampling scenario. To verify our recovery guarantees we provide several numerical results including phase transition curves and numerically-computed bounds.Comment: 13 pages and 3 figure

Compressed Sensing and Parallel Acquisition

Parallel acquisition systems arise in various applications in order to moderate problems caused by insufficient measurements in single-sensor systems. These systems allow simultaneous data acquisition in multiple sensors, thus alleviating such problems by providing more overall measurements. In this work we consider the combination of compressed sensing with parallel acquisition. We establish the theoretical improvements of such systems by providing recovery guarantees for which, subject to appropriate conditions, the number of measurements required per sensor decreases linearly with the total number of sensors. Throughout, we consider two different sampling scenarios -- distinct (corresponding to independent sampling in each sensor) and identical (corresponding to dependent sampling between sensors) -- and a general mathematical framework that allows for a wide range of sensing matrices (e.g., subgaussian random matrices, subsampled isometries, random convolutions and random Toeplitz matrices). We also consider not just the standard sparse signal model, but also the so-called sparse in levels signal model. This model includes both sparse and distributed signals and clustered sparse signals. As our results show, optimal recovery guarantees for both distinct and identical sampling are possible under much broader conditions on the so-called sensor profile matrices (which characterize environmental conditions between a source and the sensors) for the sparse in levels model than for the sparse model. To verify our recovery guarantees we provide numerical results showing phase transitions for a number of different multi-sensor environments.Comment: 43 pages, 4 figure

Auto-Calibration and Biconvex Compressive Sensing with Applications to Parallel MRI

We study an auto-calibration problem in which a transform-sparse signal is compressive-sensed by multiple sensors in parallel with unknown sensing parameters. The problem has an important application in pMRI reconstruction, where explicit coil calibrations are often difficult and costly to achieve in practice, but nevertheless a fundamental requirement for high-precision reconstructions. Most auto-calibrated strategies result in reconstruction that corresponds to solving a challenging biconvex optimization problem. We transform the auto-calibrated parallel sensing as a convex optimization problem using the idea of `lifting'. By exploiting sparsity structures in the signal and the redundancy introduced by multiple sensors, we solve a mixed-norm minimization problem to recover the underlying signal and the sensing parameters simultaneously. Robust and stable recovery guarantees are derived in the presence of noise and sparsity deficiencies in the signals. For the pMRI application, our method provides a theoretically guaranteed approach to self-calibrated parallel imaging to accelerate MRI acquisitions under appropriate assumptions. Developments in MRI are discussed, and numerical simulations using the analytical phantom and simulated coil sensitives are presented to support our theoretical results.Comment: Keywords: Self-calibration, Compressive sensing, Convex optimization, Random matrices, Parallel MR

CORE-Deblur: Parallel MRI Reconstruction by Deblurring Using Compressed Sensing

In this work we introduce a new method that combines Parallel MRI and Compressed Sensing (CS) for accelerated image reconstruction from subsampled k-space data. The method first computes a convolved image, which gives the convolution between a user-defined kernel and the unknown MR image, and then reconstructs the image by CS-based image deblurring, in which CS is applied for removing the inherent blur stemming from the convolution process. This method is hence termed CORE-Deblur. Retrospective subsampling experiments with data from a numerical brain phantom and in-vivo 7T brain scans showed that CORE-Deblur produced high-quality reconstructions, comparable to those of a conventional CS method, while reducing the number of iterations by a factor of 10 or more. The average Normalized Root Mean Square Error (NRMSE) obtained by CORE-Deblur for the in-vivo datasets was 0.016. CORE-Deblur also exhibited robustness regarding the chosen kernel and compatibility with various k-space subsampling schemes, ranging from regular to random. In summary, CORE-Deblur enables high quality reconstructions and reduction of the CS iterations number by 10-fold.Comment: 11 pages, 6 figures, 1 tabl

Calibrationless Multi-coil Magnetic Resonance Imaging with Compressed Sensing

We present a method for combining the data retrieved by multiple coils of a Magnetic Resonance Imaging (MRI) system with the a priori assumption of compressed sensing to reconstruct a single image. The final image is the result of an optimization problem that only includes constraints based on fundamental physics (Maxwell's equations and the Biot-Savart law) and accepted phenomena (e.g. sparsity in the Wavelet domain). The problem is solved using an alternating minimization approach: two convex optimization problems are alternately solved, one with the Fast Iterative Shrinkage Threshold Algorithm (FISTA) and the other with the Primal-Dual Hybrid Gradient (PDHG) method. We show results on simulated data as well as data of the knee, brain, and ankle. In all cases studied, results from the new algorithm show higher quality and increased detail when compared to conventional reconstruction algorithms

An Empirical Study of the Maximum Degree of Undersampling in Compressed Sensing for T2*-weighted MRI

International audienceMagnetic Resonance Imaging (MRI) is one of the most dynamic and safe imaging modalities used in clinical routine today. Yet, one major limitation to this technique resides in its long acquisition times. Over the last decade, Compressed Sensing (CS) has been increasingly used to address this issue and offers to shorten MR scans by reconstructing images from undersampled Fourier data. Nevertheless, a quantitative guide on the degree of acceleration applicable to a given acquisition scenario is still lacking today, leading in practice to a trial-and-error approach in the selection of the appropriate undersampling factor. In this study, we shortly point out the existing theoretical sampling results in CS and their limitations which motivate the focus of this work: an empirical and quantitative analysis of the maximum degree of undersampling allowed by CS in the specific context of T2*-weighted MRI. We make use of a generic method based on retrospective undersampling to quantitatively deduce the maximum acceleration factor R max which preserves a desired image quality as a function of the image resolution and the available signal-to-noise ratio (SNR). Our results quantify how larger acceleration factors can be applied to higher resolution images as long as a minimum SNR is guaranteed. In practice however, the maximum acceleration factor for a given resolution appears to be constrained by the available SNR inherent to the considered acquisition. Our analysis enables to take this a priori knowledge into account, allowing to derive a sequence-specific maximum acceleration factor adapted to the intrinsic SNR of any MR pipeline. These results obtained on an analytical T2*-weighted phantom image were corroborated by prospective experiments performed on MR data collected with radial trajectories on a 7 Tesla scanner with the same contrast. The proposed framework allows to study other sequence weightings and therefore better optimize sequences when accelerated using CS

Robust Algorithms for Low-Rank and Sparse Matrix Models

Data in statistical signal processing problems is often inherently matrix-valued, and a natural first step in working with such data is to impose a model with structure that captures the distinctive features of the underlying data. Under the right model, one can design algorithms that can reliably tease weak signals out of highly corrupted data. In this thesis, we study two important classes of matrix structure: low-rankness and sparsity. In particular, we focus on robust principal component analysis (PCA) models that decompose data into the sum of low-rank and sparse (in an appropriate sense) components. Robust PCA models are popular because they are useful models for data in practice and because efficient algorithms exist for solving them. This thesis focuses on developing new robust PCA algorithms that advance the state-of-the-art in several key respects. First, we develop a theoretical understanding of the effect of outliers on PCA and the extent to which one can reliably reject outliers from corrupted data using thresholding schemes. We apply these insights and other recent results from low-rank matrix estimation to design robust PCA algorithms with improved low-rank models that are well-suited for processing highly corrupted data. On the sparse modeling front, we use sparse signal models like spatial continuity and dictionary learning to develop new methods with important adaptive representational capabilities. We also propose efficient algorithms for implementing our methods, including an extension of our dictionary learning algorithms to the online or sequential data setting. The underlying theme of our work is to combine ideas from low-rank and sparse modeling in novel ways to design robust algorithms that produce accurate reconstructions from highly undersampled or corrupted data. We consider a variety of application domains for our methods, including foreground-background separation, photometric stereo, and inverse problems such as video inpainting and dynamic magnetic resonance imaging.PHDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/143925/1/brimoor_1.pd