GPU-Accelerated Algorithms for Compressed Signals Recovery with Application to Astronomical Imagery Deblurring

Compressive sensing promises to enable bandwidth-efficient on-board compression of astronomical data by lifting the encoding complexity from the source to the receiver. The signal is recovered off-line, exploiting GPUs parallel computation capabilities to speedup the reconstruction process. However, inherent GPU hardware constraints limit the size of the recoverable signal and the speedup practically achievable. In this work, we design parallel algorithms that exploit the properties of circulant matrices for efficient GPU-accelerated sparse signals recovery. Our approach reduces the memory requirements, allowing us to recover very large signals with limited memory. In addition, it achieves a tenfold signal recovery speedup thanks to ad-hoc parallelization of matrix-vector multiplications and matrix inversions. Finally, we practically demonstrate our algorithms in a typical application of circulant matrices: deblurring a sparse astronomical image in the compressed domain

Blur resolved OCT: full-range interferometric synthetic aperture microscopy through dispersion encoding

We present a computational method for full-range interferometric synthetic aperture microscopy (ISAM) under dispersion encoding. With this, one can effectively double the depth range of optical coherence tomography (OCT), whilst dramatically enhancing the spatial resolution away from the focal plane. To this end, we propose a model-based iterative reconstruction (MBIR) method, where ISAM is directly considered in an optimization approach, and we make the discovery that sparsity promoting regularization effectively recovers the full-range signal. Within this work, we adopt an optimal nonuniform discrete fast Fourier transform (NUFFT) implementation of ISAM, which is both fast and numerically stable throughout iterations. We validate our method with several complex samples, scanned with a commercial SD-OCT system with no hardware modification. With this, we both demonstrate full-range ISAM imaging, and significantly outperform combinations of existing methods.Comment: 17 pages, 7 figures. The images have been compressed for arxiv - please follow DOI for full resolutio

Performance Comparisons of Greedy Algorithms in Compressed Sensing

Compressed sensing has motivated the development of numerous sparse approximation algorithms designed to return a solution to an underdetermined system of linear equations where the solution has the fewest number of nonzeros possible, referred to as the sparsest solution. In the compressed sensing setting, greedy sparse approximation algorithms have been observed to be both able to recovery the sparsest solution for similar problem sizes as other algorithms and to be computationally efficient; however, little theory is known for their average case behavior. We conduct a large scale empirical investigation into the behavior of three of the state of the art greedy algorithms: NIHT, HTP, and CSMPSP. The investigation considers a variety of random classes of linear systems. The regions of the problem size in which each algorithm is able to reliably recovery the sparsest solution is accurately determined, and throughout this region additional performance characteristics are presented. Contrasting the recovery regions and average computational time for each algorithm we present algorithm selection maps which indicate, for each problem size, which algorithm is able to reliably recovery the sparsest vector in the least amount of time. Though no one algorithm is observed to be uniformly superior, NIHT is observed to have an advantageous balance of large recovery region, absolute recovery time, and robustness of these properties to additive noise and for a variety of problem classes. The algorithm selection maps presented here are the first of their kind for compressed sensing

CGIHT: Conjugate Gradient Iterative Hard Thresholding\ud for Compressed Sensing and Matrix Completion

We introduce the Conjugate Gradient Iterative Hard Thresholding (CGIHT) family of algorithms for the efficient solution of constrained underdetermined linear systems of equations arising in compressed sensing, row sparse approximation, and matrix completion. CGIHT is designed to balance the low per iteration complexity of simple hard thresholding algorithms with the fast asymptotic convergence rate of employing the conjugate gradient method. We establish provable recovery guarantees and stability to noise for variants of CGIHT with sufficient conditions in terms of the restricted isometry constants of the sensing operators. Extensive empirical performance comparisons establish significant computational advantages for CGIHT both in terms of the size of problems which can be accurately approximated and in terms of overall computation time

Expander ℓ0\ell_0-Decoding

We introduce two new algorithms, Serial-$\ell_0$ and Parallel-$\ell_0$ for solving a large underdetermined linear system of equations $y = Ax \in \mathbb{R}^m$ when it is known that $x \in \mathbb{R}^n$ has at most $k < m$ nonzero entries and that $A$ is the adjacency matrix of an unbalanced left $d$-regular expander graph. The matrices in this class are sparse and allow a highly efficient implementation. A number of algorithms have been designed to work exclusively under this setting, composing the branch of combinatorial compressed-sensing (CCS). Serial-$\ell_0$ and Parallel-$\ell_0$ iteratively minimise $\|y - A\hat x\|_0$ by successfully combining two desirable features of previous CCS algorithms: the information-preserving strategy of ER, and the parallel updating mechanism of SMP. We are able to link these elements and guarantee convergence in $\mathcal{O}(dn \log k)$ operations by assuming that the signal is dissociated, meaning that all of the $2^k$ subset sums of the support of $x$ are pairwise different. However, we observe empirically that the signal need not be exactly dissociated in practice. Moreover, we observe Serial-$\ell_0$ and Parallel-$\ell_0$ to be able to solve large scale problems with a larger fraction of nonzeros than other algorithms when the number of measurements is substantially less than the signal length; in particular, they are able to reliably solve for a $k$-sparse vector $x\in\mathbb{R}^n$ from $m$ expander measurements with $n/m=10^3$ and $k/m$ up to four times greater than what is achievable by $\ell_1$-regularization from dense Gaussian measurements. Additionally, Serial-$\ell_0$ and Parallel-$\ell_0$ are observed to be able to solve large problems sizes in substantially less time than other algorithms for compressed sensing. In particular, Parallel-$\ell_0$ is structured to take advantage of massively parallel architectures.Comment: 14 pages, 10 figure

Compressive Sensing Using Iterative Hard Thresholding with Low Precision Data Representation: Theory and Applications

Modern scientific instruments produce vast amounts of data, which can overwhelm the processing ability of computer systems. Lossy compression of data is an intriguing solution, but comes with its own drawbacks, such as potential signal loss, and the need for careful optimization of the compression ratio. In this work, we focus on a setting where this problem is especially acute: compressive sensing frameworks for interferometry and medical imaging. We ask the following question: can the precision of the data representation be lowered for all inputs, with recovery guarantees and practical performance? Our first contribution is a theoretical analysis of the normalized Iterative Hard Thresholding (IHT) algorithm when all input data, meaning both the measurement matrix and the observation vector are quantized aggressively. We present a variant of low precision normalized {IHT} that, under mild conditions, can still provide recovery guarantees. The second contribution is the application of our quantization framework to radio astronomy and magnetic resonance imaging. We show that lowering the precision of the data can significantly accelerate image recovery. We evaluate our approach on telescope data and samples of brain images using CPU and FPGA implementations achieving up to a 9x speed-up with negligible loss of recovery quality.Comment: 19 pages, 5 figures, 1 table, in IEEE Transactions on Signal Processin

Vanishingly Sparse Matrices and Expander Graphs, With Application to Compressed Sensing

We revisit the probabilistic construction of sparse random matrices where each column has a fixed number of nonzeros whose row indices are drawn uniformly at random with replacement. These matrices have a one-to-one correspondence with the adjacency matrices of fixed left degree expander graphs. We present formulae for the expected cardinality of the set of neighbors for these graphs, and present tail bounds on the probability that this cardinality will be less than the expected value. Deducible from these bounds are similar bounds for the expansion of the graph which is of interest in many applications. These bounds are derived through a more detailed analysis of collisions in unions of sets. Key to this analysis is a novel {\em dyadic splitting} technique. The analysis led to the derivation of better order constants that allow for quantitative theorems on existence of lossless expander graphs and hence the sparse random matrices we consider and also quantitative compressed sensing sampling theorems when using sparse non mean-zero measurement matrices.Comment: 17 pages, 12 Postscript figure

Parallelisation of greedy algorithms for compressive sensing reconstruction

Compressive Sensing (CS) is a technique which allows a signal to be compressed at the same time as it is captured. The process of capturing and simultaneously compressing the signal is represented as linear sampling, which can encompass a variety of physical processes or signal processing. Instead of explicitly identifying redundancies in the source signal, CS relies on the property of sparsity in order to reconstruct the compressed signal. While linear sampling is much less burdensome than conventional compression, this is more than made up for by the high computational cost of reconstructing a signal which has been captured using CS. Even when using some of the fastest reconstruction techniques, known as greedy pursuits, reconstruction of large problems can pose a significant burden, consuming a great deal of memory as well as compute time. Parallel computing is the foundation of the field of High Performance Computing (HPC). Modern supercomputers are generally composed of large clusters of standard servers, with a dedicated low-latency high-bandwidth interconnect network. On such a cluster, an appropriately written program can harness vast quantities of memory and computational power. However, in order to exploit a parallel compute resource, an algorithm usually has to be redesigned from the ground up. In this thesis I describe the development of parallel variants of two algorithms commonly used in CS reconstruction, Matching Pursuit (MP) and Orthogonal Matching Pursuit (OMP), resulting in the new distributed compute algorithms DistMP and DistOMP. I present the results from experiments showing how DistMP and DistOMP can utilise a compute cluster to solve CS problems much more quickly than a single computer could alone. Speed-up of as much as a factor of 76 is observed with DistMP when utilising 210 workers across 14 servers, compared to a single worker. Finally, I demonstrate how DistOMP can solve a problem with a 429GB equivalent sampling matrix in as little as 62 minutes using a 16-node compute cluster.Funded by an ICASE award from the Engineering and Physical Sciences Research Council, with sponsorship provided by Thales Research and Technology