Compressive Sensing Literature Research Articles

Spectral Gap-Based Seismic Survey Design

Seismic imaging in challenging sedimentary basins and reservoirs requires acquiring, processing, and imaging very large volumes of data (tens of terabytes). To reduce the cost of acquisition and the time from acquiring the data to producing a subsurface image, novel acquisition systems based on compressive sensing (CS), low-rank matrix recovery (LRMR), and randomized sampling have been developed and implemented. These approaches allow practitioners to achieve dense wavefield reconstruction from a substantially reduced number of field samples. However, designing acquisition surveys suited for this new sampling paradigm remains a critical and challenging role in oil, gas, and geothermal exploration. Typical random designs studied in the LRMR and CS literature are difficult to achieve by standard industry hardware. For practical purposes, a compromise between stochastic and realizable samples is needed. In this article, we propose a deterministic and computationally cheap tool to alleviate randomized acquisition design, prior to survey deployment and large-scale optimization. We consider universal and deterministic matrix completion results in the context of seismology, where a bipartite graph representation of the source–receiver layout allows for the respective spectral gap (SG) to act as a quality metric for wavefield reconstruction. We provide realistic scenarios to demonstrate the utility of the SG as a flexible tool that can be incorporated into existing survey design workflows for successful seismic data acquisition via low-rank and sparse signal recovery.

Matrix Element-Based Theory of Compressive Sensing and Its Application to Electromagnetic Imaging

We present a simple and general theory of compressive sensing (CS) that relies on elements of the sensing matrix rather than on the number of measurements. We prove the exact recovery using a dual certificate by showing that the sensing matrix satisfies an incoherence property and isotropy property if the sparsity level is kept lower than the reciprocal of the largest element of a matrix created from the sensing matrix. Unlike the CS literature, this unconventional approach does not require a linear relationship between the sparsity and the number of measurements and at the same time, can easily be evaluated. This adaptability captures anisotropic measurements appropriately as with anisotropic measurements, adding more measurements does not really imply that a signal with more nonzero elements will be recovered exactly. As an illustration, we demonstrate the theory’s ability to accurately handle the anisotropic (Green’s function-based sensing matrix) measurements and also its similarity to the existing CS literature for isotropic (Fourier) measurements. Further, we show the usefulness of the theory in comparing different sensing matrices and in generating dielectric images. The dielectric images are perfectly recovered even when there is only a single transmitter.

Efficient Detectors for Joint Compressed Sensing Detection and Channel Decoding

In slotted random access of many nodes, multi-user detection (MUD) can be applied to handle collisions. One novel PHY layer approach for jointly detecting activity and data in such a setting is Compressed Sensing based Multi-User Detection (CS-MUD). In this paper, we first summarize previous investigations on CS-MUD and subsequently propose two novel solutions for problems which have not yet been fully addressed: Firstly, we improve on previous results, by introducing a new approach which incorporates the channel decoder into the Compressed Sensing (CS) detector. Secondly, we analyze the resource efficiency of CS-MUD by adapting phase diagrams known from CS literature to the application of sporadic communication.

Pitfalls and possibilities of radar compressive sensing.

In this paper, we consider the application of compressive sensing (CS) to radar remote sensing applications. We survey a suite of practical system-level issues related to the compression of radar measurements, and we advocate the consideration of these issues by researchers exploring potential gains of CS in radar applications. We also give abbreviated examples of decades-old radio-frequency (RF) practices that already embody elements of CS for relevant applications. In addition to the cautionary implications of system-level issues and historical precedents, we identify several promising results that RF practitioners may gain from the recent explosion of CS literature.

Joint Channel Estimation and Symbol Detection for OFDM Systems in Rapidly Time-Varying Sparse Multipath Channels

In this paper, we propose a space-alternating generalized expectation maximization (SAGE) based joint channel estimation and data detection algorithm in compressive sensing (CS) framework for orthogonal frequency-division multiplexing (OFDM) systems in rapidly time-varying sparse multipath channels. Using dynamic parametric channel model, the sparse multipath channel is parameterized by a small number of distinct paths, each represented by the path delays and path gains. In our model, we assume that the path gains rapidly vary within the OFDM symbol duration while the number of paths and path delays vary symbol by symbol. Since the convergency of the SAGE algorithm needs statistically independent parameter set of interest to be estimated, we specifically choose the discrete orthonormal Karhunen---Loeve basis expansion model (DKL-BEM) to provide statistically independent BEM coefficients within one OFDM symbol duration and use just a few significant BEM coefficients to represent the rapidly time-varying path gains. The resulting SAGE algorithm that also incorporates inter-channel interference cancellation updates the data sequences and the channel parameters serially. The computer simulations show that our proposed algorithm has better channel estimation and symbol error rate performance than that of the orthogonal matching pursuit algorithm that is commonly proposed in the CS literature.

Signal Space CoSaMP for Sparse Recovery With Redundant Dictionaries

Compressive sensing (CS) has recently emerged as a powerful framework for acquiring sparse signals. The bulk of the CS literature has focused on the case where the acquired signal has a sparse or compressible representation in an orthonormal basis. In practice, however, there are many signals that cannot be sparsely represented or approximated using an orthonormal basis, but that do have sparse representations in a redundant dictionary. Standard results in CS can sometimes be extended to handle this case provided that the dictionary is sufficiently incoherent or well conditioned, but these approaches fail to address the case of a truly redundant or overcomplete dictionary. In this paper, we describe a variant of the iterative recovery algorithm CoSaMP for this more challenging setting. We utilize the \mbi D-RIP, a condition on the sensing matrix analogous to the well-known restricted isometry property. In contrast to prior work, the method and analysis are “signal-focused”; that is, they are oriented around recovering the signal rather than its dictionary coefficients. Under the assumption that we have a near-optimal scheme for projecting vectors in signal space onto the model family of candidate sparse signals, we provide provable recovery guarantees. Developing a practical algorithm that can provably compute the required near-optimal projections remains a significant open problem, but we include simulation results using various heuristics that empirically exhibit superior performance to traditional recovery algorithms.

Kronecker Compressive Sensing

Compressive sensing (CS) is an emerging approach for the acquisition of signals having a sparse or compressible representation in some basis. While the CS literature has mostly focused on problems involving 1-D signals and 2-D images, many important applications involve multidimensional signals; the construction of sparsifying bases and measurement systems for such signals is complicated by their higher dimensionality. In this paper, we propose the use of Kronecker product matrices in CS for two purposes. First, such matrices can act as sparsifying bases that jointly model the structure present in all of the signal dimensions. Second, such matrices can represent the measurement protocols used in distributed settings. Our formulation enables the derivation of analytical bounds for the sparse approximation of multidimensional signals and CS recovery performance, as well as a means of evaluating novel distributed measurement schemes.

Joint reconstruction of multiecho MR images using correlated sparsity

This works addresses the problem of reconstructing multiple T1- or T2-weighted images of the same anatomical cross section from partially sampled K-space data. Previous studies in reconstructing magnetic resonance (MR) images from partial samples of the K-space used compressed sensing (CS) techniques to exploit the spatial correlation of the images (leading to sparsity in wavelet domain). Such techniques can be employed to reconstruct the individual T1- or T2-weighted images. However, in the current context, the different images are not really independent; they are images of the same cross section and, hence, are highly correlated. We exploit the correlation between the images, along with the spatial correlation within the images to achieve better reconstruction results than exploiting spatial correlation only. For individual MR images, CS-based techniques lead to a sparsity-promoting optimization problem in the wavelet domain. In this article, we show that the same framework can be extended to incorporate correlation between images leading to group/row sparsity-promoting optimization. Algorithms for solving such optimization problems have already been developed in the CS literature. We show that significant improvement in reconstruction accuracy can be achieved by considering the correlation between different T1- and T2-weighted images. If the reconstruction accuracy is considered to be constant, our proposed group sparse formulation can yield the same result with 33% less K-space samples compared with simple sparsity-promoting reconstruction. Moreover, the reconstruction time by our proposed method is about two to four times less than the previous method.

Democracy in action: Quantization, saturation, and compressive sensing

Recent theoretical developments in the area of compressive sensing (CS) have the potential to significantly extend the capabilities of digital data acquisition systems such as analog-to-digital converters and digital imagers in certain applications. To date, most of the CS literature has been devoted to studying the recovery of sparse signals from a small number of linear measurements. In this paper, we study more practical CS systems where the measurements are quantized to a finite number of bits; in such systems some of the measurements typically saturate, causing significant nonlinearity and potentially unbounded errors. We develop two general approaches to sparse signal recovery in the face of saturation error. The first approach merely rejects saturated measurements; the second approach factors them into a conventional CS recovery algorithm via convex consistency constraints. To prove that both approaches are capable of stable signal recovery, we exploit the heretofore relatively unexplored property that many CS measurement systems are democratic, in that each measurement carries roughly the same amount of information about the signal being acquired. A series of computational experiments indicate that the signal acquisition error is minimized when a significant fraction of the CS measurements is allowed to saturate (10–30% in our experiments). This challenges the conventional wisdom of both conventional sampling and CS.

Compressed-sensing MRI with random encoding.

Compressed sensing (CS) has the potential to reduce magnetic resonance (MR) data acquisition time. In order for CS-based imaging schemes to be effective, the signal of interest should be sparse or compressible in a known representation, and the measurement scheme should have good mathematical properties with respect to this representation. While MR images are often compressible, the second requirement is often only weakly satisfied with respect to commonly used Fourier encoding schemes. This paper investigates the use of random encoding for CS-MRI, in an effort to emulate the "universal" encoding schemes suggested by the theoretical CS literature. This random encoding is achieved experimentally with tailored spatially-selective radio-frequency (RF) pulses. Both simulation and experimental studies were conducted to investigate the imaging properties of this new scheme with respect to Fourier schemes. Results indicate that random encoding has the potential to outperform conventional encoding in certain scenarios. However, our study also indicates that random encoding fails to satisfy theoretical sufficient conditions for stable and accurate CS reconstruction in many scenarios of interest. Therefore, there is still no general theoretical performance guarantee for CS-MRI, with or without random encoding, and CS-based methods should be developed and validated carefully in the context of specific applications.