Variational Bayesian Compressed Sensing for Sparse and Locally Constant Signals

In this paper, we consider the theoretical bound of the probability of error in compressed sensing (CS) with the Bayesian approach. In the detection problem, the signal is sparse and is reconstructed from a compressed measurement vector. Utilizing the oracle estimator in CS, we provide a theoretical bound of the probability of error when the noise in CS is white Gaussian noise (WGN). We show that without any additional information in CS, the probability of error obtained using the signal reconstructed by four recovery algorithms: the basis pursuit denoising (BPDN) algorithm, the Dantzig selector, the orthogonal matching pursuit (OMP) method and the compressive sampling matching pursuit (CoSaMP) algorithm is always larger than the derived theoretical bound. Simulation results demonstrate the effectiveness of our result.

Resolution analysis for the problem of signal recovery from finitely many linear measurements is the subject of this paper. The classical Rayleigh limit serves only as a lower bound on resolution since it does not assume any recovery strategy and is based only on observed data. We show that details finer than the Rayleigh limit can be recovered by simple linear processing that incorporates prior information. We first define a measure of resolution based on allowable levels of error that is more appropriate for current signal recovery strategies than the Rayleigh definition. In the practical situation in which only finitely many noisy observations are available, we have to restrict the class of signals in order to make the resolution measure meaningful. We consider the set of bandlimited and essentially timelimited signals since it describes most signals encountered in practice. For this set, we show how to precompute resolution limits from knowledge of measurement functionals, signal-to-noise ratio, passband, energy concentration regions, energy concentration factor, and a prescribed level of error tolerance. In the process, we also derive an algorithm for high-resolution signal recovery. We illustrate the results with examples in one and two dimensions.

In this paper, we study the problem of sparse vector recovery at the fusion center of a sensor network from linear sensor measurements when there is missing data. In the presence of missing data, the random sampling approach employed in compressed sensing is known to provide excellent reconstruction accuracy. However, when there is missing data, the theoretical guarantees associated with sparse recovery have not been well studied. Therefore, in this paper, we derive an upper bound on the minimum number of measurements required to ensure faithful recovery of a sparse signal when the generation of missing data is modeled using a Bernoulli erasure channel. We analyze three different network topologies, namely, star, (relay aided-)tree, and serial-star topologies. Our analysis establishes how the minimum required number of measurements for recovery scales with the network parameters, the properties of the measurement matrix, and the recovery algorithm. Finally, through numerical simulations, we show the variation of the minimum required number of measurements with different system parameters and validate our theoretical results.

The problem of sparse signal recovery from lower number of observations is formalized as a constrained minimization problem. Due to the dissimilar requirements of signal processing, the constraints and objective functions differ from each other. This paper presents an algorithm scheme base on alternating iteration to deal with the nonconvex objective function. The proposed method employs the log-penalty and proximity operator to seek the sparse solution. In each iteration, the introduction of nonconvex penalty decreases the penalization of large coefficients which contributes to a faster decrease of the function value. By making use of proximity operator, it reduces the computational complexity and achieves less number of iterations. In addition, the experimental results illustrate the effectiveness of the proposed method as well as the drawbacks of the algorithm.

Compressive sensing (CS) theory proves that signals can be recovered from far fewer random measurements than that suggested by Nyquist-Shannon theorem. This advantage of CS is extremely useful in some data processing applications, especially in video processing. Considering that nonlocal patch-based CS methodology has achieved an impressive performance in image CS field, it motivates us to generalize the idea into the video CS filed. In this paper, by taking account of the spatiotemporal information, we propose a novel strategy to reconstruct the non-key frames of a video sequence by introducing the nonlocal similar patch group from some key frames. Experimental results illustrate that our proposed nonlocal patch-based video CS recovery algorithm can extremely exploit the information of the key frames, and achieve a better performance compared with some existing popular video CS recovery algorithms. It is noted that our idea to construct a spatiotemporal comprehensive patch group may be combined with any nonlocal patch-based CS method for video reconstruction, which leads to various of nonlocal patch-based compressive sensing video recovery algorithms.

A local search enhanced differential evolutionary algorithm for sparse recovery

We consider the recovery of sparse signals, f ∈ ℝN, containing at most k ≪ N nonzero entries using linear measurements contaminated with i.i.d. Gaussian background noise. Given this measurement model, we present and analyze a computationally efficient group testing strategy for recovering the support of f and approximating its nonzero entries. In particular, we demonstrate that group testing measurement matrix constructions may be combined with statistical binary detection and estimation methods to produce efficient adaptive sequential algorithms for sparse signal support recovery. Furthermore, when f exhibits sufficient sparsity, we show that these adaptive group testing methods allow the recovery of sparse signals using fewer noisy linear measurements than possible with non-adaptive methods based on Gaussian measurement ensembles. As a result we improve on previous sufficient conditions for sparsity pattern recovery in the noisy sublinear-sparsity regime.

In practical compressive sensing (CS) communication system, Analog to Digital Converter (ADC) is a necessary component to convert analog signals into digital ones. However, nonlinear distortion in ADC is unavoidable and definitely affects the reception accuracy. Though recent works have studied various methods to combat the negative effect of ADC nonlinear distortion, few of them discuss the solution in communication system with CS. This paper studies the CS recovery method when compressive measurements suffer from ADC nonlinear distortion. A double-sparsity model is first formulated, where the original signal and the ADC nonlinear distortion are both sparse. Then, for the type of clipping ADC, we propose a corresponding algorithm based on the Alternating Direction Method of Multipliers (ADMM) strategy to solve the double-sparsity (DS) problem, named as DS-ADMM. For the type of self-reset (SR) ADC, we explore its essence of rounding operation to design an integer constraint based feedback updating (ICFU) strategy, and accordingly propose the DS-ADMM-ICFU recovery algorithm. Experiment results show that the DS-ADMM algorithm for the double-sparsity problem improves the recovery performance compared with the existing counterpart, and DS-ADMM-ICFU for SR ADC exhibits preferable advantage in typical communication systems.

Compressive sensing (CS) enables us to reconstruct a signal from a few number of measurements obtained from a random or deterministic measurement matrix. Knowledge of the sparsifying basis of the signal is required for the recovery process. In this work, we use a recently developed deterministic measurement matrix and demonstrate recovery of the original signal from compressed samples without the knowledge of the sparsifying basis or the order of sparsity. We experimented this recovery on the Biomedical images. Using smoothed l 0 -norm (SL0) as a recovery algorithm, the original images were recovered from CS measurements with high quality.

Multiphase flow measurements, based on tomographic imaging techniques, have made substantial progress in the past several years. In particular, ultrasonic tomographic reconstruction has shown its high potential in accurately measuring fluid fractions in a flow cross-section based on direct flow measurements. These types of measurements are very attractive as the system is nonradioactive, has direct contact to fluid, yet without flow obstruction. An ultrasonic tomographic multiphase flow meter (UTM) is being developed to capture the fluid flow rates and phase compositions in producing oil wells. The UTM aims to generate cross-sectional images of the flow inside the pipe. To recover these images, complex computations and high processing time are required, based on a limited amount of available data. For this purpose, compressive sensing (CS) reconstruction methods have shown great potential in overcoming the mentioned data quality and quantity issues. CS builds upon the fact that one can recover certain data of interest from potentially very few measurements. Many CS algorithms have been proposed in the literature. In this paper, we seek to find a suitable algorithm by comparing different CS algorithms in terms of the resulting recovered image quality. Specifically, we survey three algorithms in CS, namely basis pursuit, iterative soft thresholding (IST), and orthogonal matching pursuit (OMP). We also compare the results of the CS algorithms to that of a least-square based conventional algorithm. The numerical results show the superiority of CS recovery algorithms over the conventional approach, both qualitatively and quantitatively. The results also suggest that high image quality could potentially be achieved with less complex CS algorithms, i.e., the IST and OMP algorithms. As presented in the paper, our goal is to adapt CS reconstruction algorithms for images generated by the UTM, for the purpose of accurate quantification of oil, gas, and water phase fractions produced from different wells.

Compressed Sensing (CS) is an emerging theory in signal processing. The CS framework suggests that the signal can be reconstructed from the sub-samples by some recovery algorithm when the signal is sparse. We propose an improved video coding scheme to compress the depth map by Total Variation (TV) minimization in the pixel domain from the quantized/dequantized transform coefficients due to the extremely sparse gradient of depth maps. We keep all the frequency coefficients to avoid introducing noise to the reconstruction results. The TV minimization solver is integrated into the standard encoder, and the proposed coding scheme chooses the optimal recovery method by using the Rate-distortion Optimization (RDO) method. The experimental results show that our proposed scheme is highly efficient for the depth map sequences.

Source localization is an important field of research and the received signal strength (RSS)-based method is of particular interest. Further, by exploiting the sparsity of localization problem, the compressive sensing (CS) can be applied to considerably decrease the number of RSS measurements, especially in multi-source scenario. However, most existing CS-based localization methods usually neglect some practical issues. In particular, the sensor positions are assumed be known exactly, while in practice they may not be accurate. Additionally, a uniform Gaussian noise assumption is made that the noise variances of sensors are identical, but in practice the noise should be nonuniform. When these assumptions are violated, the localization performance will deteriorate dramatically. To address such issues, in this paper, we formulate the source localization based on superimposed RSS measurements as a sparse signal recovery problem. Moreover, by regarding the sensor positions as adjustable parameters, the inaccurate sensor positions can be refined through the adjustment of parameters. Following this idea, we develop a novel iterative algorithm for joint signal recovery and parameter optimization based on the variational expectation-maximization algorithm. Consequently, the sensor position uncertainty can be alleviated and thus the signal recovery performance will be improved greatly. Meanwhile, it is also capable of learning the variance of nonuniform noise. Extensive simulation results demonstrate the superiority of the proposed method.

The cloud radio access network (C-RAN) is a promising network architecture for future mobile communications, and one practical hurdle for its large scale implementation is the stringent requirement of high capacity and low latency fronthaul connecting the distributed remote radio heads (RRH) to the centralized baseband pools (BBUs) in the C-RAN. To improve the scalability of C-RAN networks, it is very important to take the fronthaul loading into consideration in the signal detection, and it is very desirable to reduce the fronthaul loading in C-RAN systems. In this paper, we consider uplink C-RAN systems and we propose a distributed fronthaul compression scheme at the distributed RRHs and a joint recovery algorithm at the BBUs by deploying the techniques of distributed compressive sensing (CS). Different from conventional distributed CS, the CS problem in C-RAN system needs to incorporate the underlying effect of multi-access fading for the end-to-end recovery of the transmitted signals from the users. We analyze the performance of the proposed end-to-end signal recovery algorithm and we show that the aggregate measurement matrix in C-RAN systems, which contains both the distributed fronthaul compression and multiaccess fading, can still satisfy the restricted isometry property with high probability. Based on these results, we derive tradeoff results between the uplink capacity and the fronthaul loading in C-RAN systems.

The algorithms based on the technique of optimal <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-thresholding (OT) were recently proposed for signal recovery, and they are very different from the traditional family of hard thresholding methods. However, the computational cost for OT-based algorithms remains high at the current stage of their development. This stimulates the development of the so-called natural thresholding (NT) algorithm and its variants in this paper. The family of NT algorithms is developed through the first-order approximation of the so-called regularized optimal <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-thresholding model, and thus the computational cost for this family of algorithms is significantly lower than that of the OT-based algorithms. The guaranteed performance of NT-type algorithms for signal recovery from noisy measurements is shown under the restricted isometry property and concavity of the objective function of regularized optimal <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-thresholding model. Empirical results indicate that the NT-type algorithms are robust and very comparable to several mainstream algorithms for sparse signal recovery.

In this paper, we study the feasibility of the exact recovery of a sparse vector from its linear measurements when there are missing data. For this setting, the random sampling approach employed in compressed sensing is known to provide excellent reconstruction accuracy. However, when there is missing data, the theoretical guarantees associated with the sparse vector recovery have not been well studied. Therefore, in this paper, we derive an upper bound on the minimum number of measurements required to ensure faithful recovery of a sparse signal when the generation of missing data is modeled using an erasure channel. We show that the number of measurements required scales as - [log(1 - p + Cp)] <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> to overcome the missing data with arbitrarily high probability, where p is the probability of observing (not missing) a measurement and 0 <; C <; 1 is a constant that depends on the properties of the measurement matrix and the recovery algorithm. Our analysis is based on the restricted isometric property of the measurement matrix whose entries as well as the dimension are random.