Distributed continuous-time accelerated neurodynamic approaches for sparse recovery via smooth approximation to [formula omitted]-minimization

Sparsity has been extensively employed in multimedia sensing and computing in consumer electronics, signal and image processing, depth video codec, adaptive sparse-type equalizer, blind speech separation, and machine learning. Throughout this paper, we propose a novel distributed projection neurodynamic approach for solving the Basis Pursuit (BP) with flexible partition methods in a distributed manner. The proposed neurodynamic approach requires only that the network is undirected and connected, and no node can access the entire matrix simultaneously. First, we equivalently formulate the BP into a standard distributed optimization problem with a flexible partition-by-blocks method to obtain global information, and discuss the equivalence of their optimality conditions. Then, we propose a distributed continuous-time neurodynamic approach on the basis of primal-dual dynamical systems and projection operators, and also study its global convergence property. Finally, numerical experiments on sparse signals and image recovery further verify the effectiveness and superiority of our proposed neurodynamic approach.

Multipath signal is often considered an interference that must be removed. The coherence between multipath and direct component makes it difficult to use conventional direction-of-arrival (DOA) estimation methods in a smart antenna system. This study demonstrates a new multipath signal DOA estimation technique. Unlike the common methods, without decoherence preprocessing, the proposed algorithm first apply a complex fast independent component analysis (cFastICA) algorithm to obtain the steering vectors with multipath information that corresponds to each source signal. Then, according to the special structure of the obtained steering vectors and spatial sparsity of the multipath signal components, the algorithm uses the solution of the sparse signal reconstruction problem in the compressive sensing (CS) theory, and the DOA estimation of the multipath signal is translated into an l1 norm minimization problem. Finally, we search the space spectrums to acquire the DOAs for each direct component and multipath component. Comparative simulation tests and analysis prove the effectiveness of the proposed algorithm in estimation accuracy in underdetermined conditions.

In the Compressed Sensing (CS) framework, the underdetermined system of linear equation (USLE) can have infinitely many possible solutions. However, we intend to find the sparsest possible solution, which is l0-norm minimization. However, finding an l0 norm solution out of infinitely many possible solutions is NP-hard problem that becomes non-convex optimization problem. It has been a practically proven fact that l0 norm penalty can be adequately estimated by l1 norm, which recasts a non-convex minimization problem to a convex problem. However, l1 norm non-differentiable and gradient-based minimization algorithms are not applicable, due to this very reason there is a need to approximate l1 norm by its smooth approximation. Iterative shrinkage algorithms provide an efficient method to numerically minimize l1-regularized least square optimization problem. These algorithms are required to induce sparsity in their solutions to meet the CS recovery requirement. In this research article, we have developed a novel recovery method that uses hyperbolic tangent function to recover undersampled signal/images in CS framework. In our work, l1 norm and soft thresholding are both approximated with the hyperbolic tangent functions. We have also proposed the criteria to tune optimization parameters to get optimal results. The error bounds for the proposed l1 norm approximation are evaluated. To evaluate performance of our proposed method, we have utilized a dataset comprised of 1-D sparse signal, compressively sampled MR image and cardiac cine MRI. The MRI is an important imaging modality for assessing cardiac vascular function. It provides the ejection fraction and cardiac output of the heart. However, this advantage comes at the cost of a slow acquisition process. Hence, it is essential to speed up the acquisition process to take the full benefits of cardiac cine MRI. Numerical results based on performance metrics, such as Structural Similarity (SSIM), Peak Signal to Noise Ratio (PSNR) and Root Mean Square Error (RMSE) show that the proposed tangent hyperbolic based CS recovery offers a much better performance as compared to the traditional Iterative Soft Thresholding (IST) recovery methods.

Inertial primal-dual projection neurodynamic approaches for constrained convex optimization problems and application to sparse recovery.

Compressed sensing (CS) is a sparse signal sampling methodology for efficiently acquiring and reconstructing a signal from relatively few measurements. Recent work shows that CS is well-suited to be applied to problems in genomics, including probe design in microarrays, RNA interference (RNAi), and taxonomic assignment in metagenomics. The principle of using different CS recovery methods in these applications has thus been established, but a comprehensive study of using a wide range of CS methods has not been done. For each of these applications, we apply three hitherto unused CS methods, namely, l1-magic, CoSaMP, and l1-homotopy, in conjunction with CS measurement matrices such as randomly generated CS m matrix, Hamming matrix, and projective geometry-based matrix. We find that, in RNAi, the l1-magic (the standard package for l1 minimization) and l1-homotopy methods show significant reduction in reconstruction error compared to the baseline. In metagenomics, we find that l1-homotopy as well as CoSaMP estimate concentration with significantly reduced time when compared to the GPSR and WGSQuikr methods.

Compressive sensing (CS) is mainly concerned with low-coherence pairs, since the number of samples needed to recover the signal is proportional to the mutual coherence between projection matrix and sparsifying matrix. Until now, papers on CS always assume the projection matrix to be a random matrix. In this paper, aiming at minimizing the mutual coherence, a method is proposed to optimize the projection matrix. This method is based on equiangular tight frame (ETF) design because an ETF has minimum coherence. It is impossible to solve the problem exactly because of the complexity. Therefore, an alternating minimization type method is used to find a feasible solution. The optimally designed projection matrix can further reduce the necessary number of samples for recovery or improve the recovery accuracy. The proposed method demonstrates better performance than conventional optimization methods, which brings benefits to both basis pursuit and orthogonal matching pursuit.

This paper introduces a nonconvex approach for sparse signal recovery, proposing a novel model termed the τ2-model, which utilizes the squared ℓ1/ℓ2 norms for this purpose. Our model offers an advancement over the ℓ0 norm, which is often computationally intractable and less effective in practical scenarios. Grounded in the concept of effective sparsity, our approach robustly measures the number of significant coordinates in a signal, making it a powerful alternative for sparse signal estimation. The τ2-model is particularly advantageous due to its computational efficiency and practical applicability. We detail two accompanying algorithms based on Dinkelbach’s procedure and a difference of convex functions strategy. The first algorithm views the model as a linear-constrained quadratic programming problem in noiseless scenarios and as a quadratic-constrained quadratic programming problem in noisy scenarios. The second algorithm, capable of handling both noiseless and noisy cases, is based on the alternating direction linearized proximal method of multipliers. We also explore the model’s properties, including the existence of solutions under certain conditions, and discuss the convergence properties of the algorithms. Numerical experiments with various sensing matrices validate the effectiveness of our proposed model.

Microwave Imaging offers safe, low-cost, and portable method for medical imaging applications. These advantages make the microwave imaging convenient for early detection of tumor or cancer. The transmission method is one of the methods in microwave imaging which provides fast measurement and simple image reconstruction. However, this method requires a great number of measurements to obtain a well-reconstructed image. In order to reduce the number of measurements, this research proposes a Compressive Sensing (CS) approach for image reconstruction on microwave imaging. Compressive Sensing allows reconstruction of a signal with fewer measurements than the conventional approach. In this research, the scanning process is conducted on Computer Simulation Technology (CST) Microwave Studio software. Two dipole antennas with 3 GHz frequency are utilized as microwave transmitter and receiver. A two-layer cube phantom acts as the scanned object. Each layer has different relative permittivity which illustrates the healthy cell and abnormal cell. To meet the framework of Compressive Sensing, a weighted matrix of Discrete Radon Transform (DRT) is created as a projection matrix which delineates the data acquisition scheme in the scanning process. Discrete Cosine Transform (DCT) is selected as sparse dictionary matrix to represent the sparse basis while Basis Pursuit is selected as sparse reconstruction algorithm to reconstruct the sparse signal from measurement data. The measured $S_{21}$ data are successfully reconstructed into an image using the Compressive Sensing approach. The reconstructed image is analyzed both qualitatively and quantitatively using image quality parameters such as Structural Similarity Index (SSIM) and Mean Squared Error (MSE).

Compressive sensing (CS) provides a new paradigm for data collection in wireless sensor networks (WSNs). In this paper, we continue to exploit the integration between CS and tree-based data gathering in WSNs. Based on a tree formed as down-stream generations from the sink or base-station (BS), each parent node stores its children's readings and the corresponding measurement vectors through the data collecting process. All sensors send their own readings only once to their parents. The parent nodes generate measurements based on the projection matrix and then forward a certain number of measurements required to the upper nodes or the BS. The CS recovery algorithm implemented at the BS reconstructs precise readings from all sensors. This significantly reduces a certain number of transmissions in routing. Two additional ideas are suggested related to sensor transmission range and the probability of non-zero elements in the projection matrix for the purpose of reducing power consumption for the networks.

This paper considers the sparse recovery problem of multiple measurement vector (MMV) model corrupted in impulsive noise. To ensure outlier-robust sparse recovery, we formulate an MMV problem that includes the generalized lp-norm (1 < p < 2) divergence data-fidelity term added to the l2,0 joint sparsity-promoting regularizer. The joint sparse penalty, however, is non-continuous and hence non-differentiable, which inevitably raises difficulty in optimization when using a gradient-based method. To address this, we build a smooth approximation for the l2,0-based sparse metric via the log-sum based sparse-encouraging surrogate function. Then, we propose a block successive upper-bound minimization algorithm for the smooth MMV problem by solving a series of subproblems based on the block coordinate descent (BCD) method. Furthermore, local convergence of the proposed algorithm to a stationary point of the smooth problem is proved. Experiments demonstrate its efficiency and robust recovery performance for suppressing impulsive noise.

Recently, compressed sensing (CS) has been widely used in ground penetrating radar (GPR) imaging due to its low sampling time, high quality performance and high quality underground imaging results. There are always a large number of matrix operations involved in CS based GPR imaging process, which takes a long time. Traditionally, all scanning area was imaged by this CS imaging method. However, with the imaging scene becoming larger, the measurement matrix and the data dictionary in CS imaging model expand, which increases the time of CS-based imaging process. Thus, this paper proposes a fast CS based GPR imaging method to reduce the imaging time. Specifically, the geometric characteristics of the B-scan image can firstly be used to extract the region of interest (ROI) where the target may exist in a layered medium. Then the GPR imaging based on CS can be realized in ROI, which can reduce the imaging area and imaging time. Experimental results present that the method can effectively reduce the time of GPR imaging based on CS.

Sparsity and incoherence are the two key ingredients in compressive sensing (CS). Given a sparsifying dictionary D, the projection matrix P must be as incoherent with D as possible for the CS system to be efficient. Thus the design of projection matrix is naturally a problem of minimizing the coherence between P andD. Unfortunately, this turns out to be a nonconvex, nonsmooth, large-scale problem even for a CS system of moderate size. In this paper, the above-mentioned problem is investigated in a formulation where the problem is converted into a sequence of nonsmooth but convex subproblems. A subgradient projection algorithm is proposed to solve the nonsmooth subproblems that converges to a projection matrix with improved performance. The performance of the proposed algorithm is evaluated by simulations and comparisons with several existing techniques.

SummaryCognitive radio (CR) is an emerging technology for optimum spectrum utilization. Spectrum sensing (SS) is the first step of CR. Important challenge in the next generations is wideband SS. Compressive sensing (CS) can be used as an SS in CR to solve this challenge. Sensing matrix (SM) plays an important role in CS. For better performance, SM must have low mutual coherence. The existing matrices in the literature are random and chaotic. All random matrices have a good performance, but they have problems of large memory storage, complexity, sharing bandwidth, and low security. Chaotic matrices (CMs) are the best but still have some problems including the use of sample distance in chaotic sequence that consumes high storage resources, low performance and compression under low signal‐to‐noise ratio (SNR), and low security. In this paper, we propose a new 1‐D chaotic map for constructing a novel SM. The proposed map has simple structure, wide range chaotic behavior, and high security. Its chaotic behavior is confirmed using Lyapunov exponent, bifurcation, and trajectory. The CS performance based on novel CM is measured using absolute error ( , mutual coherence, memory cost, computational complexity, and MSE (mean square error), while evaluated through comparisons with matrices in the literature. The simulation and mathematical results show that the proposed map is a hyperchaotic with efficient chaotic behavior in wide range of parameters and low memory storage and complexity. The results also show that it has low , MSE, and mutual coherence at low SNR with high compression.

Structural health monitoring requires collection of large number sample data and sometimes high frequent vibration data for detecting the damage of structures. The expensive cost for collecting the data is a big challenge. The recent proposed Compressive Sensing method enables a potentially large reduction in the sampling, and it is a way to meet the challenge. The Compressed Sensing theory requires sparse signal, meaning that the signals can be well-approximated as a linear combination of just a few elements from a known discrete basis or dictionary. The signal of structure vibration can be decomposed into a few sinusoid linear combinations in the DFT domain. Unfortunately, in most cases, the frequencies of decomposed sinusoid are arbitrary in that domain, which may not lie precisely on the discrete DFT basis or dictionary. In this case, the signal will lost its sparsity, and that makes recovery performance degrades significantly. One way to improve the sparsity of the signal is to increase the size of the dictionary, but there exists a tradeoff: the closely-spaced DFT dictionary will increase the coherence between the elements in the dictionary, which in turn decreases recovery performance. In this work we introduce three approaches for arbitrary frequency signals recovery. The first approach is the continuous basis pursuit (CBP), which reconstructs a continuous basis by introducing interpolation steps. The second approach is a semidefinite programming (SDP), which searches the sparest signal on continuous basis without establish any dictionary, enabling a very high recovery precision. The third approach is spectral iterative hard threshold (SIHT), which is based on redundant DFT dictionary and a restricted union-of-subspaces signal model, inhibiting closely spaced sinusoids. The three approaches are studied by numerical simulation. Structure vibration signal is simulated by a finite element model, and compressed measurements of the signal are taken to perform signal recovery. Comparison of the performance of the three approaches is made, and future work on design of compressive sampling testing system for vibration signal is proposed.

The most prominent challenges in compressive sensing are seeking the domain where an image is represented sparsely and hence be faithfully recovered to obtain high-quality results. This paper introduces an approach for image compression and recovery. The proposed approach involves two phases: the initial step is the compression phase, and the second step is the recovery phase. Initially, the medical image is subjected to the compression module wherein the self-similarity and the 3-dimensional (3D) transform are adapted for compressing the image. Then, in the recovery phase, the compressive sensing recovery is performed based on structural similarity index measure (SSIM)-based collaborative sparsity measure (S-CoSM), and the novel optimization algorithm, named Taylor-based Sunflower optimization (Taylor-SFO) algorithm. An effective S-CoSM measure is designed by modifying the CoSM using the SSIM metric. The proposed Taylor-SFO will be designed by integrating the Taylor series with the sunflower optimization (SFO) algorithm. The performance of the proposed Taylor-SFO approach is evaluated for matrices SSIM of 0.9412 and peak signal to noise ratio of 57.57 dB.