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Practitioners employ operator splitting methods—such as alternating direction method of multipliers (ADMM) and its "dual" Douglas-Rachford method (DR)—to solve many kinds of optimization problems. We provide a gentle introduction to these algorithms, and illustrations of their duality-like relationship in the context of solving basis pursuit problems for audio signal recovery. Recently, researchers have used the dynamical systems associated with the iterates of splitting methods to motivate the development of schemes to improve performance. These developments include a class of methods that act by iteratively minimizing surrogates for a Lyapunov function for the dynamical system. An exemplar of this class is currently state-of-the-art for the feasibility problem of finding wavelets with special structure. Early experimental evidence has also suggested that, when implemented in a primal-dual (ADMM and DR) framework, this exemplar may provide improved performance for basis pursuit problems. We provide a reasonable way to compute the updates for this exemplar, and we study the application of this method to the aforementioned basis pursuit audio problems. We provide experimental results and visualizations of the dynamical system for the dual DR sequence. We observe that for highly structured problems with real data, the algorithmic behavior is noticeably different than for randomly generated problems.

Sparse representation has garnered significant attention across multiple fields, including signal processing, statistics, and machine learning. The fundamental concept of this technique is that we can express the signal as a linear combination of only a few elements from a known basis. Compressed sensing (CS) is an interesting application of this technique. It is valued for its potential to improve data collection and ensure efficient acquisition and recovery from just a few measurements. In this paper, we propose a novel approach for the high-order CS problem based on the Einstein product, utilizing a tensor dictionary instead of the commonly used matrix-based dictionaries in the Tucker model. Our approach provides a more general framework for compressed sensing. We present two novel models to address the CS problem in the multidimensional case. The first model represents a natural generalization of CS to higher-dimensional signals; we extend the traditional CS framework to effectively capture the sparsity of multidimensional signals and enable efficient recovery. In the second model, we introduce a complexity reduction technique by utilizing a low-rank representation of the signal. We extend the OMP and the homotopy algorithms to solve the high-order CS problem. Through various simulations, we validate the effectiveness of our proposed method, including its application to solving the completion tensor problem in 2D and 3D colored and hyperspectral images. [ABSTRACT FROM AUTHOR]

Practitioners employ operator splitting methods--such as alternating direction method of multipliers (ADMM) and its "dual" Douglas-Rachford method (DR)--to solve many kinds of optimization problems. We provide a gentle introduction to these algorithms, and illustrations of their duality-like relationship in the context of solving basis pursuit problems for audio signal recovery. Recently, researchers have used the dynamical systems associated with the iterates of splitting methods to motivate the development of schemes to improve performance. These developments include a class of methods that act by iteratively minimizing surrogates for a Lyapunov function for the dynamical system. An exemplar of this class is currently state-of-the-art for the feasibility problem of finding wavelets with special structure. Early experimental evidence has also suggested that, when implemented in a primal-dual (ADMM and DR) framework, this exemplar may provide improved performance for basis pursuit problems. We provide a reasonable way to compute the updates for this exemplar, and we study the application of this method to the aforementioned basis pursuit audio problems. We provide experimental results and visualizations of the dynamical system for the dual DR sequence. We observe that for highly structured problems with real data, the algorithmic behavior is noticeably different than for randomly generated problems. [ABSTRACT FROM AUTHOR]

The improved Sparse Signal Reconstruction (SR) algorithm for Trusted Artificial Intelligence (AI) and Distributed Compressed Sensing (DCS) technology was thoroughly investigated. The study verified its effectiveness and advantages in trusted AI and DCS systems, which have significant implications for enhancing the credibility, security, and performance of signal processing and AI algorithms. The reconstruction performance was evaluated using Orthogonal Matching Pursuit (OMP), Basis Pursuit (BP), and Least Absolute Shrinkage and Selection Operator (LASSO). The analysis primarily focused on runtime, refactoring errors, and the number of successful reconstruction attempts. When K = 4, K = 6, K = 8, and K = 10, OMP outperformed BP and LASSO in terms of successful reconstructions, demonstrating better performance and higher reconstruction precision. [ABSTRACT FROM AUTHOR]

Compressive sensing (CS) plays a critical role in sampling, transmitting, and storing the color medical image, i.e., magnetic resonance imaging, colonoscopy, wireless capsule endoscopy, and eye images. Although CS for medical images has been extensively investigated, a challenge remains in the reconstruction time of the CS. This paper considers a reconstruction of CS using sparsity averaging (SA)-based basis pursuit (BP) for RGB color space of eye image, referred to as RGB-BPSA. Next, an enhanced RGB-BPSA (E-RGB-BPSA) is proposed to reduce the reconstruction time of RGB-BPSA using a simple SA generated by the combination of Daubechies-1 and Daubechies-8 wavelet filters. In addition, variable density sampling is proposed for the measurement of E-RGB-BPSA. The performance metrics are investigated in terms of structural similarity (SSIM) index, signal-to-noise ratio (SNR), and CPU time. The simulation results show the superior E-RGB-BPSA over the existing RGB-BPSA at an image with a resolution 512 $\times $ 512 pixels into a measurement rate 10% with SSIM of 0.9, SNR of 20 dB, and CPU time of 20 seconds. The E-RGB-BPSA can be a solution to massive data transmissions and storage for the future of medical imaging.

Basis pursuit (BP), basis pursuit deNoising (BPDN), and least absolute shrinkage and selection operator (LASSO) are popular methods for identifying important predictors in the high‐dimensional linear regression model Y=Xβ+ε. By definition, when ε=0, BP uniquely recovers β when Xβ=Xb and β≠b implies ‖b‖1>‖β‖1 (identifiability condition). Furthermore, LASSO can recover the sign of β only under a much stronger irrepresentability condition. Meanwhile, it is known that the model selection properties of LASSO can be improved by hard thresholding its estimates. This article supports these findings by proving that thresholded LASSO, thresholded BPDN, and thresholded BP recover the sign of β in both the noisy and noiseless cases if and only if β is identifiable and large enough. In particular, if X has iid Gaussian entries and the number of predictors grows linearly with the sample size, then these thresholded estimators can recover the sign of β when the signal sparsity is asymptotically below the Donoho–Tanner transition curve. This is in contrast to the regular LASSO, which asymptotically, recovers the sign of β only when the signal sparsity tends to 0. Numerical experiments show that the identifiability condition, unlike the irrepresentability condition, does not seem to be affected by the structure of the correlations in the X matrix. [ABSTRACT FROM AUTHOR]

The modern digital world comprises of transmitting media files like image, audio, and video which leads to usage of large memory storage, high data transmission rate, and a lot of sensory devices. Compressive sensing (CS) is a sampling theory that compresses the signal at the time of acquiring it. Compressive sensing samples the signal efficiently below the Nyquist rate to minimize storage and recoveries back the signal significantly minimizing the data rate and few sensors. The proposed paper proceeds with three phases. The first phase describes various measurement matrices like Gaussian matrix, circulant matrix, and special random matrices which are the basic foundation of compressive sensing technique that finds its application in various fields like wireless sensors networks (WSN), internet of things (IoT), video processing, biomedical applications, and many. Finally, the paper analyses the performance of the various reconstruction algorithms of compressive sensing like basis pursuit (BP), compressive sampling matching pursuit (CoSaMP), iteratively reweighted least square (IRLS), iterative hard thresholding (IHT), block processing-based basis pursuit (BP-BP) based on mean square error (MSE), and peak signal to noise ratio (PSNR) and then concludes with future works. [ABSTRACT FROM AUTHOR]

The analysis of Cerebral Angiographies are an essential tool for the assessment of the future of patients that underwent thrombolysis after a stroke event. Many semi-qualitative visual diagnostic scales have been developed for this purpose. Perfusion angiographies show essentially three phases: the arterial (early), the capillary (intermediate), and venous (late) phase. We call parenchymogram the image sequence corresponding to the capillary phase only. Unfortunately the parenchymogram is often under exploited in practice, despite containing many pertinent hints on the quality of reperfusion. In this paper we propose a set of methods for the extraction of the parenchymogram from raw Cerebral Angiographies. These methods rely on basis pursuit and on the representation of images with an over-complete basis arising from an redundant wavelet transform. We will show that the extraction of the parenchymogram by applying the aforementioned methods on real clinical data allows us to recover essential information for the comparison of blood flow before and after thrombolysis.

Being spectrally efficient, easily implemented, and highly immune to selective channel imperfections and multipath fading, the Orthogonal Frequency Division Multiplexing (OFDM) can provide a sufficiently robust and high data rate modulation technique for emerging wired and wireless telecommunication applications. However, a major drawback of OFDM that is represented by its high Peak-to-Average Power Ratio (PAPR) of the transmitted signal, which leads to degrade the system’s transmission accuracy. In this paper, a Compressed Sampling (CS) based approach is considered for reducing the PAPR without lowering its transmission capacity or affecting its Bit Error Rate (BER) performance. The proposed scheme adds a sampling stage after the IFFT block in the transmitter side, that is compressively represents the transmitted signal by fewer symbols that is transmitted instead of the original signal. At the receiver side, the received compressively sampled signal is then recovered before the FFT block by following the Basis Pursuit (𝓵1–norm) algorithm. The proposed scheme shows an enhanced PAPR and BER performances while preserving the rest of the system performance aspects.

• The effect of nonlinearity on the performance of Basis Pursuit (BP) is examined. • It is shown the performance impairment from nonlinearity can be notably mitigated by using the BP a set selector. • It is shown that normalization of the Jacobian to equal column norm leads to notable improvement when the likelihood of damage is uniform in the parameter space. The Basis Pursuit (BP) is an optimization statement used to search for the sparsest solution of underconstrained problems of the form Φ x = b using ℓ 1 minimization. This paper shows that approximating nonlinear problems as linear can result in considerable compromise in the capacity of the BP to attain the minimum cardinality solution and that this impairment can be mitigated by treating the strategy as a selector of a set of unknowns that renders the problem fully constrained, subsequently solving with due consideration for the nonlinearity. It is shown that gains realized by the selector approach derive primarily from the fact that the part of the nonlinearity that projects onto the right-hand side, b , has no effect on the relative values of the solution, x , and that the orthogonal residual seldom modifies the largest entries. The paper considers damage characterization as the application domain and shows that normalization of the Jacobian to equal column norm, a step that is standard when BP is applied in most fields, but not implemented so far in structural engineering, leads to large improvements when the probability of damage across the parameter space is essentially uniform. [ABSTRACT FROM AUTHOR]

Deep learning architectures have been extensively used in recent years for the classification of biomedical images to assist clinicians for diagnosis and treatment management of patients with different health conditions. These architectures have demonstrated expert level diagnosis, and in some cases, surpassed human experts in diagnosing health conditions. The automation tools based on deep learning frameworks have the potential to transform all stages of medical imaging pipeline from image acquisition to interpretation and analysis. One of the most common areas where these techniques are applied is knee MR image classification for different types of Anterior Cruciate Ligament (ACL) tears. If properly and timely managed, the diagnosis and treatment of ACL tear can avoid further degradation of patients' knee joints and can also help slow the process of subsequent knee arthritis. In this work, we have implemented a novel classification framework based on multilayered basis pursuit algorithms inspired from recent research work in the area of the theoretical foundation of deep learning with the help of celebrated sparse coding theory. We implement an optimal multi-layered Convolutional Sparse Coding (ML-CSC) framework for classification of a labelled dataset of knee MR images with the coronal view and compare the results with traditional convolutional neural network (CNN) based classifiers. Empirical results demonstrate the effectiveness of the ML-CSC framework and show that the framework can successfully learn distinct features on a small dataset and achieve a good efficiency of more than 92% without employing regularization techniques and extensive training on large datasets. In addition to 95% average accuracy on the presence and absence of ACL tears, the framework also performs well on the imbalanced and challenging classification of partial ACL tear with 85% accuracy.

Compressed sensing has empowered quality image reconstruction with fewer data samples than previously thought possible. These techniques rely on a sparsifying linear transformation. The Daubechies wavelet transform is commonly used for this purpose. In this work, we take advantage of the structure of this wavelet transform and identify an affine transformation that increases the sparsity of the result. After inclusion of this affine transformation, we modify the resulting optimization problem to comply with the form of the Basis Pursuit Denoising problem. Finally, we show theoretically that this yields a lower bound on the error of the reconstruction and present results where solving this modified problem yields images of higher quality for the same sampling patterns using both magnetic resonance and optical images. [ABSTRACT FROM AUTHOR]

We investigate the computation of a sparse solution to an underdetermined system of linear equations using the Huber loss function as a proxy for the 1-norm and a quadratic error term à la Lasso. The approach is termed "penalized Huber loss". The results of the paper allow to calculate a sparse solution using a simple extrapolation formula under a sign constancy condition that can be removed if one works with extreme points. Conditions leading to sign constancy, as well as necessary and sufficient conditions for computation of a sparse solution by penalized Huber loss, and ties among different solutions are presented. [ABSTRACT FROM AUTHOR]

Being spectrally efficient, easily implemented, and highly immune to selective channel imperfections and multipath fading, the Orthogonal Frequency Division Multiplexing (OFDM) can provide a sufficiently robust and high data rate modulation technique for emerging wired and wireless telecommunication applications. However, a major drawback of OFDM that is represented by its high Peak-to-Average Power Ratio (PAPR) of the transmitted signal, which leads to degrade the system’s transmission accuracy. In this paper, a Compressed Sampling (CS) based approach is considered for reducing the PAPR without lowering its transmission capacity or affecting its Bit Error Rate (BER) performance. The proposed scheme adds a sampling stage after the IFFT block in the transmitter side, that is compressively represents the transmitted signal by fewer symbols that is transmitted instead of the original signal. At the receiver side, the received compressively sampled signal is then recovered before the FFT block by following the Basis Pursuit (l1–norm) algorithm. The proposed scheme shows an enhanced PAPR and BER performances while preserving the rest of the system performance aspects. [ABSTRACT FROM AUTHOR]

The multiple measurement vector (MMV) problem is applicable in a wide range of applications such as photoplethysmography (PPG), remote PPG measurement, heart rate estimation, and directional arrival estimation of multiple sources. Measurements in the aforementioned applications exhibit a dependency structure, which is not considered in the general MMV algorithms. Modeling the dependency or the correlation structure of the solution matrix to MMV problems can increase the recovery performance. The solution matrix $X$ can be decomposed into a mixing matrix $A$ and a sparse matrix with independent columns $S$ . The key idea of this model is that the matrix S can be sparser than the mixing matrix $A$ . Previous MMV algorithms did not consider such a structure for $X$ . This paper proposes two algorithms, which are based on orthogonal matching pursuit and basis pursuit, and derives the exact recovery guarantee conditions for both approaches. We compare the simulation results of the proposed algorithms with the conventional algorithms and show that the proposed algorithms outperform previous algorithms especially in the case of the low number of measurements.

Powerline interference (PLI) is a major source of interference in the acquisition of electroencephalogram (EEG) signal. Digital notch filters (DNFs) have been widely used to remove the PLI such that actual features, which are weak in energy and strongly connected to brain states, can be extracted explicitly. However, DNFs are mathematically implemented via discrete Fourier analysis, the problem of overlapping between spectral counterparts of PLI and those of EEG features is inevitable. In spite of their effectiveness, DNFs usually cause distortions on the extracted EEG features, which may lead to incorrect diagnostic results. To address this problem, we investigate an adaptive sparse detector for reducing PLI. This novel approach is proposed based on sparse representation inspired by self-adaptive machine learning. In the coding phase, an overcomplete dictionary, which consists of redundant harmonic waves with equally spaced frequencies, is employed to represent the corrupted EEG signal. A strategy based on the split augmented Lagrangian shrinkage algorithm is employed to optimize the associated representation coefficients. It is verified that spectral components related to PLI are compressed into a narrow area in the frequency domain, thus reducing overlapping with features of interest. In the decoding phase, eliminating of coefficients within the narrow band area can remove the PLI from the reconstructed signal. The sparsity of the signal in the dictionary domain is determined by the redundancy factor. A selection criteria of the redundancy factor is suggested via numerical simulations. Experiments have shown the proposed approach can ensure less distortions on actual EEG features.

We propose a novel differentiable reformulation of the linearly-constrained ℓ 1 minimization problem, also known as the basis pursuit problem. The reformulation is inspired by the Laplacian paradigm of network theory and leads to a new family of gradient-based methods for the solution of ℓ 1 minimization problems. We analyze the iteration complexity of a natural solution approach to the reformulation, based on a multiplicative weights update scheme, as well as the iteration complexity of an accelerated gradient scheme. The results can be seen as bounds on the complexity of iteratively reweighted least squares (IRLS) type methods of basis pursuit. [ABSTRACT FROM AUTHOR]

This paper presents a new fast iterative shrinkage-thresholding algorithm, termed AFISTA. The essential idea is to improve the convergence rate of FISTA using a new continuation strategy leading to a less number of iterations compared to FISTA. The convergence theorem of the AFISTA is proposed. In order to further accelerate the AFISTA method, it is equipped with the Barzilai-Borwein (BB) method. Also, for applications with orthogonal sensing matrix A, we proposed a specialized version of the AFISTA method. AFISTA is tailored for solving the basis pursuit problem which can be applied successfully on a variety of problems arising in signal and image processing issues such as sparse signal recovery, signal and image denoising, image restoration, and compressive sensing. To show the efficiency of the method, we compare our results with generalizations of linearized Bregman and fixed - point continuation (FPC) methods in sparse signal recovery applications, with split Bregman method in compressive sensing for sparse MRI and with Gradient projection for sparse reconstruction (GPSR) method in image deconvolution. Numerical results demonstrate that AFISTA overcomes all of the compared methods in convergence rate and some of them in both convergence rate and quality of reconstructed results. [ABSTRACT FROM AUTHOR]

In engineering structures, such as large fluid-filled pipelines, continuous monitoring for damage detection is needed. To address this issue, we study the generation of guided waves in pipes by using a circumferential strip of macro fiber composite transducer to generate and detect torsional and flexural lower modes. The propagated elastic waves and their resulting reflected and mode-converted signals at the interaction wave discontinuity are post-processed with basis pursuit denoising using a Gabor dictionary to improve signal identification. Numerical results are obtained and experimentally tested on a stainless-steel pipe A-36 (43.6 and 48.2 mm in inner and outer diameter). It was found that the proposed method makes it possible to identify an artificial discontinuity by detecting the scattered wave and converted modes of a propagated torsional wave. [ABSTRACT FROM AUTHOR]

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Over the last decade, the sparse representation model has led to remarkable results in numerous signal and image processing applications. To incorporate the inherent structure of the data and account for the fact that not all support patterns are equally likely, this model was enriched by enforcing various structural sparsity patterns. One plausible such extension of classic sparse coding, instigated by the emergence of graph signal processing, is graph regularized sparse coding. This model explicitly considers the intrinsic geometrical structure of the data domain, and has been successfully employed in various applications. However, emphasis was given to developing algorithmic solutions, and to date, the theoretical foundations to this problem have been lagging behind. In this work, we fill this gap and present a novel theoretical analysis of the graph regularized sparse coding problem, providing worst-case guarantees for the stability of the obtained solution, as well as for the success of several pursuit techniques. Furthermore, we formulate the conditions for which the superiority of the graph regularized sparse coding solution over the structure-agnostic sparse coding counterpart is established. [ABSTRACT FROM AUTHOR]

Different types of dependent uncertainties exist in detonation system since the random vibration of physical parameters in measurement technique, and the equation of state (EOS) and the reaction rate equation are empirical modeling. And these random variables are not independent and identically distributed. Assessing the impact of these input uncertainties on the output result of system has important theoretical significance and practical value. The corner effect in detonation diffraction is studied. The non-intrusive polynomial chaos based on regression method is used for uncertainty quantification. Rosenblatt transformation is used to transform the dependent random variables into independent random variables satisfying standard uniform distribution. Under-determined linear equations are derived from the sampling method. Optimization method is chosen to solve the regression equation. The basis pursuit is applied to change the optimization problem into linear programming. The expectation and confidence interval of velocity components, horizontal positions, and pressures of two Lagrangian reference points near the corner are given by using the method mentioned. The results show that the trajectories of two Lagrangian reference points are dramatically different although they are not far from each other. It is difficult to judge the long time dynamical behavior since the uncertainty is becoming large over time. The method can also be applied to other detonation problems. [ABSTRACT FROM AUTHOR]

The time domain signal of bearing pitting/spalling fault always presents shock and modulation, and it is often submerged by strong noises, especially in the early stage. the conventional fault feature extraction method may have insufficient feature extraction accuracy, and even in some extreme cases, the fault feature frequency cannot be extracted because of the strong noise interference. Aiming at overcoming the noise interference problem encountered in this kind of weak fault feature extraction, a novel weak fault feature extraction algorithm termed as ACFSS of rolling bearing is proposed. The ACFSS is based on an overcomplete dictionary (or overcomplete atomic library) of Attenuated Cosines(AC) basis, which is highly matched to the bearing fault waveforms, and an improved Basis Pursuit algorithm with Feature Sign Search(FSS) is introduced into the ACFSS to improve the calculating speed. In order to select the suitable parameters of the attenuated cosine dictionary, some methods such as peak resonance frequency (PRF), power variation peak (PVK), time shift parameter (TSP), etc. are introduced. These parameters span the sparse overcomplete dictionary. Finally, the bearing fault data of Case-Western University and full life accelerated IMS bearing data are utilized to verify the validation of ACFSS. Compared with the ordinary envelope spectrum analysis(ESA) method /the ordinary Basis Pursuit Denoising(BPDN) method/ Wavelet package transform(WPT) Kurtogram method and Empirical Mode Decomposition(EMD) combining Singular Value Decomposition(SVD) method, The experiment show that the proposed method are more redundant and robust when facing strong noise interference, and it can be used to extract the weak fault feature frequency efficiently and accurately. • A novel sparse dictionary based on attenuated cosine basis is proposed. • Feature Sign Search algorithm is introduced to calculate the sparse coefficient. • A new algorithm ACFSS is proposed for weak fault feature extraction of rolling element bearing. • Experiments demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Three-dimensional ground penetrating radar data are often ambiguous and complex to interpret when attempting to detect only underground cavities because ground penetrating radar reflections from various underground objects can appear like those from cavities. In this study, we tackle the issue of ambiguity by proposing a system based on deep convolutional neural networks, which is capable of autonomous underground cavity detection beneath urban roads using three-dimensional ground penetrating radar data. First, a basis pursuit-based background filtering algorithm is developed to enhance the visibility of underground objects. The deep convolutional neural network is then established and applied to automatically classify underground objects using the filtered three-dimensional ground penetrating radar data as represented by three types of images: A-, B-, and C-scans. In this study, we utilize a novel two-dimensional grid image consisting of several B- and C-scan images. Cavity, pipe, manhole, and intact features extracted from in situ three-dimensional ground penetrating radar data are used to train the convolutional neural network. The proposed technique is experimentally validated using real three-dimensional ground penetrating radar data obtained from urban roads in Seoul, South Korea. [ABSTRACT FROM AUTHOR]

Here, the authors present an improved and fast sparse recovery method for joint estimation of channel errors and direction of arrival (DOA) for square kilometre antennas low-frequency array. Since this array is formed of a large number of antennas and distributed in a very large space, array calibration is a vital step to provide accurate signal information. Then, channel errors are the concerns here, which are assumed to be localised in a priori known interval. To achieve the joint estimation, an approach is proposed by using the alternative optimisation strategy, which theoretically gives convergence results. In detail, the basis pursuit de-noising (BPDN) algorithm and the proximal algorithm are used to estimate the DOA and the array errors, respectively, and the final results are obtained by alternatively solving these two algorithms. Finally, numerical simulation results are presented to demonstrate the effectiveness of their proposed method. [ABSTRACT FROM AUTHOR]

Photoplethysmography (PPG)-based heart rate (HR) estimation during physical exercise is challenging as PPG signals are often contaminated by motion artifacts (MA). This study develops a novel HR estimation method to effectively attenuate the impact of MA on PPG signals and accurately identify HR variations during physical exercise. First, a new signal reconstruction procedure is implemented based on a joint basis pursuit linear program (BPLP) to decompose PPG signal into different time series. Furthermore, an adaptive MA removal technique is developed, where the correlation between the acceleration signals and PPG time series are calculated and used as a reference to eliminate MA. Then, a new sparse spectra reconstruction method is designed to rebuild the spectrum of the current window based on the previous time frame. Furthermore, a simple HR estimation method with only one tuning parameter is designed to select the HR associated peak from the reconstructed spectra. Finally, a postprocessing technique is applied to further boost the accuracy of detection. The performance of the proposed algorithm is compared with three popular methods in recent studies using both training and testing sets from 2015 IEEE Signal Processing Cup. The proposed method provides the average absolute error of 1.79 beats per minutes (BPM) on all 22 recordings. With respect to testing datasets with stronger MA, the average absolute error is computed as 2.61BPM. The proposed HR tracking algorithm shows good robustness as it only involves a small set of parameters and can provide accurate estimations when PPG signals are contaminated by strong MA. [ABSTRACT FROM AUTHOR]

Sparse coding refers to the pursuit of the sparsest representation of a signal in a typically overcomplete dictionary. From a Bayesian perspective, sparse coding provides a maximum a posteriori estimate of the unknown vector under a sparse prior. In this paper, we suggest enhancing the performance of sparse coding algorithms by a deliberate and controlled contamination of the input with random noise, a phenomenon known as stochastic resonance. The proposed method adds controlled noise to the input and estimates a sparse representation from the perturbed signal. A set of such solutions is then obtained by projecting the original input signal onto the recovered set of supports. We present two variants of the described method, which differ in their final step. The first is a provably convergent approximation to the minimum mean square error (MMSE) estimator, relying on the generative model and applying a weighted average over the recovered solutions. The second is a relaxed variant of the former that simply applies an empirical mean. We show that both methods provide a computationally efficient approximation to the MMSE estimator, which is typically intractable to compute. We demonstrate our findings empirically and provide a theoretical analysis of our method under several different cases. [ABSTRACT FROM AUTHOR]

Inverse methods involving compressive sensing are tested in the application of two‐dimensional aperture‐synthesis imaging of radar backscatter from field‐aligned plasma density irregularities in the ionosphere. We consider basis pursuit denoising, implemented with the fast iterative shrinkage thresholding algorithm, and orthogonal matching pursuit (OMP) with a wavelet basis in the evaluation. These methods are compared with two more conventional optimization methods rooted in entropy maximization (MaxENT) and adaptive beamforming (linearly constrained minimum variance or often "Capon's Method.") Synthetic data corresponding to an extended ionospheric radar target are considered. We find that MaxENT outperforms the other methods in terms of its ability to recover imagery of an extended target with broad dynamic range. Fast iterative shrinkage thresholding algorithm performs reasonably well but does not reproduce the full dynamic range of the target. It is also the most computationally expensive of the methods tested. OMP is very fast computationally but prone to a high degree of clutter in this application. We also point out that the formulation of MaxENT used here is very similar to OMP in some respects, the difference being that the former reconstructs the logarithm of the image rather than the image itself from basis vectors extracted from the observation matrix. MaxENT could in that regard be considered a form of compressive sensing. Key Points: Compressive sensing inverse methods have been applied to aperture synthesis radar imaging of ionospheric plasma density irregularitiesPerformance of basis pursuit denoisint (BPDN) and orthogonal matching pursuit (OMP) are generally inferior to that of the maximum‐entropy method (MaxENT)Computational speed of OMP is attractive and prompts research into more suitable function library [ABSTRACT FROM AUTHOR]

The problem of exact recovery of an individual sparse vector using the Basis Pursuit (BP) model is considered. A differentiable Huber loss function (a convex quadratic spline) is used to replace the ℓ1-norm in the BP model. Using the theory of duality and classical results from quadratic perturbation of linear programs, a necessary condition for exact recovery leading to a negative result is given. An easily verifiable sufficient condition is also presented. [ABSTRACT FROM AUTHOR]

The signal processing problem has become increasingly complex and demand high acquisition system, this paper proposes a new method to reconstruct the structure phased array structural health monitoring signal. The method is derived from the compressive sensing theory and the signal is reconstructed by using the basis pursuit algorithm to process the ultrasonic phased array signals. According to the principles of the compressive sensing and signal processing method, non-sparse ultrasonic signals are converted to sparse signals by using sparse transform. The sparse coefficients are obtained by sparse decomposition of the original signal, and then the observation matrix is constructed according to the corresponding sparse coefficients. Finally, the original signal is reconstructed by using basis pursuit algorithm, and error analysis is carried on. Experimental research analysis shows that the signal reconstruction method can reduce the signal complexity and required the space efficiently. [ABSTRACT FROM AUTHOR]

In this paper, we study the solution uniqueness of an individual feasible vector of a class of convex optimization problems involving convex piecewise affine functions and subject to general polyhedral constraints. This class of problems incorporates many important polyhedral constrained ℓ1 recovery problems arising from sparse optimization, such as basis pursuit, LASSO, and basis pursuit denoising, as well as polyhedral gauge recovery. By leveraging the max-formulation of convex piecewise affine functions and convex analysis tools, we develop dual variables based necessary and sufficient uniqueness conditions via simple and yet unifying approaches; these conditions are applied to a wide range of ℓ1 minimization problems under possible polyhedral constraints. An effective linear program based scheme is proposed to verify solution uniqueness conditions. The results obtained in this paper not only recover the known solution uniqueness conditions in the literature by removing restrictive assumptions but also yield new uniqueness conditions for much broader constrained ℓ1-minimization problems. [ABSTRACT FROM AUTHOR]

This research focuses on restoring signals caused by power failures in transmission lines using the basis pursuit, matching pursuit, and orthogonal matching pursuit sensing techniques. The original signal corresponds to the instantaneous current and voltage values of the electrical power system. The heuristic known as brute force is used to find the quasi-optimal number of atoms k in the original signal. Next, we search for the minimum number of samples known as m; this value is necessary to reconstruct the original signal from sparse and random samples. Once the values of k and m have been identified, the signal restoration is performed by sampling sparse and random data at other bus bars of the power electrical system. Basis pursuit allows recovering the original signal from 70% of the random samples of the same signal. The higher the number of samples, the longer the restoration times, approximately 12 s for recovering the entire signal. Matching pursuit allows recovering the same percentage, but with the lowest restoration time. Finally, orthogonal matching pursuit recovers a slightly lower percentage with a higher number of samples with a significant increase in its recovery time. Therefore, for real-time electrical fault signal restoration applications, the best selection will be matching pursuit due to the fact that it presents the lowest machine time, but requires more samples compared with orthogonal matching pursuit. Basis pursuit and orthogonal matching pursuit require fewer sparse and random samples despite the fact that these require a longer processing time for signal recovery. These two techniques can be used to reduce the volume of data that is stored by phasor measurement systems.

In modern high resolution SAR data, due to the intrinsic side-looking geometry of SAR sensors, layover and foreshortening issues inevitably arise, especially in dense urban areas. SAR tomography provides a new way of overcoming these problems by exploiting the back-scattering property for each pixel. However, traditional non-parametric spectral estimators, e.g. Truncated Singular Value Decomposition (TSVD), are limited by their poor elevation resolution, which is not comparable to the azimuth and slant-range resolution. In this paper, the Compressive Sensing (CS) approach using Basis Pursuit (BP) and TWo-step Iterative Shrinkage/Thresholding (TWIST) are introduced. Experimental studies with real spotlight-mode TerraSAR-X dataset are carried out using both BP and TWIST, to demonstrate the merits of compressive sensing approaches in terms of robustness, computational efficiency, and super-resolution capability.