Basis Pursuit Research Articles

Partial discharge pattern recognition via sparse representation and ANN

In this study, seventeen samples were created for classifying internal, surface, and corona partial discharges (PDs) in a high voltage lab. Next, PDs were measured experimentally to provide a dictionary comprising the types. Due to the huge size of the recorded dataset, a new and straightforward preprocessing method based on signal norms was used to extract the appropriate features of various samples. The new sparse representation classifier (SRC) was computed using l1 and stable l1-norm minimization by means of Primal-Dual Interior Point (PDIP) and Basis Pursuit De-noise (BPDN) algorithms, respectively. The pattern recognition was also performed with an artificial neural network (ANN) and compared with the sparse method. It is shown that both methods have comparable performance if training process, tuning options, and other tasks for finding the best result from ANN are not taken into account. Even with this assumption, it is shown that SRC still performs better than ANN in some cases. In addition, the SRC technique presented in this paper converges to a fixed result, while the results after training the ANN vary with every run due to random initial weights.

Comparison of reconstruction accuracy of sparse recovery algorithms for gapped RCS data

ABSTRACTThis study evaluates and compares the reconstruction accuracy of sparse recovery algorithms (SRAs) when we reconstruct a two‐dimensional (2D) inverse synthetic aperture radar (ISAR) image from a gapped radar‐cross‐section (RCS) dataset. To analyze the influence of the gap on the reconstruction accuracy, we adopted a relative error (RE) in a 2D image domain, and evaluated the reconstruction accuracy of each method by calculating each RE value against the rate of missing data in the gap. The accuracy of the basis pursuit denoising method was better than that of other methods considered in this study, especially when we reconstructed the ISAR image from the gapped RCS dataset when signal‐to‐noise ratio was low. Consequently, the relaxed constraint rather than the strictly equality constraint for the SRA is more adequate for SRA applied to radar imaging. © 2015 Wiley Periodicals, Inc. Microwave Opt Technol Lett 57:1249–1255, 2015

Low complexity predistortion and equalization in nonlinear multicarrier satellite communications

Aiming to reduce the power/mass requirements in satellite transponders and to reduce mission costs, joint amplification of multiple carriers using a single high-power amplifier (HPA) is being considered. In this scenario, a careful investigation of the resulting power efficiency is essential as amplification is nonlinear, and multicarrier signals exhibit enlarged peak-to-average power ratio. Thus, operating the amplifier close to saturation vastly increases signal distortion resulting in a severe degradation of performance, especially for higher order modulations. This paper proposes a reduced-complexity digital predistortion (DPD) scheme at the transmitter and a corresponding equalizer (EQ) at the receiver to mitigate these nonlinear effects. Scenarios include both the forward as well as the return links. In particular, the paper exploits the MIMO Volterra representation and builds on a basis pursuit approach using a LASSO (least absolute shrinkage and selection operator) algorithm to achieve an efficient basis representation, avoiding large computational complexity, to describe the selection of predistorter/equalizer model. The work further compares and contrasts the two mitigation techniques taking various system aspects into consideration. The gains in performance and amplification efficiency demonstrated by the use of DPD/ EQ motivate their inclusion in next-generation satellite systems.

Sparse component analysis using time-frequency representations for operational modal analysis.

Sparse component analysis (SCA) has been widely used for blind source separation(BSS) for many years. Recently, SCA has been applied to operational modal analysis (OMA), which is also known as output-only modal identification. This paper considers the sparsity of sources' time-frequency (TF) representation and proposes a new TF-domain SCA under the OMA framework. First, the measurements from the sensors are transformed to the TF domain to get a sparse representation. Then, single-source-points (SSPs) are detected to better reveal the hyperlines which correspond to the columns of the mixing matrix. The K-hyperline clustering algorithm is used to identify the direction vectors of the hyperlines and then the mixing matrix is calculated. Finally, basis pursuit de-noising technique is used to recover the modal responses, from which the modal parameters are computed. The proposed method is valid even if the number of active modes exceed the number of sensors. Numerical simulation and experimental verification demonstrate the good performance of the proposed method.

Analysis of suitable modulation scheme for compressive sensing algorithm in wireless sensor network

Purpose – The purpose of this paper is to find out the use of compressive sensing (CS) algorithm for wireless sensor networks (WSNs). As energy-efficient algorithms are required for WSNs, CS is very much useful as less than 25 per cent of the entire input data alone is required to be transmitted, and reconstruction at the receiver with this reduced data set is of good quality. But, the usefulness of the algorithm with suitable modulation schemes is not analyzed so far in the literature. Hence, this work concentrated on the algorithm performance with different modulation schemes and different channel conditions. Design/methodology/approach – Compressive sensing encoding is performed by using suitable transform on the input signal. Here, DCT and DWT are used to generate the sparse signal. Random measurement matrix is used to generate the compressed output, which is reconstructed using the Basis Pursuit (BP) method. Also, an analysis for the energy-efficient modulation scheme is performed by modulating the compressed output using QPSK/BPSK/QAM and transmitted by considering the Gaussian and Rayleigh Channels. Energy required per bit transmission is modeled and computed for different schemes. Findings – Simulation result shows that the use of CS algorithm for data compression tremendously reduces the number of transmission bits and, hence, enhances the transmission and bandwidth efficiency in WSN. Results show that DWT is a much suitable transform to be used for sparse measurement generation. In comparison with DCT, DWT is computationally simple and takes very less time, which is expected in real-time application. The reconstruction result shows that about 25 per cent of the data sample is sufficient to recover the original image, perhaps which is the most surprising result. An extensive analysis of various modulation schemes based on the energy model shows that QPSK is in the AWGN channel, and QAM modulation in the Rayleigh channel is a much suitable modulation scheme to be used in WSN for further reduction of energy consumption. Originality/value – Compressive sensing is recently gaining importance for quantization, compression and noise removal in images. In this paper, this technique was used along with modulation schemes to analyze the suitability of the algorithm for WSN.

Semi-blind nonstationary deconvolution: Joint reflectivity and Q estimation

Source signature deconvolution and attenuation or inverse quality factor- (Q-) filtering are two challenging problems in seismic data analysis which are used for extending the temporal bandwidth of the data. The separate estimates of the wavelet and, especially, the Earth Q model are by themselves problematic and add further uncertainties to inverse problems which are clearly ill-conditioned. The two problems are formulated in the framework of polynomial extrapolation and a closed form solution is provided based on the Lagrange interpolation. Analysis of the stability issue shows that the errors in the estimated results grow exponentially with both the problem size N and the inverse of Q. In order to circumvent both the instability and uncertainty of the Q model, these problems are addressed in a unified formulation as a semi-blind nonstationary deconvolution (SeND) to decompose the observed trace into the least number of nonstationary wavelets selected from a dictionary via a basis pursuit algorithm. The dictionary is constructed from the known source wavelet with different propagation times, each attenuated with a range of possible Q values. Using the Horner's rule, an efficient algorithm is also provided for application of the dictionary and its adjoint. SeND is an extension of the conventional sparse spike deconvolution to its nonstationary form, which provides the reflectivity and Q models simultaneously without requiring a-priori Q information. Assuming that the wavelet and attenuation mechanism are both known, the numerical data SeND allows to estimate both the original reflectivity and the Q models with higher accuracy, especially with respect to conventional spectral ratio techniques. The application of the algorithm to field data finally indicates a substantial improvement in temporal resolution on a seismic record.

Image Compression Based on Compressive Sensing Using Wavelet Lifting Scheme

Many algorithms have been developed to find sparse representation over redundant dictionaries or transform. This paper presents a novel method on compressive sensing (CS)-based image compression using sparse basis on CDF9/7 wavelet transform. The measurement matrix is applied to the three levels of wavelet transform coefficients of the input image for compressive sampling. We have used three different measurement matrix as Gaussian matrix, Bernoulli measurement matrix and random orthogonal matrix. The orthogonal matching pursuit (OMP) and Basis Pursuit (BP) are applied to reconstruct each level of wavelet transform separately. Experimental results demonstrate that the proposed method given better quality of compressed image than existing methods in terms of proposed image quality evaluation indexes and other objective (PSNR/UIQI/SSIM) measurements.

Intelligent fault diagnosis of rolling element bearings using sparse wavelet energy based on overcomplete DWT and basis pursuit

This paper explores sparse time-frequency distribution (TFD) using overcomplete discrete wavelet transform (DWT) and sparse representation techniques. This distribution is discovered for characterizing the periodic transient information embedded in rolling element bearings and extracting effective features that can discriminate different fault conditions. Based on the sparse TFD, a new sparse wavelet energy (SWE) feature is obtained by three main steps: first, an overcomplete discrete DWT is employed to decompose the fault signal and construct a redundant dictionary; second, the redundant dictionary is optimized by basis pursuit to obtain the sparsest TFD; finally, SWE is calculated from the new TFD to produce a feature vector for each signal. SWE features that combine the merits of overcomplete DWT and sparse representation techniques can precisely reveal fault-induced information, thereby exhibiting valuable properties for automatic fault identification by intelligent classifiers. The effectiveness and advantages of the proposed features are confirmed by simulation and the practical fault pattern recognition of rolling bearings.

New Robust Estimators of Correlation and Weighted Basis Pursuit

The autocorrelation function is a commonly used tool in statistical time series analysis. In this paper we examine the robustness of two estimators of correlation based on memoryless nonlinear functions of the observations, the Median-of-Ratios Estimator (MRE) and the Phase-Phase Correlator (PPC), which are applicable to complex-valued Gaussian random processes. We show that they are robust to impulsive noise from a bias perspective at the expense of statistical efficiency at the Gaussian distribution. Additionally, we develop iterative versions of these estimators named the IMRE and IPPC, realizing an improved bias performance over their non-iterative counterparts and the well-known robust Schweppe-type Generalized M-estimator utilizing a Huber cost function (SHGM). We use robust Mahalanobis distances generated by the IMRE and IPPC to improve the performance of impulsive noise suppression through the use of weighted basis pursuit methods. These estimation and data-cleaning techniques are applied to both synthetic and actual collected data using an Ettus Research USRP to perform robust spectral estimation on signals from the digital television band in the presence of additive impulsive noise. Finally, the analysis reveals that if the time series is highly correlated and the contamination rate is low, the MRE outperforms the PPC estimator from a variance and bias perspective. However, the PPC is the estimator of choice when the correlation is lower and is better suited to time critical applications due to lower computation costs.

Compressed Sensing Based Encryption Approach for Tax Forms Data

In this work we investigate the possibility to use the measurement matrices from compressed sensing as secret key to encrypt / decrypt signals. Practical results and a comparison between BP (basis pursuit) and OMP (orthogonal matching pursuit) decryption algorithms are presented. To test our method, we used 10 text messages (10 different tax forms) and we generated 10 random matrices and for distortion validate we used the PRD (the percentage root-mean-square difference), its normalized version (PRDN) measures and NMSE (normalized mean square error). From the practical results we found that the time for BP algorithm is much higher than for OMP algorithm and the errors are smaller and should be noted that the OMP does not guarantee the convergence of the algorithm. We found that it is more advantageous, for tax forms (or other templates that show no interest for encryption) to encrypt only the recorded data. The time required for decoding is significantly lower than the decryption for the entire form

Alternating Direction Method of Multipliers for Separable Convex Optimization of Real Functions in Complex Variables

The alternating direction method of multipliers (ADMM) has been widely explored due to its broad applications, and its convergence has been gotten in the real field. In this paper, an ADMM is presented for separable convex optimization of real functions in complex variables. First, the convergence of the proposed method in the complex domain is established by using the Wirtinger Calculus technique. Second, the basis pursuit (BP) algorithm is given in the form of ADMM in which the projection algorithm and the soft thresholding formula are generalized from the real case. The numerical simulations on the reconstruction of electroencephalogram (EEG) signal are provided to show that our new ADMM has better behavior than the classic ADMM for solving separable convex optimization of real functions in complex variables.

Identification of Equation Error Models from Small Samples using Compressed Sensing Techniques

System identification (SI), especially from small samples, is a challenging problem and of interest in several applications. Standard prediction-error minimization methods (PEM), under these conditions, generally result in estimates with higher variance. Moreover, in the identification of parametric models, one often needs prior knowledge of the input-output delay, obtaining estimates of which, is not possible using classical methods when the delay is either comparable or greater than the sample size. In this work, we develop a compressed sensing (CS)-based method for identifying sparse equation-error models that includes both auto-regressive eXogenous (ARX) and AR moving average eXogenous (ARMAX) structures with large delays, small orders and small delays with large orders, but with missing coefficients. The outcome is an iterative basis pursuit de noising (IBPDN) algorithm for solving non-linear CS problems. In addition, we propose a semi-rigorous method to lower the mutual coherence of the regressor matrix so as to obtain lower variance parameter estimates with the CS techniques. Errors in parameter estimates are computed using the bootstrapping method. Simulation studies on three diverse examples are presented to demonstrate the efficacy of the proposed methodology.

Sparse channel estimation of underwater acoustic orthogonal frequency division multiplexing based on basis pursuit denoising

To solve the problem of poor performance of the traditional l2-norm channel estimation, a sparse channel estimation approach based on basis pursuit denoising (BPDN) is proposed in orthogonal frequency division multiplex underwater acoustic communication. Owing to the sparsity of the underwater acoustic channel, only a few observations are needed to recover the channel impulse response with a high accuracy. Compared with greedy pursuit algorithm, BPDN algorithm has the globally excellentest solution. The signal is estimated based on the l2-l1 norm rule and the observations containing the noise are considered. The regularization parameter can be changed to balance the signal's sparsity against the residual error. The influences of the pilot distribution and the regularization parameter on the BPDN algorithm are discussed in the simulation. The BPDN channel estimator is compared with the least square (LS) and also with orthogonal matching pursuit (OMP). The data collected from lake experiment show that the BPDN channel estimator outperforms the LS and OMP channel estimator over spare underwater acoustic channel.

Choice of regularization weight in basis pursuit reflectivity inversion

Seismic inverse problem of estimating P- and S-wave reflectivity from seismic traces has recently been revisited using a basis pursuit denoising inversion (BPI) approach. The BPI uses a wedge dictionary to define model constraints, which has been successful in resolving thin beds. Here we address two fundamental problems associated with BPI, namely, the uniqueness of the estimate and the choice of regularization weight λ to be used in the model norm. We investigated these using very fast simulated re-annealing (VFSR) and gradient projection sparse reconstruction (GPSR) approaches. For a synthetic model with two reflectors separated by one time sample, we are able to demonstrate convergence of VFSR to the true model with different random starting models. Two numerical approaches to estimating the regularization weight were investigated. One uses λ as a hyper-parameter and the other uses this as a temperature-like annealing parameter. In both cases, we were able to obtain λ fairly rapidly. Finally, an analytic formula for λ that is iteration adaptive was also implemented. Successful applications of our approach to synthetic and field data demonstrate validity and robustness.

Dispersion curve recovery with orthogonal matching pursuit

Dispersion curves characterize many propagation mediums. When known, many methods use these curves to analyze waves. Yet, in many scenarios, their exact values are unknown due to material and environmental uncertainty. This paper presents a fast implementation of sparse wavenumber analysis, a method for recovering dispersion curves from data. This approach, based on orthogonal matching pursuit, is compared with a prior implementation, based on basis pursuit denoising. In the results, orthogonal matching pursuit provides two to three orders of magnitude improvement in speed and a small average reduction in prediction capability. The analysis is demonstrated across multiple scenarios and parameters.

On Sparse Representation in Fourier and Local Bases

We consider the classical problem of finding the sparse representation of a signal in a pair of bases. When both bases are orthogonal, it is known that the sparse representation is unique when the sparsity K of the signal satisfies K <; 1/μ(D), where μ(D) is the mutual coherence of the dictionary. Furthermore, the sparse representation can be obtained in polynomial time by basis pursuit (BP), when K <; 0.91/μ(D). Therefore, there is a gap between the unicity condition and the one required to use the polynomial-complexity BP formulation. For the case of general dictionaries, it is also well known that finding the sparse representation under the only constraint of unicity is NP-hard. In this paper, we introduce, for the case of Fourier and canonical bases, a polynomial complexity algorithm that finds all the possible K-sparse representations of a signal under the weaker condition that K <; √2/μ(D). Consequently, when K <; 1/μ(D), the proposed algorithm solves the unique sparse representation problem for this structured dictionary in polynomial time. We further show that the same method can be extended to many other pairs of bases, one of which must have local atoms. Examples include the union of Fourier and local Fourier bases, the union of discrete cosine transform and canonical bases, and the union of random Gaussian and canonical bases.

Sparsity Properties of Compressive Video Sampling Generated by Coefficient Thresholding

We study the compressive sampling (CS) and its application in video encoding framework. The video input is firstly transformed into suitable domain in order to achieve sparser configuration of coefficients. Then, we apply coefficient thresholding to classify which frames to be sampled compressively or conventionally. For frames chosen to undergo compressive sampling, the coefficient vectors will be projected into smaller vectors using random measurement matrix. As CS requires two main conditions, i.e. sparsity and matrix incoherence, this research is emphasized on the enhancement of sparsity property of the input signal. It was empirically proven that the sparsity enhancement could be reached by applying motion compensation and thresholding to the non-significant coefficient count. At the decoder side, the reconstruction algorithm can employ basis pursuit or L1 minimization algorithm.

Basis pursuit denoising-based image superresolution using a redundant set of atoms

Digital investigations are very difficult to conduct from low-quality images generated by low-quality sensors. Therefore, we present a novel superresolution (SR) scheme that applies SR and denoising simultaneously, using the concept of sparse representation. For SR, a low-resolution (LR) input image is scaled up using our recently described adaptive interpolation scheme, and for each patch of the LR input, a vector of the sparse coefficients is then sought using a basis pursuit denoising sparse-coding algorithm instead of orthogonal matching pursuit. A high-resolution output is generated from the given LR input using the recovered vector of the sparse coefficients over a redundant set of atoms, i.e., an overcomplete dictionary. For the proposed technique, we modified the sparse-coding method of the K-SVD dictionary training approach by incorporating an efficient \(l_{1}\)-regularized least-squares method, i.e., a feature-sign search algorithm. Experimental evaluations validate the effectiveness of the proposed SR scheme.

Basis Pursuit Receiver Function

Abstract Receiver functions (RFs) are derived by deconvolution of the horizontal (radial or transverse) component of ground motion from the vertical component, which segregates the PS phases. Many methods have been proposed to employ deconvolution in frequency as well as in time domain. These methods vary in their approaches to impose regularization that addresses the stability problem. Here, we present application of a new time‐domain deconvolution technique called basis pursuit deconvolution (BPD) that has recently been applied to seismic exploration data. Unlike conventional deconvolution methods, the BPD uses an L1 norm constraint on model reflectivity to impose sparsity. In addition, it uses an overcomplete wedge dictionary based on a dipole reflectivity series to define model constraints, which can achieve higher resolution than that obtained by the traditional methods. We demonstrate successful application of BPD based RF estimation from synthetic data for a crustal model with a near‐surface thin layer of thickness 5, 7, 10, and 15 km. The BPD can resolve these thin layers better with much improved signal‐to‐noise ratio than the conventional methods. Finally, we demonstrate application of the BPD receiver function (BPRF) method to a field dataset from Kutch, India, where near‐surface sedimentary layers are known to be present. The BPRFs are able to resolve reflections from these layers very well.

Hierarchical Bayesian Approach For Jointly-Sparse Solution Of Multiple-Measurement Vectors.

It is well-known that many signals of interest can be well-estimated via just a small number of supports under some specific basis. Here, we consider finding sparse solution for Multiple Measurement Vectors (MMVs) in case of having both jointly sparse and clumpy structure. Most of the previous work for finding such sparse representations are based on greedy and sub-optimal algorithms such as Basis Pursuit (BP), Matching Pursuit (MP), and Orthogonal Matching Pursuit (OMP). In this paper, we first propose a hierarchical Bayesian model to deal with MMVs that have jointly-sparse structure in their solutions. Then, the model is modified to account for clumps of the neighbor supports (block sparsity) in the solution structure, as well. Several examples are considered to illustrate the merit of the proposed hierarchical Bayesian model compared to OMP and a modified version of the OMP algorithm.