Basis Pursuit Method Research Articles

Signal Recovery from Randomly Quantized Data Using Neural Network Approach.

We present an efficient scheme based on a long short-term memory (LSTM) autoencoder for accurate seismic deconvolution in a multichannel setup. The technique is beneficial for compressing massive amounts of seismic data. The proposed robust estimation ensures the recovery of sparse reflectivity from acquired seismic data that have been under-quantized. By adjusting the quantization error, the technique considerably improves the robustness of data to the quantization error, thereby boosting the visual saliency of seismic data compared to the other existing algorithms. This framework has been validated using both field and synthetic seismic data sets, and the assessment is carried out by comparing it to the steepest decent and basis pursuit methods. The findings indicate that the proposed scheme outperforms the other algorithms significantly in the following ways: first, in the proposed estimation, fraudulently or overbearingly estimated impulses are significantly suppressed, and second, the proposed guesstimate is much more robust to the quantization interval changes. The tests on real and synthetic data sets reveal that the proposed LSTM autoencoder-based method yields the best results in terms of both quality and computational complexity when compared with existing methods. Finally, the relative reconstruction error (RRE), signal-to-reconstruction error ratio (SRER), and power spectral density (PSD) are used to evaluate the performance of the proposed algorithm.

Decomposed structured subsets for semidefinite and sum-of-squares optimization

Semidefinite programs (SDPs) are standard convex problems that are frequently found in control and optimization applications. Interior-point methods can solve SDPs in polynomial time up to arbitrary accuracy, but scale poorly as the size of matrix variables and the number of constraints increases. To improve scalability, SDPs can be approximated with lower and upper bounds through the use of structured subsets (e.g., diagonally-dominant and scaled-diagonally dominant matrices). Meanwhile, any underlying sparsity or symmetry structure may be leveraged to form an equivalent SDP with smaller positive semidefinite constraints. In this paper, we present a notion of decomposed structured subsets to approximate an SDP with structured subsets after an equivalent conversion. The lower/upper bounds found by approximation after conversion become tighter than the bounds obtained by approximating the original SDP directly. We apply decomposed structured subsets to semidefinite and sum-of-squares optimization problems with examples of H∞ norm estimation and constrained polynomial optimization. An existing basis pursuit method is adapted into this framework to iteratively refine bounds.

Enhanced Sparse Representations of Spike Waveforms Obtained by using the Basis Pursuit Approach

In the extracellular neural recordings, the spike waveforms formed by the neurons nearby the recording electrode must be sorted according to their morphology. This process is called as spike sorting and it is an important prerequisite in neural decoding algorithms. Low Q-factor wavelet transforms are frequently being used as feature extractors to detect the discriminative patterns between adjacent neurons’ activity. However, the wavelet coefficients are highly sensitive to noise that may occur due to the employed instrumentation system and the local field potentials defined as the total activity of nearby neurons. However, enhanced sparse representations of the spike wave forms, having reduced noise activity, can be attained by using the basis pursuit method that is applied to the tunable Q-factor wavelet transform coefficients. In the tunable Q-factor wavelet transform, the Q-factor of the wavelet filters can be tuned according to the signal of interest with a controllable redundancy. In the proposed study, enhanced sparse representations of the spike waveforms were obtained by using the basis pursuit approach. Later, the energy values of the decomposed subbands were employed as features that can discriminate morphological differences in spike shapes. Finally, the obtained features were fed to k-nearest neighbors and decision trees learning models in an unbiased cross-validation scheme to objectively measure the effect of the enhanced sparsity decomposition. The qualitative and quantitative results show that the enhanced sparsity-based energy features are superior to the traditional low Q-factor based wavelet decomposition in terms of the accuracy metric.

Algorithm for constructing trajectories of maneuvering object based on bearing-only information using the Basis Pursuit method

The possibility of using the Basis Pursuit signal recovery method in the object tracking algorithm as a method of filtering and extrapolating the trajectory of the observed object in a one-dimensional measurement space is considered. The simulation results are presented for the trajectory processing algorithm using the proposed filtering method for the data obtained during the object direction-finding in comparison with the block algorithm based on the maximum likelihood method and the Kalman filter.

Seismic reflectivity inversion using an L1-norm basis-pursuit method and GPU parallelisation

Abstract Seismic reflectivity inversion problem can be formulated using a basis-pursuit method, aiming to generate a sparse reflectivity series of the subsurface media. In the basis-pursuit method, the reflectivity series is composed by large amounts of even and odd dipoles, thus the size of the seismic response matrix is huge and the matrix operations involved in seismic inversion are very time-consuming. In order to accelerate the matrix computation, a basis-pursuit method-based seismic inversion algorithm is implemented on Graphics Processing Unit (GPU). In the basis-persuit inversion algorithm, the problem is imposed with a L1-norm model constraint for sparsity, and this L1-norm basis-pursuit inversion problem is reformulated using a linear programming method. The core problems in the inversion are large-scale linear systems, which are resolved by a parallelised conjugate gradient method. The performance of this fully parallelised implementation is evaluated and compared to the conventional serial coding. Specifically, the investigation using several field seismic data sets with different sizes indicates that GPU-based parallelisation can significantly reduce the computational time with an overall factor up to 145. This efficiency improvement demonstrates a great potential of the basis-pursuit inversion method in practical application to large-scale seismic reflectivity inversion problems.

Trend-cycles of vegetation dynamics as a tool for land degradation assessment and monitoring

Abstract The use of time series of Normalized Difference Vegetation Index (NDVI), obtained from satellite sensors has become frequent in studies for land degradation assessment and monitoring. Linear trends of NDVI are usually considered as indicators of vegetation dynamics and widely used as proxies for land degradation. Yet, long-term trends of NDVI often exhibit unidirectional (monotonic) but also cyclic (non-monotonic) dynamics, including mid-term oscillations, both of which are poorly captured by linear trends. Trend-cycle is a time series analysis that represents a smoothed version of a seasonally adjusted time series, which provides information on long-term movements while including changes in direction underlying the series. We assessed NDVI trend-cycles in Patagonia (Argentina) as proxies for land dynamics, integrating trend and medium-term cycles (>4 years). We used MODIS images between years 2000 and mid-2018; trend-cycles were analysed using the Basis Pursuit method. We observed that trend-cycles explained a significant portion of total temporal variability (reaching almost 20%), from which most patterns were explained by non-monotonic behaviour. We identified five major patterns in vegetation dynamics: decreasing (0.1% of area), increasing (0.6%), recovery (48.8%), relapsing (36.8%) and no trend-cycle (13.8%). Contrary to what is generally seen in the literature, monotonic patterns and particularly decreasing trend-cycles were marginally recorded in the last 18 years of NDVI records in Patagonia. Instead, the greater proportion of the area was classified as initial or advanced recovery and initial relapsing patterns, which refer to phases of a cyclic behaviour. We call for the need to revisit the conceptualization of land degradation assessment by means of remote sensing, and to critically review the ability of linear trends to reflect vegetation dynamics. Finally, we discuss the potential use of trend-cycle as a tool to monitor land dynamics and progress towards land degradation neutrality.

Data-driven diagnosis for compressed sensing algorithms

A data-driven diagnosis approach is developed for compressed sensing algorithms to objectively detect the failure of each algorithm and the insufficiency of data. Compressed sensing enables us to reduce the amount of data to a threshold; however, the problem is that no one can tell where the threshold is in practice. The approach is based on the statistical technique of cross-validation, in which part of the data is set aside as a validation dataset to assess the results of the analysis of the remaining data with an objective algorithm. The relation between cross-validation error and the size ratio of the whole dataset and validation dataset is evaluated and compared with a power-law relation to determine whether data analysis was successfully conducted. After summarizing our recent theoretical work on the first application of the approach to the basis pursuit method, it is shown by numerical experiments that the approach is universally applicable to several other algorithms.

Sparse Representation Based on Tunable Q‐Factor Wavelet Transform for Whale Click and Whistle Extraction

Whale sounds may mix several elements including whistle, click, and creak in the same vocalization, which may overlap in time and frequency, so it leads to conventional signal separation techniques challenging to be applied for the signal extraction. Unlike conventional signal separation techniques which are based on the frequency bands, such as WT and EMD, tunable‐Q wavelet transform (TQWT) can separate the objected signal into particular components with different structures according to its oscillation property and eliminate in‐band noise using the basis pursuit method. Considering the characteristics of oscillatory and transient impulse, we propose a novel signal separation method for whale whistle and click extraction. The proposed method is performed by the following two steps: first, TQWT is used to construct the dictionary for sparse representation. Secondly, the whale click and whistle construction are performed by designing the basis pursuit denoising (BPD) algorithm. The proposed method has been compared with one of the popular signal decomposition techniques, i.e., the EMD method. The experimental results show that the proposed method has a better performance of click and whistle signal separation in comparison with the EMD algorithm.

Sparse spikes super-resolution on thin grids II: the continuous basis pursuit

This article analyzes the performance of the continuous basis pursuit (C-BP) method for sparse super-resolution. The C-BP has been recently proposed by Ekanadham, Tranchina and Simoncelli as a refined discretization scheme for the recovery of spikes in inverse problems regularization. One of the most well known discretization scheme, the basis pursuit (BP, also known as ) makes use of a finite dimensional norm on a grid. In contrast, the C-BP uses a linear interpolation of the spikes positions to enable the recovery of spikes between grid points. When the sought-after solution is constrained to be positive, a remarkable feature of this approach is that it retains the convexity of the initial problem. The present paper shows how the C-BP is able to recover the spikes locations with sub-grid accuracy in the favorable case. We also prove that this regime generally breaks when the grid is too thin, and we describe precisely the artifacts that appear: each spike is approximated by a pair of Dirac masses. We show numerical illustrations of these phenomena, and evaluate numerically the validity of the technical assumptions of our analysis.

Design and utilisation of protocols to characterise dynamic PET uptake of two tracers using basis pursuit

Imaging using more than one biological process using PET could be of great utility, but despite previously proposed approaches to dual-tracer imaging, it is seldom performed. The alternative of performing multiple scans is often infeasible for clinical practice or even in research studies. Dual-tracer PET scanning allows for multiple PET radiotracers to be imaged within the same imaging session. In this paper we describe our approach to utilise the basis pursuit method to aid in the design of dual-tracer PET imaging experiments, and later in separation of the signals. The advantage of this approach is that it does not require a compartment model architecture to be specified or even that both signals are distinguishable in all cases. This means the method for separating dual-tracer signals can be used for many feasible and useful combinations of biology or radiotracer, once an appropriate scanning protocol has been decided upon.Following a demonstration in separating the signals from two consecutively injected radionuclides in a controlled experiment, phantom and list-mode mouse experiments demonstrated the ability to test the feasibility of dual-tracer imaging protocols for multiple injection delays. Increases in variances predicted for kinetic macro-parameters VD and KI in brain and tumoral tissue were obtained when separating the synthetically combined data. These experiments confirmed previous work using other approaches that injections delays of 10–20 min ensured increases in variance were kept minimal for the test tracers used. On this basis, an actual dual-tracer experiment using a 20 min delay was performed using these radio tracers, with the kinetic parameters (VD and KI) extracted for each tracer in agreement with the literature.This study supports previous work that dual-tracer PET imaging can be accomplished provided certain constraints are adhered to. The utilisation of basis pursuit techniques, with its removed need to specify a model architecture, allows the feasibility of a range of imaging protocols to be investigated via simulation in a straight-forward manner for a wide range of possible scenarios. The hope is that the ease of utilising this approach during feasibility studies and in practice removes any perceived technical barrier to performing dual-tracer imaging.

Multi-frequency sparse Bayesian learning for robust matched field processing.

The multi-snapshot, multi-frequency sparse Bayesian learning (SBL) processor is derived and its performance compared to the Bartlett, minimum variance distortionless response, and white noise constraint processors for the matched field processing application. The two-source model and data scenario of interest includes realistic mismatch implemented in the form of array tilt and data snapshots not exactly corresponding to the range-depth grid of the replica vectors. Results demonstrate that SBL behaves similar to an adaptive processor when localizing a weaker source in the presence of a stronger source, is robust to mismatch, and exhibits improved localization performance when compared to the other processors. Unlike the basis or matching pursuit methods, SBL automatically determines sparsity and its solution can be interpreted as an ambiguity surface. Because of its computational efficiency and performance, SBL is practical for applications requiring adaptive and robust processing.

Identification-While-Scanning of a Multi-Aircraft Formation Based on Sparse Recovery for Narrowband Radar.

It is known that the identification performance of a multi-aircraft formation (MAF) of narrowband radar mainly depends on the time on target (TOT). To realize the identification task in one rotated scan with limited TOT, the paper proposes a novel identification-while-scanning (IWS) method based on sparse recovery to maintain high rotating speed and super-resolution for MAF identification, simultaneously. First, a multiple chirp signal model is established for MAF in a single scan, where different aircraft may have different Doppler centers and Doppler rates. Second, based on the sparsity of MAF in the Doppler parameter space, a novel hierarchical basis pursuit (HBP) method is proposed to obtain satisfactory sparse recovery performance as well as high computational efficiency. Furthermore, the parameter estimation performance of the proposed IWS identification method is analyzed with respect to recovery condition, signal-to-noise ratio and TOT. It is shown that an MAF can be effectively identified via HBP with a TOT of only about one hundred microseconds for IWS applications. Finally, some numerical experiment results are provided to demonstrate the effectiveness of the proposed method based on both simulated and real measured data.

Variable Chandler and Annual Wobbles in Earth’s Polar Motion During 1900–2015

The Chandler wobble (CW) and annual wobble (AW) are the two main components of polar motion, which are difficult to separate because of their very close periods. In the light of Fourier dictionary and basis pursuit method, a Fourier basis pursuit (FBP) spectrum is developed, which can reduce spectral smearing and leakage caused by the finite length of the time series. Further, a band-pass filtering method based on FBP spectrum (FBPBPF), which can effectively suppress the edge effect, is proposed in this paper. The simulation test results show that the FBPBPF method can effectively suppress the edge effect caused by spectral smearing and leakage and that its reconstruction accuracy at the boundary is approximately three times higher than the Fourier transform band-pass filtering method, which is based on Hamming windowed FFT spectrum, in extracting quasi-harmonic signals. The FBPBPF method is then applied to Earth’s polar motion data during 1900–2015. Through analyzing the amplitude and period variations of CW and AW, and calculating the eccentricity variation of the AW, we found that: (1) the amplitude of the CW is currently at a historic minimum level, and it is even possible to diminish further until a complete stop; and (2) the eccentricity of the AW has a gradually decreased fluctuation during the last 116 years.

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.

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.

Stable Signal Recovery via Randomly Enhanced Adaptive Subspace Pursuit Method

This letter addresses the stable signal recovery problem in the presence of noisy perturbations, and presents a variant of Subspace Pursuit (SP) method, called Randomly Enhanced Adaptive SP (REASP) method. REASP first adds correlative and irrelative atoms with an adaptive technique and a random strategy respectively, next detects the overall reliability, and then deletes unreliable atoms per iteration. Through these modifications, REASP is expected to achieve superior performance. The complexity analysis shows that the computational complexity of REASP is a bit higher but acceptable. The experimental results illustrate the superior performance of REASP, compared to the frequently used greedy pursuit (GP) and l1 basis pursuit (BP) methods.

Compressive sampling and wavelet-based multi-view image compression scheme

A multi-view image codec using a disparity compensated lifting based wavelet transform and ‘compressive sampling (CS)’ is presented. The input images are de-correlated into their sub-bands, using disparity compensated view filtering lifting based wavelet transforms. A wavelet transform is then applied to the baseband view, de-composing it into its sub-bands. High-frequency sub-bands are separately hard threshold. Wavelet-weights for high-frequency sub-bands are calculated and used to adjust threshold values for different sub-bands. The CS algorithm is then employed to generate measurements for each resulting sub-band. In the decoder side, the Basis Pursuit method is used to recover the high-frequency sub-bands. Results indicate that the proposed codec significantly outperforms the state-of-the-art CS-based multi-view image codecs.

PSD Map Construction Scheme Based on Compressive Sensing in Cognitive Radio Networks

of late, many researchers have been interested in sparse representation of signals and its applications such as Compressive Sensing in Cognitive Radio (CR) networks as a way of overcoming the issue of limited bandwidth. Compressive sensing based wideband spectrum sensing is a novel approach in cognitive radio systems. Also in these systems, using spatial-frequency opportunistic reuse is emerged interestingly by constructing and deploying spatial-frequency Power Spectral Density (PSD) maps. Since the CR sensors are distributed in the region of support, the sensed PSD by each sensor should be transmitted to a master node (base-station) in order to construct the PSD maps in space and frequency domains. When the number of sensors is large, this data transmission which is required for construction of PSD map can be challenging. In this paper, in order to transmit the CR sensors' data to the master node, the compressive sensing based scheme is used. Therefore, the measurements are sampled in a lower sampling rate than of the Nyquist rate. By using the proposed method, an acceptable PSD map for cognitive radio purposes can be achieved by only 30% of full data transmission. Also, simulation results show the robustness of the proposed method against the channel variations in comparison with classical methods. Different solution schemes such as Basis Pursuit, Lasso, Lars and Orthogonal Matching Pursuit are used and the quality performance of them is evaluated by several simulation results over a Rician channel with respect to several different compression and Signal to Noise Ratios. It is also illustrated that the performance of Basis Pursuit and Lasso methods outperform the other compression methods particularly in higher compression rates.

Quantifying the distribution of probes between subcellular locations using unsupervised pattern unmixing

Motivation: Proteins exhibit complex subcellular distributions, which may include localizing in more than one organelle and varying in location depending on the cell physiology. Estimating the amount of protein distributed in each subcellular location is essential for quantitative understanding and modeling of protein dynamics and how they affect cell behaviors. We have previously described automated methods using fluorescent microscope images to determine the fractions of protein fluorescence in various subcellular locations when the basic locations in which a protein can be present are known. As this set of basic locations may be unknown (especially for studies on a proteome-wide scale), we here describe unsupervised methods to identify the fundamental patterns from images of mixed patterns and estimate the fractional composition of them.Methods: We developed two approaches to the problem, both based on identifying types of objects present in images and representing patterns by frequencies of those object types. One is a basis pursuit method (which is based on a linear mixture model), and the other is based on latent Dirichlet allocation (LDA). For testing both approaches, we used images previously acquired for testing supervised unmixing methods. These images were of cells labeled with various combinations of two organelle-specific probes that had the same fluorescent properties to simulate mixed patterns of subcellular location.Results: We achieved 0.80 and 0.91 correlation between estimated and underlying fractions of the two probes (fundamental patterns) with basis pursuit and LDA approaches, respectively, indicating that our methods can unmix the complex subcellular distribution with reasonably high accuracy.Availability: http://murphylab.web.cmu.edu/softwareContact: murphy@cmu.edu

An Empirical Bayesian Strategy for Solving the Simultaneous Sparse Approximation Problem

Given a large overcomplete dictionary of basis vectors, the goal is to simultaneously represent L>1 signal vectors using coefficient expansions marked by a common sparsity profile. This generalizes the standard sparse representation problem to the case where multiple responses exist that were putatively generated by the same small subset of features. Ideally, the associated sparse generating weights should be recovered, which can have physical significance in many applications (e.g., source localization). The generic solution to this problem is intractable and, therefore, approximate procedures are sought. Based on the concept of automatic relevance determination, this paper uses an empirical Bayesian prior to estimate a convenient posterior distribution over candidate basis vectors. This particular approximation enforces a common sparsity profile and consistently places its prominent posterior mass on the appropriate region of weight-space necessary for simultaneous sparse recovery. The resultant algorithm is then compared with multiple response extensions of matching pursuit, basis pursuit, FOCUSS, and Jeffreys prior-based Bayesian methods, finding that it often outperforms the others. Additional motivation for this particular choice of cost function is also provided, including the analysis of global and local minima and a variational derivation that highlights the similarities and differences between the proposed algorithm and previous approaches.