Basis Pursuit Denoising Research Articles

Sampling numbers of smoothness classes via ℓ1-minimization

Using techniques developed recently in the field of compressed sensing we prove new upper bounds for general (nonlinear) sampling numbers of (quasi-)Banach smoothness spaces in L2. In particular, we show that in relevant cases such as mixed and isotropic weighted Wiener classes or Sobolev spaces with mixed smoothness, sampling numbers in L2 can be upper bounded by best n-term trigonometric widths in L∞. We describe a recovery procedure from m function values based on ℓ1-minimization (basis pursuit denoising). With this method, a significant gain in the rate of convergence compared to recently developed linear recovery methods is achieved. In this deterministic worst-case setting we see an additional speed-up of m−1/2 (up to log factors) compared to linear methods in case of weighted Wiener spaces. For their quasi-Banach counterparts even arbitrary polynomial speed-up is possible. Surprisingly, our approach allows to recover mixed smoothness Sobolev functions belonging to SprW(Td) on the d-torus with a logarithmically better rate of convergence than any linear method can achieve when 1<p<2 and d is large. This effect is not present for isotropic Sobolev spaces.

A Distributed Projection Neurodynamic Approach for Solving BP Denoising Problem in Sparse Recovery

This paper investigates a novel projection neurodynamic approach for solving the basis pursuit denoising (BPDN) in a distributed manner. First, by using the distributed consensus theorem over undirected graph and supplementary variables method, the distributed version of BPDN is obtained. Second, with the help of projection operators, primal-dual dynamical system and derivative feedback terms, a novel distributed neurodynamic approach is proposed to deal with the distributed version of BPDN for sparse recovery. Moreover, the optimality and convergence properties of the proposed distributed projection neurodynamic approach (DPNA) are analyzed rigorously. Finally, we apply DPNA to sparse signal reconstruction which demonstrates the effectiveness of DPNA through numerical experiments. In addition, inspired by the role of image reconstruction technology in the field of defense against adversarial attack, we use DPNA as a preprocessing method to enhance the robustness of the deep model. Compared with known defense schemes such as JEPG, ComDefend, and OMP, our DPNA is better than them.

Sparsity-Driven Inverse Synthetic Aperture Radar Imaging Using Accelerated Meta-Heuristic Optimization

When the inverse synthetic aperture radar (ISAR) system have sparse aperture (SA) dataset, which are non-uniform and less than Nyquist sampling rate along the cross-range axis, it can significantly blur the ISAR image of target having complex motion. To overcome this, we propose an improved ISAR imaging method based on dictionary estimation using parametric signal reconstruction (DEPSR) using accelerated meta-heuristic optimization (MHO). The proposed method reconstructs a motion-compensated and full-aperture (FA) dataset from a sparse-aperture (SA) dataset using MHO, which can be accelerated by a graphics processing unit (GPU). After then, we can obtain the motion-compensated FA dataset and by using this, we can exploit the conventional range-Doppler (R-D) processing. Simulation and real-data-based ISAR imaging results validate the superiority of the proposed method in terms of image quality in comparison with conventional methods, such as simple R-D processing, basis pursuit denoising, and DEPSR. Particularly, the proposed method exhibits a significantly lower computational burden than that of the traditional DEPSR approach.

Exploiting Tunable Q-Factor Wavelet Transform Domain Sparsity to Denoise Wrist PPG Signals

Wearable health monitoring devices based on photoplethysmogram (PPG) can estimate valuable physiological parameters and are often regarded as a touchstone for indicating the cardiovascular status of a person. Severe motion artifacts are generally observed in the acquired PPG signals from the biosensors in the wearables when the PPG acquisition is performed during different physical activities. Classical filters can remove thermal noise and electromagnetic interference from the raw PPG signals, but they fail to handle the motion artifacts that cover a wider range of signal frequencies. This work introduces a new motion artifact removal scheme that exploits the sparsity of PPG signals in the tunable-Q factor wavelet transform (TQWT) domain using the basis pursuit denoising (BPDN) scheme. The proposed scheme deftly removes noise while preserving the original morphology of the PPG signal without using any reference accelerometer sensor data. In particular, a regularized SALSA-based BPDN algorithm is employed for sparse recovery, which utilizes a soft thresholding approach to denoise the noisy PPG signals. TQWT can be tuned according to the oscillatory behavior of the raw PPG signal. At the same time, BPDN exploits the sparsity of the wavelet coefficients to yield an artifact-free reconstruction of PPG. Further, heart rate estimation is performed using the denoised PPG signal with support vector regression to demonstrate the clinical information preservation during the proposed denoising process. The proposed denoising scheme is evaluated on two publicly available databases, and an overall minimum mean absolute error (MAE) of 1.03 and 0.76 beats per minute (bpm) is achieved. The suggested denoising approach has demonstrated better performance with the lowest MAE compared to the previously reported methods.

A dual active set method for &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;$ \ell $&lt;/tex-math&gt;&lt;/inline-formula&gt;1-regularized problem

We consider an $ \ell $1-regularized problem, which has many applications such as compressed sensing, MRI, and seismic images. Although there have been some methods for the problem, they cannot obtain a highly accurate solution. To overcome this drawback, we propose a dual active set method, which is an active set method for the Fenchel dual problem of the $ \ell $1-regularized problem. Our method finds a highly accurate solution quickly because of the following features of the dual problem: (i) the dimension of the decision variables is smaller; (ii) since the optimal solution of the primal problem is sparse, the active constraints of the optimal solution of the dual problem are few and can be identified quickly. Then, we show that the method outputs the solution with finite iterations when it solves certain subproblems exactly. Moreover, we give its efficient and concrete implementation for Basis Pursuit Denoising (BPDN), which is a typical $ \ell $1-regularized problem. In numerical experiments for BPDN, our method provides more accurate solutions than all baselines in comparable computing time.

Denoising and Inpainting of Sonar Images Using Convolutional Sparse Representation

Recently sparse representation (SR) techniques have been extensively applied for sonar images restoration tasks such as denoising and inpainting. However, the fine details such as edges and texture information present in sonar images are not fully restored using these techniques. This is due to ineffective dictionary formation and learning of sparse components and dictionaries. In this paper, the author proposes the convolutional sparse representation (CSR)-based methods for sonar image denoising and inpainting tasks. To minimize the Gaussian noise from sonar images, an alternative direction method of the multiplier (ADMM) optimization algorithm is used for solving the convolutional basis pursuit denoising (CBPDN) problem with joint sparsity through an L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2,1</sub> norm term. For the removal of impulse noise from sonar images, an ADMM algorithm is used to solve a CSR problem with an L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> data fidelity term and both L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> and L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> gradient regularization terms. For the inpainting of sonar images, the additive mask simulation technique is used for solving CSR problems related to boundary masking. From the experimental results, it is found that the proposed sonar image denoising and inpainting methods have created better visual quality sonar images, higher peak signal-to-noise ratio (PSNR), and structure similarity (SSIM) values, as compared to state-of-the-art sonar image denoising and inpainting methods.

A Lagrange Programming Neural Network Approach with an ℓ0-Norm Sparsity Measurement for Sparse Recovery and Its Circuit Realization

Many analog neural network approaches for sparse recovery were based on using ℓ1-norm as the surrogate of ℓ0-norm. This paper proposes an analog neural network model, namely the Lagrange programming neural network with ℓp objective and quadratic constraint (LPNN-LPQC), with an ℓ0-norm sparsity measurement for solving the constrained basis pursuit denoise (CBPDN) problem. As the ℓ0-norm is non-differentiable, we first use a differentiable ℓp-norm-like function to approximate the ℓ0-norm. However, this ℓp-norm-like function does not have an explicit expression and, thus, we use the locally competitive algorithm (LCA) concept to handle the nonexistence of the explicit expression. With the LCA approach, the dynamics are defined by the internal state vector. In the proposed model, the thresholding elements are not conventional analog elements in analog optimization. This paper also proposes a circuit realization for the thresholding elements. In the theoretical side, we prove that the equilibrium points of our proposed method satisfy Karush Kuhn Tucker (KKT) conditions of the approximated CBPDN problem, and that the equilibrium points of our proposed method are asymptotically stable. We perform a large scale simulation on various algorithms and analog models. Simulation results show that the proposed algorithm is better than or comparable to several state-of-art numerical algorithms, and that it is better than state-of-art analog neural models.

Communication Bistatic RCS Estimation Using Monostatic Scattering Centers With Compressive Sensing

This communication presents a novel technique for obtaining bistatic radar cross sections (RCSs) extracted from a monostatic scattering field dataset using a discrete scattering center model and basis pursuit denoising (BPDN) algorithm, which is compressive sensing (CS) technique. With the high-frequency assumption, a complex vector of independent point sources is formulated in a monostatic configuration for a geometrical relationship among the transmitter, receiver, and equivalent scatterers. A matrix equation for the scattering problem of an underdetermined form is effectively calculated by an iterative BPDN solver. Then, compensating the phase difference in the bistatic condition can derive the bistatic RCS at specific frequency points and observation angles. The accuracy of the proposed method for extracting bistatic results is verified using two numerical examples, and the performance of the method in terms of measurement efficiency and data resolution in frequency and angle domains is compared with that of conventional method.

A Priori Denoising Strategies for Sparse Identification of Nonlinear Dynamical Systems: A Comparative Study

Abstract In recent years, identification of nonlinear dynamical systems from data has become increasingly popular. Sparse regression approaches, such as Sparse Identification of Nonlinear Dynamics (SINDy), fostered the development of novel governing equation identification algorithms assuming the state variables are known {\it a priori} and the governing equations lend themselves to sparse, linear expansions in a (nonlinear) basis of the state variables. In the context of the identification of governing equations of nonlinear dynamical systems, one faces the problem of identifiability of model parameters when state measurements are corrupted by noise. Measurement noise affects the stability of the recovery process yielding incorrect sparsity patterns and inaccurate estimation of coefficients of the governing equations. In this work, we investigate and compare the performance of several local and global smoothing techniques to {\it a priori} denoise the state measurements and numerically estimate the state time-derivatives to improve the accuracy and robustness of two sparse regression methods to recover governing equations: Sequentially Thresholded Least Squares (STLS) and Weighted Basis Pursuit Denoising (WBPDN) algorithms. We empirically show that, in general, global methods, which use the entire measurement data set, outperform local methods, which employ a neighboring data subset around a local point. We additionally compare Generalized Cross Validation (GCV) and Pareto curve criteria as model selection techniques to automatically estimate near optimal tuning parameters, and conclude that Pareto curves yield better results. The performance of the denoising strategies and sparse regression methods is empirically evaluated through well-known benchmark problems of nonlinear dynamical systems.

Accuracy and robustness of sparse reconstruction techniques for azimuthal mode analysis of in-duct sound fields

Sparse reconstruction techniques enable the detection of the dominant azimuthal mode content in ducts with a limited number of sensors. This is of great importance in various fields of application, such as fan noise analyses. The analysis of weaker azimuthal modes has hardly been considered or looked at, although they can contribute considerably to the total sound field and can provide important information on sound propagation effects. To address this shortcoming, the Enhanced Orthogonal Matching Pursuit (EOMP) algorithm presented in this paper was developed. It is based on the established Compressed Sensing Orthogonal Matching Pursuit (OMP) algorithm, which is used to reliably detect the dominant azimuthal modes. In an extension step, the weaker modes are determined by applying a Least Squares Fit or a Discrete Fourier Transform to the deconvolved measurement vector. The capabilities and limitations of the different versions of the EOMP algorithm are tested on experimental data of multiple rotor-coherent sound field components generated by a fan stage. Furthermore, the EOMP algorithm is explored in detail for typical practical application scenarios using Monte-Carlo simulations. In both cases, the performance is compared with an alternative method, the Basis Pursuit Denoising (BPDN) algorithm. Within the Monte-Carlo framework, extensive parameter variations for the signal-to-noise ratio, mode sparsity (number of dominant modes) and mode field sparsity-level (magnitude ratio of dominant modes to weak modes), are performed over a wide frequency range. These Monte-Carlo simulations allow the comprehensive evaluation of the accuracy of the algorithms under investigation. Sensor ring designs and the considered analysis mode ranges can have a large impact on the overall reconstruction accuracy. Therefore, both aspects are also addressed in the presented study to provide helpful insights for future applications of the azimuthal mode analysis. Additionally, the EOMP algorithm is further improved by a regularised Least Squares Fit for the enhanced step and is shown to provide the best accuracy and robustness among the investigated algorithms.

On the sign recovery by least absolute shrinkage and selection operator, thresholded least absolute shrinkage and selection operator, and thresholded basis pursuit denoising

AbstractBasis 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 . By definition, when , BP uniquely recovers when and implies (identifiability condition). Furthermore, LASSO can recover the sign of 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 in both the noisy and noiseless cases if and only if 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 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 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.

Convolutional analysis operator learning for multifocus image fusion

Sparse representation (SR), convolutional sparse representation (CSR) and convolutional dictionary learning (CDL) are synthetic-based priors that have proven to be successful in signal inverse problems (such as multifocus image fusion). Unlike “synthesis” formulas, “analysis” model assigns probabilities to signals through various forward measurements of signals. Analysis operator learning (AOL) is a classical analysis-based learning method. Convolutional analysis operator learning (CAOL) is convolutional form of AOL. CAOL uses unsupervised learning method to train autocoded convolutional neural network (CNN) to more accurately solve inverse problem. From the perspective of CAOL, this paper introduces learned convolutional regularizers into multifocus image fusion and proposes CAOL-based multifocus image fusion algorithm. In the CDL stage, convergent block proximal extrapolated gradient method with majorizer (BPEG-M) and adaptive momentum restarting scheme are used. In the sparse fusion stage, alternating direction method of multipliers (ADMM) approach with convolutional basis pursuit denoising (CBPDN) and l1 norm maximum strategy are employed for high-frequency and low-frequency component, respectively. 3 types of multifocus images (static gray images, gray images in sports and color images) are tested to verify performance of the proposed method. A comparison with representative methods demonstrates superiority of our method in terms of subjective observation and objective evaluation.

Compressive Sensing to Reduce the Number of Elements in a Linear Antenna Array With a Phased Array Weather Radar

For weather radar, although a phased array system is useful for rapid scanning, its development cost is several times higher than that of radar with a dish-type antenna, and it requires relatively more antenna elements, analog-to-digital converters, and phase shifters. To reduce antenna elements with the devices of phased array systems, array geometry in a random or triangular pattern is often used. However, relatively fewer antenna elements result in observation accuracy degradation due to the increased sidelobe level or the wider beamwidth. We employ compressive sensing (CS) processing [i.e., L1 minimization (L1) and basis pursuit denoising (BPDN)] to reduce the number of antenna elements. We conducted numerical simulations for point and distributed like targets and discussed the observation accuracy using L1 and BPDN. Compared to the estimations made using a full array antenna, where BPDN can be estimated with extremely high accuracy given a 25% reduction in antenna elements, we also applied CS to real measurement data obtained using the PAWR. The BPDN was also very effective for measurement data. The novelty of the study is highlighted in its discussion of the feasibility of applying CS in observation data with the PAWR. The study findings provide a reference for development cost reduction and the mass production of PAWR.

Basis Pursuit Denoising via Recurrent Neural Network Applied to Super-Resolving SAR Tomography

Finding sparse solutions of underdetermined linear systems commonly requires the solving of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> regularized least squares minimization problem, which is also known as the basis pursuit denoising (BPDN). They are computationally expensive since they cannot be solved analytically. An emerging technique known as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">deep unrolling</i> provided a good combination of the descriptive ability of neural networks, explainable, and computational efficiency for BPDN. Many unrolled neural networks for BPDN, e.g. learned iterative shrinkage thresholding algorithm and its variants, employ shrinkage functions to prune elements with small magnitude. Through experiments on synthetic aperture radar tomography (TomoSAR), we discover the shrinkage step leads to unavoidable information loss in the dynamics of networks and degrades the performance of the model. We propose a recurrent neural network (RNN) with novel sparse minimal gated units (SMGUs) to solve the information loss issue. The proposed RNN architecture with SMGUs benefits from incorporating historical information into optimization, and thus effectively preserves full information to the final output. Taking TomoSAR inversion as an example, extensive simulations demonstrated that the proposed RNN outperforms the state-of-the-art deep learning-based algorithm in terms of super-resolution power as well as generalization ability. It achieved 10% to 20% higher double scatterers detection rate and is less sensitive to phase and amplitude ratio difference between scatterers. Test on real TerraSAR-X spotlight images also shows high-quality 3-D reconstruction of test site.

A linearly convergent proximal ADMM with new iterative format for BPDN in compressed sensing problem

&lt;abstract&gt;&lt;p&gt;In recent years, compressive sensing (CS) problem is being popularly applied in the fields of signal processing and statistical inference. The alternating direction method of multipliers (ADMM) is applicable to the equivalent forms of basis pursuit denoising (BPDN) in CS problem. However, the solving speed and accuracy are adversely affected when the dimension increases greatly. In this paper, a new iterative format of proximal ADMM, which has fast solving speed and pinpoint accuracy when the dimension increases, is proposed to solve BPDN problem. Global convergence of the new type proximal ADMM is established in detail, and we exhibit a $ R- $ linear convergence rate under suitable condition. Moreover, we apply this new algorithm to solve different types of BPDN problems. Compared with the state-of-the-art of algorithms in BPDN problem, the proposed algorithm is more accurate and efficient.&lt;/p&gt;&lt;/abstract&gt;

A Compressed Sensing Recovery Algorithm Based on Support Set Selection

The theory of compressed sensing (CS) has shown tremendous potential in many fields, especially in the signal processing area, due to its utility in recovering unknown signals with far lower sampling rates than the Nyquist frequency. In this paper, we present a novel, optimized recovery algorithm named supp-BPDN. The proposed algorithm executes a step of selecting and recording the support set of original signals before using the traditional recovery algorithm mostly used in signal processing called basis pursuit denoising (BPDN). We proved mathematically that even in a noise-affected CS system, the probability of selecting the support set of signals still approaches 1, which means supp-BPDN can maintain good performance in systems in which noise exists. Recovery results are demonstrated to verify the effectiveness and superiority of supp-BPDN. Besides, we set up a photonic-enabled CS system realizing the reconstruction of a two-tone signal with a peak frequency of 350 MHz through a 200 MHz analog-to-digital converter (ADC) and a signal with a peak frequency of 1 GHz by a 500 MHz ADC. Similarly, supp-BPDN showed better reconstruction results than BPDN.

Uncertainty Quantification and Bayesian Inference of Cloud Parameterization in the NCAR Single Column Community Atmosphere Model (SCAM6)

Uncertainty quantification (UQ) in weather and climate models is required to assess the sensitivity of their outputs to various parameterization schemes and thereby improve their consistency with observations. Herein, we present an efficient UQ and Bayesian inference for the cloud parameters of the NCAR Single Column Atmosphere Model (SCAM6) using surrogate models based on a polynomial chaos expansion. The use of a surrogate model enables to efficiently propagate uncertainties in parameters into uncertainties in model outputs. We investigated eight uncertain parameters: the auto-conversion size threshold for ice to snow (dcs), the fall speed parameter for stratiform cloud ice (ai), the fall speed parameter for stratiform snow (as), the fall speed parameter for cloud water (ac), the collection efficiency of aggregation ice (eii), the efficiency factor of the Bergeron effect (berg_eff), the threshold maximum relative humidity for ice clouds (rhmaxi), and the threshold minimum relative humidity for ice clouds (rhmini). We built two surrogate models using two non-intrusive methods: spectral projection (SP) and basis pursuit denoising (BPDN). Our results suggest that BPDN performs better than SP as it enables to filter out internal noise during the process of fitting the surrogate model. Five out of the eight parameters (namely dcs, ai, rhmaxi, rhmini, and eii) account for most of the variance in predicted climate variables (e.g., total precipitation, cloud distribution, shortwave and longwave cloud radiative effect, ice, and liquid water path). A first-order sensitivity analysis reveals that dcs contributes ~40–80% of the total variance of the climate variables, ai around 15–30%, and rhmaxi, rhmini, and eii around 5–15%. The second- and higher-order effects contribute ~7 and 20%, respectively. The sensitivity of the model to these parameters was further explored using response curves. A Markov chain Monte Carlo (MCMC) sampling algorithm was also implemented for the Bayesian inference of dcs, ai, as, rhmini, and berg_eff using cloud distribution data collected at the Southern Great Plains (USA). The inferred parameters suggest improvements in the global Climate Earth System Model (CESM2) simulations of the tropics and sub-tropics.

Efficient Tuning-Free l1-Regression of Nonnegative Compressible Signals

In compressed sensing the goal is to recover a signal from as few as possible noisy, linear measurements with the general assumption that the signal has only a few non-zero entries. The recovery can be performed by multiple different decoders, however most of them rely on some tuning. Given an estimate for the noise level a common convex approach to recover the signal is basis pursuit denoising. If the measurement matrix has the robust null space property with respect to the ℓ2-norm, basis pursuit denoising obeys stable and robust recovery guarantees. In the case of unknown noise levels, nonnegative least squares recovers non-negative signals if the measurement matrix fulfills an additional property (sometimes called the M+-criterion). However, if the measurement matrix is the biadjacency matrix of a random left regular bipartite graph it obeys with a high probability the null space property with respect to the ℓ1-norm with optimal parameters. Therefore, we discuss non-negative least absolute deviation (NNLAD), which is free of tuning parameters. For these measurement matrices, we prove a uniform, stable and robust recovery guarantee. Such guarantees are important, since binary expander matrices are sparse and thus allow for fast sketching and recovery. We will further present a method to solve the NNLAD numerically and show that this is comparable to state of the art methods. Lastly, we explain how the NNLAD can be used for viral detection in the recent COVID-19 crisis.

Sensor position optimization for flux mapping in a nuclear reactor using compressed sensing

The paper presents a novel approach to sensor position optimisation based on compressed sensing (CS) and its application to neutron flux estimation in a nuclear reactor. An online flux mapping system (FMS) with a suitable algorithm is needed to continuously monitor neutron flux distribution from the measurements of in-core detectors in large nuclear reactors. In the present work, sensor positions are optimized by designing a sensing matrix Φ for the CS system in which the dictionary is known in advance. The sensing matrix is optimized by maximising the incoherence of the dictionary matrix and incorporating the estimation error in the optimization process. The proposed method uses the CS recovery technique, Basis Pursuit De-Noising (BPDN), for flux mapping along the vertical direction at the detector housing locations (1-D). In the end, the method suggests the optimized positions and number of sensors for efficient estimation of neutron flux inside the reactor core.

The Importance of Phase in Complex Compressive Sensing

We consider the question of estimating a real low-complexity signal (such as a sparse vector or a low-rank matrix) from the phase of complex random measurements. We show that in this phase-only compressive sensing (PO-CS) scenario, we can perfectly recover such a signal with high probability and up to global unknown amplitude if the sensing matrix is a complex Gaussian random matrix and the number of measurements is large compared to the complexity level of the signal space. Our approach proceeds by recasting the (non-linear) PO-CS scheme as a linear compressive sensing model built from a signal normalization constraint, and a phase-consistency constraint imposing any signal estimate to match the observed phases in the measurement domain. Practically, stable and robust estimation of the signal direction is achieved from any instance optimal algorithm of the compressive sensing literature (such as basis pursuit denoising). This is ensured by proving that the matrix associated with this equivalent linear model satisfies with high probability the restricted isometry property under the above condition on the number of measurements. We finally observe experimentally that robust signal direction recovery is reached at about twice the number of measurements needed for signal recovery in compressive sensing.