Random Measurement Matrices Research Articles

Semi-tensor product-based one-bit compressed sensing

The area of one-bit compressed sensing (1-bit CS) focuses on the recovery of sparse signals from binary measurements. Over the past decade, this field has witnessed the emergence of well-developed theories. However, most of the existing literature is confined to fully random measurement matrices, like random Gaussian and random sub-Gaussian measurements. This limitation often results in high generation and storage costs. This paper aims to apply semi-tensor product-based measurements to 1-bit CS. By utilizing the semi-tensor product, this proposed method can compress high-dimensional signals using lower-dimensional measurement matrices, thereby reducing the cost of generating and storing fully random measurement matrices. We propose a regularized model for this problem that has a closed-form solution. Theoretically, we demonstrate that the solution provides an approximate estimate of the underlying signal with upper bounds on recovery error. Empirically, we conduct a series of experiments on both synthetic and real-world data to demonstrate the proposed method’s ability to utilize a lower-dimensional measurement matrix for signal compression and reconstruction with enhanced flexibility, resulting in improved recovery accuracy.

Compressed sensing for ECG signal compression using DWT based sensing matrices

ABSTRACT In this article, we have investigated the 1-D discrete wavelet transform (DWT)-based measurement matrices for electrocardiogram (ECG) compression. Moreover, the current work examines the suitability of the diverse DWT matrices, namely Symlets, Battle, Coiflets, Vaidyanathan, and Beylkin wavelet families, for ECG compression. Furthermore, this article shows the comparative performance study of the proposed DWT matrices with conventional deterministic and random measurement matrices. Overall, the Battle1 wavelet-based measurement matrices demonstrate the enhanced performance against the db3, coif5, and sym6 based measurement matrices in terms of Percentage Root-Mean Squared Difference (PRD), Root Mean Square Error (RMSE), and Signal-to-Noise Ratio (SNR). Finally, it was seen that the proposed Battle1 matrix demonstrates the improved performance against the conventional measurement matrices such as the Karhunen–Loeve transform (KLT), Discrete Cosine Transform (DCT) matrix, and random Hadamard measurement matrix. Thus, the result shows the adequacy of DWT measurement matrices for the compression of ECG.

Construction of Structured Random Measurement Matrices in Semi-Tensor Product Compressed Sensing Based on Combinatorial Designs.

A random matrix needs large storage space and is difficult to be implemented in hardware, and a deterministic matrix has large reconstruction error. Aiming at these shortcomings, the objective of this paper is to find an effective method to balance these performances. Combining the advantages of the incidence matrix of combinatorial designs and a random matrix, this paper constructs a structured random matrix by the embedding operation of two seed matrices in which one is the incidence matrix of combinatorial designs, and the other is obtained by Gram-Schmidt orthonormalization of the random matrix. Meanwhile, we provide a new model that applies the structured random matrices to semi-tensor product compressed sensing. Finally, compared with the reconstruction effect of several famous matrices, our matrices are more suitable for the reconstruction of one-dimensional signals and two-dimensional images by experimental methods.

GenMod: A generative modeling approach for spectral representation of PDEs with random inputs

We propose a method for quantifying uncertainty in high-dimensional PDE systems with random parameters, where the number of solution evaluations is small. Parametric PDE solutions are often approximated using a spectral decomposition based on polynomial chaos expansions. For the class of systems we consider (i.e., high dimensional with limited solution evaluations) the coefficients are given by an underdetermined linear system in a regression formulation. This implies additional assumptions, such as sparsity of the coefficient vector, are needed to approximate the solution. Here, we present an approach where we assume the coefficients are close to the range of a generative model that maps from a low to a high dimensional space of coefficients. Our approach is inspired be recent work examining how generative models can be used for compressed sensing in systems with random Gaussian measurement matrices. Using results from PDE theory on coefficient decay rates, we construct an explicit generative model that predicts the polynomial chaos coefficient magnitudes. The algorithm we developed to find the coefficients, which we call GenMod, is composed of two main steps. First, we predict the coefficient signs using Orthogonal Matching Pursuit. Then, we assume the coefficients are within a sparse deviation from the range of a sign-adjusted generative model. This allows us to find the coefficients by solving a nonconvex optimization problem, over the input space of the generative model and the space of sparse vectors. We obtain theoretical recovery results for a Lipschitz continuous generative model and for a more specific generative model, based on coefficient decay rate bounds. We examine three high-dimensional problems and show that, for all three examples, the generative model approach outperforms sparsity promoting methods at small sample sizes.

Deep Unfolding Basis Pursuit: Improving Sparse Channel Reconstruction via Data-Driven Measurement Matrices

For massive multiple-input multiple-output systems operating in frequency-division duplex mode, downlink channel state information (CSI) acquisition will incur large overhead. This overhead is substantially reduced when sparse channel estimation techniques are employed, owing to the channel sparsity in the angular domain. When a sparse channel estimation method is implemented, the measurement matrix, which is related to the pilot matrix, is essential to the channel estimation performance. Existing sparse channel estimation schemes widely adopt random measurement matrices, which have been criticized for their suboptimal reconstruction performance. This paper proposes novel data-driven solutions to design the measurement matrix. Model-based autoencoders are customized to optimize the measurement matrix by unfolding the classical basis pursuit algorithm. The obtained data-driven measurement matrices are applied to existing sparse reconstruction algorithms, leading to flexible hybrid data-driven implementations for sparse channel estimation. Numerical results show that the proposed data-driven measurement matrices can achieve more accurate reconstructions and use fewer measurements than the existing random matrices, thereby leading to a higher achievable rate for CSI acquisition. Moreover, compared with existing pure deep learning-based sparse reconstruction methods, the proposed hybrid data-driven scheme, which uses the novel data-driven measurement matrices with conventional sparse reconstruction algorithms, can achieve higher reconstruction accuracy.

Sparsity-Free Compressed Sensing With Applications to Generative Priors

We study the problem of reconstructing a high-dimensional signal xRn from a low-dimensional noisy linear measurement y=Mx+eRℓ, assuming x admits a certain structure. We model the measurement matrix as M=BA, with arbitrary BRℓ×m and sub-gaussian ARm×n; therefore allowing for a family of random measurement matrices which may have heavy tails, dependent rows and columns, and a large dynamic range for the singular values. The structure is either given as a non-convex cone TRn, or is induced via minimizing a given convex function f(∙), hence our study is sparsity-free. We prove, in both cases, that an approximate empirical risk minimizer robustly recovers the signal if the effective number of measurements is sufficient, even in the presence of a model mismatch, i.e. the signal not exactly admitting the model’s structure. While in classical compressed sensing the number of independent (sub)-gaussian measurements regulates the possibility of a robust reconstruction, in our setting the effective number of measurements depends on the properties of B. We show that, in this model, the stable rank of B indicates the effective number of measurements, and an accurate recovery is guaranteed whenever it exceeds, to within a constant factor, the effective dimension of the structure set. We apply our results to the special case of generative priors, i.e. when x is close to the range of a Generative Neural Network (GNN) with ReLU activation functions. Also, if the GNN has random weights in the last layer, our theory allows a partial Fourier measurement matrix, thus taking the first step towards a theoretical analysis of compressed sensing MRI with GNN. Our work relies on a recent result in random matrix theory by Jeong, Li, Plan, and Yılmaz [1].

Dictionary-sparse recovery from heavy-tailed measurements

Abstract The recovery of signals that are sparse not in a given basis, but rather sparse with respect to an over-complete dictionary is one of the most flexible settings in the field of compressed sensing with numerous applications. As in the standard compressed sensing setting, it is possible that the signal can be reconstructed efficiently from a few linear measurements, for example, by the so-called $\ell _1$-synthesis method. However, it has been less well understood which measurement matrices provably work for this setting. Whereas in the standard setting, it has been shown that even certain heavy-tailed measurement matrices can be used in the same sample complexity regime as Gaussian matrices, comparable results are only available for the restrictive class of sub-Gaussian measurement matrices as far as the recovery of dictionary-sparse signals via $\ell _1$-synthesis is concerned. In this work, we fill this gap and establish optimal guarantees for the recovery of vectors that are (approximately) sparse with respect to a dictionary via the $\ell _1$-synthesis method from linear, potentially noisy measurements for a large class of random measurement matrices. In particular, we show that random measurements that fulfill only a small-ball assumption and a weak moment assumption, such as random vectors with i.i.d. Student-$t$ entries with a logarithmic number of degrees of freedom, lead to comparable guarantees as (sub-)Gaussian measurements. Our results apply for a large class of both random and deterministic dictionaries. As a technical tool, we show a bound on the expectation of the sum of squared order statistics under very general assumptions, which might be of independent interest. As a corollary of our results, we also obtain a slight improvement on the weakest assumption on a measurement matrix with i.i.d. rows sufficient for uniform recovery in standard compressed sensing, improving on results by Lecué and Mendelson and by Dirksen, Lecué and Rauhut.

A Remote Security Computational Ghost Imaging Method Based on Quantum Key Distribution Technology

Computational ghost imaging (CGI) is a method of acquiring object information by measuring light field intensity, which would be used to achieve imaging in a complicated environment. The main issue to be addressed in CGI technology is how to achieve rapid and high-quality imaging while assuring the secure transmission of detection data in practical distant imaging applications. In order to address the mentioned issues, this paper proposes a remote secure CGI method based on quantum communication technology. Using the quantum key distribution (QKD) network, the CGI system can be reconstructed while solving the problem of information security transmission between the detector and the reconstructed computing device. By exploring the influence of different random measurement matrices on the quality of image reconstruction, it is found that the randomness of the numerical sequence constituting the matrix is positively correlated with the imaging quality. Based on this discovery, a new type of quantum cryptography measurement matrix is constructed using quantum cryptography with good randomness. In addition, through further orthogonalization and normalization of the matrix, the matrix has both good randomness and orthogonality, and high-quality imaging results can be obtained at a low sampling rate. The feasibility and effectiveness of the method are verified by simulation imaging experiments. Compared with the traditional GI system, the method proposed in this paper has higher transmission security and high-quality imaging under this premise, which provides a new idea for the practical development of CGI technology.

Exploiting 2D compressed sensing and information entropy for secure color image compression and encryption

Compared to 1D compressed sensing (CS), 2D CS is more efficient for compressing the plaintext image from two directions, but security level of current 2D CS-based ciphers is unsatisfactory. To solve this problem, this paper presents a novel color image compression and encryption algorithm by combining 2D CS, information entropy and chaos. Firstly, the color image is decomposed into red, green and blue components, then they are sparsely transformed by the discrete wavelet transform (DWT) to get three sparse matrices. Next, the obtained matrices are observed by two asymptotical deterministic random measurement matrices based on information entropy and counter (ADMMIC), which not only encrypts the plaintext image, but also compresses it in proportion to reduce the transmission bandwidth and storage space. Subsequently, the corresponding measurement value matrices are shuffled by a double random scrambling based on Arnold map and index vector (DRSAIV) to eliminate the correlation between adjacent pixels. Furthermore, the obtained permutated matrices are diffused by a simultaneous multiple random diffusion of inter–intra components (SMRDIC) to obtain the final cipher image, the plaintext pixel to be diffused, the key matrix involved in diffusion and the position of the obtained ciphertext pixel are all unpredictable, which makes statistical attack invalid. In addition, information entropy values of plaintext image are obtained to generate the initial values of the used chaotic systems, which greatly improve the ability to resist the known-plaintext and chosen-plaintext attacks. Simulation results and security analyses verify that this algorithm has good compression and high security.

Universal Measurement Matrix Design for Sparse and Co-Sparse Signal Recovery

Compressed Sensing (CS) avails mutual coherence metric to choose the measurement matrix that is incoherent with dictionary matrix. Random measurement matrices are incoherent with any dictionary, but their highly uncertain elements necessitate large storage and make hardware realization difficult. In this paper deterministic matrices are employed which greatly reduce memory space and computational complexity. To avoid the randomness completely, deterministic sub-sampling is done by choosing rows deterministically rather than randomly, so that matrix can be regenerated during reconstruction without storing it. Also matrices are generated by orthonormalization, which makes them highly incoherent with any dictionary basis. Random matrices like Gaussian, Bernoulli, semi-deterministic matrices like Toeplitz, Circulant and full-deterministic matrices like DFT, DCT, FZC-Circulant are compared. DFT matrix is found to be effective in terms of recovery error and recovery time for all the cases of signal sparsity and is applicable for signals that are sparse in any basis, hence universal.

Sparse recovery in bounded Riesz systems with applications to numerical methods for PDEs

We study sparse recovery with structured random measurement matrices having independent, identically distributed, and uniformly bounded rows and with a nontrivial covariance structure. This class of matrices arises from random sampling of bounded Riesz systems and generalizes random partial Fourier matrices. Our main result improves the currently available results for the null space and restricted isometry properties of such random matrices. The main novelty of our analysis is a new upper bound for the expectation of the supremum of a Bernoulli process associated with a restricted isometry constant. We apply our result to prove new performance guarantees for the CORSING method, a recently introduced numerical approximation technique for partial differential equations (PDEs) based on compressive sensing.

Reconstruction of bandlimited graph signals from measurements

Graph signal reconstruction is a fundamental topic in graph signal processing. Recently, graph signal reconstruction in three typical sampling schemes has been studied in the literature. In this paper, we study the reconstruction of bandlimited graph signals in a general measurement scheme, which includes the three existing sampling schemes as its special cases. Theorems for perfect reconstruction are established, and reconstruction by random measurement matrices is studied. An iterative reconstruction algorithm based on convex projection is proposed, and numerical experiments are implemented to demonstrate the effectiveness of the algorithm.

Construction of a Class of Logistic Chaotic Measurement Matrices for Compressed Sensing

The construction of the measurement matrix is the key technology for accurate recovery of compressed sensing. In this paper, we demonstrated correlation properties of nonpiecewise and piecewise logistic chaos system to follow Gaussian distribution. The correlation properties can generate a class of logistic chaotic measurement matrices with simple structure, easy hardware implementation and ideal measurement efficiency. Specifically, spread spectrum sequences generated by the correlation properties follow Gaussian distribution. Thus, the proposed algorithm constructs chaos-Gaussian matrices by the sequences. Simulation results of one-dimensional signals and two-dimensional images show that chaos-Gaussian measurement matrices can provide comparable performance against common random measurement matrices. In addition, chaos-Gaussian matrices are deterministic measurement matrices.

Infrared Image Reconstruction Based on Archimedes Spiral Measurement Matrix

It is a new research direction to realize infrared (IR) image reconstruction using compressed sensing (CS) theory. In the field of CS, the construction of measurement matrix is very principal. At present, the types of measurement matrices are mainly random and deterministic. The random measurement matrix can well satisfy the property of measurement matrix, but needs a large amount of storage space and has an inconvenient in hardware implementation. Therefore, a deterministic measurement matrix construction method is proposed for IR image reconstruction in this paper. Firstly, a series of points are collected on Archimedes spiral to construct a definite sequence; then the initial measurement matrix is constructed; finally, the deterministic measurement matrix is obtained according to the required sampling rate. Simulation results show that the IR image could be reconstructed by the measured values obtained through the proposed measurement matrix. Moreover, the proposed measurement matrix has better reconstruction performance compared with the Gaussian and Bernoulli random measurement matrices.

A robust image encryption algorithm based on Chua's circuit and compressive sensing

In terms of Chua's circuit system, compressive sensing (CS) and Haar wavelet, a novel image compression-encryption scheme (CES) is proposed in this paper. Firstly, the plaintext image is decomposed into approximate component and detail components through Haar wavelet. Then the approximate component is diffused by the threshold processing of local binary patterns (LBP) operator-based chaotic sequence which is produced by the combination of Chua's circuit and Logistic map. Next, the Lissajous map is applied to generate the chaos-combined asymptotic deterministic random measurement matrices (CADRMM) which are employed to measure the detail components in different compression ratios. In addition, the combination of mapped approximate and detail components is shuffled by the Logistic map. The experimental results and simulation analysis prove that the proposed cryptosystem is capable of reducing data for transmission and has good security performance under various attacks, especially for the shear and noise attacks.

Compressive Imaging Using RIP-Compliant CMOS Imager Architecture and Landweber Reconstruction

In this paper, we present a new image sensor architecture for fast and accurate compressive sensing (CS) of natural images. Measurement matrices usually employed in CS CMOS image sensors are recursive pseudo-random binary matrices. We have proved that the restricted isometry property of these matrices is limited by a low sparsity constant. The quality of these matrices is also affected by the non-idealities of pseudo-random number generators (PRNG). To overcome these limitations, we propose a hardware-friendly pseudo-random ternary measurement matrix generated on-chip by means of class III elementary cellular automata (ECA). These ECA present a chaotic behavior that emulates random CS measurement matrices better than other PRNG. We have combined this new architecture with a block-based CS smoothed-projected Landweber reconstruction algorithm. By means of single value decomposition, we have adapted this algorithm to perform fast and precise reconstruction while operating with binary and ternary matrices. Simulations are provided to qualify the approach.

Compressed Sensing Based UWB Receiver Using Signal-Matched Sparse Measurement Matrix

In this paper, we propose a deterministic signal-matched sparse measurement matrix for ultra wideband (UWB) systems. The proposed matrix is easy to implement and is computationally efficient as needed in a practical system. The proposed matrix captures higher signal energy in the compressive sensing (CS) domain as compared to the commonly used random matrices. Furthermore, a simple OneShot signal recovery of a full (Nyquist rate) signal from its sub-Nyquist rate measurement samples is proposed for symbol detection. Theoretical analysis and simulation results demonstrate that the detection performance of a CS-based UWB receiver using the proposed matrix is better than the random measurement matrices and is close to a conventional system sampled at full Nyquist rate.

Distributed fusion filtering for multi-sensor systems with correlated random transition and measurement matrices

ABSTRACTThis paper is concerned with the distributed fusion estimation problem for discrete-time linear stochastic systems with measurements coming from different sensors and correlated random parameter matrices in both the state and measurement equations. At each sampling time, the random state transition parameter matrices are assumed to be correlated at the same sampling time with the measurement random parameter matrices of each sensor. Moreover, the random parameter matrices in the observation equations are one-step auto-correlated and cross-correlated between the different sensors. The additive noises are also assumed to be correlated. Under these assumptions, the distributed fusion filter is designed as the matrix-weighted linear combination of the local least-squares linear filters obtained at every single sensor, using the linear minimum variance optimality criterion. A numerical simulation example considering a two-sensor system with randomly delayed measurements is used to illustrate the applicability of multi-sensor systems with correlated random parameter matrices and analyse the performance of the proposed filtering estimators.

Secure Wireless Communications Based on Compressive Sensing: A Survey

Compressive sensing (CS) has become a popular signal processing technique and has extensive applications in numerous fields such as wireless communications, image processing, magnetic resonance imaging, remote sensing imaging, and anology to information conversion, since it can realize simultaneous sampling and compression. In the information security field, secure CS has received much attention due to the fact that CS can be regarded as a cryptosystem to attain simultaneous sampling, compression and encryption when maintaining the secret measurement matrix. Considering that there are increasing works focusing on secure wireless communications based on CS in recent years, we produce a detailed review for the state-of-the-art in this paper. To be specific, the survey proceeds with two phases. The first phase reviews the security aspects of CS according to different types of random measurement matrices such as Gaussian matrix, circulant matrix, and other special random matrices, which establishes theoretical foundations for applications in secure wireless communications. The second phase reviews the applications of secure CS depending on communication scenarios such as wireless wiretap channel, wireless sensor network, Internet of Things, crowdsensing, smart grid, and wireless body area networks. Finally, some concluding remarks are given.

Energy‐efficient data reconstruction algorithm for spatially‐ and temporally correlated data in wireless sensor networks

This study introduces a novel algorithm for reconstruction of wireless sensor networks data which inherently have spatial and temporal correlations. The authors' algorithm is based on compressed sensing (CS) and benefits from sliding window processing. This new algorithm rearranges the data in form of a cube and uses this representation to extract more information about the data. There are two optimisation loops which are solved simultaneously and periodically reconstruct one part of the whole signal from measurements that arrive at the sink. In particular, the first reconstruction loop, which uses a modified version of basis pursuit reconstruction algorithm, is meant for reconstruction of a temporal data which is extracted from the data cube, and the second loop which uses a modified version of reweighted l 1 -norm algorithm is for reconstruction of data windows. The authors used a special kind of binary sparse random measurement matrices for sampling which is equipped with a condition to get samples as variously as possible and this, in turn, balances the duty among sensors and provides more information from the field. Simulation results verify that the proposed algorithm achieves better reconstruction accuracy and less energy consumption in comparison with state-of-the-art CS reconstruction methods.