Optimized Projection Matrix for Compressive Sensing

Compressive (or namely compressed) sensing (CS) exhibits superior performance in sparse spatial spectrum estimation from a predefined vandermonde based dictionary. The CS theory requires that dictionary is as incoherent (orthogonal) as possible since the number of atoms is generally much more than observation vectors, and, namely, the dictionary is over-complete or redundant. Previous researches focus on designing sensing matrix to reduce the mutual coherence of dictionary. However, according to Grassmannian frames, it is still a problem that the coherence of a given dictionary is hard to break through an equiangular tight frame (ETF). To address the problem, we proposed and proved a KR-KSVD method to break through the original lower bound of mutual coherence. In the Khatri-Rao subspace, measurement matrix is designed by minimizing the cost function between the Gram matrix of the equivalent dictionary and an identity matrix with the KSVD method. Simulations demonstrate that the method can produce a better performance in terms of mutual coherence property and sparse recovery accuracy.Compressive (or namely compressed) sensing (CS) exhibits superior performance in sparse spatial spectrum estimation from a predefined vandermonde based dictionary. The CS theory requires that dictionary is as incoherent (orthogonal) as possible since the number of atoms is generally much more than observation vectors, and, namely, the dictionary is over-complete or redundant. Previous researches focus on designing sensing matrix to reduce the mutual coherence of dictionary. However, according to Grassmannian frames, it is still a problem that the coherence of a given dictionary is hard to break through an equiangular tight frame (ETF). To address the problem, we proposed and proved a KR-KSVD method to break through the original lower bound of mutual coherence. In the Khatri-Rao subspace, measurement matrix is designed by minimizing the cost function between the Gram matrix of the equivalent dictionary and an identity matrix with the KSVD method. Simulations demonstrate that the method can produce a better...

Multiphase flow measurements, based on tomographic imaging techniques, have made substantial progress in the past several years. In particular, ultrasonic tomographic reconstruction has shown its high potential in accurately measuring fluid fractions in a flow cross-section based on direct flow measurements. These types of measurements are very attractive as the system is nonradioactive, has direct contact to fluid, yet without flow obstruction. An ultrasonic tomographic multiphase flow meter (UTM) is being developed to capture the fluid flow rates and phase compositions in producing oil wells. The UTM aims to generate cross-sectional images of the flow inside the pipe. To recover these images, complex computations and high processing time are required, based on a limited amount of available data. For this purpose, compressive sensing (CS) reconstruction methods have shown great potential in overcoming the mentioned data quality and quantity issues. CS builds upon the fact that one can recover certain data of interest from potentially very few measurements. Many CS algorithms have been proposed in the literature. In this paper, we seek to find a suitable algorithm by comparing different CS algorithms in terms of the resulting recovered image quality. Specifically, we survey three algorithms in CS, namely basis pursuit, iterative soft thresholding (IST), and orthogonal matching pursuit (OMP). We also compare the results of the CS algorithms to that of a least-square based conventional algorithm. The numerical results show the superiority of CS recovery algorithms over the conventional approach, both qualitatively and quantitatively. The results also suggest that high image quality could potentially be achieved with less complex CS algorithms, i.e., the IST and OMP algorithms. As presented in the paper, our goal is to adapt CS reconstruction algorithms for images generated by the UTM, for the purpose of accurate quantification of oil, gas, and water phase fractions produced from different wells.

Time delay estimation (TDE) is a hot research topic in wireless location technology. Compressed sensing (CS) theory has been widely applied to image reconstruction and direction of arrival estimation since it was proposed in 2004. The sparse model can be constructed in time domain for estimating the time delay by using the CS theory. The measurement matrix plays a crucial role in the processing of signal reconstruction which is the core problem of CS theory. Therefore the research in the measurement matrix has becomes a hotspot in recent years. The existing measurement matrix is mainly divided into two categories, i.e., random measurement matrix and deterministic measurement matrix. The performance of random measurement matrix has bottlenecks. Firstly, because of the redundant measurement matrix data, the generation and storage of the random number put forward a high requirement for hardware. Secondly the random matrix can only satisfy the restricted isometry property in a statistical sense. The research of the deterministic measurement matrix is of great value under this background. The parity check matrix of low density parity check (LDPC) code has good performance in CS theory. However, the method of randomly selecting non-zero element position has a certain probability to generate a measurement matrix with a short loop structure during generating LDPC code measurement matrix. The robustness of the reconstruction performance decreases with the increase of iteration times. A novel quasi-cyclic CS algorithm based on progressive edge-growth is constructed to estimate the time delay. The purpose of this article is to deal with the need to store a large number of data in existing measurement matrix during time delay, by using the CS theory. The algorithm presented here can achieve TDE in a high precision. First, the theoretical bridge between CS and the maximum likelihood decoding is established. And the design criterion of measurement matrix based on the LDPC code is derived. The sparse measurement matrix with quasi-cyclic structure is constructed by introducing the idea of progressive edge-growth. Finally, the orthogonal matching pursuit algorithm is used to estimate the time delay. Furthermore, the computational complexity of the algorithm and the data storage of the measurement matrix are analyzed theoretically. Simulations show that the correct reconstruction probability of the proposed approach is higher than those of the Gauss random matrix and random LDPC matrix under the same dimension. Compared with the random LDPC matrix, the proposed method can improve performance at the expense of less complexity under the condition of the same data storage.

Microwave Imaging offers safe, low-cost, and portable method for medical imaging applications. These advantages make the microwave imaging convenient for early detection of tumor or cancer. The transmission method is one of the methods in microwave imaging which provides fast measurement and simple image reconstruction. However, this method requires a great number of measurements to obtain a well-reconstructed image. In order to reduce the number of measurements, this research proposes a Compressive Sensing (CS) approach for image reconstruction on microwave imaging. Compressive Sensing allows reconstruction of a signal with fewer measurements than the conventional approach. In this research, the scanning process is conducted on Computer Simulation Technology (CST) Microwave Studio software. Two dipole antennas with 3 GHz frequency are utilized as microwave transmitter and receiver. A two-layer cube phantom acts as the scanned object. Each layer has different relative permittivity which illustrates the healthy cell and abnormal cell. To meet the framework of Compressive Sensing, a weighted matrix of Discrete Radon Transform (DRT) is created as a projection matrix which delineates the data acquisition scheme in the scanning process. Discrete Cosine Transform (DCT) is selected as sparse dictionary matrix to represent the sparse basis while Basis Pursuit is selected as sparse reconstruction algorithm to reconstruct the sparse signal from measurement data. The measured $S_{21}$ data are successfully reconstructed into an image using the Compressive Sensing approach. The reconstructed image is analyzed both qualitatively and quantitatively using image quality parameters such as Structural Similarity Index (SSIM) and Mean Squared Error (MSE).

In compressed sensing, the Toeplitz sensing matrices are generated by randomly drawn entries and further optimizes them with suitable optimization methods. However, during an optimization process, state-of-the-art optimization methods tend to lose control over the structure of measurement matrices. In this paper, we proposed the novel approach for optimization of Toeplitz sensing matrices based on evolutionary algorithms such as Genetic Algorithm (GA), Simulated Annealing (SA), and Particle Swarm Optimization (PSO) for compression of an image signal. Furthermore, we investigated the performance of Basis Pursuit (BP) and Orthogonal Matching Pursuit (OMP) algorithms for the reconstruction of the images. The proposed optimized Toeplitz sensing matrices based on evolutionary algorithms such as GA, SA, and PSO exhibit a significant reduction in the mutual coherence (μ) and thus improved the recovery performance of 2D images compared to state-of-the-art non-optimized Toeplitz sensing matrices. The result reveals that the optimized Toeplitz sensing matrices with Basis Pursuit (BP) achieved more accurate results with a robust and uniform reconstruction guarantee compared to the OMP algorithm. However, BP shows the slow reconstruction performance of the image signal. On the other hand, an optimized Toeplitz sensing matrix with OMP shows a fast reconstruction guarantee, but at the cost of a reduction in the PSNR. Furthermore, the proposed approach retains the structure of Toeplitz sensing matrices and improves the image recovery performance of compressed sensing. Finally, the experimental results validate the effectiveness of the proposed method based on evolutionary algorithms for image compression.

In image fusion techniques based on joint sparse representation (JSR), the composite image is calculated from the fusion of features, which are represented with sparse coefficients. Orthogonal matching pursuit (OMP) and basis pursuit (BP) are the main candidates to estimate the coefficients. Previously OMP is utilized for the advantage of low complexity. However, noticeable errors occur when the dictionary of JSR cannot ensure the coefficients are sparse enough. Alternatively, BP is more robust than OMP in such cases (though suffered from larger complexity). Unfortunately, it has never been studied in image fusion tasks. In this paper, we investigate JSR based on BP for image fusion. The target is to verify that 1) to what extent can BP outperform OMP; and 2) what is the trade-off between BP and OMP. Finally, we conclude, in some cases, fusion with BP obviously outperforms the one with OMP under an affordable computational complexity.

With the pervasive infrastructures of WLAN, user’s privacy has emerged as an important security and privacy problem. Rogue Access Points (AP), as one of the threat, is expected to be detected and located accurately. Therefore, in this paper, we propose a novel rogue AP localization method leveraging compressive sensing (CS) via kernel optimization. Although the CS based technique has been widely used in mobile user localization system, this is the first time to apply it to reversely localize AP. In addition, designing an appropriate kernel is the key to successful application of CS technique, however, traditional Gaussian or Bernoulli random kernels could not be utilized in rogue AP localization system, due that the kernel is related to the number and distribution of monitors, which could not randomly change every time. Hence, for CS kernel optimization, we firstly deduce the minimum number of monitors required in this system through a theoretical analysis which aims at justifying the validity of problem formulation. Then an Equiangular Tight Frame (ETF) based monitors distribution scheme is presented to achieve higher location accuracy. Finally, we perform both simulations and experiments to demonstrate the superiority of our approach as compare to other methods theoretically and practically.

It is of great significance to construct deterministic measurement matrices with good practical characteristics in Compressed Sensing (CS), including good reconstruction performance, low memory cost and low computing resources. Low-density-parity check (LDPC) codes and CS codes can be closely related. This paper presents a method of constructing compressed sensing measurement matrices based on quasi-cyclic (QC) LDPC codes and arithmetic sequence sets. The cyclic shift factor in each submatrix of QC-LDPC is determined by arithmetic sequence sets. Compared with random matrices, the proposed method has great advantages because it is generated based on a cyclic shift matrix, which requires less storage memory and lower computing resources. Because the restricted isometric property (RIP) is difficult to verify, mutual coherence and girth are used as computationally tractable indicators to evaluate the measurement matrix reconstruction performance. Compared with several typical matrices, the proposed measurement matrix has the minimum mutual coherence and superior reconstruction capability of CS signal according to one-dimensional (1D) signals and two-dimensional (2D) image simulation results. When the sampling rate is 0.2, the maximum (minimum) gain of peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM) is up to 2.89 dB (0.33 dB) and 0.031 (0.016) while using 10 test images. Meanwhile, the reconstruction time is reduced by nearly half.

In this paper, channel overhead is reduced by exploiting channel sparsity for multiple input multiple output-orthogonal frequency division multiplexing (MIMO-OFDM) system. Where, compressive sensing (CS) based dictionary design algorithms has been adopted as a channel estimation technique in high mobile systems with minimal number of pilots, such as high-speed train (HST) systems. A novel framework design of the dictionary-based CS was proposed considering both delay and Doppler effects in order to correctly recover the channel response. The channel under consideration is a 2 by 2 space-time block code (STBC) MIMO channel. Simulation tests according to the international telecommunication union (ITU) channel model demonstrated the suitability of the proposed dictionary for estimating the channel impulse response (CIR) of a liner time varying (LTV) channel with a mobility approaches 675 Km/h related to a Doppler frequency of 1500 Hz and 2.4 GHz carrier frequency. Two CS recovery algorithms were applied; orthogonal matching pursuit (OMP) and basis pursuit (BP), where by about 7 dB gain in signal to noise ratio (SNR) was achieved with mobility of 675 Km/h using OMP as compared to BP at a bit error rate (BER) of with 128 OFDM subcarriers.

Deterministic construction of sparse binary matrices via incremental integer optimization

Random measurement matrices play a critical role in successful recovery with the compressive sensing (CS) framework. However, due to its randomly generated elements, these matrices require massive amounts of storage space to implement a random matrix in CS applications. To effectively reduce the storage space of the random measurement matrix for CS, we propose a random sampling approach for the CS framework based on the semi-tensor product (STP). The proposed approach generates a random measurement matrix, where the dimensions of the random measurement matrix are reduced to a quarter (or 1/16, 1/64, and even 1/256) of the number of dimensions, which are used for conventional CS. We then estimate the values of the sparse vector with a modified iteratively re-weighted least-squares (IRLS) algorithm. The results of numerical simulations showed that the proposed approach can reduce the storage space of a random matrix to at least a quarter while maintaining quality of reconstruction. All results confirmed that the proposed approach significantly influences the physical implementation of the CS in images, especially on embedded system and field programmable gate array (FPGA), where storage is limited.

For signals reconstruction based on compressive sensing, to reconstruct signals of higher accuracy with lower compression rates, it is required that there is a smaller mutual coherence between the measurement matrix and the sparsifying matrix. Mutual coherence between the measurement matrix and sparsifying matrix can be expressed indirectly by the property of the Gram matrix. On the basis of the Gram matrix, a new optimization algorithm of acquiring a measurement matrix has been proposed in this paper. Firstly, a new mathematical model is designed and a new method of initializing measurement matrix is adopted to optimize the measurement matrix. Then, the loss function of the new algorithm model is solved by the gradient projection-based method of Gram matrix approximating an identity matrix. Finally, the optimized measurement matrix is generated by minimizing mutual coherence between measurement matrix and sparsifying matrix. Compared with the conventional measurement matrices and the traditional optimization methods, the proposed new algorithm effectively improves the performance of optimized measurement matrices in reconstructing one-dimensional sparse signals and two-dimensional image signals that are not sparse. The superior performance of the proposed method in this paper has been fully tested and verified by a large number of experiments.

Nonuniform support recovery from noisy random measurements by Orthogonal Matching Pursuit

SummaryCognitive radio (CR) is an emerging technology for optimum spectrum utilization. Spectrum sensing (SS) is the first step of CR. Important challenge in the next generations is wideband SS. Compressive sensing (CS) can be used as an SS in CR to solve this challenge. Sensing matrix (SM) plays an important role in CS. For better performance, SM must have low mutual coherence. The existing matrices in the literature are random and chaotic. All random matrices have a good performance, but they have problems of large memory storage, complexity, sharing bandwidth, and low security. Chaotic matrices (CMs) are the best but still have some problems including the use of sample distance in chaotic sequence that consumes high storage resources, low performance and compression under low signal‐to‐noise ratio (SNR), and low security. In this paper, we propose a new 1‐D chaotic map for constructing a novel SM. The proposed map has simple structure, wide range chaotic behavior, and high security. Its chaotic behavior is confirmed using Lyapunov exponent, bifurcation, and trajectory. The CS performance based on novel CM is measured using absolute error ( , mutual coherence, memory cost, computational complexity, and MSE (mean square error), while evaluated through comparisons with matrices in the literature. The simulation and mathematical results show that the proposed map is a hyperchaotic with efficient chaotic behavior in wide range of parameters and low memory storage and complexity. The results also show that it has low , MSE, and mutual coherence at low SNR with high compression.

Compressive sensing (CS) is a new technique that give an approach to reconstruct the signal with few numbers of observations or measurements. The CS is based on L1 -norm minimizations to find the sparse solutions and it is known as basis pursuit. In this investigation, CS scheme with different transforms is proposed. The three utilized transform techniques are: Discrete Fourier Transform(DFT), Discrete Cosine Transform(DCT) and Discrete Wavelet Transform(DWT) with daubechies1(DB1) and coiflets1(coif1) basis. The proposed system is tested by employing the following signals: Blocks, Heavy Sine, ‘Bumps’ and ‘Doppler’ which cover wide range of applications. The four testing signals are represented in sparse domain using different transforms. In order to threshold the coefficients of the signals in sparse domain, the universal threshold is utilized in the case of CS with FFT and DWT whereas, the universal threshold is modified to prune the DCT coefficients. The main aim of this study is to investigate the differences among CS with DFT, CS with DCT and CS with DWT, and consequently a suitable transform domain used with CS to be selected. The comparative study is established by assessing the performance of the proposed system using Root Mean Square Error (RMSE), output SNR, and the time required to reconstruct approximated signals. Simulation results have shown that the CS with DWT outperforms the CS with FFT and DCT. CS with DWT has achieved good RMSE values about (0.0014 to 3.359e-8) even when half of the signal elements are removed. CS with FFT and DCT enhanced the noisy Blocks and Bumps signals by 3dB and 1dB respectively, while it is failed to enhance noisy Heavy Sine and Doppler signals. CS with DWT of two basis and for single decomposition level have improved the noisy Blocks, Bumps, Heavy Sine and Doppler signals by 5dB, 4dB, 3dB, and 3dB respectively.