Low storage space for compressive sensing: semi-tensor product approach

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.

Compressive sensing (CS) takes advantage of the signal's sparseness in some domain, allowing the entire signal to be efficiently acquired and reconstructed from relatively few measurements. A proper measurement matrix for compressive sensing is significance in above processions. In most compressive sensing frameworks, random measurement matrix is employed. However, the random measurement matrix is hard to implement by hardware. So the randomness of the measurement matrix leads to the poor performance of signal reconstruction. In this paper, Toeplitz matrix is employed and optimized as a deterministic measurement matrix. A hardware platform for signal efficient acquisition and reconstruction is built by field programmable gate arrays (FPGA). Experimental results demonstrate the proposed approach, compare with the existing state-of-the-art method, and have the highest technical feasibility, lowest computational complexity, and least amount of time consumption in the same reconstruction quality.

BackgroundThe construction of measurement matrix becomes a focus in compressed sensing (CS) theory. Although random matrices have been theoretically and practically shown to reconstruct signals, it is still necessary to study the more promising deterministic measurement matrix.MethodsIn this paper, a new method to construct a simple and efficient deterministic measurement matrix, sparse kronecker pascal (SKP) measurement matrix, is proposed, which is based on the kronecker product and the pascal matrix.ResultsSimulation results show that the reconstruction performance of the SKP measurement matrices is superior to that of the random Gaussian measurement matrices and random Bernoulli measurement matrices.ConclusionsThe SKP measurement matrix can be applied to reconstruct high-dimensional signals such as natural images. And the reconstruction performance of the SKP measurement matrix with a proper pascal matrix outperforms the random measurement matrices.

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

Compressed Sensing (CS), as a new rapidly growing research field, promises to effectively recover a sparse signal at the rate of below Nyquist rate. This revolutionary technology strongly relies on the sparsity of the signal and incoherency between sensing basis and representation basis. Exact recovery of a sparse signal will be occurred in a situation that the signal of interest sensed randomly and the measurements are also taken based on sparsity level and log factor of the signal dimension. In this paper, compressed sensing method is proposed to reduce the noise and reconstruct the image signal. Noise reduction and image reconstruction are formulated in the theoretical framework of compressed sensing using Basis Pursuit (BP) and Compressive Sampling Matching Pursuit (CoSaMP) algorithm when random measurement matrix is utilized to acquire the data. In this research we have evaluated the performance of our proposed image enhancement methods using the quality measure peak signal-to-noise ratio (PSNR).

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.

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.

Wideband spectrum sensing is regarded as one of the key features in cognitive radio systems. Compressive sensing (CS) has recently become one of the promising techniques to deal with the Nyquist sampling rate bottleneck of wideband spectrum sensing. Theoretical analyses and simulation have shown that CS could achieve high detection probability and low false alarm for wideband spectrum sensing. However, implementation of CS on the real-time signals and real-time processing poses significant challenges due to the iterative nature of the CS algorithms. In this paper, we propose a novel adaptively regularized iterative reweighted least squares (AR-IRLS) algorithm to implement the real-time signal recovery on the CS based wideband spectrum sensing. The proposed algorithm moves estimated solutions along an exponential-linear path by regularizing weights with a series of non- increasing penalty terms, which significantly speeds up the convergence of reconstruction and provides high fidelity guarantee to cope with the varying bandwidths and power levels of occupied channels. The proposed algorithm presents robustness against different sparsity levels at low compressive ratio without degradation on the reconstruction performance, and is tested on the real-time signals over TV white space spectrum after having been validated on the simulated signals. Both the simulation and real-time experiments show that the proposed algorithm outperforms the conventional iterative reweighted least squares (IRLS) algorithms in terms of convergence speed, reconstruction accuracy, and compressive ratio requirement.

Compressive sensing (CS) enables us to reconstruct a signal from a few number of measurements obtained from a random or deterministic measurement matrix. Knowledge of the sparsifying basis of the signal is required for the recovery process. In this work, we use a recently developed deterministic measurement matrix and demonstrate recovery of the original signal from compressed samples without the knowledge of the sparsifying basis or the order of sparsity. We experimented this recovery on the Biomedical images. Using smoothed l 0 -norm (SL0) as a recovery algorithm, the original images were recovered from CS measurements with high quality.

Deterministic construction of sparse binary matrices via incremental integer optimization

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].

This paper provides a random complex measurement matrix and an algorithm for complex phase retrieval of sparse or approximately sparse signals from the noisy magnitudes of the measurements obtained with this matrix. We compute explicit error bounds for the recovery which depend on the noise-to-signal ratio, the sparsity s, the number of measured quantitites m, and the dimension of the signal N. This requires m to be of the order of s ln(N/s). In comparison with sparse recovery from complex linear measurements, our phase retrieval algorithm requires six times the number of measured quantities.

The compressive sensing (CS) theory provides a new slight to the big-data problem led by the Shannon sampling theorem in rolling element bearings condition monitoring, where the measurement matrix of CS tends to be designed by the random matrix (RM) to preserve the integrity of signal roughly. However, when the signal to be analyzed is infected with strong noise, not only does the signal become insufficiently sparse, but the randomness of the measurement matrix will bring down the sensing efficiency, resulting in the loss of fault feature. Thus, a sensing-enhanced CS scheme based on a series of modes after VMD decomposition is proposed under this paper. The core of this scheme is as follows: (1) the principal mode of VMD with better sparsity replaces the raw signal for compressive sensing; (2) all these modes contain the time-frequency characteristics of the raw signal; (3) a new measurement matrix called mode-circulant matrix (MCM) is defined by circulating the mode matrix, and when the amount of samples is shrunk, the sensing efficiency can be enhanced greatly. Besides, considering the fault signal of rolling bearings under variable speed, there is a need to use order tracking to overcome the nonstationarity of the signal before applying CS theory. The analysis results of simulation and experiment prove that the VMD- and MCM-based CS can successfully extract the weak fault feature of rolling bearings with operating speed changing.

Aimed at scene matching problem for taking infrared image as the actual data and the visible image as the referenced data, a multi-resolution matching algorithm which fuses compressive sensing Scale Invariant Feature Transform (SIFT) feature is presented based on Bandelet transform. Two kinds of images are separately transformed into Bandelet domain to compress the feature search space of scene matching based on the best sparse representation of natural images by Bandelet transform. On the basis of adaptive Bayes threshold denoising for infrared image, the concept of sparse feature representation of compressive sensing theory is introduced into SIFT algorithm. For low-frequency image in Bandelet domain, high-dimensional SIFT key point feature description vector is projected on compressive sensing random measurement matrix to achieve dimensionality reduction. Then, the improved Genetic Algorithm (GA) to overcome premature phenomena is used as the search strategy, and the L1 distance measure of SIFT feature vectors of compressive sensing for two kinds of images is applied to the search similarity criterion to match low-frequency image of high scale in Bandelet domain. The matching result is used as the guidance of the matching process for low-frequency image of low scale, and the matching result of full-resolution image is obtained iteratively. Experimental results show that the proposed method has not only high matching accuracy and fast matching speed, but also better robustness in comparison with some classic matching algorithms, which can resist the geometric distortion of rotation for actual image.

Compressed sensing (CS) has gained much interests in the speech processing community and especially in compression. The success of this appealing paradigm relies, heavily, on the sparsity of speech signals in a given dictionary. Thanks to machine learning, it is possible to go beyond the limits of analytical dictionaries by designing learned dictionaries, which are more able to fit the nature of data. In this paper, we propose speech compression scheme based on CS using K-singular value decomposition (K-SVD) algorithm to learn a dictionary for sparse representation and iteratively reweighted least square (IRLS) algorithm for signal reconstruction from randomly generated measurements. Different sizes of dictionaries are trained and compared with discrete cosine transform (DCT). Signal quality results, measured with perceptual evaluation speech quality (PESQ) revealed that learned dictionary using K-SVD improves the performance of speech compressed sensing comparing to DCT.