Construction Of Measurement Matrices Research Articles

Flexible construction of measurement matrices in compressed sensing based on extensions of incidence matrices of combinatorial designs

In signal processing, compressed sensing (CS) can be used to acquire and reconstruct sparse signals. This paper presents a method of combining vertical expansions and horizontal expansions to construct measurement matrices. Firstly, we give a construction of (n,n,n−1,n−1,n−2)-BIBD based on finite set. It is important to estimate recovery performance of measurement matrices in terms of coherence, and it is found that the incidence matrix H of (n,n,n−1,n−1,n−2)-BIBD is not suitable as a measurement matrix in CS. We present an optimal method of combining vertical expansions and horizontal expansions for addressing this problem. These two extensions provide a new perspective for the construction of measurement matrices. Vertical expansions ensure that the matrix has low coherence. Horizontal expansions ensure that the matrix is more suitable as a measurement matrix in CS because of sizes and coherence. Finally, compared with several typical matrices, our matrices have better recovery performance under OMP and IST by the simulation experiments.

Deterministic Constructions of Compressed Sensing Matrices From Unitary Geometry

Compressed sensing is an emerging theory of signal processing and it has wide applications in many frontier fields. The construction of the measurement matrices is still a central problem in compressed sensing. In this paper, two types of deterministic constructions of binary measurement matrices are presented via unitary geometry. Then, the lower bounds of the spark of unitary geometry measurement matrices are theoretically analyzed, and an asymptotic comparison between unitary geometry measurement matrices and projective geometry measurement matrices is given via the worst-case recovery capability. After that, a clipping-embedding operation is proposed for binary matrices to generate measurement matrices with more sizes, which can strongly extend the applicability of the deterministic binary matrices in practice. Finally, simulation results demonstrate that the performance of our measurement matrices is comparable to, sometimes even better than, that of the corresponding Gaussian random matrices under OMP and BP.

A Novel Optimization Method for Bipolar Chaotic Toeplitz Measurement Matrix in Compressed Sensing

In this paper, a bipolar chaotic Toeplitz measurement matrix optimization algorithm for alternating optimization is presented. The construction of measurement matrices is one of the key techniques for compressive sensing from theory to engineering applications. Recent studies have shown that bipolar chaotic Toeplitz matrices, constructed by combining the intrinsic determinism of bipolar chaotic sequences with the advantages of Toeplitz matrices, have significant advantages over other measurement matrices in terms of memory overhead, computational complexity, and hard implementation. However, problems such as strong correlation and large interdependence coefficients between measurement matrices and sparse dictionaries may still exist in practical applications. To address this problem, we propose a new bipolar chaotic Toeplitz measurement matrix alternating optimization algorithm. Firstly, by introducing the structure matrix, the optimization problem of the measurement matrix is transformed into the optimization problem of the generating sequence, thus ensuring that the optimization process does not destroy the structural properties of the matrix; then, constraints are added to the values of the generating sequence during the optimization process, so that the optimized measurement matrix still maintains the bipolar properties. Finally, the effectiveness of the optimization algorithm in this paper is verified by simulation experiments. The experimental results show that the optimized bipolar chaotic Toeplitz measurement matrix can effectively reduce the reconstruction error and improve the reconstruction probability.

Flexible construction of compressed sensing matrices with low storage space and low coherence

This paper considers the construction of measurement matrices with low storage space and good performance. Aiming at source-limited devices, this paper proposes a novel block sampling model based on the embedding operation. In the sampling procedure, the proposed model saves a lot of storage space by storing two smaller seed matrices. Meanwhile, the block sampling mechanism makes the proposed model have a lower sampling complexity. Since the embedding operation is oriented to binary matrices, the construction of binary matrices plays a crucial role in the proposed model. In order to construct binary matrices with low coherence, this paper proposes a modified progressive edge growth algorithm by removing some limited condition of the traditional progressive edge growth algorithm. Based on the modified progressive edge growth algorithm and the embedding operation, this paper proposes a flexible method to construct sparse measurement matrices with low coherence. Simulation experiments demonstrate that our measurement matrices give better performance than several typical measurement matrices.

Deterministic construction of array QC CS measurement matrices based on Singer perfect difference sets

Low-density parity-check (LDPC) codes and compressed sensing (CS) share many common environments. In this study, a novel approach for constructing a new class of deterministic sparse sensing matrices based on array quasi-cyclic (QC) LDPC codes via Singer perfect difference sets is proposed. In contrast to random and the other deterministic matrices, the proposed framework would be highly desirable as it is generated based on circulant permutation matrices, which requires less memory for storage and lower computational cost for sensing. Since the restricted isometric property is very difficult to verify, then the mutual coherence and the girth are two computationally tractable criteria that the authors used to assess the CS recovery capabilities of sensing matrices. In addition, inspired by LDPC codes, they extract a necessary condition for the proposed measurement matrix to have effective values for girth as large as g ≥ 6 and 8 . Comprehensive one-dimensional (1D) and 2D simulations verify that their proposed sensing matrix has minimum coherence and superior CS recovery abilities in comparison with the corresponding random Gaussian, Bernoulli, and the other deterministically generated matrices. Furthermore, the required physical storage space and the complexity of the hardware implementation are greatly reduced due to being sparse and QC in structure.

General approach for construction of deterministic compressive sensing matrices

In this study, deterministic construction of measurement matrices in compressive sensing is considered. First, by employing the column replacement concept, a theorem for construction of large minimum distance linear codes containing all-one codewords is proposed. Then, by applying an existing theorem over these linear codes, deterministic sensing matrices are constructed. To evaluate this procedure, two examples of constructed sensing matrices are presented. The first example contains a matrix of size and coherence , and the second one comprises a matrix with the size and coherence , where p is a prime integer. Based on the Welch bound, both examples asymptotically achieve optimal results. Moreover, by presenting a new theorem, the column replacement is used for resizing any sensing matrix to a greater-size sensing matrix whose coherence is calculated. Then, using an example, the outperformance of the proposed method is compared to a well-known method. Simulation results show the satisfying performance of the column replacement method either in created or resized sensing matrices.

Deterministic Construction of Measurement Matrices Based on Bose Balanced Incomplete Block Designs

Compressed sensing is a novel information collection theory. Compared with the Nyquist sampling theorem, compressed sensing can obtain all the information of a signal with very few samples. The deterministic construction of the measurement matrix is an important research area in compressed sensing. Inspired by the observation that the parity-check matrix of low-density parity-check code can be used as a deterministic measurement matrix, in this paper, a special balanced incomplete block design proposed by Bose is exploited to construct the deterministic measurement matrix. The incidence matrix of the balanced incomplete block design proposed by Bose is used as the deterministic measurement matrix. The experimental results show that the proposed measurement matrix has lower mutual coherence than some widely used measurement matrices and shows better performance than the progressive edge-growth measurement matrix. Moreover, by using the embedding operation, the modified measurement matrix with a more flexible size and improved performance is constructed. In our simulation, the modified measurement matrix has lower mutual coherence and better performance than some widely used measurement matrices.

Deterministic construction of sparse binary matrices via incremental integer optimization

A central problem in compressed sensing (CS) is the design of measurement matrices. Compared with the conventional random matrices, sparse binary matrices have some attractive properties, such as lower storage/encoding cost and easy hardware implementation. In this paper, we formulate the construction of sparse binary measurement matrices from an optimization standpoint. A new algorithm is presented to construct arbitrary-size sparse binary measurement matrices through relaxing the resultant optimization model to an incremental integer programming problem. The proposed method in general outputs sparse binary matrices with optimal mutual coherence. In addition, we prove that the constructed matrices can be almost completely incoherent with the conventional wavelet dictionary. Extensive simulation results show that the sparse binary matrices constructed by the proposed algorithm significantly outperform Gaussian random matrices, random sparse binary matrices and two well-performing deterministic sparse binary matrices.

A hierarchical framework for recovery in compressive sensing

A combinatorial framework for the construction of measurement matrices for compressive sensing is shown to exhibit great flexibility in signal recovery. The deterministic column replacement technique is hierarchical: Given as input a pattern matrix and ingredient measurement matrices, it produces a larger measurement matrix by replacing elements of the pattern matrix with columns from the ingredient matrices. Recovery for the measurement matrix produced does not rely on any fixed algorithm; rather it employs the recovery schemes of the ingredient matrices, which may differ from ingredient to ingredient. Because ingredient matrices can be much smaller than the measurement matrix produced, one can employ more computationally intensive recovery methods, sometimes resulting in fewer measurements. Noise can be accommodated in signal recovery by imposing additional conditions both on the pattern matrix and on the ingredient measurement matrices.

Finite projective spaces in deterministic construction of measurement matrices

In this study, the authors concentrate on the designing of a deterministic measurement matrix. Unlike most of the studies, they employ the points on finite projective spaces rather than finite fields. These spaces provide more choices for the size of the proposed matrix. A new group of binary measurement matrices is presented through generalising DeVore's construction. For this purpose, homogenous polynomials over finite projective spaces are applied. To investigate the performance of the proposed matrix they provide an example on projective lines. It can be observed that the coherence of the result matrix is lower than DeVore's construction. The simulation results show that the proposed matrix outperforms the Gaussian matrix and DeVore's matrix in terms of noiseless and noisy signal recovery.

The construction of measurement matrices based on block weighing matrix in compressed sensing

In this paper, we propose a new structured measurement matrix for practical compressed sensing based on block weighing matrix, called partial Random Block Weighing Matrix (pRBWM). The proposed pRBWM is universal with a variety of sparse signals and provides high reconstruction performance simultaneously. In addition, with the sparse and circulant block structure, these new measurement matrices feature low-memory requirement and low computational complexity in reconstruction. Moreover, it can be more easily implemented in hardware thanks to its sample elements and the application of Chaos-based permutation operator in construction of pRBWM. Simulation results demonstrate that the proposed pRBWM performs comparably to, or even better than completely random matrices and many other structured matrices. And the proposed pRBWM forms a high balance between reconstruction performance,storage and computational complexity and hardware implementation.

Deterministic Constructions of Binary Measurement Matrices From Finite Geometry

Deterministic constructions of measurement matrices in compressed sensing (CS) are considered in this paper. The constructions are inspired by the recent discovery of Dimakis, Smarandache and Vontobel which says that parity-check matrices of good low-density parity-check (LDPC) codes can be used as provably good measurement matrices for compressed sensing under l1-minimization. The performance of the proposed binary measurement matrices is mainly theoretically analyzed with the help of the analyzing methods and results from (finite geometry) LDPC codes. Particularly, several lower bounds of the spark (i.e., the smallest number of columns that are linearly dependent, which totally characterizes the recovery performance of l0-minimization) of general binary matrices and finite geometry matrices are obtained and they improve the previously known results in most cases. Simulation results show that the proposed matrices perform comparably to, sometimes even better than, the corresponding Gaussian random matrices. Moreover, the proposed matrices are sparse, binary, and most of them have cyclic or quasi-cyclic structure, which will make the hardware realization convenient and easy.

图像视频信号的压缩采样与稀疏重建

Vision sensors usually do not account for the physical process of imaging and they acquire image/video samples at the Nyquist rate. The Nyquist rate is significantly higher than the effective dimensions of an image/video, and consequently compression is essential for the image/video prior to storage or transmission. The emerging Compressive Sensing (CS) theory states that a signal can be perfectly reconstructed, or can be robustly approximated in the presence of noise, using a few random measurements, provided that it is sparse in some linear transform domain. CS is the theoretical foundation for capturing a signal with effective information dimensions, and thus represents an unprecedented breakthrough in many fields such as sampling, processing, and recognition of image/video. We review the fundamental problems of CS for image/video including compressive sampling, sparse reconstruction models, and algorithms for the models. For compressive sampling, the construction of random and structural measurement matrices are considered separately and the performance of these two kinds of matrices is evaluated. For sparse reconstruction, models are classified as analysis-based or synthesisbased reconstruction models by the sparse representation prior, features of which are presented. The optimization models can be considered as constrained and unconstrained optimization problems. Some feasible algorithms for these two kinds of optimization problems are explained in detail and the performance of the algorithms is given. In addition, several challenges of compressive sensing technology are presented and future work is discussed.

Strengthening hash families and compressive sensing

The deterministic construction of measurement matrices for compressive sensing is a challenging problem, for which a number of combinatorial techniques have been developed. One of them employs a widely used column replacement technique based on hash families. It is effective at producing larger measurement matrices from smaller ones, but it can only preserve the strength (level of sparsity supported), not increase it. Column replacement is extended here to produce measurement matrices with larger strength from ingredient arrays with smaller strength. To do this, a new type of hash family, called a strengthening hash family, is introduced. Using these hash families, column replacement is shown to increase strength under two standard notions of recoverability. Then techniques to construct strengthening hash families, both probabilistically and deterministically, are developed. Using a variant of the Stein–Lovász–Johnson theorem, a deterministic, polynomial time algorithm for constructing a strengthening hash family of fixed strength is derived.

Compressive Sensing Matrices and Hash Families

Deterministic construction of measurement matrices for compressive sensing can be effected by first constructing a relatively small matrix explicitly, and then inflating it using a column replacement technique to form a large measurement matrix that supports at least the same level of sparsity. In particular, using easily developed null space conditions for l0- and l1-recoverability, properties of the pattern matrix used to select columns lead to well-studied matrices, separating and distributing hash families. Two-stage compression and recovery techniques are developed that employ more computationally intensive l0-recoverability for small matrices and simpler l1-recoverability for one larger matrix; this can reduce the number of measurements required.

Sparse Recovery of Nonnegative Signals With Minimal Expansion

We investigate the problem of reconstructing a high-dimensional nonnegative sparse vector from lower-dimensional linear measurements. While much work has focused on dense measurement matrices, sparse measurement schemes can be more efficient both with respect to signal sensing as well as reconstruction complexity. Known constructions use the adjacency matrices of expander graphs, which often lead to recovery algorithms which are much more efficient than minimization. However, prior constructions of sparse measurement matrices rely on expander graphs with very high expansion coefficients which make the construction of such graphs difficult and the size of the recoverable sets very small. In this paper, we introduce sparse measurement matrices for the recovery of nonnegative vectors, using perturbations of the adjacency matrices of expander graphs requiring much smaller expansion coefficients, hereby referred to as minimal expanders. We show that when minimization is used as the reconstruction method, these constructions allow the recovery of signals that are almost three orders of magnitude larger compared to the existing theoretical results for sparse measurement matrices. We provide for the first time tight upper bounds for the so called weak and strong recovery thresholds when minimization is used. We further show that the success of optimization is equivalent to the existence of a “unique” vector in the set of solutions to the linear equations, which enables alternative algorithms for minimization. We further show that the defined minimal expansion property is necessary for all measurement matrices for compressive sensing, (even when the non-negativity assumption is removed) therefore implying that our construction is tight. We finally present a novel recovery algorithm that exploits expansion and is much more computationally efficient compared to minimization.

Hadamard Spectroscopy

The basic concept of Hadamard spectroscopy is presented. General methods are given for the construction of cyclic measurement matrices. A new algorithm, which permits adaptation of the fast Hadamard transform to the calculation of spectral intensities when the measurement matrices are cyclic, is introduced. Some applications are also briefly discussed.