Nonadaptive Linear Measurements Research Articles

Image Denoising Using a Compressive Sensing Approach Based on Regularization Constraints.

In remote sensing applications and medical imaging, one of the key points is the acquisition, real-time preprocessing and storage of information. Due to the large amount of information present in the form of images or videos, compression of these data is necessary. Compressed sensing is an efficient technique to meet this challenge. It consists in acquiring a signal, assuming that it can have a sparse representation, by using a minimum number of nonadaptive linear measurements. After this compressed sensing process, a reconstruction of the original signal must be performed at the receiver. Reconstruction techniques are often unable to preserve the texture of the image and tend to smooth out its details. To overcome this problem, we propose, in this work, a compressed sensing reconstruction method that combines the total variation regularization and the non-local self-similarity constraint. The optimization of this method is performed by using an augmented Lagrangian that avoids the difficult problem of nonlinearity and nondifferentiability of the regularization terms. The proposed algorithm, called denoising-compressed sensing by regularization (DCSR) terms, will not only perform image reconstruction but also denoising. To evaluate the performance of the proposed algorithm, we compare its performance with state-of-the-art methods, such as Nesterov’s algorithm, group-based sparse representation and wavelet-based methods, in terms of denoising and preservation of edges, texture and image details, as well as from the point of view of computational complexity. Our approach permits a gain up to 25% in terms of denoising efficiency and visual quality using two metrics: peak signal-to-noise ratio (PSNR) and structural similarity (SSIM).

An enhanced block-based Compressed Sensing technique using orthogonal matching pursuit

The theory of compressed sensing asserts that one can recover signals in $$\mathbb {R}^n$$ from far fewer samples or measurements, if the signal has a sparse representation in some orthonormal basis; from non-adaptive linear measurements by solving a $$\mathbb {L}_1$$ norm minimisation problem. The non-adaptive measurements have the character of random linear combinations of the basis or frame elements. However, for large-scale 2D image signals, the randomized sensing matrix consumes enormous computational resources that makes it impractical. The problem has been addressed in the paper as a block compressed sensing (BCS) with sparsity normalization in the transformed domain in the preprocessing stage. The blocks obtained are converted to non-adaptive measurements using identically independent weighted Gaussian random matrices. The feasibility of reconstruction is verified using orthogonal matching pursuit. Simulation results show that better reconstruction performance can be achieved by the proposed technique in comparison with the existing BCS approaches.

Welch bound-achieving compressed sensing matrices from optimal codebooks

Compressed sensing (CS) is a new data acquisition theory taking full use of the sparsity of signals. It reveals that higher-dimensional sparse signals can be reconstructed from fewer nonadaptive linear measurements. The construction of CS matrices in CS is the key problem. In this paper, the deterministic CS matrices from optimal codebooks are constructed. Furthermore, the maximum sparsity of recovering the sparse signals by using our CS matrices are obtained. Meanwhile, a comparison is made with the CS matrices constructed by DeVore based on polynomials over finite fields. In the numerical simulations, our CS matrix outperforms DeVore[Formula: see text]s matrix in the process of recovering sparse signals.

Compressed-Sensing-Based Periodic Impulsive Feature Detection for Wind Turbine Systems

Identifying impulsive features from massive amounts of dynamic signals for wind turbine systems is like finding needles in haystacks, leading to a major challenge for the Shannon-sampling-theorem-based fault detection techniques. Therefore, this paper describes and analyzes a novel impulsive feature identification technique based on compressed sensing model and convex optimization techniques. One important point of this work is to establish the prior information that periodic impulsive component is frequency compressible. On the other hand, based on a small set of nonadaptive linear measurements, a convex optimization algorithm generated from a popular algorithmic framework (alternating direction of multiplier method) is developed to recover the impulsive features. Consequently, the main highlight of this work is to enable the high-accuracy recovery of impulsive feature signals from far few measurements than the Shannon sampling theory requires. Extensive numerical studies are implemented to quantitatively evaluate the performance of the proposed technique, and its feasibility and superiority are verified simultaneously. More importantly, the validity and applicability of our impulsive feature detection technique is comprehensively investigated and confirmed based on a practical engineering dataset from a wind turbine gearbox in a wind farm.

Feature Identification With Compressive Measurements for Machine Fault Diagnosis

Machine fault diagnosis collects massive amounts of vibration data about complex mechanical systems. Performing feature detection from these data sets has already led to a major challenge. Compressive sensing theory is a new sampling framework that provides an alternative to the well-known Shannon sampling theory. This theory enables the recovery of sparse or compressible signals from a small set of nonadaptive linear measurements. However, it is suboptimal to recover the whole signals from the compressive measurements and then solve feature identification problems through traditional DSP techniques. Thus, a novel mechanical feature identification method is proposed in this paper. Its main advantage is that fault features are extracted directly in the compressive measurement domain without sacrificing accuracy, while a significant reduction in the dimensionality of the measurement data is achieved. Moreover, Gaussian white noises are significantly alleviated, which dramatically enhances the reliability of machine fault diagnosis. Parameter analysis is also profoundly investigated through a set of numerical experiments. Numerical simulations and experiments are further performed to prove the reliability and effectiveness of the proposed method.

Exact reconstruction of the nonnegative measures using model sets

In this paper we are concerned with the reconstruction of a class of measures on the square from the sampling of its Fourier coefficients on some sparse set of points. We show that the exact reconstruction of a weighted Dirac sum measure is still possible when one knows a finite number of non-adaptive linear measurements of the spectrum. Surprisingly, these measurements are defined on a model set, i.e. quasicrystal.

ON THE BLOCK-SPARSITY OF MULTIPLE-MEASUREMENT VECTORS.

Based on the compressive sensing (CS) theory, it is possible to recover signals, which are either compressible or sparse under some suitable basis, via a small number of non-adaptive linear measurements. In this paper, we investigate recovering of block-sparse signals via multiple measurement vectors (MMVs) in the presence of noise. In this case, we consider one of the existing algorithms which provides a satisfactory estimate in terms of minimum mean-squared error but a non-sparse solution. Here, the algorithm is first modified to result in sparse solutions. Then, further modification is performed to account for the unknown block sparsity structure in the solution, as well. The performance of the proposed algorithm is demonstrated by experimental simulations and comparisons with some other algorithms for the sparse recovery problem.

Adaptive compressed sampling based on extended wavelet trees.

The theory of compressed sensing (CS) indicates that a signal that is sparse or compressible can be recovered from a relatively small number of nonadaptive linear measurements that is far below the Nyquist-Shannon limit. However, CS suffers from a huge stored and computational overhead when dealing with images of high resolution, taking tens of minutes or longer. In this work, we extend the concept of wavelet trees by adding the sibling relationship and propose an imaging strategy named adaptive compressed sampling based on extended wavelet trees (EWT-ACS). Exploiting both parent-children relationship and sibling relationship in extended wavelet trees, EWT-ACS predicts the locations of significant coefficients adaptively and samples the significant coefficients using a binary digital micromirror device directly. The simulation and experimental results reveal that the proposed strategy breaks through the limitation in CS, and the reconstruction time is reduced significantly. Due to its single-pixel detection mechanism, EWT-ACS shows great potential in many imaging applications.

Compressive sensing of sparse tensors.

Compressive sensing (CS) has triggered an enormous research activity since its first appearance. CS exploits the signal's sparsity or compressibility in a particular domain and integrates data compression and acquisition, thus allowing exact reconstruction through relatively few nonadaptive linear measurements. While conventional CS theory relies on data representation in the form of vectors, many data types in various applications, such as color imaging, video sequences, and multisensor networks, are intrinsically represented by higher order tensors. Application of CS to higher order data representation is typically performed by conversion of the data to very long vectors that must be measured using very large sampling matrices, thus imposing a huge computational and memory burden. In this paper, we propose generalized tensor compressive sensing (GTCS)-a unified framework for CS of higher order tensors, which preserves the intrinsic structure of tensor data with reduced computational complexity at reconstruction. GTCS offers an efficient means for representation of multidimensional data by providing simultaneous acquisition and compression from all tensor modes. In addition, we propound two reconstruction procedures, a serial method and a parallelizable method. We then compare the performance of the proposed method with Kronecker compressive sensing (KCS) and multiway compressive sensing (MWCS). We demonstrate experimentally that GTCS outperforms KCS and MWCS in terms of both reconstruction accuracy (within a range of compression ratios) and processing speed. The major disadvantage of our methods (and of MWCS as well) is that the compression ratios may be worse than that offered by KCS.

Research on Compressive Sensing‐Based 3D Imaging Method Applied to GPR

ABSTRACTResults in theory of compressive sensing enable the reconstruction of sparse signals from a small set of nonadaptive linear measurements by solving a convex optimization problem. Considering the sparse structure of actual target space in ground penetrating radar (GPR) application, three‐dimensional imaging method based on random aperture compressive sensing (RACS) is brought up, which requires the GPR transceiver to record only a minimum amount of samples through incoherent measurement at each aperture point, and only to measure a small number of random apertures in x‐y plane of interested target space. Simulation and experimental results verified that the proposed RACS method allow much fewer measurement aperture points, and has much sparse imaging results with much higher resolution. © 2013 Wiley Periodicals, Inc. Microwave Opt Technol Lett 55:2896–2901, 2013

Stable Image Reconstruction Using Total Variation Minimization

This paper presents near-optimal guarantees for stable and robust image recovery from undersampled noisy measurements using total variation minimization. In particular, we show that from $O(s\log(N))$ nonadaptive linear measurements, an image can be reconstructed to within the best $s$-term approximation of its gradient up to a logarithmic factor, and this factor can be removed by taking slightly more measurements. Along the way, we prove a strengthened Sobolev inequality for functions lying in the null space of a suitably incoherent matrix.

Stability and Instance Optimality for Gaussian Measurements in Compressed Sensing

In compressed sensing, we seek to gain information about a vector x∈ℝN from d ≪ N nonadaptive linear measurements. Candes, Donoho, Tao et al. (see, e.g., Candes, Proc. Intl. Congress Math., Madrid, 2006; Candes et al., Commun. Pure Appl. Math. 59:1207–1223, 2006; Donoho, IEEE Trans. Inf. Theory 52:1289–1306, 2006) proposed to seek a good approximation to x via l 1 minimization. In this paper, we show that in the case of Gaussian measurements, l 1 minimization recovers the signal well from inaccurate measurements, thus improving the result from Candes et al. (Commun. Pure Appl. Math. 59:1207–1223, 2006). We also show that this numerically friendly algorithm (see Candes et al., Commun. Pure Appl. Math. 59:1207–1223, 2006) with overwhelming probability recovers the signal with accuracy, comparable to the accuracy of the best k-term approximation in the Euclidean norm when k∼d/ln N.

Compressive sensing for subsurface imaging using ground penetrating radar

The theory of compressive sensing (CS) enables the reconstruction of sparse signals from a small set of non-adaptive linear measurements by solving a convex ℓ 1 minimization problem. This paper presents a novel data acquisition system for wideband synthetic aperture imaging based on CS by exploiting sparseness of point-like targets in the image space. Instead of measuring sensor returns by sampling at the Nyquist rate, linear projections of the returned signals with random vectors are used as measurements. Furthermore, random sampling along the synthetic aperture scan points can be incorporated into the data acquisition scheme. The required number of CS measurements can be an order of magnitude less than uniform sampling of the space–time data. For the application of underground imaging with ground penetrating radars (GPR), typical images contain only a few targets. Thus we show, using simulated and experimental GPR data, that sparser target space images are obtained which are also less cluttered when compared to standard imaging results.

Reduce and Boost: Recovering Arbitrary Sets of Jointly Sparse Vectors

The rapid developing area of compressed sensing suggests that a sparse vector lying in a high dimensional space can be accurately and efficiently recovered from only a small set of nonadaptive linear measurements, under appropriate conditions on the measurement matrix. The vector model has been extended both theoretically and practically to a finite set of sparse vectors sharing a common sparsity pattern. In this paper, we treat a broader framework in which the goal is to recover a possibly infinite set of jointly sparse vectors. Extending existing algorithms to this model is difficult due to the infinite structure of the sparse vector set. Instead, we prove that the entire infinite set of sparse vectors can be recovered by solving a single, reduced-size finite-dimensional problem, corresponding to recovery of a finite set of sparse vectors. We then show that the problem can be further reduced to the basic model of a single sparse vector by randomly combining the measurements. Our approach is exact for both countable and uncountable sets, as it does not rely on discretization or heuristic techniques. To efficiently find the single sparse vector produced by the last reduction step, we suggest an empirical boosting strategy that improves the recovery ability of any given suboptimal method for recovering a sparse vector. Numerical experiments on random data demonstrate that, when applied to infinite sets, our strategy outperforms discretization techniques in terms of both run time and empirical recovery rate. In the finite model, our boosting algorithm has fast run time and much higher recovery rate than known popular methods.