Non-local compressive sampling recovery

Shannon’s Nyquist theorem has always dictated the conventional signal acquisition policies. Power system is not an exception to this. As per this theory, the sampling rate must be at least twice the maximum frequency present in the signal. Recently, compressive sampling (CS) theory has shown that the signals can be reconstructed from samples obtained at sub-Nyquist rate. Signal reconstruction in this theory is exact for “sparse signals” and is near exact for compressible signals provided certain conditions are satisfied. CS theory has already been applied in communication, medical imaging, MRI, radar imaging, remote sensing, computational biology, machine learning, geophysical data analysis, and so forth. CS is comparatively new in the area of computer based power system monitoring. In this paper, subareas of computer based power system monitoring where compressive sampling theory has been applied are reviewed. At first, an overview of CS is presented and then the relevant literature specific to power systems is discussed.

The emerging compressive sampling (CS) theory makes processing ultra-wide-band (UWB) signal at a low sampling rate possible if the underlying signal has a sparse representation in a certain basis. The feasibility of model based compressive sampling for ultra-wide-band (UWB) signal is investigated. In this paper, a multichannel compressive sampling architecture is developed to capture UWB signal at a rate much lower than Nyquist rate. The proposed framework considers sub-Nyquist sampling stream of delayed and weighted versions of a known signal with finite support in time domain. A basis function is constructed to realize sparse signal representation. To reduce the hardware cost, a segmented architecture is suggested. In addition, a joint signal recovery algorithm is presented. Experimental results indicate that, with this system, a UWB signal sampled at about 4% of Nyquist rate still can be recovered with overwhelming probability.

The theory of compressive sampling (CS) was reintroduced by Candes, Romberg and Tao, and D. Donoho in 2006. Using a priori knowledge that a signal is sparse, it has been mathematically proven that CS can defY Nyquist sampling theorem. Theoretically, reconstruction of a CS image relies on the minimization and optimization techniques to solve this complex almost NP-complete problem. There are many paths to consider when compressing and reconstructing an image but these methods have remained untested and unclear on natural images, such as underwater sonar images. The goal of this research is to perfectly reconstruct the original sonar image from a sparse signal while maintaining pertinent information, such as mine-like object, in Side-scan sonar (SSS) images. Goldstein and Osher have shown how to use an iterative method to reconstruct the original image through a method called Split Bregman's iteration. This method decouples the energies using portions of the energy from both the !1 and !2 norm. Once the energies are split, Bregman iteration is used to solve the unconstrained optimization problem by recursively solving the problems simultaneously. The faster these two steps or energies can be solved then the faster the overall method becomes. While the majority of CS research is still focused on the medical field, this paper will demonstrate the effectiveness of the Split Bregman's methods on sonar images.

The emerging theory of compressed sensing (CS) has led to the remarkable result that signals having a sparse representation in some known basis can be represented (with high probability) by a small sample set, taken from random projections of the signal. Notably, this sample set can be smaller than that required by the ubiquitous Nyquist sampling theorem. Much like the generalized Nyquist sampling theorem dictates that the sampling rate can be further reduced for the representation of bandlimited signals, this paper points to similar results for the sampling density in CS. In particular, it is shown that if additional spectral information of the underlying sparse signals is known, colored random projections can be used in CS in order to further reduce the number of measurements needed. Such a priori information is often available in signal processing applications and communications. Algorithms to design colored random projection vectors are developed. Further, an adaptive CS sampling method is developed for applications where non-uniform spectral characteristics of the signal are expected but are not known a priori.

The research of frame synchronisation, which processing is based on the sampling at Nyquist rate in traditional method, has been become the hot topic over the past few decades. In the light of compressed sensing theory that sampling received signals below the Nyquist rate, a frame synchronisation method with correlation rule is proposed in this paper. Firstly, the synchronisation metrics of frame synchronisation is expressed as compressed mode by developing the sparsity of the synchronisation metric amplitudes over observation window. The measurement matrix for compressive sampling is then projected, and compressive sampling for the received signal is employed according to the projected measurement matrix. Finally, we reconstruct the synchronisation metrics from the compressive sampling by utilising compressive sampling matching pursuit (CoSaMP) algorithm and obtain the estimate of frame synchronisation with the reconstructed synchronisation metrics. Compared to the traditional methods, the analysis and simulation results show that the sampling rate in proposed method is significantly reduced while obtaining comparative correct synchronisation probability when received signal-to-noise ratio (SNR) is relatively high. Simultaneously, for the reason that the correct synchronisation probability is not particularly sensitive to the iteration number of the CoSaMP algorithm for frame synchronisation, low reconstruction complexity is also obtained in the proposed method.

Classic signal processing theory dictates that, in order to faithfully reconstruct a band-limited signal (e.g., an image), the sampling rate must be at least twice the maximum frequency contained within the signal, i.e., the Nyquist frequency. Recent developments in applied mathematics, however, have shown that it is often possible to reconstruct signals sampled below the Nyquist rate. This new method of compressed sensing (CS) requires that the signal have a concise and extremely dense representation in some mathematical basis. Magnetic resonance imaging (MRI) is particularly well suited for CS approaches, owing to the flexibility of data collection in the spatial frequency (Fourier) domain available in most MRI protocols. With custom CS acquisition and reconstruction strategies, one can quickly obtain a small subset of the full data and then iteratively reconstruct images that are consistent with the acquired data and sparse by some measure. Successful use of CS results in a substantial decrease in the time required to collect an individual image. This extra time can then be harnessed to increase spatial resolution, temporal resolution, signal-to-noise, or any combination of the three. In this article, we first review the salient features of CS theory and then discuss the specific barriers confronting CS before it can be readily incorporated into clinical quantitative MRI studies of cancer. We finally illustrate applications of the technique by describing examples of CS in dynamic contrast-enhanced MRI and dynamic susceptibility contrast MRI.

Compressed sensing based multi-rate sub-Nyquist sampling system

In wideband cognitive radio networks, Nyquist sampling rate is very challenging problem. It required expensive high speed analog to digital converter and large storage spaces. Lately, compressive sensing has been emerged as significant solution to crack the conventional sampling rate requirements. It proved the ability to sample below Shannan-Nyquist criteria and reconstructing back the signal after considerable dimensional reduction. Mostly in cognitive radio networks, energy detection is widely used due to its simple implementation and blind detection property. However, regardless that energy detection is subject to noise uncertainty as well as shadowing and fading which deteriorate its detection performance. Several articles have been published to improve energy detection performance using large number of measurements. In this paper, since, the detection performance using small number of measurements or compressed measurements achieved significant performance using energy detection under additive white Gaussian noise channel. This motivated us to investigate the performance of compressed measurements-based detection over fading channels which has not been studied yet. The proposed algorithm has been implemented using MATLAB. We also studied the tradeoff between the compression ratios and using fraction of transmitted signal and its impact on detection performance and threshold choice. In comparison with the ordinary compressed energy detection over the Rayleigh fading channel the results reveal that the proposed enhanced compressed measurements-based energy detection is better in performance of detection.

Compressive sampling (CS) is an attractive method to implement analog-to-information conversion (AIC) for a sub-Nyquist radar, where random demodulation (RD) is the most successful AIC. However, RD only considers the sparse characteristic of one single radar pulse, and its mixing circuit still works at the Nyquist sampling rate. To exploit the strong correlation of radar echo pulses, this letter presents a pulse repetition rate compressive sampler for radar target detection. It can acquire these highly correlated pulses by setting a different sampling start time for each pulse and compresses each pulse into one single sample, which means that the low compressive sampling rate is equal to the pulse repetition rate. Compared with RD, this new system reduces the complexity of the hardware circuit and improves the signal compression ratio. Experimental results verify that the proposed radar system outperforms RD by achieving target detection from compressive samples without signal recovery.

The paradigm of using a very simple encoder and a sophisticated decoder for compression of signals became popular with the theory of distributed coding and it has been exercised for the compression of various types of signals such as images and video. The theory of compressive sampling later introduced a similar concept but with the focus on guarantees of signal recovery using sparse and low rank priors lying in an incoherent domain to the domain of sampling. In this paper, we bring together the concepts introduced in distributed coding and compressive sampling with the informed source separation, in which the goal is to efficiently compress the audio sources so that they can be decoded with the knowledge of the mixture of the sources. The proposed framework uses a very simple time domain sampling scheme to encode the sources, and a sophisticated decoding algorithm that makes use of the low rank non-negative tensor factorization model of the distribution of short-time Fourier transform coefficients to recover the sources, which is a direct application of the principles of both compressive sampling and distributed coding.

Reconstruction of atomic force microscopy image using compressed sensing

Power allocation and measurement matrix design for block CS-based distributed MIMO radars

This paper proposes an efficient image reconstruction for compressive sensing (CS) that combines the Lifting Wavelet Transform (LWT) using Symlet 8 (sym8) wavelets with three reconstruction algorithms: Subspace Pursuit (SP), Compressive Sampling Matching Pursuit (CoSaMP), and the Analytic Learned Iterative Shrinkage Thresholding Algorithm (ALISTA). Unlike the conventional Discrete Wavelet Transform (DWT) which relies on computationally intensive convolution operations the LWT provides a faster sparse representation while preserving the sparsity crucial for CS. The proposed approach leverages a key insight: among the four subbands produced by the LWT namely the approximation (CA) and the detail coefficients (LH, HL, HH) only the latter three are inherently sparse. Therefore, compressive sensing is applied exclusively to these detail subbands, while the CA subband is left uncompressed to retain essential low-frequency information. Experiments were conducted on both a natural test image (Lena) and a medical MRI scan, across image resolutions ranging from 200×200 to 512×512 pixels and sampling rates from 10% to 80%. Performance was assessed using the Structural Similarity Index (SSIM) and reconstruction time. Results consistently demonstrate that ALISTA significantly outperforms SP and CoSaMP in both reconstruction fidelity and computational efficiency. At an 80% sampling rate, ALISTA achieves SSIM values of 0.99346 for Lena and 0.98 for the MRI image, compared to approximately 0.97 and 0.94, respectively, for the other two methods. Furthermore, ALISTA maintains remarkably low reconstruction times under 4 seconds even for 512×512-pixel images. These findings confirm that the ALISTA + LWT/sym8 combination offers the best trade-off between image quality and speed, exhibiting robustness across different image types and scales.

This paper proposes an efficient image reconstruction for compressive sensing (CS) that combines the Lifting Wavelet Transform (LWT) using Reverse Biorthogonal 6.8 (rbio6.8) wavelets with three reconstruction algorithms: Subspace Pursuit (SP), Compressive Sampling Matching Pursuit (CoSaMP), and the Analytic Learned Iterative Shrinkage Thresholding Algorithm (ALISTA). Unlike the conventional Discrete Wavelet Transform (DWT) which relies on computationally intensive convolution operations the LWT provides a faster sparse representation while preserving the sparsity crucial for CS. The proposed approach leverages a key insight: among the four subbands produced by the LWT namely the approximation (CA) and the detail coefficients (LH, HL, HH) only the latter three are inherently sparse. Therefore, compressive sensing is applied exclusively to these detail subbands, while the CA subband is left uncompressed to retain essential low-frequency information. Experiments were conducted on both a natural test image (Lena) and a medical MRI scan, across image resolutions ranging from 200×200 to 512×512 pixels and sampling rates from 10% to 80%. Performance was assessed using the Structural Similarity Index (SSIM) and reconstruction time. Results consistently demonstrate that ALISTA significantly outperforms SP and CoSaMP in both reconstruction fidelity and computational efficiency. At an 80% sampling rate, ALISTA achieves SSIM values of 0.99314 for Lena and 0.97688 for the MRI image, compared to approximately 0.97402 and 0.93577, respectively, for the other two methods. Furthermore, ALISTA maintains remarkably low reconstruction times under 4 seconds even for 512×512-pixel images. These findings confirm that the ALISTA + LWT/rbio6.8 combination offers the best trade-off between image quality and speed, exhibiting robustness across different image types and scales.

Most recently, an emerging Compressed Sensing (CS) theory has brought a major breakthrough for data acquisition and recovery. It asserts that a signal, which is highly compressible in a known basis, can be reconstructed with high probability through sampling frequency which is well below Nyquist Sampling Frequency. When applying CS to Remote Sensing (RS) Video imaging, it can directly and efficiently acquire compressed image data by randomly projecting original data to obtain linear and non-adaptive measurements. In this paper, with the help of distributed video coding scheme which is a low-complexity technique for resource limited sensors, the frames of a RS video sequence are divided into Key frames (K frames) and Non-Key frames (CS frames). In other words, the input video sequence consists of many groups of pictures (GOPs) and each GOP consists of one K frame followed by several CS frames. Both of them are measured based on block, but at different sampling rates. In this way, the major encoding computation burden will be shifted to the decoder. At the decoder, the Side Information (SI) is generated for the CS frames using traditional Motion-Compensated Interpolation (MCI) technique according to the reconstructed key frames. The over-complete dictionary is trained by dictionary learning methods based on SI. These learning methods include ICA-like, PCA, K-SVD, MOD, etc. Using these dictionaries, the CS frames could be reconstructed according to sparse-land model. In the numerical experiments, the reconstruction performance of ICA algorithm, which is often evaluated by Peak Signal-to-Noise Ratio (PSNR), has been made compared with other online sparse representation algorithms. The simulation results show its advantages in reducing reconstruction time and robustness in reconstruction performance when applying ICA algorithm to remote sensing video reconstruction.