Sparse Optimization of Vibration Signal by ADMM

Circuits, Systems, and Signal Processing, 2019

The sparse signal reconstruction of compressive sensing can be accomplished by l 1-norm minimization, but in many existing algorithms, there are the problems of low success probability and high computational complexity. To overcome these problems, an algorithm based on the alternating direction method of multipliers is proposed. First, using variable splitting techniques, an additional variable is introduced, which is tied to the original variable via an affine constraint. Then, the problem is transformed into a non-constrained optimization problem by means of the Augmented Lagrangian Multiplier method, where the multipliers can be obtained using the gradient ascent method according to dual optimization theory. The l 1-norm

The sparse signal reconstruction of compressive sensing can be accomplished by l 1-norm minimization, but in many existing algorithms, there are the problems of low success probability and high computational complexity. To overcome these problems, an algorithm based on the alternating direction method of multipliers is proposed. First, using variable splitting techniques, an additional variable is introduced, which is tied to the original variable via an affine constraint. Then, the problem is transformed into a non-constrained optimization problem by means of the augmented Lagrangian multiplier method, where the multipliers can be obtained using the gradient ascent method according to dual optimization theory. The l 1-norm minimization can finally be solved by cyclic iteration with concise form, where the solution of the original variable could be obtained by a projection operator, and the auxiliary variable could be solved by a soft threshold operator. Simulation results show that a higher signal reconstruction success probability is obtained when compared to existing methods, while a low computational cost is required.

2012

The classical Shannon Nyquist theorem tells us that, the number of samples required for a signal to reconstruct must be at least twice the bandwidth of the highest frequency for the signal of interest. In fact, this principle is used in all signal processing applications. Unfortunately, in most of the practical cases we end up with far too many samples. In such cases a new sampling method has been developed called Compressive Sensing (CS) or Compressive Sampling, where one can reconstruct certain signals and images from far fewer samples or measurements when compared to that of samples in classical theorem. CS theory primarily relies on sparsity principle and it exploits the fact that many natural signals or images are sparse in the sense that they have concise representations when expressed in the proper basis. Since CS theory relies on sparsity, we focused on reconstructing a sparse signal or sparse approximated image from its corresponding few measurements. In this document we focused on 1 l norm minimization problem (convex optimization problem) and its importance in recovering a sparse signal or sparse approximated image in CS. To sparse approximate the image we have transformed the image form standard pixel domain to wavelet domain, because of its concise representation. The algorithms we used to solve the 1 l norm minimization problem are primal-dual interior point method and barrier method. We came up with certain examples in Matlab to explain the differences between barrier method and primal-dual interior point method in solving a 1 l norm minimization problem i.e. recovering a sparse signal or image from very few measurements. While recovering the images the approach we used is block wise approach and treating each block as vector.

International Journal of Innovative Research in Computer and Communication Engineering, 2015

Compressive Sensing acquire sparse signal significantly at very lower rate than Nyquist sampling rate. For this, a low complexity compressed sensing operation is defined and it is the combination of sampling and compression. The signals formed from compressed sensing operation are compressible signals and a set of random linear measurements accurately reconstructs compressible signals with the use of nonlinear or convex reconstruction algorithms. Basis Pursuit algorithm is one of the convex optimization algorithms to reconstruct the sparse signal. The l1 minimization theory for linear programming problems is used to formulate the compressive sensing method. Interior point method is used to solve the basis pursuit algorithm for sparse signal reconstruction. In this paper, the methodology of reconstructing sparse signal using basis pursuit algorithm is discussed.

Global Scientific Journals, 2022

In this paper, sparse signal problem is defined and formulated. Constrained optimization and unconstrained optimization problems that solve sparse signal problem are also modeled. The main objective of the paper is to reconstruct sparse signal that was corrupted by a white noise. A white Gaussian noise was added to the original signal for an observed sparse signal and the split Bregman iterations is an optimization method for regularization of inverse problem used to solve and recover the sparse signal which is closer to the original signal. This is obtained by varying the regularization parameters and choose the best one to reconstruct the signal as required comparatively to the original signal.

arXiv (Cornell University), 2013

In this paper, we propose a time-frequency analysis method to obtain instantaneous frequencies and the corresponding decomposition by solving an optimization problem. In this optimization problem, the basis to decompose the signal is not known a priori. Instead, it is adapted to the signal and is determined as part of the optimization problem. In this sense, this optimization problem can be seen as a dictionary learning problem. This dictionary learning problem is solved by using the Augmented Lagrangian Multiplier method (ALM) iteratively. We further accelerate the convergence of the ALM method in each iteration by using the fast wavelet transform. We apply our method to decompose several signals, including signals with poor scale separation, signals with outliers and polluted by noise and a real signal. The results show that this method can give accurate recovery of both the instantaneous frequencies and the intrinsic mode functions.

International Journal of Communications, Network and System Sciences, 2015

In digital signal processing (DSP), Nyquist-rate sampling completely describes a signal by exploiting its bandlimitedness. Compressed Sensing (CS), also known as compressive sampling, is a DSP technique efficiently acquiring and reconstructing a signal completely from reduced number of measurements, by exploiting its compressibility. The measurements are not point samples but more general linear functions of the signal. CS can capture and represent sparse signals at a rate significantly lower than ordinarily used in the Shannon's sampling theorem. It is interesting to notice that most signals in reality are sparse; especially when they are represented in some domain (such as the wavelet domain) where many coefficients are close to or equal to zero. A signal is called K-sparse, if it can be exactly represented by a basis, { } 1 ψ N i i = , and a set of coefficients k x , where only K coefficients are nonzero. A signal is called approximately K-sparse, if it can be represented up to a certain accuracy using K non-zero coefficients. As an example, a K-sparse signal is the class of signals that are the sum of K sinusoids chosen from the N harmonics of the observed time interval. Taking the DFT of any such signal would render only K non-zero values k x. An example of approximately sparse signals is when the coefficients k x , sorted by magnitude, decrease following a power law. In this case the sparse approximation constructed by choosing the K largest coefficients is guaranteed to have an approximation error that decreases with the same power law as the coefficients. The main limitation of CS-based systems is that they are employing iterative algorithms to recover the signal. The sealgorithms are slow and the hardware solution has become crucial for higher performance and speed. This technique enables fewer data samples than traditionally required when capturing a signal with relatively high bandwidth, but a low information rate. As a main feature of CS, efficient algorithms such as 1 -minimization can be used for recovery. This paper gives a survey of both theoretical and numerical aspects of compressive sensing technique and its applications. The theory of CS has many potential applications in signal processing, wireless communication, cognitive radio and medical imaging.

2013

Compressed sensing (CS) is a rapidly growing field, attracting considerable attention in many areas from imaging to communication and control systems. This signal processing framework is based on the reconstruction of signals, which are sparse in some domain, from a very small data collection of linear projections of the signal. The solution to the underdetermined linear system, resulting from these data, allows us to estimate the original signal. Finding the solution is an optimization problem which is mainly based on the minimization of the l1-norm. For this purpose, fast algorithms have been developed, and the aim of this paper is to make a comparative analysis of some of these algorithms. Specifically, we study the performance of sparse reconstructions with five commonly used algorithms implemented in Matlab (CVX, L1magic, SPGL1, UnlocBoX and YALL1) in order to determine the viability of implementation of a large number of applications proposed in the last years using CS. The ob...

International Journal of Engineering and Technology

Compressive Sensing is a novel technique where reconstruction of an image can be done with less number of samples than conventional Nyquist theorem suggests. The signal will pass through sensing matrix wavelet transformation to make the signal sparser enough which is a criterion for compressive sensing. The low frequency and high frequency components of an image have different kind of information. So, these have to be processed separately in both measurements and reconstruction techniques for better image compression. The performance further can be improved by using DARC prediction method. The reconstructed image should be better in both PSNR and visual quality. In medical field, especially in MRI scanning, compressive sensing can be utilized for less scanning time. Keyword-Compressive Sensing, Wavelet transform, Sparsity, DARC prediction I. INTRODUCTION Compressive sensing (CS) is a new compression technique where fewer samples of measurement are enough to reconstruct the image with good visual quality. The samples required are much lesser than Nyquist criterion suggests. But the non-linear reconstruction used in CS is more complex compared to linear reconstruction in conventional compression. CS will measure finite dimensional vectors. Now CS is actively researched in applications like MRI, RADAR, single pixel camera, etc. as in [1]. We consider the application in medical field. MR images are sparse in Wavelet domain. The medical images will undergo CS so that the image reconstructed will have good visual quality but with less number of measurements. MRI is slow process due to the large number of data needed to be collected while scanning a patient as in [2]. With the help of CS we can reduce the number of samples, thus reducing scan time, which will benefit patient with less radiation exposure. In CS, there are three main principles-Sparsity, Measurements taking and Nonlinear reconstruction. The signal should be sparse-Information rate contained in the image should be much less than bandwidth-to undergo CS. If it's not sparse enough; we need to undergo the transformation of the image to make it sparse. We took wavelet transform as sparsity inducing matrix in this paper. The reconstruction of signals from lesser samples can only be possible if the chosen sparsity matrix and measurement matrix follows Restricted Isometry Property. The incoherence between these matrices is necessary for this. There are two approaches for reconstructing image at receiver side-basis pursuit and greedy algorithm. These nonlinear techniques will result in good quality reconstructed image as in [3]. In short, CS helps to reduce sampling and computation costs for sensing signals that have a sparse or compressible representation. In paper [4], the authors introduced a novel compressive sensing based prediction measurement (CSPM) encoder. The sparse image undergoes CS by using Gaussian matrix and these measured values pass to CSPM. In CSPM, the measured matrix undergoes linear prediction and entropy encoding. Since the sparsity level of prediction residual is higher than its original image block, the performance of CS image reconstruction algorithm will be better. This CSPM encoder can achieve significant reduction in data storage and saves transmission energy. The bandwidth consumption of CSPM based CS will be considerably less which in turn increases the lifetime of sensors. The wavelet transformed image has high frequency coefficients that are sparse and the low frequency coefficients that are not sparse. The low-frequency coefficients contain most of the energy of the image and have coherent nature. In paper [5], they measured (CS applied) the high-frequency sub band coefficients, and kept the low-frequency sub-band coefficients unchanged. In this paper we have chosen deterministic matrices such as Hadamard matrix, random matrices such as Gaussian matrix, as measurement matrices and to attain sparsity the Daubechies wavelets are used. The nonlinear reconstruction methods used are Orthogonal Matching Pursuit (OMP) and L1 minimisation technique. The prediction methods used are Linear and DARC.

Przegląd Elektrotechniczny, 2019

The paper presents the application of the compressive sensing technique to reconstruct a non-stationary signal based on compressed samples in the time-frequency domain. A greedy algorithm with different dictionaries to seek sparse atomic decomposition of the signal was applied. The results of the simulation confirm that the use of compressive sensing allows reconstruction of the non-stationary signal from a reduced number of randomly acquired samples, with slight loss of reconstruction quality. Streszczenie. Przedstawiono zastosowanie techniki oszczędnego próbkowania do rekonstrukcji sygnału niestacjonarnego na podstawie skompresowanych próbek w dziedzinie czas-częstotliwość. Zastosowano nadmiarowy algorytm z różnymi słownikami aby znaleźć rzadką reprezentację sygnału. Wyniki symulacji potwierdzają, że zastosowanie oszczędnego próbkowania pozwala na rekonstrukcję sygnału niestacjonarnego z małej liczby losowo pobranych próbek, z niewielką utratą jakości rekonstrukcji. (Rzadka reprezentacja sygnału niestacjonarnego w technice oszczędnego próbkowania).