Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 7616393, 16 pages

http://dx.doi.org/10.1155/2016/7616393

## Compressive Sensing in Signal Processing: Algorithms and Transform Domain Formulations

^{1}Faculty of Electrical Engineering, University of Montenegro, Džordža Vašingtona bb, 81000 Podgorica, Montenegro^{2}Faculty of Electrical Engineering, Mechanical Engineering & Naval Architecture, University of Split, Split, Croatia^{3}Grenoble Institute of Technology, GIPSA-Lab, Saint-Martin-d’Hères, France^{4}School of Information Science and Engineering, Hangzhou Normal University, Zhejiang, China

Received 26 March 2016; Revised 23 July 2016; Accepted 2 August 2016

Academic Editor: Francesco Franco

Copyright © 2016 Irena Orović et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Compressive sensing has emerged as an area that opens new perspectives in signal acquisition and processing. It appears as an alternative to the traditional sampling theory, endeavoring to reduce the required number of samples for successful signal reconstruction. In practice, compressive sensing aims to provide saving in sensing resources, transmission, and storage capacities and to facilitate signal processing in the circumstances when certain data are unavailable. To that end, compressive sensing relies on the mathematical algorithms solving the problem of data reconstruction from a greatly reduced number of measurements by exploring the properties of sparsity and incoherence. Therefore, this concept includes the optimization procedures aiming to provide the sparsest solution in a suitable representation domain. This work, therefore, offers a survey of the compressive sensing idea and prerequisites, together with the commonly used reconstruction methods. Moreover, the compressive sensing problem formulation is considered in signal processing applications assuming some of the commonly used transformation domains, namely, the Fourier transform domain, the polynomial Fourier transform domain, Hermite transform domain, and combined time-frequency domain.

#### 1. Introduction

The fundamental approach for signal reconstruction from its measurements is defined by the Shannon-Nyquist sampling theorem stating that the sampling rate needs to be at least twice the maximal signal frequency. In the discrete case, the number of measurements should be at least equal to the signal length in order to be exactly reconstructed. However, this approach may require large storage space, significant sensing time, heavy power consumption, and large number of sensors. Compressive sensing (CS) is a novel theory that goes beyond the traditional approach [1–4]. It shows that a sparse signal can be reconstructed from much fewer incoherent measurements. The basic assumption in CS approach is that most of the signals in real applications have a concise representation in a certain transform domain where only few of them are significant, while the rest are zero or negligible [5–7]. This requirement is defined as signal sparsity. Another important requirement is the incoherent nature of measurements (observations) in the signal acquisition domain. Therefore, the main objective of CS is to provide an estimate of the original signal from a small number of linear incoherent measurements by exploiting the sparsity property [3, 4].

The CS theory covers not only the signal acquisition strategy, but also the signal reconstruction possibilities and different algorithms [8–17]. Several approaches for CS signal reconstruction have been developed and most of them belong to one of three main approaches: convex optimizations [8–11] such as basis pursuit, Dantzig selector, and gradient-based algorithms; greedy algorithms like matching pursuit [14] and orthogonal matching pursuit [15]; and hybrid methods such as compressive sampling matching pursuit [16] and stage-wise OMP [17]. When comparing these algorithms, convex programming provides the best reconstruction accuracy, but at the cost of high computational complexity. The greedy algorithms bring about low computation complexity, while the hybrid methods try to provide a compromise between these two requirements [18].

The proposed work provides a survey of the general compressive sensing concept supplemented with the several existing approaches and methods for signal reconstruction, which are briefly explained and summarized in the form of algorithms with the aim of providing the readers with an easier and practical insight into the state of the art in this field. Apart from the standard CS algorithms, a few recent solutions have been included as well. Furthermore, the paper provides an overview of different sparsity domains and the possibilities of employing them in the CS problem formulation. Additional contribution is provided through the examples showing the efficiency of the presented methods in practical applications.

The paper is organized as follows. In Section 2, a brief review of the general compressive sensing idea is provided together with the conditions for successful signal reconstruction from reduced set of measurements and the signal recovery formulations using minimization approaches. In Section 3, the commonly used CS algorithms are reviewed. The commonly used domains for CS strategy implementation are given in Section 4, while some of the examples in real applications are provided in Section 5. The concluding remarks are given in Section 6.

#### 2. Compressive Sensing: A General Overview

##### 2.1. Sparsity and Compressibility

Reducing the sampling rate using CS is possible for the case of sparse signals that can be represented by a small number of significant coefficients in an appropriate transform basis. A signal having* K* nonzero coefficients is called -sparse. Assume that signal exhibits sparsity in certain orthonormal basis defined by the basis vectors . The signal can be represented using its sparse transform domain vector as follows:In matrix notation, the previous relation can be written asCommonly, the sparsity is measured using the -norm, which represents the cardinality of the support of : In real applications, the signals are usually not strictly sparse but only approximately sparse. Therefore, instead of being sparse, these signals are often called compressible, meaning that the amplitudes of coefficients decrease rapidly when arranged in descending order. For instance, if we consider coefficients , then the magnitude decays with a power law if there exist constants and satisfying [19]where larger means faster decay and consequently more compressible signal. The signal compressibility can be quantified using the minimal error between the original and sparsified signal (obtained by keeping only largest coefficients):

##### 2.2. Conditions on the CS Matrix: Null Space Property, Restricted Isometry Property, and Incoherence

Instead of acquiring a full set of signal samples of length , in CS scenario, we deal with a quite reduced set of measurements of length , where . The measurement procedure can be modeled by projections of the signal onto vectors constituting the measurement matrix :Using the sparse transform domain representation of vector** s** given by (2), we havewhere will be referred to as CS matrix.

In order to define some requirements for the CS matrix , which are important for successful signal reconstruction, let us introduce the null space of matrix. The null space of CS matrix contains all vectors that are mapped to 0:In order to provide a unique solution, it is necessary to provide the notion that two* K*-sparse vectors and do not result in the same measurement vector. In other words, their difference should not be part of the null space of CS matrix :Since the difference between two* K*-sparse vectors is at most 2*K*-sparse, then a* K*-sparse vector is uniquely defined if null space of contains no 2*K*-sparse vectors. This corresponds to the condition that any 2*K* columns of are linearly independent; that is, and since , we obtain a lower bound on the number of measurements:In the case of strictly sparse signals, the spark can provide reliable information about the exact reconstruction possibility. However, in the case of approximately sparse signals, this condition is not sufficient and does not guarantee stable recovery. Hence, there is another property called null space property that measures the concentration of the null space of matrix . The null space property is satisfied if there is a constant such that [19]for all sets with cardinality* K* and their complements . If the null space property is satisfied, then a strictly -sparse signal can be perfectly reconstructed by using -minimization. For approximately -sparse signals, an upper bound of the -minimization error can be defined as follows [20]:where is defined in (5) for as the minimal error induced by the best -sparse approximation.

The null space property is necessary and sufficient for establishing guarantees for recovery. A stronger condition is required in the presence of noise (and approximately sparse signals). Hence, in [8], the restricted isometry property (RIP) of CS matrix has been introduced. The CS matrix satisfies the RIP property with constant iffor every -sparse vector . This property shows how well the distances are preserved by a certain linear transformation. We might now say that if the RIP is satisfied for 2*K* with , then there are no two* K*-sparse vectors that can correspond to the same measurement vector .

Finally, the incoherence condition mentioned before, which is also related to the RIP of matrix , refers to the incoherence of the projection basis and the sparsifying basis . The mutual coherence can be simply defined by using the combined CS matrix as follows [21]:The mutual coherence is related to the restricted isometry constant using the following bound [22]:

##### 2.3. Signal Recovery Using Minimization Approach

The signal recovery problem is defined as the reconstruction of vector from the measurements . This problem can be generally seen as a problem of solving an underdetermined set of linear equations. However, in the circumstances when is sparse, the problem can be reduced to the following minimization:The -minimization requires an exhaustive search over all possible sparse combinations, which is computationally intractable. Hence, the -minimization is replaced by convex -minimization, which will provide the sparse result with high probability if the measurement matrix satisfies the previous conditions. The -minimization problem is defined as follows:and it has been known as the basis pursuit.

In the situation when the measurements are corrupted by the noise of level and , the reconstruction problem can be defined in a form: called basis pursuit denoising. The error bound for the solution of (19), where** A** satisfies the RIP of order with and , is given bywhere the constants and are defined as [19] For a particular regularization parameter , the minimization problem (19) can be defined using the unconstrained version as follows:which is known as the Lagrangian form of the basis pursuit denoising. These algorithms are commonly solved using primal-dual interior-point methods [22].

Another form of basis pursuit denoising is solved using the least absolute shrinkage and selection operator (LASSO), and it is defined as follows: where is a nonnegative real parameter. The convex optimization methods usually require high computational complexity and high numerical precision.

When the noise is unbounded, one may apply the convex program based on Dantzig selector (it is assumed that the noise variance is per measurement, i.e., the total variance is ): which (for enough measurements) reconstructs a signal with the error bound: The norm is infinite norm called also supreme (maximum) norm.

Besides the -norm minimization, there exist some approaches using the -norm minimization, with :or using -norm minimization, in which case the solution is not rigorously sparse enough [23]:

#### 3. Review of Some Signal Reconstruction Algorithms

The -minimization problems in CS signal reconstruction are usually solved using the convex optimization methods. In addition, there exist greedy methods for sparse signal recovery which allow faster computation compared to -minimization. Greedy algorithms can be divided into two major groups: greedy pursuit methods and thresholding-based methods. In practical applications, the widely used ones are the orthogonal matching pursuit (OMP) and compressive sampling matching pursuit (CoSaMP) from the group of greedy pursuit methods, while from the thresholding group the iterative hard thresholding (IHT) is commonly used due to its simplicity, although it may not be always efficient in providing an exact solution. Some of these algorithms are discussed in detail in this section.

##### 3.1. Matching Pursuit

The matching pursuit algorithm has been known for its simplicity and was first introduced in [14]. This is the first algorithm from the class of iterative greedy methods that decomposes a signal into a linear set of basis functions. Through the iterations, this algorithm chooses in a greedy manner the basis functions that best match the signal. Also, in each iteration, the algorithm removes the signal component having the form of the selected basis function and obtains the residual. This procedure is repeated until the norm of the residual becomes lower than a certain predefined threshold value (halting criterion) (Algorithm 1).