Compressive sensing for efficient health monitoring and effective damage detection of structures

Real world Structural Health Monitoring (SHM) systems consist of sensors in the scale of hundreds, each sensor generating extremely large amounts of data, often arousing the issue of the cost associated with data transfer and storage. Sensor energy is a major component included in this cost factor, especially in Wireless Sensor Networks (WSN). Data compression is one of the techniques that is being explored to mitigate the effects of these issues. In contrast to traditional data compression techniques, Compressive Sensing (CS) a very recent development introduces the means of accurately reproducing a signal by acquiring much less number of samples than that defined by the Nyquist’s theorem. CS achieves this task by exploiting the sparsity of the signal. By the reduced amount of data samples, CS may help reduce the energy consumption and storage costs associated with SHM systems. This paper investigates CS based data acquisition in SHM, in particular, the implications of CS on damage detection and localization. CS is implemented in a simulation environment to compress structural response data from a Reinforced Concrete (RC) structure. Promising results were obtained from the compressed data reconstruction process as well as the subsequent damage identification process using the reconstructed data. A reconstruction accuracy of 99% could be achieved at a Compression Ratio (CR) of 2.48 using the experimental data. Further analysis using the reconstructed signals provided accurate damage detection and localization results using two damage detection algorithms, showing that CS has not compromised the crucial information on structural damages during the compression process.


Introduction
Transmitting and maintaining large amounts of data in SHM systems have been issues addressed in recent researches due to the large scale real-world structures in use. Traditionally, uncompressed structural response signals are acquired by each sensor and transmitted to a central server for processing and damage decision making. This task unnecessarily consumes one of the most valuable resource in a WSN-based SHM system -energy, and also wastes memory space of sensor nodes in both wired and wireless systems. In a wireless-based SHM system, such wastage results in reduced system lifetime and increased maintenance costs, since data transmission in a WSN is carried out at the cost of limited battery power of sensors. For both types of systems, transmitting a lot of data also increases the network traffic and collisions, reducing the reliability of data communication process. Even though memory is becoming cheaper, even the newest version of the most widely used commercially available sensor node -Imote2 -has only a combined memory of 64 MB. This amount of memory will barely be sufficient to a network as large as the SHM system installed on Tsing Ma, Ting Kau and Kap Shui Mun bridges in Hong Kong [1,2]. This system, known as the "Wind And Structural Health Monitoring System" (WASHMS) consists of approximately 800 sensors and operates 24 hours a day, 7 days a week, where each sensor node measures temperature, strains in structural members, wind speed deflection and rotation of the kilometres of cables in the bridges and any movement of the bridge decks and towers, generating an enormous amount of data [3].
There have been numerous proposals to mitigate the effects of these issues including decentralized data processing in SHM [4], power saving strategies [5] and data compression techniques [6]. In traditional data compression techniques, compression is applied on the already sampled signal. However, this usually leads to wastage of processing power and memory as the important information in most signals is carried only by a few samples compared to the total number of samples. The novel concept of CS appears a potential alternative to solve this issue. By exploiting the sparsity of the signal, CS provides the means of accurately reconstructing the signal with much less number of acquired samples than that defined by the Nyquist's theorem [7,8]. This concept is believed to have tremendous potential in the field of WSNs [7]. In fact, already there have been a few attempts to use this method in WSNs in the literature [9,10,11]. Being an application of WSN, SHM may also benefit from the compressing abilities of CS. With CS used for data acquisition in SHM systems, the energy and storage costs associated will be vastly reduced [2]. Thus, CS can be an area with high potential for research and development in the field of SHM. A preliminary study on the feasibility of adopting CS for SHM was presented in [12].
In this paper, CS based data acquisition in SHM is explored, in particular for damage detection and localization. The attempt is to preserve the energy and storage associated with SHM systems through data reduction, without losing the confidence of accurate decisionmaking. The effectiveness of CS for data compression in SHM is analyzed using a set of experimental data from a series of impact tests carried out on an RC slab. Promising results were obtained from this application in terms of CRs and reconstruction accuracy.
Further analyses on the reconstructed signal using two damage detection and localization algorithms (the ARD method [13] and the Wiener filter based method [14,15]) provided successful damage detection and localization results. The main contributions of this paper are as follows: • Proposing the use of CS based data acquisition for reliable SHM.
• Successful reconstruction of the compressed responses.
• Comparison of the data compression performance using CS and other compression techniques.
• Successful damage detection and localization using the reconstructed responses.
The remainder of this paper is organized as follows: First, the current work in literature on the applications of CS are discussed. Then the theory of CS is presented followed by the application of CS on the available experimental data. The success of the CS based data acquisition is analyzed using the achieved CRs and the accuracy of signal reconstruction in terms of a frequency content based metric that was developed in this study. Next, a brief comparison is carried out between two other data compression techniques and CS. Finally, damage detection and localization results using the CS reconstructed data are presented followed by the conclusions.

Related work
CS is a quite recent development in communication and data processing technologies which has been recently introduced to WSNs. CS has the potential to serve as alternatives to both Nyquists rate sampling and data compression techniques. Being a novel concept, applications of CS are largely under-explored in the field of sensor networks. Only a limited number of studies exist on CS in SHM applications in the literature. Bajwa et al. introduced the concept of CS for WSNs [9]. This work discusses the power-distortion trade-off associated with the data acquisition through CS in a centralized WSN, and the latency involved in the information retrieval. Then, a distributed matched source-channel communication scheme is proposed for the estimation of sensed data at the fusion center or the central server. Feng et al. investigate the problem of sensor localization in WSNs and proposes a novel method called Compressive Sensing for Manifold Learning [10]. The measurement matrix for data acquisition is chosen to be the pair-wise distance matrix on which CS is applied. This pair-wise measurement matrix is constructed by each node measuring the distance to all its neighbouring nodes. A central node retrieves the compressive measurements and reconstructs the full matrix in order to construct the location map of the sensors. This work shows the high accuracy of the achieved localization results even though the distance measurement matrix is reconstructed at the central node with only a limited number of samples from the original measurement. The communication cost reduction associated with the method is also proven. A distributed algorithm for sparse signal recovery is introduced by Ling and Tian [11]. While a fraction of the sensors are turned off by a sleeping strategy, compressive sampling is performed on the awake sensors. These sensors collaborate with their active neighbours through one-hop communication to recover the compressed measurements and iteratively improves the local estimates until reaching a global maximum.
The performance of this algorithm is illustrated in terms of scalability and optimal data acquisition.
Cortial et al. discuss the application of CS in structural damage detection [16]. They demonstrate the effectiveness of the application by using CS data to successfully locate the damage in a F-16 fighter jet simulation. Bao et al. investigate the performance of CS for compression of vibration data obtained from an actual SHM system [2,17]. Reconstruction of the signal is done using both wavelet and Fourier orthogonal bases. Their results show that the CRs achieved using CS in this case are not very high. They account this poor result to the fact that the vibration data of civil structures used for SHM are not naturally sparse in the chosen bases. In their later work Bao et al. apply CS to recover lost data in a WSN used in SHM. Their results show that the signal recovery is good when the signal is sparse regardless of continuous or random packet loss and noise, demonstrating the significance of sparsity for CS [18]. Wang and Hao were the first to propose a damage identification scheme based on CS for SHM [19]. Mapping damage identification as pattern classification problems, they construct a feature matrix based on the sparse representation results of time domain structural responses. They demonstrate the success of the method using numerical and experimental data. However, the application of this method to complex large civil structures could be a challenging task as the time taken to train the model on damage cases may be too high. Mascarenas et al. address the challenges of extracting information on damage condition from large heterogeneous sets of data from structures and the costeffective collection of relevant data, respecting the limitations of the system in their work [20]. They propose a two-stage CS framework with potential to solve both these challenges.
They evaluate the implementation of the compression filter named smashed filter for damage classification in the structure with successful results. Yang and Nagarajaiah present a novel algorithm for both locating and assessing structural damage using Blind Source Separation (BSS), sparse representation and CS [21]. In this algorithm they first extract modal features of the structure blindly using an unsupervised Complexity Pursuit (CP) algorithm and then use the sparse nature of the classification network established based on CS to classify damage.
In a later study they work on denoising of structural responses through a new technique named Principal Component Pursuit (PCP) which is inspired by CS [22]. In [23] they propose a new method output-only modal identification of structures using a combination of BSS and CS and show that the method can accurately identify the modes using low-rate sampling than that required by the Nyquist theorem.
As the theory states, when applied appropriately, CS can be expected to acquire undersampled structural responses from sensors in an SHM system and accurately reconstruct them at the central server. This will save sensor energy spent on data transmission, save sensor storage space and data maintenance costs by reducing the amount of data, while still enabling accurate reconstruction of the data providing the ability to process the data at convenience. This overall picture, as well as the successful application of CS in recent researches in communication and networking inspired the application of CS for SHM.

Compressive Sensing for Structural Health Monitoring
Traditional sampling exploits the band-limitedness of a signal, sampling it at twice (or more) as fast as its bandwidth (the Nyquist rate), to ensure accurate reconstruction [24].
CS exploits the sparsity of a signal during its sampling [7]. The concept of CS [2,7,9,8] is summarized in the following section using this principle of sparsity.

Compressive sensing
The applicability of CS for successful data compression can be explained with the aid of two principles: Sparsity and Incoherence, which are important in the understanding of the theory of CS.
Sparsity: Many natural and man-made signals are sparse or compressible in the sense that, there exists a base Ψ where the representation in Equation (1) has just a few large coefficients and many small coefficients. In other words, they have a concise representation when expressed in an appropriate basis Ψ. Another word to explain this situation is redundancy of a signal when expressed in an appropriate basis.
Incoherence: This states the idea that a signal having a sparse representation in Ψ, must be spread out in the domain in which they are acquired. An example is the impulse response in the time domain, which spreads out in the frequency domain.
A real-valued, finite length, one-dimensional, discrete time signal can be expressed as x(t), t = 1, 2, ..., n in where n is the signal length. In this study, x(t) is the data to be compressed (e.g. acceleration, strain) and can be represented in terms of a basis of n × 1 vectors ψ i , i = 1, 2, ..., n. For convenience, the basis is assumed to be orthonormal. Some where, the basis coefficients α j are given by α = Ψ T x.
By a linear projection of x, a vector y of length m(< n) can be acquired as follows: where, Φ is the m × n measurement matrix and Θ = ΦΨ is an m × n matrix.
Since m < n in Φ, recovering the original response x of length n from the m measurements in y is ill-posed. If the signal x is known to be sparse in the basis Ψ, i.e., many of the coefficients in α are zero, then under certain conditions, α can be reconstructed exactly by solving the following convex optimization: α = arg min α 1 such that, Θα = y.
Given that; 1) the signal x is s-sparse (it has at most s non-zero entries) and 2) the matrix Θ obeys the Restricted Isometric Property (RIP) described below, α can be precisely recovered by solving Equation (3).
The RIP [8] states that, there exists an isometric constant δ s for the matrix Θ, where δ s is defined as the smallest number such that for all s-sparse vectors v s , the following condition holds: Therefore, the s-sparse signals can be exactly recovered by solving the optimization in Equation (3), if Θ satisfies Equation (4). In other words, suppose that Ψ is an orthonormal basis (for simplicity) such as a wavelet basis or a Fourier sinusoids basis. Then, a measurement matrix Φ needs to be constructed such that, the matrix Θ(Θ = ΦΨ) obeys Equation (4).
In CS, Φ is selected as a random m × n matrix, such as: 1. by sampling independent and identically distributed (i.i.d.) entries φ i,j from a normal distribution with zero mean and 1/m variance 2. by sampling i.i.d. entries φ i,j from a symmetric Bernoulli distribution (P (φ i,j = 1/m)) 3. by sampling n column vectors uniformly at random on the unit sphere of m 4. by sampling a random projection P and normalizing; Φ = n/mP The matrix Φ obtained from either of the above methods obeys the RIP with overwhelming probability given the following condition is fulfilled [7]. m cs log 10 (n/s) , where, c is a specified constant depending on each instance.
When the above condition is satisfied, the matrix Θ also obeys the RIP with m ≥ cs/ log 10 (n/s) regardless of the orthonormal basis matrix Ψ. And it also becomes incoherent with Ψ. Therefore, the perfect reconstruction of an s-sparse response signal x of length n is possible independently of its basis Ψ, if the ratio n/m does not exceed c −1 (n/s) log 10 (n/s).
This makes n/m a measure of sparsity of the signal x [2].
As a summary, the ultimate aim is to obtain a compressed signal y through a linear projection of x as shown in Equation (2), and be able to accurately reconstruct x by solving the convex optimization in Equation (3). In CS, this is rendered possible by selecting the measurement matrix as described above, given that the constraint in Equation (5) is fulfilled.
Then, the original signal x can be reconstructed using the measured signal y by solving the convex optimization given in Equation (3).

Compressing noisy signals
When responses are contaminated with noise, those signals may not be naturally sparse in any basis. Therefore, de-noising the response is required to make the signal sparse. In such a case, the measured signal y can be represented as follows: where, e is the error associated with noise, which is uncertain and bounded as e 2 with being a chosen bound on the size of the measurement error.
The basis coefficients α can then be reconstructed by solving the following convex opti- where, |Θα − y|| 2 is the l 2 norm of Θα − y.
Similar to the optimization in Equation (3), Equation (7) also has a unique solution for α which can be used to represent the data. But in this case, since the unknown noise is only assumed to be bound on its norm, the exact recovery of the original response x cannot be guaranteed [2]. Having discussed the theory of CS, the focus will now be on adopting CS for data acquisition in SHM systems, which will be discussed in the following sections.

Experimental evaluation
In order to verify the effectiveness and accuracy of the process, a set of experimental data is used from a series of experiments carried out on an RC slab. These data will be briefly described in this section before proceeding to the details of CS application: CS requirements and how the data fits with the requirements, data compression with CS and analysis of the reconstruction. The simulation work in this paper has been carried out using MATLAB.

Experimental setup
A two span RC slab has been used as the test specimen in this experiment [25]. Each span of the slab was loaded with a two-point loading system as shown in Figure 1a, which was also connected to the supports in order to reduce the effect on the supports. Damages were created on the slab by incrementally loading it via a static load test in 13 loading levels while monitoring damage locations and lengths. A dynamic load test was carried out on the slab using an impact hammer, in order to record structural responses. The slab was equipped with nine sensors evenly distributed along the length of the slab on its central axis.
A data acquisition system was used to acquire six acceleration measurements at each loading stage at 500Hz sampling rate giving six test samples of structural responses, each of length 4100. Figure

Compressive sensing pre-requisites
For feasible reconstruction of the signal, CS requires the data to be compressed, to be sparse in some basis. Therefore, the first step in adopting CS for data acquisition in SHM systems is to analyze the experimental data for their sparsity. Several commonly used methods such as the wavelet transform, Fourier transform and Discrete Cosine Transform (DCT) were initially applied on the structural responses. However, the responses exhibited no sparsity in either of the above bases. This can be further examined looking at the zoomed portion of the signal -from data point 3000 to 3050, in the 4100 length original sample shown in Figure 2a. It can be seen in this figure that although the magnitude of the signal appears to be zero in the latter part of the sample, it always has a negligibly small but non-zero value. Nevertheless, looking at the signal, it can be reasonably considered that most of the information in this structural response is carried by only a few data points. In other words, most of the data points appear to be negligibly small such as noise. Therefore, de-noising techniques were adopted in order to make the signal sparse in some basis.
Wavelet de-noising technique was selected to de-noise the signal in this case due to its simplicity and versatility [26,27]. Wavelet de-noising is also the most common method used for signal de-noising [26,27]. There are two main de-noising techniques used in wavelet de- Due to its compression abilities and also because of its popular use in CS applications in the literature [2,7,8], wavelets were selected as the preferred basis functions for this study.
Further, Daubechies wavelet family was selected due to its sophistication compared to other wavelet families in use [30]. Figure 3 shows a comparison of the original signal and the de-noised signals with different threshold selection methods indicated above. These results were obtained using Hard-thresholding, which is a choice that will be justified later in the section. It should be noted that although the complete signal of length 4100 is used in all the actual analyses throughout this study, only the first half of it that contains the impulse response (a length of 2050) is plotted in most of the graphs included in this paper for better clarity.
The de-noised signals using the above mentioned threshold selection methods show following sparsities (number of non-zero points) in the Daubechies wavelet basis: 1) rigrsure, s

Signal compression and reconstruction
As the de-noised structural response using Stein's Unbiased Risk threshold selection and Hard thresholding technique exhibited sparsity in the Daubechies wavelet basis, accurate signal reconstruction should now be feasible. However, due to the noise term involved in the signal, the measurement signal to be acquired in the current scenario resembles Equation (6) and the convex optimisation problem to be solved is of the form shown in Equation (7).
The bound on the measurement noise is ideally estimated by taking the norm of the noise measured either using a sensor that is not programmed for CS or by prior knowledge. In this study, it was initially estimated by optimizing the accuracy of reconstruction (discussed later in this section). However, the reconstruction quality of the signal was found to be independent of this choice in the current analysis which simplifies the situation, as knowledge of the complete signal would not be available in a real-world CS scenario.
When the sparsity of the de-noised signal is known, the measurement matrix Φ of Equation (6) can be selected. As mentioned in Section 3, in CS, this measurement matrix is required to be random in order to accurately reconstruct the compressive sampled signal.
It also needs to fulfil the condition in Equation (5) in order to satisfy the RIP in Equation  Figure 5 illustrates the compressed signals at these six compression levels. In the framework of this study, the compressed signal is obtained by a linear projection of the original data using the chosen measurement matrix, since the original data is already available. However, in an actual system, the signal needs to be obtained directly in the compressed form at sampling. For this, the measurement matrix needs to be built into the sensor before sensor installation, which is not discussed in this study. It can be seen from Figure 5 that as the compressed length increases, more information from the original sample is retained in the compressed signal at the cost of CR.
At reconstruction, the receiver or the central server in case of an SHM system, has knowledge of the utilized measurement matrix and the sparse basis. In order to solve the modified convex optimization problem in Equation (7), CVX, a package built on MATLAB for specifying and solving convex problems [31] is utilized.  (5)) the quality of the reconstructed signal will improve with better resemblance to the original signal. However, this will reduce the CR, thereby reducing the advantage provided by the compression. Therefore, the CR and the reconstruction quality presents a trade-off.
Analyzing the accuracy of the reconstruction is vital when validating a data compression technique, as the important information carried by the signal should not be compromised by the compression process. Especially in SHM measurements, retaining the frequency content of the measurements is crucial as they carry important information required for structural condition analysis in the data post-processing. Therefore, a reconstruction accuracy metric RA CS to quantify the accuracy of the reconstructed signal in this analysis is computed as follows: where, x Orig is the original signal and x CSrecon is the CS reconstructed signal with 'fft' indicating the fast Fourier transform.
is the correlation coefficient of data samples x and y, where M is the data length with k = 1, 2, .., M and µ x and µ y are the means of x and y respectively. Signal compression and reconstruction with varying de-noising thresholds are analyzed next in order to observe the effects of the threshold estimation methods on the CS process.
The threshold estimation criteria discussed previously, a) rigrsure, b) heursure, c) sqtwolog and d) minimaxi are considered in this analysis. Figure 8 illustrates the compressed samples obtained by compressing the de-noised data. It can be seen how the Stein's Unbiased Risk (rigrsure) has more length -and thereby more information -compared to the other three techniques. A compressed length of 10×s which depends on the sparsity (s) of the de-noised signals from each of the above threshold selections is used in this analysis.
The corresponding reconstructed data and their frequency spectra are shown in Figure 9. Further into the analysis, Figure 11 shows the combined CS-reconstruction accuracy results from the two previous discussions. The RA CS is plotted against the de-noising sample length (or the CR), and the four different threshold estimation methods. It can be seen that each of the de-noising methods exhibit similar results with varying CR. That is, as the compressed length increases, de-noised signals from all four threshold estimation methods can be reconstructed with better accuracy at the cost of CR. Stein's Unbiased Risk (rigrsure) method for threshold estimation proves to be the most effective in this process in terms of the reconstruction accuracy.
Nevertheless, as the most essential task required by an SHM system is monitoring and deciding on the structural condition, the data available for post-processing and decisionmaking should be reliable enough. Therefore, if the important information embedded in the original signal is not lost in the process of compression and reconstruction, the compression technique can be considered as good enough to be employed in an SHM system for decision making. This possibility will be explored using damage detection and localization algorithms in Section 5.

Sensor energy and memory space saving with compressive sensing
Energy is not an issue for a wired sensor network as the sensors could have a never ending supply of power through the connecting cables. However, limited energy of wireless sensors is a major issue which has attracted the attention of researchers worldwide. Research is still on-going for a sustainable solution to this problem. Limited sensor memory space can be an issue to both these network types. However, since the sensors could be programmed to transmit the data continuously to the central server in a wired network, sensor memory space can be freed often, making its impact somewhat less in a wired network. Nevertheless, the cost associated with data transmission and management is a problem for any such network.
Hence, a discussion on the ability of CS for sensor energy and memory saving is presented in this section.
It is said that transmitting one bit through wireless communication may consume as much energy as executing a few thousand instructions in a processor [32,33]. It is clear that the transmission energy increases with increasing data amount to be transmitted. As defined previously, CR is the ratio between the original signal length and the compressed signal length. It is a fact that some control information are added to a data packet before it is transmitted to the central server. However, compared to a data sample of length 4100 as with this analysis, CR could be considered a good indicator of the energy savings in the data transmission process. For sensor memory saving, CR is an obvious indicator. Therefore, for the purpose of this discussion, CR is used as an approximate indicator of both sensor energy and memory saved in the CS process. Although the energy/memory saving decreases with increasing compressed sample length, in this study, CS has been able to save more than 50% sensor energy/memory with an impressive reconstruction accuracy of 99% at CR=2.48. Thus, the application of CS for compressed data sampling in SHM systems shows promise in saving energy and memory space in sensors.

Comparison with other compression techniques
The main advantage of using CS for data compression is that in CS, the data is acquired in an already compressed form right at the moment of sampling, eliminating the need to first sample the signal at a higher rate, store it in the sensor and then process it for compression.
This saves sensor energy for sampling and processing as well as sensor memory for data storage. As mentioned previously, the energy spent on data transmission by a wireless sensor is much more than that spent on data processing [32,33]. Therefore, the CR that can be achieved by a compression method is also an important deciding factor. In this section, the performance of CS for data compression and reconstruction are compared against a few other commonly used compression techniques. For this analysis, the basic method of wavelet compression is chosen using different wavelet families and levels. Wavelet compression is a widely used technique for image compression [34]. In this analysis the signal is denoised and wavelet decomposed at different levels using different wavelets. Then the decomposed signal compression and compared against the original signal. In addition to the quantitative analysis above, a visual comparison of the reconstruction quality of the two wavelet compression techniques against the original signal are illustrated in Figure 13. It shows the reconstructed signals and their frequency spectra. This figure shows that the reconstructed signals from the two wavelet techniques are almost the same (if not exactly the same) as the original signal. These results show that traditional lossless compression techniques can achieve equal or better signal reconstruction accuracy than CS under same conditions. The advantage in CS only comes in the case when exact or near perfect reconstruction can be achieved, it will save sensor memory and energy compared to traditional compression techniques, as in CS the samples are already achieved in a compressed manner.

Damage detection and localization
The main purpose of a real-world SHM system is accurate detection and localization of damages in the structure. Achieving both of these tasks with sufficient accuracy has been a challenge faced by researchers for many years [4]. Accomplishing such a task with a reconstructed signal from a set of under-sampled data can be even more difficult. This section focuses on achieving these tasks -i.e. accurate damage detection and localization using the reconstructed data from under-sampled CS data. This analysis uses the de-noised data from Stein's Unbiased Risk threshold selection with Hard thresholding and a CR of 3.73.
Two damage detection and localization techniques are used in this analysis: 1) ARD method [13], which uses time-series models estimated from the structural responses, 2) Wiener filter based method, from the work described in [14,15].
The ARD method is a statistical time-series based structural damage detection method which makes use of the Auto-Regressive (AR) model [13]. In this method, the structural response data is first fitted to an AR model with a suitable model order. The AR coefficients from this model are used in the damage detection process. A damage detection index called D index computed using the undamaged data's AR coefficients and the current data's AR coefficients is used for damage detection. This D index is represented as: where, p is the AR model order and φ y i and φ x i are the i th AR model coefficients of the current data y (from unknown state of the structure) and reference data x, respectively. The AR model order p, is determined by exploring the autocorrelation function of the model residual errors [35,36].
For damage localization, Fisher criterion f is computed using the above D indexes for each sensor as follows: Wiener filter e(n) as follows: where e(n) = d(n) − y(n) with d(n) being an estimation of the desired signal expected from the Wiener filter and y(n) being the resulting noise-reduced filter output which is targeted to be an estimation of d(n), respectively.
In the Wiener filter method, the damage localization is carried out at sensor pair level utilizing the spatial information associated with the sensors unlike in the ARD method.
Cross-correlation function (CCF) coefficients are computed from the Wiener filter MSEs of a sensor pair, and the variance of these coefficients is used as the damage location indicator. Compared with the results from the original data in Figure 14a, D indexes from the CSreconstructed data show a loss in magnitude. This can be due to the information lost during the data de-noising process, which shows the implications of the practical situation in contrast to an ideal set of data. However, it is evident that equally-good and clear damage detection results have been achieved using CS-reconstructed data.  Figure 15a shows that CS-reconstructed data can be relied-upon for equally accurate damage detection. Similar to the ARD comparison, the magnitude reduction of the Wiener filter MSE values of CS-reconstructed data compared to those of the original data can be a consequence of information loss during the wavelet de-nosing process. Figure 16 illustrates the damage localization results from the ARD method. The computed Fisher criterion is plotted against the sensor number for damage localization. Figure   16a, Figure 16b and Figure 16c show the localization results of One damage zone, Two damage zones and Three damage zones respectively as per the experiment records in Figure   1b. Figure   and mid-support of the structure; and finally three damage zones in right and left spans and mid-support of the structure respectively. Therefore, it is evident from Figure 17 that the method has successfully located damage in all three cases using the CS-reconstructed data.

Damage localization
Although the results of only three loading stages are presented here due to limited space, all 13 loading stages of the experiment showed accurate damage detection and localization results. The above results confirm that the ARD method and the Wiener filter method have successfully detected and located damages in the structure using the CS-reconstructed signal.
Therefore, it can be concluded that the data compression and the reconstruction using CS has not been carried out at the expense of the important information of the structural measurements. In other words, CS has successfully captured the important information of the original signal sufficient for damage detection and localization with the used damage detection methods.
Application of CS for an SHM system requires pre-knowledge of the sparsity of the signal.
The measurement matrix has to be determined according to the sparsity and built-in to the sensors prior to installation. Any SHM system has to undergo some pre-testing phases before it is fully operational. Thus, in this pre-testing phase, the structural responses can be acquired using the intended sensors and tested for their sparsity. The measurement matrix can be determined accordingly. Once the sensors are installed and the SHM system is online, the signal can be acquired in an already compressed form during sampling itself, which gives the main advantage of saving sensor energy and memory space using CS for compressed data acquisition. These data are first stored in the sensors as usual, and then transmitted to the central server for processing and decision making.
The heavy computations such as convex optimization required for the reconstruction process are carried out at the central server where the compressed data are received. Thereby, CS offers another advantage by not placing a computational burden on the sensors. Using CS in an SHM system will not only provide very good reconstruction of the original structural response, impressive CRs and the above mentioned advantages, but also facilitates the system with the means of collecting the full signal at the central server. This will enable more studies and monitoring work to be carried out at convenience, without compromising the important information contained in the structural responses.

Conclusions
CS is a recent breakthrough in communication technology that introduces the means of reproducing a signal with a significantly lower number of acquired samples than the Nyquist's rate by exploiting the sparsity of a signal. Sampling the signal in an already compressed form, CS saves sensor memory space and power spent on data storage and compression processes. In this paper, CS is explored for its applicability in effective and efficient SHM using experimental structural response data from an RC structure. Using CS with SHM measurements, successful reconstruction of the signal was achieved with reconstruction accuracies as good as 99% at considerable compression levels. Since CR is a direct indication of energy and memory saving in the system, these results suggest that CS could be used for energy efficient and memory efficient SHM. Further analysis were carried out on the reconstructed signal for accuracy in structural damage identification. These analyses resulted in successful damage detection and localization in the structure using two structural damage identifica-