Sparse representation approach to data compression for strain-

based traffic load monitoring: a comparative study

Helder SOUSA (1,2), Ying WANG (1)

(1) University of Surrey, Guildford, Surrey GU2 7XH United Kingdom

(2) BRISA S.A., 2785-599 S. Domingos de Rana, Lisbon, Portugal

Abstract

Among different Structural Health Monitoring (SHM) systems applied on bridges, Bridge Weight-

in-Motion (BWIM) is probably the one with widest applications worldwide. Briefly, BWIM uses on-

structure sensors that are able to acquire signals sensitive to traffic load events, which can be used as an

indirect indicator of the load magnitude. The sampling rate required for this is relatively high (at least

10 Hz), which usually lead to databases with sizes that might reach the order of gigabytes. It is

impractical to process this volume of information in the context of infrastructure asset management.

Hence, an effective and efficient method for the compression and storage of BWIM data is becoming

mandatory. In this paper, sparse representation algorithms have been innovatively applied to the

BWIM data compression. A comparative study is performed based on measurements collected from a

real bridge, by exploring different methods including Discrete Fourier Transform (DFT), Discrete

Cosine Transform (DCT), Discrete Wavelet Transform (DWT), and two dictionary learning methods,

i.e. Compressive Sensing (CS) and K-means Singular Value Decomposition (K-SVD). It has been

found that the K-SVD method shows the best performance when applied to this specific type of data,

while the DWT method using Haar wavelet is the most computationally efficient. Nearly lossless

reconstruction of the signal is achieved by using K-SVD with less than 0.1 % reserved coefficients,

which gives evidence that dictionary learning technologies are feasible to guarantee the same level of

information even with much smaller databases. Therefore, the utilization of dictionary learning is a

clear step forward towards higher levels of efficiency in the compression and storage of data collected

by SHM systems.

Keywords: Data compression, Sparse representation, Transform coding, Dictionary learning, K-

SVD, Monitoring data

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Introduction

The development of Structural Health Monitoring (SHM) systems applied on bridges has

spread worldwide. Although initially the attention was addressed with the focus on development

and improvement of sensors, nowadays it has been extended to the practical implications related

to the acquisition, collection and processing of data (Van Der Auweraer and Peeters 2003).

Current SHM systems allow for data acquisition with sampling rates up to 1 kHz.

Furthermore, in complement to the common use of these systems, i.e. for the assessment of the

structural integrity, there is a potential for marginal benefits, such as the characterization of the

effective crossing traffic on a bridge. If properly designed, a SHM system can act as a weighting

device – Bridge-Weigh-In-Motion (BWIM) – as long as linear elastic behaviour is assured and

consequently traffic parameters may be quantified (Karoumi, Wiberg et al. 2005). One of the

first works, related to the identification of traffic loads on bridges based on strain

measurements, was presented by Moses (Moses 1979). In addition, the utilization of strain

measurements were extended to derive other parameters, for example, stress influence line

(Chen, Cai et al. 2016). Indeed, several analyses gave evidence that a good correlation exists

between gross load and peak strains (Moses 1979, Lijencrantz, Karoumi et al. 2007,

Liljencrantz and Karoumi 2009). This fact may contribute to the improvement of the knowledge

of real traffic loads on bridges and consequently, funds might be saved by avoiding unnecessary

rehabilitation and replacement procedures (Getachew and Obrien 2007).

On the other hand, signal processing plays an important role in modern structural

engineering fields. For example, a modern SHM system yields massive data volume, which by

consequence makes the storage and processing of such volume of data difficult and time

consuming. Indeed, this is what happens when characterizing traffic events based on strain

measurements collected at a high sampling rate. Therefore, the compression of monitoring data

becomes a critical issue, which has attracted increasing research attention (Spanias, Jonsson et

al. 1991, Zhang and Li 2006, Bao, Beck et al. 2011, Yang and Nagarajaiah 2014). Moreover, an

effective data compression scheme can enable efficient transmission of the data in real time,

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which is a requirement of online SHM systems devoted to support the analysis and assessment

of the structure. For offline SHM systems, data can be stored with a much smaller size for

further interpretation. For wireless sensor network, data compression can contribute to less

communication bandwidth and thus more energy can be saved (Zhou and Yi 2013).

Regarding to the format of the monitoring data, the most common one is time series. That is,

at each time point, which is reciprocal of the sampling rate, a measurement is collected (e.g.

1,000 data points are generated per second for 1kHz sampling rate). However, these 1,000 data

may not contain, in many cases, any significant information. As demonstrated in (Wang and

Hao 2010), only 20 % information of vibration data or 5 % information of guided wave data can

be used to reconstruct the original signals without losing significant original information.

Hence, and under such conditions, the data might contain redundant information and

consequently, it can be represented with fewer data points. One of the most widely-used

methods to achieve this is transform coding. The fundamental concept is to transform the data

from a domain (e.g. time domain) into another domain, in which the data become sparser. In

other words, if a proper domain is used, where only a few values are non-zero and the remaining

ones are zero values, only the non-zero elements need to be saved, transmitted and processed.

One of the most widely used methods is the Discrete Fourier Transform (DFT) (Spanias,

Jonsson et al. 1991), but others exist with the aim to achieve more efficient data compression

such as Discrete Cosine Transform (DCT) (Spanias, Jonsson et al. 1991) and Wavelet

Transform (WT) (Zhang and Li 2006). It is worth to mention that most of these methods have

been applied in the compression of seismic data (Spanias et al. 1991).

Although the abovementioned fixed transforms can achieve very good results, the efficiency

in data compression remains an open question. As demonstrated by Rubinstein et al.

(Rubinstein, Bruckstein et al. 2010), the compression results using an adaptive dictionary can

usually outperform those from any fixed transform. In this kind of scheme, instead of

performing a transform on the data set, a dictionary learning algorithm is used to learn from the

data and to form a dictionary to represent the data, which is a popular sparse representation

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problem. Recently, both Compressive Sensing (CS) (Bao, Beck et al. 2011) and Independent

Component Analysis (ICA) (Yang and Nagarajaiah 2014) have been proposed for monitoring

data compression. In signal processing field, K-means Singular Value Decomposition (K-SVD)

algorithm, which is a generalization of the k-means clustering method and a dictionary learning

algorithm for creating a dictionary for sparse representations via a Singular Value

Decomposition (SVD) approach, has been regarded as one of the most successful sparse

representation algorithms (Aharon, Elad et al. 2006). Although the underlined mathematical

basis can be used for other types of data, the majority of the published work focusses on image

and video data compression and no application to BWIM data has been found by the authors.

This paper presents a novel application of this advanced technology to the compression of

data from SHM monitoring systems, more precisely to BWIM data. Because of the difference of

data formats, scales and characteristics, the adaption and further development of the existing

algorithms are necessary to be set. Further, this paper compares its compression performance

with those of several existing methods, mainly DFT, DCT, Haar WT, and CS. Firstly, the

relevant concepts related to data compression using different methods are introduced. A real

case study is used – Lezíria Bridge –, from which strain measurements are collected with a high

sampling rate in order to assess the performance of the abovementioned methods. Then, the data

is compressed and a statistical analysis is performed focussing on the discussion of the

reduction of the volume of data to be stored without losing significant information. Finally, the

main conclusions obtained from the comparison analysis are outlined to help the reader to easily

identify the most suitable method depending on the requirements of a specific problem in the

identification of traffic load events based on BWIM systems.

Theory and methodology review of sparse representation

The data compression efficiency is dependent on both the employed method and the data

specifications. The existing references on data compression methods are mainly for image/video

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applications, which needs to be re-evaluated for the aim of this paper – i.e. the compression of

strain monitoring data for the storage of traffic load events.

From transform to dictionary

Essentially, almost every signal might have a more concise representation, when expressed

in a convenient domain. In other words, they might be set as a linear combination of a few

atoms (i.e. the essential columns) from a dictionary. Mathematically speaking, a signal vector f

with length N can be transformed to a domain where the data is sparse according to Eq. (1),

where x is the transform coefficient vector with length N and W is the transform basis (N  N

matrix).

x=W ∙ f(1)

The typical transform bases includes DFT (W m ,n=1

√Ne

j 2πm nN , where m and n are the row

number and column number of the element, respectively, the same in the following formula),

DCT (W m ,n=α (m )cos (πm 2n+12 N

), where α (0 )= 1√N

,α (m )=√ 2N

, m=1 , 2,…, N−1), etc.

Then, the simple inverse transform is given by Eq. (2), where W−1 is the inverse transform, also

an N  N matrix. More generally, Eq. (2) can be rewritten according to Eq. (3), where D is the

dictionary matrix.

f =W−1 ∙ x (2)

f =D ∙ x (3)

The difference between Eq. (2) and Eq. (3) is that D might be a more complex operator (e.g.

matrices), rather than an inverse transform. In such case, the signal vector f is expressed as the

product of a well-designed or well-trained dictionary operator D and a coefficient vector x. As x

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contains more zero elements, the sparsity of x increases, which means that the efficiency of the

data compression becomes higher. In this context, it becomes clear that the objective of data

compression is to design the sparsest coefficient vector x associated with D.

Regarding Discrete Wavelet Transform (DWT), two coefficients are needed in above

formulae, e.g.f =φ ∙ xφ+ψ ∙ xψ, where xφ and xψ represent approximation coefficients and

detailed coefficients; whereas φ and ψ represent scaling function and wavelet function,

respectively. For the specific case of Haar WT which is used in this paper, the wavelet function

is ψ={ 1 0≤ t< 12

−1 12

≤ t<1

0 otherwise

and its scaling function is φ={ 10≤ t<10otherwise . Here, Eq. (1) to Eq. (3) are

still applicable when considering D=[φ ψ ] and x=[xφ xψ ]' . Regarding its implementation in

MATLAB, the only difference from DCT and DFT methods is that two sets of coefficients are

calculated and used by DWT, whereas the other two methods only require one set of

coefficients.

As reviewed in (Rubinstein, Bruckstein et al. 2010), the operator can be from a simple linear

operator to others more complex containing non-linear features, multi-resolution capabilities

and adaptivity. Usually, these methods can be classified according to the way that the dictionary

matrix, D, is set, mainly: (i) by using a fixed transform of the data (e.g. DCT, DFT, WT), which

results into the traditional transform coding method for data compression; and (ii) by using a

training set (e.g. K-SVD), which results into the well-known dictionary learning method.

Data compression based on transform coding method

Simply, the transform coding method is described in two main steps. Firstly, the original

signal is transformed into a sparse domain according to Eq. (1). Then, all the elements with

values lower than a predefined threshold are assigned as zero, which means that only some

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elements of the coefficient vector, x, are kept. Once this is concluded, the updated coefficient

vector, x̂, becomes sparser and therefore, the objective of data compression is achieved because

only non-zero elements need to be considered.

More specifically, the following procedure for data compression is used: (i) the original

signal is coded into another domain using a fixed transform (DFT, DCT, and Haar WT; (ii) the

elements of the coefficient vector x, of the original signal in the transformed domain, are

ranked; (iii) a percentage of the maximum coefficients (i.e. from 5 % to 50 %, based on a

specific threshold) are kept, whereas the remaining ones are set to zero in order to build a new

compressed coefficient vector x̂; (iv) the inverse transform, W−1, is calculated based on the new

coefficient vector, x̂, and thus, the signal after data compression f̂ , is finally obtained; (v) the

compressed signal f̂ is compared with the original signal f, by using a reconstruction accuracy

function. Figure 1 schematically shows the main steps that are required to obtain the

compressed signal, based on the fixed transform of the data, which was implemented in

MATLAB by using DFT, DCT and Haar WT, respectively.

Figure 1 –Signal compression based on transform coding.

Along this procedure, the threshold value is the critical aspect for a good performance in data

compression. For instance, if the threshold value increases, the updated coefficient vector, x̂,

becomes sparser and consequently, the reconstruction accuracy might become poorer. Thus,

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parametric analysis is usually performed to determine the threshold value on a rational basis.

For further details, additional information can be found elsewhere (Sayood 2012).

It is worth to note that the formulation of the reconstruction accuracy is usually dependent on

the parameter that is being monitored. In the case of BWIM data, the peaks in the collected

measurements, usually strain data, is the relevant information. Hence, measurements, at each

point in time i, fi (f = [f1…fi… fN]), that are lower than a specific amplitude, S0, they are labelled

as noise. The function that sets the reconstruction accuracy is defined according to Eq. (4),

where N1 is the number of data points, that satisfies f i≥ S0; where S0 is the minimum amplitude

considered for the identification of strain peaks. For the case of BWIM based on strain

measurements, S0 is set equal to 2 , by considering the sensor precision (1 ).

Acc=1− 1N 1

∑i=1

N1 |( f̂ i−f i

f i )|, for f i≥ S0 (4)

Data compression based on dictionary learning method

Figure 2 schematically shows the main steps that are required to obtain the compressed

signal by using the dictionary learning method. Compared with the transform coding method,

the main difference lies in the fact that the dictionary learning method requires a training

process, against the transforming process in the former. Accordingly, the symbols in Eq. (3)

need to be updated, i.e. F is a collection of signals (m  N matrix, where usually m < N), D is a

dictionary matrix (m  K matrix) with K prototype signal atoms as columns. The original

signals are used to train a sparse coefficient matrix, X̂ , and the associated dictionary matrix, D.

Then, the reconstructed signals, F̂, are obtained by Eq. (5), which, mathematically, can be

written as presented in Eq. (6), where ‖∙‖p is the Lp-norm, defined as

‖x‖p=(|x1|p+|x2|

p+⋯+|xn|

p)1p for p = 1, 2, …, ∞.

F̂ ≈ D ∙ X̂ (5)

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‖F̂−D ∙ X̂‖p ≤ ε (6)

Figure 2 – Methodology to obtain the compressed signal based on dictionary learning

An infinite number of solutions can be obtained, when m < K and D is a full-rank matrix.

Therefore, constraints on the solution must be set. The most strategic one is by imposing that

the solution with the fewest number of nonzero coefficients is the most suitable. Consequently,

the data compression problem can be converted to an optimization problem, as formulated by

Eq. (7).

minx

‖X̂‖0 subject ¿‖F̂−D X̂‖2 ≤ ε (7)

where, ‖x‖0 is the L0 norm, which is defined as the number of non-zero entries of the vector x

(Candes, Romberg et al. 2006). Due to the difficulty to solve a L0 norm optimization problem, a

great amount of research attention has been attracted and many methods have been developed.

For example, CS is proposed as a breakthrough theory in signal processing field (Candes,

Romberg et al. 2006). It converts the L0 norm problem shown in Eq. (7) into a L1 norm problem

which allows to solve the optimization problem through the usage of a measurement matrix and

a sparsity basis. It has been proved that a random measurement matrix can achieve very good

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results in different fields. A popular CS package L1-magic (Candes and Romberg 2006),

together with DFT basis and DCT basis, is used to perform the data compression.

Alternatively, K-SVD converts the above problem formulated by Eq. (7) into a problem

expressed by Eq. (8), where T0 is a preset integer indicating the maximum number of non-zero

elements in each coefficient vector. Due to the evidence of good performance of K-SVD

(Aharon, Elad & Bruckstein 2006), it has been also considered in this study. The operator ‖∙‖F2

is the Frobenius norm (according to Eq. (9)), where A* is the conjugate transpose of A.

Inherently, the lower T 0, the higher the level of data compression.

minD, x

‖f̂ −Dx‖F2

subject ¿ ∀ i ,‖x i‖0≤ T 0 (8)

‖A‖F2 =√trace(A ¿ A ) (9)

To solve Eq. (8), an iterative procedure is used. Firstly, an initial dictionary matrix, D, is set,

where the columns are the measured signals with the same lengths (i.e. the same duration).

Secondly, by fixing the dictionary matrix D, the coefficient vector x is updated by using any

pursuit algorithm. In this paper, an orthogonal matching pursuit (OMP) algorithm is used.

Thirdly, the dictionary matrix D is optimized, column by column, through SVD decomposition.

More detailed procedure can be found elsewhere (Aharon, Elad et al. (2006). The prototype

algorithm developed by Elad (Elad 2005) is used. Adaptations are made to suit the requirements

for the compression of BWIM data.

Lezíria Bridge – Case study

Description

Lezíria Bridge is part of the A10 motorway in Portugal. The main structure of this bridge has

a total length of 970 m, with eight spans of 95 + 127 + 133 + 4×130 + 95 m length, respectively,

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and seven piers supported by pile caps over the riverbed. The bridge deck is a box girder with

variable inertia - approximately 30.00 m wide with a height ranging from 4.00 m to 8.00 m. The

concrete piers are supported by pile caps and are set by four walls of constant thickness and

variable width. Figure 3-a) shows the elevation of the Lezíria Bridge for the first three spans,

whereas Figure 3-b) shows the typical mid-span cross section (Coba-PC&A-Civilser-Arcadis

2006).

Figure 3 - Lezíria Bridge: a) bridge elevation and location of the instrumented section, b)

location of the strain gauge in the mid-span section P1P2.

Strain gauge measurements

The Lezíria Bridge has an integrated monitoring system devoted to its management and

surveillance. Several cross sections are instrumented with embedded and external sensors. The

capability of the implemented monitoring system to acquire strain measurements at a high rate

level enables the detection of traffic events. The sensors selected for this analysis are a part of

the 30 fibre optic strain gauges specifically developed for this bridge (Rodrigues, Sousa et al.

2007, Sousa, Félix et al. 2011). They were installed in 15 cross sections along the bridge deck,

seven of which are positioned near the support piers, and the remaining eight located at the

middle of the spans. In each instrumented section, two fibre optic strain gauges were installed,

each one in the bottom and top slabs of the box girder, both aligned with the vertical symmetry

axis of the section and the longitudinal axis of the bridge. The location of the strain gauge for

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this work is illustrated in Figure 3, i.e. the strain gauge SG-3B that is in the bottom slab of the

mid span cross-section P1P2.

In order to compare the performances of different data compression methods, historical

strain measurement data during the 24-hour of May 14th, 2009 were used. Although a sampling

rate of 50 Hz is originally used, a rate of 12.5 Hz was adopted for this case (i.e. BWIM data).

Among the 33 sets of data collected by the sensor, Figure 4-a shows one of them with 45,000

data points that corresponding to one hour of observation. In complement, a shorter time

window is showed in Figure 4-b (in correspondence to the gray column in Figure 4-a), in which

it is clear to observe that traffic load events are well captured with a sampling rate of 12.5 Hz.

a) overview of a data set b) detailed view on one peak

Figure 4 – A data set of the time-series of the strain gauges SG-3I (section P1P2).

Results

Fixed transform coding

For this case, different percentage values for the maximum transform coefficients, x, are

explored. Percentages starting from 50% (i.e. 22,500 coefficients) to 5% (2,250 coefficients)

with a 5% decrement is considered. Based on this, the reconstruction accuracy can be derived

for each signal set based on Eq. 4. Moreover, statistical information related to the accuracy of

the results for all the 33 signal sets is calculated, mainly the mean value and standard deviation

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with respect to different percentages of reserved coefficients. Figure 5 to Figure 7 show the

obtained results for the DFT, DCT and Harr WT with 20% coefficients reserved, respectively.

As it can be clearly seen, the performance of Haar WT is the best. If the reconstruction accuracy

is set as the objective for the data compression, Haar WT delivers the most compressed results

(the minimum number of reserved coefficients). For example, if 95% reconstruction accuracy is

set, 20% of reserved coefficients will be needed for Haar WT. For the same criterion, DFT

requires 40% to 45% coefficients, whereas DCT needs more than 50%. In addition, the lowest

standard deviation is achieved by using the Haar WT, whilst the highest value is obtained with

the DFT (Figure 8). Hence, this shows the better performance of the Haar WT for data

compression using transform coding methods. Figure 5-b) to Figure 7-b) show a detailed view

on one of the peak strains (indicated in Figure 5-a) to Figure 7-a) with the label “time window”)

where it is possible to compare the original signal and the respective reconstructed data.

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a) overview of a data set b) detailed view on one peak

Figure 6 – Original signal vs. reconstructed signal using DCT.

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a) mean value b) Standard deviationFigure 8 – Statistics related to the accuracy of data compression.

Dictionary learning

Two dictionary learning methods are considered in this study, i.e. CS and K-SVD. The

results from the former are not satisfactory, where the reconstruction accuracy is only around

90% when using 50% of reserved data. Moreover, the computation speed when using this

algorithm is low. For example, in a standard desktop (e.g. 8Gb RAM memory and an i5-4590

3.3G processor), the calculation procedure stops with error messages when the sample size of

the data set is greater than 10,000. The possible reason is the randomness of the measurement

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matrix, which decrease the efficiency of the algorithm. This is in line with previous studies

(Bao, Beck et al. 2011). Even so, it should be clear that this does not mean necessarily that the

method fails but indeed, the code implementation needs further improvements towards higher

levels of efficiency.

Regarding to the K-SVD method, only 5% reserved coefficients can achieve nearly lossless

reconstruction results (with less than 1 ×10−8 total error), based on the initial trial calculation.

This is mainly due to the adaptivity of the dictionaries used in this method, which increase the

efficiency of the data compression. In this context, trial tests were conducted by considering

only 15, 20, 23, 25, 28, 30, 33 reserved coefficients (which are all lower than 0.1% of the total

data points, i.e. 45) to further investigate the efficiency in data compression by using the K-

SVD method. It has been found that the almost lossless reconstruction results can be achieved,

if the number of coefficients is equal to, or greater than, 33 (i.e. 0.073% of total data).

According to the results presented in Figure 9, the reconstruction results are comparable with

those using 50% maximum WT coefficients (22,500) with only 30 non-zero coefficients.

Moreover, with only 20 non-zero coefficients, the reconstruction results are comparable to those

by using: (i) 5% to 10% maximum WT coefficients, (ii) 25% to 30% maximum DCT

coefficients and (iii) 35% to 40% maximum DFT coefficients. Figure 9-b) shows a detailed

view on one of the peak strains (indicated in Figure 9-a) with the label “time window”) where it

is possible to compare the original signal and the reconstructed signal, with only 30 reserved

coefficients. When comparing the results showed in Figure 9-b) with the respective ones in

Figure 5-b) to Figure 7-b), it is possible to conclude that the signal reconstructed, ~f , with the K-

SVD method is practically identical to the original signal, f. Regarding the standard deviation

values when using the K-SVD method, a less variance is observed (Figure 10). Moreover, a

higher variation is observed (for different values of the threshold) in the evolution both the

average accuracy and the standard deviation, when compared with the results presented for the

DFT, DCT and Harr WT. This means that preliminary analyses are needed to find the optimal

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number of reserved coefficients, before the application of the K-SVD method and other

dictionary learning algorithm to real cases.

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a) overview of a data set b) detailed view on one peak

Figure 9 – Original signal vs. reconstructed signal using the K-SVD method.

a) Average accuracy b) Standard deviation

Figure 10 – Statistics related to the accuracy of data compression using the K-SVD method.

Comparative discussion

In order to obtain a better comparison regarding the data compression efficiency, the

quantitative and qualitative evaluations are summarised in Table 1 and Table 2, respectively.

In Table 1, the reconstruction accuracies are calculated based on mean values

μ=1n∑i=1

n

Acc (i) and standard deviations σ=√∑i=1

n

¿¿¿¿ of the total 33 reconstructed signals

(n=33) by using different methods. The compression ratio is defined as the percentage of the

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reserved coefficients used in the different methods that can achieve a certain reconstruction

accuracy. The computation duration is calculated automatically by MATLAB. In Table 2, the

ease of use is also considered, which is relevant to the efforts for adaptation and further

development, given the existing programs.

Table 1 – Quantitative evaluation of data compression methods

Category MethodAccuracy* Compression

ratio**Calculation duration#

(s)

Transform coding

DFT 75% 16% N/A 2.84

DCT 79% 14% 50% 1.17

Haar WT

93% 8% 20% 0.94

Dictionary learning

CS N/A N/A N/A N/A

K-SVD 100%

0% 0.05% 215.53

* 10% of coefficients is used as the threshold.** The percentage of reserved coefficients when 95% data reconstruction accuracy is achieved. N/A means that the reconstruction accuracy is lower than 95% with a maximum of 50% reserved coefficients.# The calculation duration is determined by the time difference before and after running the programs, which is obtained from the “cputime” in MATLAB.

Table 2 – Qualitative evaluation of data compression methods

Category Method Performance

Computation speed Ease of use* Recommendation

Transform coding

DFT OK Fast Easy No

DCT Good Fast Easy For checking

Haar WT Very good Fast Slightly

difficultFor quick

compression

Dictionary learning

CS Poor Slow Difficult No

K-SVD Excellent Slow Slightly difficult

For high performance

* This is evaluated by the work on the implementation of MATLAB programming. Based on these results, the K-SVD clearly shows the best performance, achieving an

accuracy of 95 % with only 0.05 % reserved coefficients. However, the code runs significantly

slower, when compared to the fixed transform codes. On the other hand, the fastest code is the

one using DWT, which can reconstruct all the 33 sets of signals within 1 sec. Therefore, the

Haar WT method is most suitable in situations that quick data compression is a mandatory

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requirement (e.g. for potential online BWIM applications). Furthermore, due to the easiest code

implementation of the DCT transform, it can be used in scenarios where data compression

performance is less important (e.g. a first identification of the most severe peak strains). The

second limitation of the K-SVD method is related to its performance, which is to some extent

dependent on the quality and quantity of original data. This explains why a large amount of data

is normally needed for training the algorithm. Besides, the dictionary based on the K-SVD

method is adaptive, resulting in two-folded outcomes. On one hand it is more efficient, whereas

on the other hand it may show less stability.

These results strengthen the criticality of the storage and processing of massive monitoring

data. Therefore, data compression technologies are indeed a cutting-edge subject and should be

further investigated and continuously improved in terms of efficiency. It should be noted that

the data compression techniques here explored are not constrained to the nature of the

measurement (in this case to strains). Therefore, the data compression techniques can also be

applied to other measurements, including piezoelectric transducers (Song, Olmi et al. 2007),

optic fibre sensors (Li, Li et al. 2004), and GPS-based sensors (Yi, Li et al. 2010). This reveals

an enormous potential of the compressive techniques in the field of SHM.

Future efforts should be placed, mainly in the following two directions: (i) to design a more

efficient analytic dictionary, because the current trend in signal processing field is to integrate

transform coding methods and dictionary learning methods (Rubinstein, Bruckstein et al. 2010),

and (ii) to interpret monitoring data using the features that can be derived from learned

dictionary, which avoids transforming the dictionary coefficients back to the original signal,

which is similar to WT-based method (Li and Hao 2014). Some initial attempts have been

conducted in this direction (Wang and Hao 2013, Wang and Hao 2013, Wang, Zhang et al.

2014).

Conclusions

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Based on strain measurements collected on a real bridge, this work focuses on the suitability

of technologies for BWIM data compression, mainly DFT, DCT, Haar WT, CS and K-SVD.

The results herein presented allow drawing some relevant conclusions:

1. The original BWIM monitoring data could be transformed into a sparser domain, based on a

well-trained dictionary. Moreover, the data for storage was significantly reduced without

losing significant information in the peak strains. The novel application of dictionary

learning methods to strain measurements collected with sampling rate of 12.5 Hz, has been

successful and therefore, a new domain of application has been found with huge potential in

promoting SHM to support and complement infrastructure asset management.

2. The method performance needs to be assessed in a multi-dimensional basis, i.e. by

considering not only the degree of compression but also how easy is to use the method and

the required computational time.

3. The K-SVD technology shows to be the most advantageous in terms of performance due to

the ability of this method to adapt to the data subjected to compression. Specifically, it can

achieve nearly lossless data compression with only 0.073% reserved coefficients.

4. Although with a lower performance in the data compression level, the advantages of fixed

transform coding methods are quick computation speed and easy implementation.

5. The adoption of a specific method for BWIM data compression depends on the

requirements that should be defined in advance. For high accuracy, the K-SVD method is

the most suitable, whereas for a faster outcome, the WT method is recommended.

Acknowledgements

The first author acknowledges the support from the European Commission through

the Marie Curie Grant Fellowship grant 660275 - LostPreCon under the program

H2020-MSCA-IF-2014. In addition, the COST Action TU1402 on Quantifying the

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Value of Structural Health Monitoring is gratefully acknowledged for networking

support.

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