Compressive sensing for high-speed rail condition monitoring using
redundant dictionary and joint reconstruction
Si-Xin Chen* and Yi-Qing Ni
Hong Kong Branch of National Rail Transit Electrification and Automation
Engineering Technology Research Center, The Hong Kong Polytechnic University
Department of Civil and Environmental Engineering, The Hong Kong Polytechnic
University
Hong Kong, 999077, China
+852 67364534
Abstract
In high-speed rail (HSR) condition monitoring, the conflict between the resolution of
defect detection and the amount of recorded data is usually an issue due to the Nyquist
theorem. As an emerging technique, compressive sensing (CS) creates the opportunity
of sub-Nyquist sampling when target signals have a sparse representation in a known
domain. However, many studies have shown that the lack of sparsity limits the
applicability of CS. In addition, when multiple compressed measurement vectors are
available, conventional CS algorithms recover target signals one at a time independently
without exploiting the correlation among their sparse representations. This study applies
CS to HSR condition monitoring and employs two methods to improve the recovery
accuracy. Specifically, the process of CS is simulated using the axle box acceleration
data acquired from a high-speed train ran on one section of railway in China. After the
investigation of recovery results, the same experiments are conducted, except that the
discrete cosine transform (DCT) matrix is replaced by a redundant dictionary. Another
series of experiments assume that the signals have a joint sparsity in the DCT domain
and reconstruct them simultaneously. The results show that the HSR condition
monitoring data can be obtained through sub-Nyquist sampling and reconstructed with
small errors when they are sufficiently sparse. Even if the compressed measurements
are the same, both methods are proved effective to improve the recovery performance,
in which joint reconstruction has better performance than the other.
1. Introduction
As safety and reliability are primary concerns of high-speed rail (HSR) system, the
knowledge of track conditions is critical for making a suitable maintenance plan and
performing grinding operations where and when required (1, 2)
. Measurement systems
implemented on special inspection vehicles can give a description of the track status for
a long distance but only work when traffic is stopped (3)
. Another more foolproof way is
to use the acceleration signals measured on the axle boxes of a standard operating train
by the condition monitoring system implemented on it (4, 5)
. Axle box accelerations
permit detecting and identifying some singular track defects such as squats (6, 7)
, bolt
tightness of fish-plated joints (8)
and rail corrugations (9, 10)
. Theoretically, a highest
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possible sampling frequency is desired because those track defects whose excitation
frequency is higher than half of the sampling frequency cannot be detected. In practice,
however, the sampling frequency is usually lowered down to several kHz (11)
for speeds
up to 300 km/h to avoid a vast amount of data acquired.
As a new data acquisition paradigm, compressive sensing (CS) creates the opportunity
of sub-Nyquist sampling (12, 13)
. This theory confirms that one signal can be recovered
from a small number of random measurements as long as it has a sparse or compressible
representation in a known domain, and the number of measurements required is
governed by the sparsity level rather than the bandwidth of the signal. A number of
researchers have studied CS for structural health monitoring (SHM) in order to address
data loss issue (14, 15)
or reduce power assumption (16, 17)
in wireless sensor networks.
Even though CS provide potential solutions for the conflict between data volume and
detection resolution mentioned above, limited studies have applied this theory to HSR
condition monitoring. One of the purposes of this study is to bridge this gap.
The theory of CS has been developed only for signals that have a sparse representation
in an orthogonal basis, which is a rather stringent restriction because the lack of sparsity
level usually hinders the applicability of CS (17, 18)
. Indeed, allowing the signal to be
sparse with respect to a redundant dictionary adds a lot of flexibility and still guarantees
the probability of successful recovery (19)
. Apart from that, when multiple compressed
measurement vectors are available, conventional CS reconstruct target signals one at a
time independently without exploiting the correlation of their sparse representations.
Indeed, this single measurement vector (SMV) model can be extended to the multiple
measurement vector (MMV) model, where a key assumption is that the support (i.e.,
indexes of nonzero entries) of every sparse signal is identical (20)
. It has been shown that
compared to the SMV case, the successful recovery rate can be greatly improved using
MMV (20–22)
. This study aims at overcoming the two limitations mentioned above and
improving the recovery accuracy by replacing the orthogonal basis with a redundant
dictionary and utilising joint sparsity to reconstruct signals simultaneously.
2. Principles of compressive sensing
This chapter would briefly introduce the principles of conventional CS. Basically, CS
obtain a small number of compressed measurements by projecting the discrete-time
signal onto a specific random matrix and guarantee successful reconstruction if the
original signal is sparse in a known domain.
The measurement vector My R∈ is acquired by a linear projection of the discrete-time
signal ( )Nf R N M∈ > :
y f=Φ (1)
where Φ is an M by N measurement matrix that represents the measuring method.
When f is represented in terms of an N by N orthogonal basis matrix Ψ with the
basis vectors k
ψ as columns, the model becomes
y = ΦΨx = !Φx (2)
where !Φ = ΦΨ is an M by N sensing matrix, and x stands for the coefficient vector
in the orthogonal basis Ψ .
3
Typical orthogonal bases include the wavelet basis (23)
, the discrete Fourier transform
(DFT) matrix, the discrete cosine transform (DCT) matrix (24)
, the curvelet basis (25)
and
so on.
Although it is underdetermined, the problem of recovering N-dimension x from M-
dimension y can be solved under the following conditions (26, 27)
:
• The representation x is sufficiently sparse, where a vector is defined as S -sparse
if it has at most S non-zero entries;
• The matrix !Φ obeys the restricted isometry property (RIP) to bound the singular
values of its submatrices (28, 29)
.
Although Ψ is fixed, it is known that a random matrix Φ is largely incoherent with
any basis matrix and will enable !Φ to satisfy RIP with overwhelming probability (12)
.
Examples include Gaussian random matrices (where entries of Φ are independently
sampled from a normal distribution with mean 0 and variance 1/M) and binary random
matrices. !Φ can also be constructed by selecting M rows from an N by N orthogonal
basis matrix uniformly at random, where Φ randomly sub-samples the target signal,
and Ψ maps the time domain and the selected domain (12)
.
In the case where x is sparse, it is desirable to find the sparsest solution of !Φx = y .
This 0l -problem is hard combinatorial and generally computationally intractable.
Under some favourable conditions (30)
, the solution of the 0l -problem can be obtained
by solving an 1l -problem, also named Basis Pursuit (BP)
(31):
1ˆ argminx x= subject to !Φx = y (3)
In most practical applications where x is not sparse but only compressible, common
relaxations to (3) include the constrained basis pursuit denoising (BPDN):
1ˆ ˆargminx x= subject to !Φx̂-y
2
2
≤ ε (4)
where ε is the bound of the noise.
Finally, the original signal can be recovered using the optimal coefficient vector x̂ :
ˆ ˆf x= Ψ .
The BP and the BPDN can be recast as a linear program and a quadratic program
respectively and solved by standard methods such as interior-point algorithm (32)
even if
they are not quite efficient. There exist also algorithms specially designed to handle the
1l -problem in CS, such as Bregman iterative algorithm
(33), root-finding approach
(34),
Nesterov’s algorithm (23) and alternating direction algorithm (35)
.
Apart from the 1l -regularization, there are at least two common classes of computational
techniques for solving sparse solutions for linear inverse problems: greedy pursuit (36, 37)
and Bayesian framework (38, 39)
.
3. Compressive sensing for high-speed rail condition monitoring
3.1 Data acquisition
4
CNERC-Rail (Hong Kong branch) was authorised to monitor the bogies of an operating
train ran on Lanzhou−Xinjiang HSR line. The work lasted for about one month. Figure
1 shows the accelerometers installed at the frames, vertical-stop components and axle
boxes. These sensors, with ranges of ± 1000 g and sampling rates of 5000 Hz, formed
an important part of the system to monitor the vibration responses of the bogies.
Figure 1 Location of accelerometers
In this study, the vertical accelerations of an axle box were used for the simulation of
CS. Totally 177 segments of acceleration were recorded at about 16:30, 18:02 and 19:45
(59 for each time) when the train achieved the cruising speed (approximately 200km/h)
so that the influence of train speed was normalised. Each segment lasts for 1 second
with 5000 components. It should be noted the purpose of using signals from different
times is to increase the representativeness rather than to discuss the influence of time on
the simulation of CS.
3.2 Procedures and implementations
The procedures of simulating CS are summarised as follows. Firstly, the compressed
measurements y were obtained by projecting the target signal f onto the measurement
matrix Φ . The sensing matrix !Φ was obtained after determining the orthogonal basis
Ψ based on the characteristics of target signal. With the compressed measurements and
the sensing matrix, the original signal was reconstructed in both the sparse domain and
time domain. Finally, the reconstructed signal f̂ was compared with that obtained in
conventional way for verification. The 177 segments of acceleration were analysed.
5
In this study, the measuring method was random sub-sampling, and the compression
ratio /M N was set as 40%, 50%, 60% for different trials. The measurement matrix Φ
composes of sequential but randomly chosen 2000/2500/3000 rows from a 5000 by
5000 identity matrix, corresponding to random samples from the original signal.
The -DCT Ⅱ matrix was selected as the orthogonal basis ( 5000N = ). The function for
the thk basis vector is
10
[ ]= , 0,1,..., 11 2
cos( ( )) 1,2,..., 12
k
kN
n n N
kn k N
N N
ψπ
⎧=⎪
⎪= −⎨
⎪+ × = −⎪⎩
(5)
The signal of interest can be represented by a linear combination of the columns in the
5000 by 5000 DCT matrix Ψ .
The MATLAB package called YALL1 (40)
was used in this study, which solves the 1l
regularization problem based on the alternating direction method (35)
.
3.3 Results
The reconstructed signal f̂ is compared with the original uniformly sampled signal f .
The metric of residual sum-of-squares (RSS) normalised by f is used to assess the
reconstruction error: 2
2
2
2
ˆf fRSS
f
−
= (6)
Figure 2 shows the reconstruction errors. Due to the varying reconstruction errors, RSS
is presented using box and whisker plots. The bottom and top of the box represent the
first and third quartiles, and the interior line represents the median value. The extreme
outliers are ignored. It can be observed that the reconstruction errors decrease when the
number of measurements M increases. When /M N is 40%, 50% and 60%, the
median reconstruction error is 42.9%, 35.9% and 29.5% respectively.
Figure 3 shows the relationship between the reconstruction error and sparsity level of
target signal when the compression ratio /M N is 0.5. The sparsity level is calculated
as the ratio between the number of zeros and the segment length (5000). Indeed, none of
the coefficients is originally zero due to the contamination of noise, so the coefficients
that have a value smaller than 1% of the maximum are set to zero. The results show that
the sparser the signal is in the orthogonal basis, the better it can be reconstructed, which
correspond to previous studies (17, 18)
.
Figure 4 illustrates the random samples and the reconstructed signal within 0.1 second
out of the 1-second segment with reconstruction error of 0.222. It can be observed that
the reconstructed signal is consistent with the original one even if some samples are not
recorded at all. Figure 5 shows the zoom view of another segment with an error of
0.309. The observation that the signal is less regular corresponds to the fact that it is less
sparse in the DCT domain, and that is why it has larger reconstruction error. Generally,
signals from the accelerometers can be obtained through sub-Nyquist sampling rates
with small reconstruction errors when their sparsity levels are larger than 0.85.
6
Figure 2 (Left) Reconstruction error versus compression ratio
Figure 3 (Right) Reconstruction error versus sparsity level
Figure 4 Random samples and reconstructed signal versus target signal
(Time: 16:30:02; M/N = 0.5; RSS = 0.222)
Figure 5 Random samples and reconstructed signal versus target signal
(Time: 19:44:29; M/N = 0.5; RSS = 0.309)
7
4. Improving reconstruction accuracy
4.1 Redundant dictionary
The signal Nf R∈ can also be expressed as f Dx= , where N KD R
×∈ is a redundant
dictionary, which contains a complete set of basis vectors plus additional vectors not in
the orthogonal basis and leads to non-unique representations of a given signal. In this
case, as the sensing matrix DΦ may no longer satisfy the requirements imposed by
traditional CS assumptions, Candes et al. introduce a condition on the sensing matrix
which naturally generalises the concept of RIP (19)
. This study guarantees accurate
recovery via an 1l -synthesis optimisation when signals are represented by truly
redundant dictionaries.
In this study, an N by 2N redundant dictionary was employed:
[ ]D = Ψ ΨⅡ Ⅳ
(7)
where ΨⅡ
is the -DCT Ⅱ matrix previously employed, and ΨⅣ
is a variation of the
-DCT Ⅱ matrix named -DCT Ⅳ with basis vectors:
1( + )
22[ ]=cos( ) 0,1,..., 1, 0,1,..., 1k
k
n n k N n NN N
πψ × = − = −, (8)
This dictionary contains vectors with finer frequencies than standard DCT matrix and
thus widens the selection range for sparse representation.
On average, 1108 atoms of the dictionary rather than 1166 vectors of the orthogonal
basis can represent the target signals. It is expected that this 5% decrease leads to better
reconstruction.
Based on the same measurement vectors, which are independent of Ψ or D , the 177
experiments were conducted again using the redundant dictionary.
Figure 6 illustrates the slight decrease of average reconstruction errors. For example,
when /M N is 60%, the average error decreases from 30.4% to 28.8%.
Figure 6 Reconstruction errors with and without using redundant dictionary
8
4.2 Joint reconstruction
Conventional CS finds the sparsest solution of !Φx = y . Even when multiple
compressed measurement vectors 1 2, ,..., ly y y are available, the coefficient vectors
1 2, ,...,
lx x x are still solved one at a time independently. When coefficient vectors are
transformed from correlated discrete-time signals, they may share a similar sparse
structure like common non-zero support. This profile can be leveraged to improve the
reconstruction accuracy. A favourable approach is to encode the joint sparsity by the
2,1l -regularization
(41):
X̂ = argmin X2,1= argmin x
i
2i=1
N
∑ subject to !ΦX = Y (9)
where 1 2[ , ,..., ]
N L
lX x x x R
×= ∈ denotes a collection of L coefficient vectors;
ix and
jx denote the th
i row and thj column of X respectively;
M LY R
×∈ denotes a collection of L compressed measurement vectors.
Several efficient first-order algorithms have been proposed for this 2,1l -regularization
problem, such as accelerated gradient method (42)
, SpaRSA approach (43)
and block-
coordinate descent algorithm (44)
. This study used the YALL1-group algorithm (45)
based
on a variable splitting strategy and the classic alternating direction method (ADM).
Three 1-second segments of acceleration within sequent 3 seconds were assumed to be
temporally relevant, specifically, share a common nonzero support in the DCT domain.
Under this assumption, Y composed of 1y ,
0y and
2y , measurement vectors of the
target signal, its previous one, and its posterior one was used to jointly reconstruct
0 1 2[ , , ]X x x x= . After that,
1x was used to reconstruct
1f for the comparison with
original one. It should be noted that 0x and
2x were by-products and out of the
comparison. Another 177 experiments were conducted.
Figure 7 shows one example of reconstructed 0 1 2
[ , , ]X x x x= . It can be seen that main
components of the coefficient vectors have clustered indexes although their amplitudes
are different. The 1262nd
, 1266th
and 1267th
components contribute most to the first,
second and third segments with values of -192.1, -176.5 and -142.8 respectively.
9
Figure 7 One example of jointly reconstructed X
Figure 8 shows the significant improvement of accuracy by joint reconstruction. When
/M N is 50%, the average reconstruction error decreases from 36.6% to 32.5%. There
is also a reduction of 4.5% and 3.7% when /M N is 40% and 60% respectively.
Figure 8 Reconstruction errors with and without utilising joint sparsity
5. Conclusions
This study bridges the gap between CS and HSR condition monitoring. The process of
CS is simulated using the axle box acceleration data recorded by the monitoring system
installed on a high-speed train. After the investigation of recovery results, two methods
are induced to improve the reconstruction accuracy. The first is to replace the DCT
matrix with a redundant dictionary aiming to widen the selection range for sparse
representation. The second is to utilise the joint sparsity of sequent signals in the DCT
domain to reconstruct signals simultaneously. The following conclusions can be drawn:
• The reconstruction error decreases with the increase of or sparsity level or the
number of measurements. Generally, the axle box acceleration data can be
obtained through sub-Nyquist sampling rates with small reconstruction errors
10
when the sparsity levels are larger than 0.85. This provides the opportunity of
achieving higher resolution of defect detection with the same sampling rate.
• Employing the redundant dictionary reduces the number of vectors to represent
the signal and thus leads to a 2% decrease of average error.
• Sequent signals share a similar structure when transformed in the DCT domain.
Utilising this profile can reduce the average reconstruction error by around 4%.
As there already exists sensors that can directly acquire compressed measurements from
analogue signals, CS is expected to be implemented via hardware rather than simulation.
Apart from that, other kinds of redundant dictionaries should be investigated as they
may better represent the signals. The spatial correlations between the acceleration data
from two axle boxes of the same bogie should also be exploited to further improve the
reconstruction accuracies.
Acknowledgements
The work described in this paper was (in part) supported by a grant from the Research
Grants Council of the Hong Kong Special Administrative Region, China (Grant No.
PolyU 152767/16E). The authors would also like to appreciate the funding support by
the Innovation and Technology Commission of Hong Kong SAR Government to the
Hong Kong Branch of National Transit Electrification and Automation Engineering
Technology Research Center (Project No.: K-BBY1).
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