Compressive sensing for vibration signals in high-speed rail
monitoring
Yi-Qing Ni* and Si-Xin Chen
Hong Kong Branch of National Rail Transit Electrification and Automation
Engineering Technology Research Center and Department of Civil and Environmental
Engineering, The Hong Kong Polytechnic University
Hong Kong
+852 27666004
Abstract
The safety and reliability are critical for the high-speed rail system, and axle box
accelerations are often utilised for inspections of railways. Due to the Nyquist theorem,
there is a compromise between the resolution of defect detection and the amount of
recorded data. As an emerging technique, compressive sensing creates the opportunity
for sub-Nyquist sampling as long as the target signal has a sparse representation in a
known domain. To make use of this advantage, this study proposes a compressive
sensing framework for high-speed rail monitoring. In particular, the process of
compressive sensing is simulated using the axle box acceleration data acquired from a
high-speed train ran on one section of railway in China. The compressed measurements
are received by random projection, and the original signals are reconstructed using
convex optimisation algorithm. Based on the reconstruction results, the influence of
different measuring methods as well as orthogonal bases is evaluated. In addition, a
regression model is formulated to give a recommend equivalent sampling rate according
to the sparsity and the desired accuracy requirement of the target signal. It is found that
the vibration signals are sparser in the discrete cosine transform matrix, leading to better
reconstruction, and the performance of different measuring methods is almost identical.
More importantly, this study proves that the high-speed rail monitoring data can be
satisfactorily obtained through proper sampling rates lower than the Nyquist theorem
requires.
1. Introduction
High-speed rail (HSR) has become an essential component of transportation systems
due to its economic, environmental, and quality-of-life benefits (1)
. It is supported by
European Union, Korea, and China and viewed as the next growth economy wave (2)
.
As the safety and reliability are primary concerns of the high-speed rail system, the
knowledge of track conditions is critical for making a suitable maintenance plan and
performing grinding operations where and when required (3, 4)
. Among various methods
for the track inspection, making use of the axle box accelerations is one of the most
promising, which enables detecting and identifying some singular track defects such as
squats (5, 6)
, bolt tightness of fish-plated joints (7)
and other short track defects (8)
. Other
works also applied axle box accelerations for the detection of rail corrugation (9, 10)
.
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In practice, sampling frequencies are usually lowered down to several kHz (11)
for
speeds up to 300 km/h to avoid a vast amount of data acquired. However, the Nyquist
theorem presents the drawback that those track defects whose excitation frequency is
higher than half of the sampling frequency are not detected. Therefore, there must be a
compromise between the resolution and the amount of recorded data. As an emerging
technique, Compressive Sensing (CS) creates the opportunity for sub-Nyquist sampling.
CS theory confirms that the target signal can be recovered as long as it has a sparse or
compressible representation in a known domain (12, 13)
, and the number of measurements
required is governed by the sparsity degree rather than the bandwidth of the signal.
A number of researchers have studied CS for structural health monitoring (SHM). Bao
et al. explored the use of CS to address data loss common in wireless sensor networks
and achieved more robust signal collection when compressed measurements rather than
time-history samples are lost (14)
. More recently, they demonstrated physical realisation
of CS on the Imote2 wireless sensor using a random demodulator (15)
. O’Connor et al.
explored using CS to reduce power consumption in wireless fatigue life monitoring of
ship hulls (16)
and achieved significant energy reductions in acquiring mode shapes of
the Telegraph Road Bridge (17)
. Studies that combine CS with damage detection have
also existed: Mascarenas et al. showed a CS application for SHM damage detection
using a laboratory testbed structure with a digital prototype of a compressed sensor
embedded into a microcontroller (18)
while Haile et al. studied the use of CS in SHM for
damage detection in composite materials (19)
.
However, there hasn’t been any research that applies this technique to high-speed rail
monitoring. Therefore, this study aims to propose a compressive sensing framework for
HSR monitoring, so that through lower sampling rates the same resolution of defect
detection can be achieved. The process of compressive sensing is simulated using the
axle box acceleration data acquired from a high-speed train ran on one section of
railway in China. The effectiveness of this method is verified by the comparison of the
reconstructed signals and the original ones. Based on the results, recommendations on
the orthogonal basis, the measuring method and the compression ratio in engineering
practice are given.
2. Compressive sensing
The measurement vector My R∈ is acquired by a linear projection of the discrete-time
signal ( )Nf R N M∈ > :
y f=Φ . (1)
When f is represented in terms of an N by N orthogonal basis matrix Ψ with the
basis vectors { }i
ψ as columns, the problem becomes
y = ΦΨx = !Φx (2)
where !Φ = ΦΨ is an M by N matrix called sensing matrix.
Typical orthogonal bases include the wavelet basis (20)
, the discrete Fourier basis (21)
, the
discrete cosine basis (22)
, the curvelet basis (23)
and so on. Besides, redundant dictionaries
also work well in compressive sensing (24–26)
.
Although the problem of recovering the representation x from y with M components
is under-determined, it has been proved that x can be recovered exactly under the
following conditions (27, 28)
:
3
1. The representation x is sufficiently sparse, where a vector is defined as S -sparse
if it has at most S non-zero entries;
2. The matrix !Φ obeys a “restricted isometry property” (RIP).
RIP impose a restricted orthogonality condition on !Φ which enables exact recovery of
sparse x from y . To explain RIP, the S -restricted isometry constant S
δ of the matrix
!Φ given in (29)
is defined as the smallest number such that
(1−δS) c
l2
2
≤ !ΦTcl2
2
≤ (1+δS) c
l2
2
(3)
holds for all subsets !ΦT
with {1,..., }T N⊂ , T S≤ and coefficient sequences ( )j j Tc
∈.
The S-restricted isometry constant shows how close subsets of !ΦT
are to an orthogonal
system when restricted to sparse linear combinations ( T S≤ ).
Focusing on the case where x is sparse, it is desired to find the sparsest solution of !Φx = y and solve
0ˆ argminx x= subject to !Φx = y . (4)
Although this 0l problem has been proved to have a unique S sparse solution if the
minimum requirement that 2
1S
δ < is satisfied (29)
, it is a hard combinatorial problem
that cannot be solved by any algorithm other than brute force search. If a stronger
condition that 2
2 1S
δ < − is satisfied, the convex relaxation is exact and the solution
of the 0l problem can be obtained by solving an
1l problem
(29):
1ˆ argminx x= subject to !Φx = y . (5)
The 1l problem, also named Basis Pursuit
(30) can be recast as a linear programming
problem with computational complexity ( )3O N(31)
.
When the compressed measurements are contaminated with noise Me R∈ bounded
2le ε≤ , the problem can be rewritten as
1ˆ ˆargminx x= subject to !Φx̂-y
2
2
≤ ε (6)
and solved as Second-Order Cone Programs (SOCP) (32)
.
The original signal can be recovered using the optimal basis coefficients x̂ : ˆ ˆf x= Ψ .
However, the convex optimisation is not the only way to reconstruct sparse solutions.
There are at least other two common classes of computational techniques for solving
sparse approximation problems:
1. Greedy pursuit iteratively refines a sparse solution by successively identifying
one or more components that yield the greatest improvement in quality (33, 34)
;
2. Bayesian framework assumes a prior distribution for the unknown coefficients
that favours sparsity first, then develops a maximum a posteriori estimator that
incorporates the observation and finally averages over most probable models (35)
or identifies a region of significant posterior mass (36)
.
Although Ψ is fixed according to the characteristic of the target signal, it is known that
trivial randomised constructions of Φ will enable !Φ to satisfy RIP with overwhelming
probability (12)
. Examples include Gaussian random matrix (where entries of Φ are
4
independently sampled from a normal distribution with mean 0 and variance 1/M ) and
binary random matrix (where entries independently come from a symmetric Bernoulli
distribution ( , / ) 1/ 2P i j Mφ = ± = ).
!Φ can also be constructed by selecting M rows from an N by N orthogonal matrix
uniformly at random, where Φ randomly sub-samples the target signal and Ψ maps the
time domain and the selected domain (12)
.
3. Compressive sensing for high-speed rail monitoring
3.1 Data acquisition
CNERC-Rail (Hong Kong branch) was authorised to monitor the vibration response of
an operating train on Lanzhou−Xinjiang high-speed rail line. The monitoring work
lasted for about one month and mainly inspected the vibration of bogie and car body.
The monitored bogies located in the axis 5, axis 7, axis 8 of car 3 and axle 6 of car 4 and
the accelerometers are installed at the frame, vertical-stop component and axle box
(Figure 1) with the range of ± 1000 g and the sampling frequency of 5000 Hz.
Figure 1 Locations of the accelerometers
As the purpose of this study is to achieve the same resolution of defect detection with
fewer measurements, compressive sensing was applied to the vertical acceleration of
two axle boxes. To ensure the representativeness, 32 signal segments with 5000
components were selected from different time (morning, afternoon and evening) of two
days and under various scenarios (ordinary railway, crossing bridges and crossing
5
tunnels). The signals were obtained when the train achieved the cruising speed
(approximately 200 km/h) so that the influence of train speed was normalised.
3.2 Procedures and implementations
Although the process of acquiring the compressed measurements was simulated on the
computer rather than realised at the sensor, the data received in conventional ways can
be utilised for comparison. Firstly, the compressed measurements were obtained by
projecting the original vibration signal onto the selected M by N matrix Φ . With the
prior knowledge that the vibration signal is sparse or compressible in a specific
orthogonal basis, convex optimisation techniques were then applied to reconstruct the
target signal in the sparse domain and the time domain. Finally, the reconstructed signal
was compared with the original one for verification purposes.
One way to obtain the compressed measurements is to project the target signal onto the
Gaussian or Bernoulli random matrix Φ . These measuring methods have been achieved
physically using special sensors, so further experiments are feasible. Gaussian random
projection has been performed by a method called random demodulator (RD) (37, 38)
and
Bernoulli projection has been achieved in single-pixel camera (39)
by spatial light
modulators. Another way is to randomly sub-sample the target signal, which can be
realised by randomly triggering the ADC (17)
.
As the sampling rate of the data acquisition system is 5000 Hz, it has been determined
that the vibration responses are band-limited to 2500 Hz. However, by inducing the
technique of compressive sensing, signals with the same bandwidth can be acquired by
a sampling rate equivalently lower than the Nyquist rate.
The number of compressed measurements M was set as 40%, 50%, 60%, 70%, 80% of
N for different trials and /M N was defined as the compression ratio. The discrete
cosine basis (DCT) and the discrete Fourier basis (DFT) were selected as orthogonal
bases Ψ so that the sensing matrix !Φ can be obtained. Based on the compressed
measurements, the l1-magic package (40)
was used to reconstruct the target signal.
4. Results
4.1 Sparsity level of signals in different bases
The sparsity level of the signal in different domains can be known beforehand as the
complete time history is recorded, although in real life the orthogonal basis is chosen
according to the characteristic of the target signal without knowing the exact sparsity
level. In this study, the sparsity level is calculated as the ratio between the number of
zeros and the segment length. Indeed, as the vibration signal is contaminated with noise,
none of the coefficients of each data segment is originally zero. To generate spectrally
sparse signals, those coefficients that have a value smaller than 1% of the maximum are
eliminated from the spectrum. The statistics of sparsity level is defined as 0.01/
sL N N= .
Figure 2 gives a 1-second segment of an acceleration signal in the time domain as well
in other domains. The sparsity level is 63.6% and 82.2% when the signal is represented
by the DCT and DFT bases, respectively.
6
Figure 2 One segment of the acceleration signal in three domains
The same investigations are also done for other signals. Figure 3 shows the comparison
of average sparsity level in two orthogonal bases. It is shown that the axle box vertical
acceleration signals are sparser in the discrete cosine transform domain, where at least
60% of coefficients are close to zero.
The reason why the DCT general performs better for sparse coding of signals than the
DFT is explained as follows. The DCT implies different boundary conditions from the
DFT or other related transforms: the DCT implies an even extension at both left and
right boundaries while the DFT extend the original signal periodically to positive and
negative infinity. As any random segment of a signal is unlikely to have the same value
at both the left and right boundaries, discontinuities usually occur using the DFT. It is
well known that the smoother a discrete signal is, the fewer coefficients in its DFT or
DCT domain are required to represent it accurately.
Figure 3 Sparsity levels of vertical accelerations in different bases
4.2 Influence of sparsity level on reconstruction accuracy
The metric of residual sum-of-squares (RSS), normalised by the traditional Nyquist
sampled signal f , is used to assess the reconstruction error:
7
2
2
2
2
ˆ
l
l
f fRSS
f
−
= . (7)
To investigate the influence of sparsity level on the reconstruction quality, 32 vibration
segments are reconstructed using convex optimisation based on 2500 random sub-
samples of the target signals with 5000 entries. Figure 4 shows the relationship between
sparsity level and reconstruction error and an obvious trend: the sparser the signal is in
the orthogonal basis, the better it can be reconstructed.
Figure 4 Influence of sparsity levels on reconstruction errors
4.3 Reconstruction accuracy via different measuring method
It has been mentioned that the compressed measurements can be obtained by projecting
the target signal onto random matrices or randomly sub-sampling. To investigate the
influence of adopting different measuring approaches, 32 segments of acceleration are
assumed to be sparse in the DCT domain and recovered using convex optimisation and
50% of samples. As shown in Figure 7, the average reconstruction errors using different
methods are almost identical, which means that the selection of Φ does not make a
difference to the reconstruction as long as it is sufficiently incoherent with Ψ . Among
these methods, randomly sub-sampling is preferred as it is more intuitive and easier to
implement.
Figure 5 Reconstruction errors via different measuring methods
8
4.4 Recommended equivalent sampling ratio via regression
The Nyquist sampling theorem states that a signal must be sampled at least two times
faster than its bandwidth in order not to lose information. The theory of compressive
sensing provides a new data acquisition paradigm that the equivalent sampling rate is
determined by the sparsity level in the selected domain instead of the bandwidth. Thus,
a trade-off between reconstruction accuracy and sampling rate reduction can be
achieved. In this study, to estimate the recovery error according to the compression ratio
/M N (i.e. /se sf f when measuring method is randomly sub-sampling) and the sparsity
level of the target signal in selected domain, a linear regression model is formulated:
1 2( / )se se f f sα β β= + × + × (8)
where e : reconstruction error;
s : sparsity level of the target signal;
/se sf f : equivalent sampling rate/ sampling rate of recovery signal.
To learn the parameter α and β in this regression formula, 32 signal segments are
investigated, and their sparsity levels in the DCT basis are calculated. The compression
ratio /se sf f is set as 40%, 50%, 60%, 70% and 80%, which means that sef equals to
2000Hz, 2500Hz, 3000 Hz, 3500Hz and 4000Hz respectively. Signals are recovered
based on the compressed measurements. In this way, 160 experiments with different
configurations are conducted, and 160 results of reconstruction are obtained for the
regression. Finally, the formula becomes:
1.548 0.78 (4 1.00( ) ( / ) 7)se se f f s= + − × + ×− . (9)
This formula gives a recommend equivalent sampling rate based on the sparsity and the
desired accuracy requirement of the target signal. Once the accepted error is determined,
the equivalent sampling rate depends on the sparsity level instead of the bandwidth.
Typical reconstruction results with two error levels (0.20 and 0.30) are illustrated in the
following two figures. The results prove that when the proper compression ratio is set,
signals can be satisfactorily obtained through sampling rates lower than the Nyquist
theorem requires. The adoption of compressive sensing can reduce the sampling rate
and the communication payload of a data acquisition system for high-speed rail
monitoring.
Figure 6 Reconstructed signal versus target signal (M/N = 0.5; RSS = 0.20)
9
Figure 7 Reconstructed signal versus target signal (M/N = 0.5; RSS = 0.30)
Figure 8 illustrates the actual and expected reconstruction errors corresponding to
different compression ratios.
Figure 8 Actual and expected reconstruction errors of different compression ratios
5. Conclusions
This study is the first application of compressive sensing to high-speed rail monitoring
area. Specifically, the process of compressive sensing is simulated on the vibration
signals collected from two axle boxes of a high-speed train, and reconstruction results
are used to evaluate different configurations. It is found that the axle box acceleration
signals are sparser in the discrete cosine transform basis and can be better reconstructed
if they are sparser in the selected orthogonal basis. Additionally, all measuring methods
including projecting signals onto random matrices and randomly sub-sampling enable
similar reconstruction results. The findings are useful for engineering practice.
10
A regression model is also formulated to give a recommend equivalent sampling rate
based on the sparsity and the desired accuracy requirement of the target signal. Thus, a
trade-off between reconstruction accuracy and sampling rate reduction can be achieved.
When the proper compression ratio is set, signals from high-speed rail monitoring with
the same bandwidth can be satisfactorily obtained through sampling rates lower than the
Nyquist theorem requires. This new data acquisition paradigm has the potential to
improve the resolution of defect detection, save the energy consumption and deal with
the issue of data loss in high-speed rail monitoring.
As sensors that can directly acquire compressed measurements from analogue signals
have already existed, it is expected that compressive sensing can be implemented in the
real practice of monitoring rather than simulations. Apart from that, this study only
utilises the prior knowledge that signals are sparse and reconstruct them one by one. To
make use of the common characteristics that they could learn from each other,
dictionary learning technique will be employed in the future work. Most signals are
expected to be sparser in the dictionary learned from a training set, and thus the
reconstruction performance can be improved.
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).
References
1 A Ryder, 'High Speed Rail', Journal of Transport Geography, Vol. 22, pp. 303–
305, 2012.
2 S Tierney, 'High-Speed Rail, the Knowledge Economy and the next Growth
Wave', Journal of Transport Geography, Vol. 22, pp. 285–287, May 2012.
3 M Bocciolone, A Caprioli, A Cigada, and A Collina, 'A Measurement System for
Quick Rail Inspection and Effective Track Maintenance Strategy', Mechanical
Systems and Signal Processing, Vol. 21, No. 3, pp. 1242–1254, April 2007.
4 A Caprioli, A Cigada, and D Raveglia, 'Rail Inspection in Track Maintenance: A
Benchmark between the Wavelet Approach and the More Conventional Fourier
Analysis', Mechanical Systems and Signal Processing, Vol. 21, No. 2, pp. 631–
652, February 2007.
5 M Molodova, Z Li, A Núñez, and R Dollevoet, 'Automatic Detection of Squats in
Railway Infrastructure', IEEE Transactions on Intelligent Transportation
Systems, Vol. 15, No. 5, pp. 1980–1990, October 2014.
6 Z Li, M Molodova, A Núñez, and R Dollevoet, 'Improvements in Axle Box
Acceleration Measurements for the Detection of Light Squats in Railway
Infrastructure', IEEE Transactions on Industrial Electronics, Vol. 62, No. 7, pp.
4385–4397, July 2015.
7 Z Li, M Oregui, R Carroll, S Li, and J Moraal, 'Detection of Bolt Tightness of
Fish-Plated Joints Using Axle Box Acceleration', Proceedings of the 1st
11
International Conference on Railway Technology: Research, Development and
Maintenance, 18-20 April 2012, Las Palmas, Spain.
8 M Oregui, Z Li, and R Dollevoet, 'Identification of Characteristic Frequencies of
Damaged Railway Tracks Using Field Hammer Test Measurements', Mechanical
Systems and Signal Processing, Vol. 54, pp. 224–242, March 2015.
9 S L Grassie, 'Rail Corrugation: Advances in Measurement, Understanding and
Treatment', Wear, Vol. 258, No. 7–8, pp. 1224–1234, March 2005.
10 P T Torstensson and M Schilke, 'Rail Corrugation Growth on Small Radius
curves—Measurements and Validation of a Numerical Prediction Model', Wear,
Vol. 303, No. 1–2, pp. 381–396, June 2013.
11 H Tsunashima, Y Naganuma, A Matsumoto, T Mizuma, and H Mori, 'Condition
Monitoring of Railway Track Using in-Service Vehicle', Reliability and safety in
railway. InTech, 2012.
12 E J Candes, 'Compressive Sampling', Proceedings of the International Congress
of Mathematicians 2006, 22-30 August 2006, Madrid, Spain.
13 D L Donoho, 'Compressed Sensing', IEEE Transactions on Information Theory,
Vol. 52, No. 4, pp. 1289–1306, April 2006.
14 Y Bao, H Li, X Sun, Y Yu, and J Ou, 'Compressive Sampling–based Data Loss
Recovery for Wireless Sensor Networks Used in Civil Structural Health
Monitoring', Structural Health Monitoring: An International Journal, Vol. 12, No.
1, pp. 78–95, November 2012.
15 Z Zou, Y Bao, H Li, B F Spencer, and J Ou, 'Embedding Compressive Sensing-
Based Data Loss Recovery Algorithm Into Wireless Smart Sensors for Structural
Health Monitoring', IEEE Sensors Journal, Vol. 15, No. 2, pp. 797–808, February
2015.
16 S M O’Connor, J P Lynch, and A C Gilbert, 'Compressive Sensing Approach for
Structural Health Monitoring of Ship Hulls', Proceedings of the 8th International
Workshop on Structural Health Monitoring, 13-15 September 2011, Stanford,
CA, USA.
17 S M O’Connor, J P Lynch, and A C Gilbert, 'Compressed Sensing Embedded in
an Operational Wireless Sensor Network to Achieve Energy Efficiency in Long-
Term Monitoring Applications', Smart Materials and Structures, Vol. 23, No. 8,
July 2014.
18 D Mascareñas, A Cattaneo, J Theiler, and C Farrar, 'Compressed Sensing
Techniques for Detecting Damage in Structures', Structural Health Monitoring:
An International Journal, Vol. 12, No. 4, pp. 325–338, July 2013.
19 M Haile and A Ghoshal, 'Application of Compressed Sensing in Full-Field
Structural Health Monitoring', Smart Sensor Phenomena, Technology, Networks,
and Systems Integration, Vol. 8346, p. 834618, 2012.
20 S Mallat, A wavelet tour of signal processing. Academic press, 1999.
21 Y Bao, J L Beck, and H Li, 'Compressive Sampling for Accelerometer Signals in
Structural Health Monitoring', Structural Health Monitoring: An International
Journal, Vol. 10, No. 3, pp. 235–246, May 2011.
22 N Ahmed, T Natarajan, and K R Rao, 'Discrete Cosine Transform', IEEE
transactions on Computers, Vol. 100, No. 1, pp. 90–93, January 1974.
23 E Candes, D L Donoho, E J Candès, and D L Donoho, 'Curvelets: A Surprisingly
Effective Nonadaptive Representation of Objects with Edges', Curves and
Surface Fitting, Vol. C, No. 2, pp. 1–10, April 2000.
12
24 M Aharon, M Elad, and A Bruckstein, 'K-SVD: An Algorithm for Designing
Overcomplete Dictionaries for Sparse Representation', Signal Processing, IEEE
Transactions on, Vol. 54, No. 11, pp. 4311–4322, November 2006.
25 H Rauhut, K Schnass, and P Vandergheynst, 'Compressed Sensing and
Redundant Dictionaries', IEEE Transactions on Information Theory, Vol. 54, No.
5, pp. 2210–2219, May 2008.
26 E J Candes, Y C Eldar, D Needell, and P Randall, 'Compressed Sensing with
Coherent and Redundant Dictionaries', Applied and Computational Harmonic
Analysis, Vol. 31, No. 1, pp. 59–73, July 2011.
27 E J Candes and T Tao, 'Near-Optimal Signal Recovery From Random
Projections: Universal Encoding Strategies?', IEEE Transactions on Information
Theory, Vol. 52, No. 12, pp. 5406–5425, December 2006.
28 E J Candes and T Tao, 'Decoding by Linear Programming', IEEE Transactions on
Information Theory, Vol. 51, No. 12, pp. 4203–4215, December 2005.
29 E J Candes, 'The Restricted Isometry Property and Its Implicationsfor
Compressed Sensing', Comptes Rendus Mathematique, Vol. 346, No. 9–10, pp.
589–592, May 2008.
30 S S Chen, D L Donoho, and M A Saunders, 'Atomic Decomposition by Basis
Pursuit', SIAM Journal on Scientific Computing, Vol. 20, No. 1, pp. 33–61, 1998.
31 P Bloomfield and W L Steiger, Least absolute deviations: Theory, applications
and algorithms. Springer, 1984.
32 E J Candes, J K Romberg, and T Tao, 'Stable Signal Recovery from Incomplete
and Inaccurate Measurements', Communications on Pure and Applied
Mathematics, Vol. 59, No. 8, pp. 1207–1223, August 2006.
33 S G Mallat and Z Zhang, 'Matching Pursuits With Time-Frequency Dictionaries',
IEEE Transactions on Signal Processing, Vol. 41, No. 12, pp. 3397–3415,
December 1993.
34 D Needell and J A Tropp, 'CoSaMP: Iterative Signal Recovery from Incomplete
and Inaccurate Samples', Applied and Computational Harmonic Analysis, Vol.
26, No. 3, pp. 301–321, May 2009.
35 P Schniter, L C Potter, and J Ziniel, 'Fast Bayesian Matching Pursuit',
Proceedings of 2008 Information Theory and Applications Workshop, 27
January-1 Feb 2008, San Diego, CA, USA.
36 D P Wipf and B D Rao, 'Sparse Bayesian Learning for Basis Selection', IEEE
Transactions on Signal Processing, Vol. 52, No. 8, pp. 2153–2164, August 2004.
37 S Kirolos et al., 'Analog-to-Information Conversion via Random Demodulation',
Proceedings of 2006 IEEE Dallas/CAS Workshop on Design, Applications,
Integration and Software, 29-30 October 2006, Richardson, TX, USA.
38 J N Laska, S Kirolos, M F Duarte, T S Ragheb, R G Baraniuk, and Y Massoud,
'Theory and Implementation of an Analog-to-Information Converter Using
Random Demodulation', Proceedings of 2007 IEEE International Symposium on
Circuits and Systems, 27-30 May 2007, New Orleans, LA, USA.
39 M F Duarte et al., 'Single-Pixel Imaging via Compressive Sampling', IEEE
Signal Processing Magazine, Vol. 25, No. 2, pp. 83–91, March 2008.
40 E J Candès and J K Romberg, 'l1-Magic: Recovery of Sparse Signals via Convex
Programming', October 2005.