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

si-xin.chen@connect.polyu.hk

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

Mor

e in

fo a

bout

this

art

icle

: ht

tp://

ww

w.n

dt.n

et/?

id=

2344

5

2

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).

References

1 R M Goodall and C Roberts, 'Concepts and Techniques for Railway Condition

Monitoring', Proceedings of Institution of Engineering and Technology

International Conference on Railway Condition Monitoring 2006, 29-30

November 2006, Birmingham, UK.

2 S Bruni, R Goodall, T X Mei, and H Tsunashima, 'Control and Monitoring for

Railway Vehicle Dynamics', Vehicle System Dynamics, Vol. 45, No. 7–8, pp.

743–779, June 2007.

3 H-M Thomas, T Heckel, and G Hanspach, 'Advantage of a Combined Ultrasonic

and Eddy Current Examination for Railway Inspection Trains', Insight-Non-

Destructive Testing and Condition Monitoring, Vol. 49, No. 6, pp. 341–344, June

2007.

4 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.

5 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.

6 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.

7 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.

11

4385–4397, July 2015.

8 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

International Conference on Railway Technology: Research, Development and

Maintenance, 18-20 April 2012, Las Palmas, Spain.

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 Mori, H Tsunashima, T Kojima, A Matsumoto, and T Mizuma, 'Condition

Monitoring of Railway Track Using In-Service Vehicle', Journal of mechanical

systems for transportation and logistics, Vol. 3, No. 1, pp. 154–165, February

2010.

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, 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 Y Bao, J L Beck, and H Li, 'Compressive Sampling for Accelerometer Signals in

Structural Health Monitoring', Structural Health Monitoring, Vol. 10, No. 3, pp.

235–246, May 2011.

19 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.

20 S F Cotter, B D Rao, and K Kreutz-Delgado, 'Sparse Solutions to Linear Inverse

Problems with Multiple Measurement Vectors', IEEE Transactions on Signal

Processing, Vol. 53, No. 7, pp. 2477–2488, July 2005.

21 Y C Eldar and M Mishali, 'Robust Recovery of Signals from a Structured Union

of Subspaces', IEEE Transactions on Information Theory, Vol. 55, No. 11, pp.

5302–5316, November 2009.

22 Y C Eldar and H Rauhut, 'Average Case Analysis of Multichannel Sparse

Recovery Using Convex Relaxation', IEEE Transactions on Information Theory,

12

Vol. 56, No. 1, pp. 505–519, January 2010.

23 S Mallat, A wavelet tour of signal processing. Academic press, 1999.

24 N Ahmed, T Natarajan, and K R Rao, 'Discrete Cosine Transform', IEEE

transactions on Computers, Vol. 100, No. 1, pp. 90–93, January 1974.

25 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.

26 E J Candes and T Tao, 'Decoding by Linear Programming', IEEE Transactions on

Information Theory, Vol. 51, No. 12, pp. 4203–4215, December 2005.

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, J Romberg, and T Tao, 'Robust Uncertainty Principles: Exact Signal

Reconstruction from Highly Incomplete Frequency Information', IEEE

Transactions on Information Theory, Vol. 52, No. 2, pp. 489–509, February

2006.

29 E J Candes, 'The Restricted Isometry Property and Its Implications for

Compressed Sensing', Comptes Rendus Mathematique, Vol. 346, No. 9–10, pp.

589–592, May 2008.

30 D L Donoho, 'For Most Large Underdetermined Systems of Linear Equations the

Minimal ℓ1-Norm Solution Is Also the Sparsest Solution', Communications on

Pure and Applied Mathematics, Vol. 59, No. 6, pp. 797–829, June 2006.

31 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,

January 1998.

32 S-J Kim, K Koh, M Lustig, S Boyd, and D Gorinevsky, 'An Interior-Point

Method for Large-Scale ℓ1-Regularized Least Squares', IEEE journal of selected

topics in signal processing, Vol. 1, No. 4, pp. 606–617, December 2007.

33 W Yin, S Osher, D Goldfarb, and J Darbon, 'Bregman Iterative Algorithms for

ℓ1-Minimization with Applications to Compressed Sensing', SIAM Journal on

Imaging sciences, Vol. 1, No. 1, pp. 143–168, March 2008.

34 E Van Den Berg and M P Friedlander, 'Probing the Pareto Frontier for Basis

Pursuit Solutions', SIAM Journal on Scientific Computing, Vol. 31, No. 2, pp.

890–912, November 2008.

35 J Yang and Y Zhang, 'Alternating Direction Algorithms for ℓ1-Problems in

Compressive Sensing', SIAM journal on scientific computing, Vol. 33, No. 1, pp.

250–278, February 2011.

36 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.

37 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.

38 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.

39 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.

13

40 'YALL1: Your ALgorithms for L1'. [Online]. Available:

http://yall1.blogs.rice.edu/. [Accessed: 21-May-2018].

41 J A Tropp, 'Algorithms for Simultaneous Sparse Approximation. Part II: Convex

Relaxation', Signal Processing, Vol. 86, No. 3, pp. 589–602, March 2006.

42 J Liu, S Ji, and J Ye, 'SLEP: Sparse Learning with Efficient Projections', Arizona

State University, Vol. 6, No. 491, p. 7, 2009.

43 S J Wright, R D Nowak, and M A T Figueiredo, 'Sparse Reconstruction by

Separable Approximation', IEEE Transactions on Signal Processing, Vol. 57, No.

7, pp. 2479–2493, July 2009.

44 Z Qin, K Scheinberg, and D Goldfarb, 'Efficient Block-Coordinate Descent

Algorithms for the Group Lasso', Mathematical Programming Computation, Vol.

5, No. 2, pp. 143–169, June 2013.

45 W Deng, W Yin, and Y Zhang, 'Group Sparse Optimization by Alternating

Direction Method', Proceedings of SPIE Optical Engineering + Applications

2013, 25-29 August 2013, San Diego, CA, USA.