Research Article Sparse Signal Recovery by Stepwise ...downloads.hindawi.com/journals/ijdsn/2013/798537.pdfSparse Signal Recovery by Stepwise Subspace Pursuit in Compressed Sensing

Hindawi Publishing CorporationInternational Journal of Distributed Sensor NetworksVolume 2013 Article ID 798537 5 pageshttpdxdoiorg1011552013798537

Research ArticleSparse Signal Recovery by Stepwise Subspace Pursuit inCompressed Sensing

ZheTao Li12 JingXiong Xie1 DengBiao Tu3 and Young-June Choi4

1 The College of Information Engineering Xiangtan University Hunan 411105 China2 School of Computer National University of Defense Technology Hunan 410073 China3 National Computer Network Emergency Response Technical TeamCoordination Center of China Beijing 100029 China4Department of Information and Computer Engineering Ajou University Suwon 443749 Republic of Korea

Correspondence should be addressed to DengBiao Tu tudengbiao163com

Received 26 May 2013 Accepted 10 July 2013

Academic Editor Fu Xiao

Copyright copy 2013 ZheTao Li et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper an algorithmnamed stepwise subspace pursuit (SSP) is proposed for sparse signal recovery Unlike existing algorithmsthat select support set from candidate sets directly our approach eliminates useless information from the candidate throughthreshold processing at first and then recovers the signal through the largest correlation coefficients We demonstrate that SSPsignificantly outperforms conventional techniques in recovering sparse signals whose nonzero values have exponentially decayingmagnitudes or distribution of119873(0 1) Experimental results of Lena show that SSP is better than CoSaMP OMP and SP in terms ofpeak signal to noise ratio (PSNR) by 55 db 41 db and 42 db respectively

1 Introduction

In many applications such as statistical regression [1] digitalcommunications [2] image processing [3 4] multimediasensor networks [5 6] interpolationextrapolation [7] andsignal deconvolution [8 9] recovering high-dimensionalsignals from relatively fewer measurements is a challengingtask Fortunately in the real world many signals are or canbe transformed (such as DCT wavelet packet transform [10])to sparse such that only a small part of signal coefficients arenonzero values And compressed sensing [11 12] allows usto recover sparse signal from high-dimensional signals withvery fewmeasurements In fact some works in the real worldshow that one can recover exactly 119870 sparse signal of length119872 with only119874(119870 log119872) randommeasurements Let y isin 119877119873be an observed signal and let Φ isin 119877119873times119872 be a dictionary ofatoms then the standard of the formulation of sparse follows

x = argmin x0 st y = Φx (1)

where 1205790is the 120579rsquos ℓ

0-norm which counts the number of

nonzero elements of vector 120579

Finding the exact solution of (1) is known as NP-hard[7] It is intractable for combinatorial approaches to solvethe problems of moderate-to-high dimensionality and thusone needs to resort to heuristic procedures However if thedictionaryΦ satisfies nearly orthogonality (1) then becomes

x = argmin 1205791 st y = Φx = ΦΨ120579 = Θ120579 (2)

where 1205791is the 120579rsquos ℓ

1-norm Θ=ΦΨ and there is a sparse

vector such that x = Ψ120579The rest of the paper is organized as follows Section 2

summarizes the related work of recover algorithm in com-pressed sensing Section 3 contains the description of step-wise subspace pursuit algorithm Section 4 makes a compar-ison of our work with the related papers through simulationSection 5 presents our main conclusion

2 Related Work

Existing recovery algorithms are roughly classified into threemain families convex relaxation algorithms Bayesian algo-rithms and pursuit algorithms Our algorithm presented inthis paper belongs to the pursuit family

2 International Journal of Distributed Sensor Networks

The convex relaxation algorithms approximate the nons-mooth and nonconvex ℓ

0-norm by functions that are easier

to handle The resulting problem can be solved by means ofstandard optimization techniques Well-known instances ofalgorithms based on such an approach are Basis Pursuit (BP)[13] and FOCUSS [14] which approximate the ℓ

0-norm by

the ℓ1-norm and ℓ

119901-norm (119901 lt 1) respectively

The Bayesian algorithms express the problem as thesolution of a Bayesian inference problem and apply statisticaltools to solve it that is assuming a prior distribution forthe unknown coefficients that favors sparsity They developa maximum a posteriori estimator that incorporates theobservation There are many algorithms that incorporatesome of these features For example identify a region ofsignificant posterior mass [15] or average over most probablemodels [16] One key ingredient of Bayesian algorithms is thechoice of a proper prior on the sought sparse vector

The pursuit algorithms for sparse signal recovery are agreedy approach that iteratively refines the current estimatefor the coefficient vector x by modifying one or severalcoefficients chosen to yield a substantial improvement inapproximating the signal The family of pursuit algorithmsincludes several approaches according to the way of updatingthe support set single or multiple algorithmsThe algorithmsof single updating support set gradually increase the supportby sequentially adding new atoms The complexity of thesealgorithms is lower than the complexity of BP However theyrequire more measurements 119872 for accurate reconstructionand they often have an effect on empirical work and donot offer the strong theoretical guarantees Single algorithmsinclude matching pursuit (MP) [17] orthogonal matchingpursuit (OMP) [18] However for many applications singleupdating support set does not offer adequate performanceso researchers have developed more sophisticated pursuitmethods that work better in practice and yield essentiallyoptimal theoretical guarantees called multiple algorithmsThese techniques depend on several enhancements to thebasic greedy framework (1) selecting multiple columns periteration (2) pruning the set of active columns at eachstep (3) solving the least squares problems iteratively (4)theoretical analysis using the RIP bound There are manyalgorithms that incorporate some of these features Forexample stagewise orthogonal matching pursuit (StOMP)[19] hard thresholding pursuit (HTP) [20] regularizedorthogonal matching pursuit algorithm (ROMP) [21] whichwas the first greedy technique whose analysis was sup-ported by a RIP bound compressive sampling matchingpursuit (CoSaMP) [22] which was the first algorithm toassemble these ideas to obtain essentially optimal perfor-mance guarantees and the subspace pursuit (SP) [23] andso forth Sparsity-predict in these algorithms bring to theperfect performance However once the sparsity is falsepredicted many signals cannot be reconstructed accuratelyAll in all pursuit algorithms have often been considerednaive in part because there are contrived examples wherethe approach fails spectacularly However recent researchhas clarified that greedy pursuits succeed empirically andtheoretically in many situations where convex relaxationworks

NoYes

Input Φ y

Threshold processing

i | |c (i)| gt ts

T985747= K indices whit the largestcorrelation in solved x

Proj (y ΦT

985747)

T985747good

enough

T985747= T

985747minus1 ⋃K indices whitthe largest correlation

Solve x with (Φ998400

T985747lowast Φ )

minus1lowast Φ998400

T985747

T985747

lowast y

Correlation cal Φ lowast y 985747minus1

r

Figure 1 Description of reconstruction algorithms for 119870-sparsesignals

3 Sparse Signal Recovery by StepwiseSubspace Pursuit

Figure 1 depicts the schematic representation of the proposedSSP algorithm The ℓth iteration applies matched filtering tothe current residual and gets a candidate setΦ lowast yℓminus1

119903 which

contains a small number of significant nonzero values Thenwe eliminate useless information in the candidate throughthreshold processing 119894 | |119888(119894)| gt 119905

119904 and select 119870 indices

which are considered to be reliable on some iteration stepswith the largest correlation by interim estimate We mergethe 119870 indices of newly selected coordinates with the pre-vious support estimate thereby updating the intermediateestimate ℓ We have the new approximation x supportedin ℓ with coefficients given by (Φ1015840

ℓ lowast Φℓ)


ℓ lowast

y The updated support estimate can be gained from approx-imation x by the largest correlation We then project thevector y on the columns of Φ

119879ℓ belonging to the updated

support and check the stopping condition and if it is not yettime to stop we set ℓ = ℓ + 1 and go to the next iteration

The algorithm is depicted in Algorithm 1 The maincontribution of the SSP reconstruction algorithm is that itgenerates a list of candidates sequentially and incorporates asimple method for re-evaluating the reliability of all candi-dates at iteration thus gaining the correlation of candidatesbefore the operation of SP with the correlation at iteration ofthe method

International Journal of Distributed Sensor Networks 3

Algorithm stepwise sub space pursuitInput

An119873 times 119889measurement matrixΦAn119873 dimensional data vector yThe sparsity level119870 of the idea signal

ProcedureInitialization(1) 1198790 = 119870 indices with the largest correlation entries in the vector Φ lowast y(2) y0 = resid (yΦ

1198790)

Iteration when 119904th the iteration execute the following steps(1) threshold processing 119910119904minus1

(2) 119904 = 119879119904minus1 cup 119870 indices with largest correlation in Φ119879119904minus1 lowast y119904minus1

(3) slove x with x = (Φ1015840119904 lowastΦ119904)

minus1

lowastΦ1015840

119904 lowast y

(4) 119879119904 = 119870 indices with largest correlation in solved x(5) y119904 = resid (yΦ

119879119904)

(6) if 1003817100381710038171003817y1199041003817100381710038171003817 gt1003817100381710038171003817y119904minus11003817100381710038171003817 then break the iteration else go next iteration

OutputThe approximate solve x

Algorithm 1 Algorithm of SSP

4 Simulation and Results

In this section we show the performance of the proposedalgorithm through simulation from two aspects (1) for 119870-sparse 1-dimensional signal we compare the reconstructionprobabilities of OMP CoSaMP SP and SSP algorithmies(2) for sparse 2-dimensional images signal with DCT wecompare the effectiveness and accuracy of signal recoverywith OMP CoSaMP SP and SSP algorithms under the sametest conditions

In Figure 2 we compare the performance of SSP algo-rithm with that of OMP CoSaMP and SP algorithms Theoriginal signal in Figure 2(a) is obtained when the 119870-sparsesignal is set by random nonzero values drawn from 119873(0 1)where the 119870 nonzero coefficients are set by iid 119873(0 1) andthe remaining coefficients of x are set by 0 The originalsignal in Figure 2(b) is obtained when the 119870 nonzero coef-ficients are a random permutation of 119870 exponentially decayand the remaining coefficients of x are set to 0

Figure 2(a) shows that SSP performs better than OMPCoSaMP and SP when the nonzero entries of the sparsesignal are drawn according to zero-mean Gaussian withvariance 1 We discover that the recovery probability is 1 inlow sparsity level but the recovery signal is not accuratewhen the sparsity increases to a certain level And thereforemore measurements are needed for a better signal recoveryFurthermore the results in Figure 2(a) show that theCoSaMPOMP and SP can accurately recover signal in sparsity level in15 17 and 18 respectively while the SSP algorithm can reach21 As depicted in Figure 2(b) SSP significantly outperformsexisting methods for the exponential case

Two 256 times 256 tested images (Lena and Cameraman) areused to illustrate the quality of reconstructed image Our sim-ulation experiments were performed in the MATLAB2010benvironment using an AMDAthlon II X2 245 processor with2GB of memory The Gaussian random matrix was applied

Table 1 PSNR and reconstructed time of different algorithms forimages

CoSaMP OMP SP SSPLena

PSNR 198831 213572 212388 254261TIME 39 05 24 39

CameramanPSNR 163795 181784 173851 211282TIME 44 05 25 43

to measure the coefficients of OMP CoSaM PSP and SSPalgorithms In order to validate the effectiveness of OMPCoSaMP SP and SSP algorithms compression ratio of testimages was set 03333 Figure 3 shows that the reconstructedquality of SSP is better than that of the OMP CoSaMP andSP in the same experimental condition Table 1 is the PSNRof reconstructed images and the reconstructed times of OMPCoSaMP SP and SSP algorithms for test images

As shown in Table 1 SSP has the maximum PSNR andthe largest time consumed in reconstruction compared withother algorithms It shows that searching for maximum cor-relation set from candidate in SSP is a double-edged swordThe maximum correlation of SSP explains the possibility ofits relative higher quality compared to OMP CoSaMP andSP in this example

5 Conclusion

In this paper a stepwise subspace pursuit algorithm for signalreconstruction is proposed by using the largest correlationof the candidate set It can obtain accurate solutions thatpreserve more important coefficients as well as recover moredata than other existing algorithmsThe experimental results


10 15 20 25 30 35 40 450

01

02

03

04

05

06

07

08

09

1

Sparsity level

Freq

uenc

y of

exac

t rec

onstr

uctio

n

SPOMP

COSAMPSSP

Entries in the sparse signal are drawn from N (0 1)

(a) Nonzero entries of x are distributed119873(0 1)

10 15 20 25 30 35 40 450

01

02

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04

05

06

07

08

09

1

Sparsity level

Freq

uenc

y of

exac

t rec

onstr

uctio

n

Magnitude of entries in the sparse signal falls exponentially

SPOMP

COSAMPSSP

(b) Nonzero entries of x are decay exponentially

Figure 2 The probability of 256-length and 119870-sparse signal is recovered from 128 random projections exactly by different reconstructionalgorithms

Original CoSaMP OMP SP SSP

Figure 3 Performance of different algorithms for two images

of Lena demonstrated that SSP is a more effective algorithmfor signal recovery from random measurement than OMPCoSaMP and SP algorithms in peak signal to noise ratio by55 db 41 db and 42 db respectively In future work we needto investigate how to reduce the reconstruction time whileimproving the quality of signal

Acknowledgments

This work was supported in part by the National NaturalScience Foundation of China under Grants no 61100215no 612111198 and no 61070180 Hunan Provincial NaturalScience Foundation of ChinawithGrant no 12JJ9021 Science

and Technology Planning Project of Hunan Provincial Sci-ence amp Technology Department with Grant no 2011GK3200Natural Science Foundation for Doctor Xiangtan Universitywith Grant no 10QDZ30 and the construct program of thekey discipline in Hunan province

References

[1] A Miller Subset Selection in Regression Chapman ampHallCRCNew York NY USA 2nd edition 2002

[2] E J Candes and T Tao ldquoDecoding by linear programmingrdquoIEEE Transactions on Information Theory vol 51 no 12 pp4203ndash4215 2005


[3] B D Jeffs and M Gunsay ldquoRestoration of blurred star fieldimages by maximally sparse optimizationrdquo IEEE Transactionson Image Processing vol 2 no 2 pp 202ndash211 1993

[4] J Bobin J L Starck JM Fadili YMoudden andD LDonoholdquoMorphological component analysis an adaptive thresholdingstrategyrdquo IEEE Transactions on Image Processing vol 16 no 11pp 2675ndash2681 2007

[5] F Xiao J K Liu J Guo and L F Liu ldquoNovel side informationgeneration algorithm of multi-view distributed video codingfor multimedia sensor networksrdquo International Journal of Dis-tributed Sensor Networks vol 2012 Article ID 582403 7 pages2012

[6] F Xiao J Wang L J Sun R H Wang and J Guo ldquoCoverageenhancement strategy based on novel perception and co-evolution for multimedia sensor networksrdquo Chinese Journal ofElectronics vol 22 no 1 pp 135ndash140 2013

[7] B K Natarajan ldquoSparse approximate solutions to linear sys-temsrdquo SIAM Journal on Computing vol 24 no 2 pp 227ndash2341995

[8] J J Kormylo and J MMendel ldquoMaximum likelihood detectionand estimation of Bernoulli-Gaussian processesrdquo IEEE Transac-tions on InformationTheory vol IT-28 no 3 pp 482ndash488 1982

[9] C Soussen J Idier D Brie and J Duan ldquoFrom Bernoulli-Gaussian deconvolution to sparse signal restorationrdquo IEEETransactions on Signal Processing vol 59 no 10 pp 4572ndash45842011

[10] Z T Li J X Xie Z G Yu and Y J Choi ldquoCompressed sensingbased on best wavelet packet basis for image processingrdquoJournal of Computers vol 8 no 8 pp 1947ndash1950 2013

[11] E J Candes J Romberg and T Tao ldquoRobust uncertainty prin-ciples exact signal reconstruction from highly incomplete fre-quency informationrdquo IEEE Transactions on InformationTheoryvol 52 no 2 pp 489ndash509 2006

[12] D L Donoho ldquoCompressed sensingrdquo IEEE Transactions onInformation Theory vol 52 no 4 pp 1289ndash1306 2006

[13] S S Chen D L Donoho and M A Saunders ldquoAtomic decom-position by basis pursuitrdquo SIAM Journal on Scientific Comput-ing vol 20 no 1 pp 33ndash61 1998

[14] I F Gorodnitsky and B D Rao ldquoSparse signal reconstructionfrom limited data using FOCUSS a re-weighted minimumnorm algorithmrdquo IEEE Transactions on Signal Processing vol45 no 3 pp 600ndash616 1997

[15] D PWipf and B D Rao ldquoSparse Bayesian learning for basis sel-ectionrdquo IEEETransactions on Signal Processing vol 52 no 8 pp2153ndash2164 2004

[16] P Schniter L C Potter and J Ziniel ldquoFast bayesian matchingpursuitrdquo in Proceedings of the Information Theory and Applica-tions Workshop pp 326ndash332 San Diego Calif USA February2008

[17] S G Mallat and Z Zhang ldquoMatching pursuits with time-frequency dictionariesrdquo IEEE Transactions on Signal Processingvol 41 no 12 pp 3397ndash3415 1993

[18] J A Tropp and A C Gilbert ldquoSignal recovery from randommeasurements via orthogonal matching pursuitrdquo IEEE Trans-actions on Information Theory vol 53 no 12 pp 4655ndash46662007

[19] D L Donoho Y Tsaig I Drori and J Starck ldquoSparse solutionof underdetermined systems of linear equations by stagewiseorthogonal matching pursuitrdquo IEEE Transactions on Informa-tion Theory vol 58 no 2 pp 1094ndash1121 2012

[20] MA Efroymson ldquoMultipleAnalysis Regressionrdquo inMathemat-ical Methods for Digital Computers vol 1 pp 191ndash203 1960

[21] D Needell and R Vershynin ldquoUniform uncertainty principleand signal recovery via regularized orthogonal matching pur-suitrdquo Foundations of Computational Mathematics vol 9 no 3pp 317ndash334 2009

[22] P M T Broersen ldquoSubset regression with stepwise directedsearchrdquo Journal of the Royal Statistical Society C vol 35 no 2pp 168ndash177 1986

[23] W Dai and O Milenkovic ldquoSubspace pursuit for compressivesensing signal reconstructionrdquo IEEE Transactions on Informa-tion Theory vol 55 no 5 pp 2230ndash2249 2009

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VLSI Design



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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



The convex relaxation algorithms approximate the nons-mooth and nonconvex ℓ

0-norm by functions that are easier

to handle The resulting problem can be solved by means ofstandard optimization techniques Well-known instances ofalgorithms based on such an approach are Basis Pursuit (BP)[13] and FOCUSS [14] which approximate the ℓ

0-norm by

the ℓ1-norm and ℓ

119901-norm (119901 lt 1) respectively

The Bayesian algorithms express the problem as thesolution of a Bayesian inference problem and apply statisticaltools to solve it that is assuming a prior distribution forthe unknown coefficients that favors sparsity They developa maximum a posteriori estimator that incorporates theobservation There are many algorithms that incorporatesome of these features For example identify a region ofsignificant posterior mass [15] or average over most probablemodels [16] One key ingredient of Bayesian algorithms is thechoice of a proper prior on the sought sparse vector

The pursuit algorithms for sparse signal recovery are agreedy approach that iteratively refines the current estimatefor the coefficient vector x by modifying one or severalcoefficients chosen to yield a substantial improvement inapproximating the signal The family of pursuit algorithmsincludes several approaches according to the way of updatingthe support set single or multiple algorithmsThe algorithmsof single updating support set gradually increase the supportby sequentially adding new atoms The complexity of thesealgorithms is lower than the complexity of BP However theyrequire more measurements 119872 for accurate reconstructionand they often have an effect on empirical work and donot offer the strong theoretical guarantees Single algorithmsinclude matching pursuit (MP) [17] orthogonal matchingpursuit (OMP) [18] However for many applications singleupdating support set does not offer adequate performanceso researchers have developed more sophisticated pursuitmethods that work better in practice and yield essentiallyoptimal theoretical guarantees called multiple algorithmsThese techniques depend on several enhancements to thebasic greedy framework (1) selecting multiple columns periteration (2) pruning the set of active columns at eachstep (3) solving the least squares problems iteratively (4)theoretical analysis using the RIP bound There are manyalgorithms that incorporate some of these features Forexample stagewise orthogonal matching pursuit (StOMP)[19] hard thresholding pursuit (HTP) [20] regularizedorthogonal matching pursuit algorithm (ROMP) [21] whichwas the first greedy technique whose analysis was sup-ported by a RIP bound compressive sampling matchingpursuit (CoSaMP) [22] which was the first algorithm toassemble these ideas to obtain essentially optimal perfor-mance guarantees and the subspace pursuit (SP) [23] andso forth Sparsity-predict in these algorithms bring to theperfect performance However once the sparsity is falsepredicted many signals cannot be reconstructed accuratelyAll in all pursuit algorithms have often been considerednaive in part because there are contrived examples wherethe approach fails spectacularly However recent researchhas clarified that greedy pursuits succeed empirically andtheoretically in many situations where convex relaxationworks

NoYes

Input Φ y

Threshold processing

i | |c (i)| gt ts

T985747= K indices whit the largestcorrelation in solved x

Proj (y ΦT

985747)

T985747good

enough

T985747= T

985747minus1 ⋃K indices whitthe largest correlation

Solve x with (Φ998400

T985747lowast Φ )


T985747

T985747

lowast y

Correlation cal Φ lowast y 985747minus1

r

Figure 1 Description of reconstruction algorithms for 119870-sparsesignals

3 Sparse Signal Recovery by StepwiseSubspace Pursuit

Figure 1 depicts the schematic representation of the proposedSSP algorithm The ℓth iteration applies matched filtering tothe current residual and gets a candidate setΦ lowast yℓminus1

119903 which

contains a small number of significant nonzero values Thenwe eliminate useless information in the candidate throughthreshold processing 119894 | |119888(119894)| gt 119905

119904 and select 119870 indices

which are considered to be reliable on some iteration stepswith the largest correlation by interim estimate We mergethe 119870 indices of newly selected coordinates with the pre-vious support estimate thereby updating the intermediateestimate ℓ We have the new approximation x supportedin ℓ with coefficients given by (Φ1015840

ℓ lowast Φℓ)


ℓ lowast

y The updated support estimate can be gained from approx-imation x by the largest correlation We then project thevector y on the columns of Φ

119879ℓ belonging to the updated

support and check the stopping condition and if it is not yettime to stop we set ℓ = ℓ + 1 and go to the next iteration

The algorithm is depicted in Algorithm 1 The maincontribution of the SSP reconstruction algorithm is that itgenerates a list of candidates sequentially and incorporates asimple method for re-evaluating the reliability of all candi-dates at iteration thus gaining the correlation of candidatesbefore the operation of SP with the correlation at iteration ofthe method





1198790)




minus1

lowastΦ1015840

119904 lowast y


119879119904)











PSNR 198831 213572 212388 254261TIME 39 05 24 39




5 Conclusion



10 15 20 25 30 35 40 450

01

02

03

04

05

06

07

08

09

1

Sparsity level

Freq

uenc

y of

exac

t rec

onstr

uctio

n

SPOMP

COSAMPSSP



10 15 20 25 30 35 40 450

01

02

03

04

05

06

07

08

09

1

Sparsity level

Freq

uenc

y of

exac

t rec

onstr

uctio

n


SPOMP

COSAMPSSP






Acknowledgments



References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













1198790)




minus1

lowastΦ1015840

119904 lowast y


119879119904)











PSNR 198831 213572 212388 254261TIME 39 05 24 39




5 Conclusion



10 15 20 25 30 35 40 450

01

02

03

04

05

06

07

08

09

1

Sparsity level

Freq

uenc

y of

exac

t rec

onstr

uctio

n

SPOMP

COSAMPSSP



10 15 20 25 30 35 40 450

01

02

03

04

05

06

07

08

09

1

Sparsity level

Freq

uenc

y of

exac

t rec

onstr

uctio

n


SPOMP

COSAMPSSP






Acknowledgments



References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










10 15 20 25 30 35 40 450

01

02

03

04

05

06

07

08

09

1

Sparsity level

Freq

uenc

y of

exac

t rec

onstr

uctio

n

SPOMP

COSAMPSSP



10 15 20 25 30 35 40 450

01

02

03

04

05

06

07

08

09

1

Sparsity level

Freq

uenc

y of

exac

t rec

onstr

uctio

n


SPOMP

COSAMPSSP






Acknowledgments



References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article Sparse Signal Recovery by Stepwise ...downloads.hindawi.com/journals/ijdsn/2013/798537.pdfSparse Signal Recovery by Stepwise Subspace Pursuit in Compressed Sensing