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Research ArticleCompressed Sensing Based Apple ImageMeasurement Matrix Selection
Ying Xiao1 Wanlin Gao1 Ganghong Zhang1 and Han Zhang2
1College of Information and Electrical Engineering China Agriculture University Beijing 100083 China2Institute of Acoustics Chinese Academy of Sciences Beijing 100190 China
Correspondence should be addressed to Wanlin Gao gaowlincaueducn
Received 27 December 2014 Revised 22 April 2015 Accepted 14 May 2015
Academic Editor Jinjun Chen
Copyright copy 2015 Ying Xiao et alThis is an open access article distributed under theCreativeCommonsAttributionLicense whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The purpose of this paper is to design a measurement matrix of apple image based on compressed sensing to realize low costsampling apple image Compressed sensing based apple image sampling method makes a breakthrough to the limitation of theNyquist sampling theorem By investigating the matrix measurement signal the method can project a higher dimensional signalto a low-dimensional space for data compression and reconstruct the original image using less observed values But this methodrequires that the measurement matrix and sparse transformation base satisfy the conditions of RIP or incoherence Real timeacquiring and transmitting apple image has great importance for monitoring the growth of fruit trees and efficiently picking appleThis paper firstly chooses sym5wavelet base as apple image sparse transformation base and then it uses Gaussian randommatricesBernoulli random matrices Partial Orthogonal random matrices Partial Hadamard matrices and Toeplitz matrices to measureapple images respectively Using the same measure quantity we select the matrix that has best reconstruction effect as the appleimagemeasurementmatrixThe reconstruction PSNRvalues and runtimewere used to compare and contrast the simulation resultsAccording to the experiment results this paper selects Partial Orthogonal random matrices as apple image measurement matrix
1 Introduction
Conventional approaches to sample signal or images includefour steps sampling compression transmission and decom-pression In order to reduce the cost of sensor acquisitiontransmission and storage a large number of researchersfocus on data compressing and dimension reduction andso forth But prior studies require users to first obtain theoriginal data and store it and then compress and transmitthe data This process brings a great burden to the sensorIn practical applications it is hard to satisfy the demandof collecting data in real time On the contrary if weuse ldquosampling and compressionrdquo technology and obtain thecompressed signal directly it will enormously reduce thetime and store space demand of acquisition data Randommatrix is common and easy to analyze The process of thesignal acquisition can be projected to dimension reductionwhich can greatly realize data compression Thereby CShas gathered widespread attention from researchers Forexample Toeplitzmatrix and circulantmatrix were applied to
the linear dynamic systems [1ndash4] The block diagonal matrixwas applied to the distributed sensor system [5] and the initialforecast linear systems [6] These random matrices acquiregood effects in practice
Compressed sensing is a new sampling theory proposedin recent years this theory makes a breakthrough on thelimitation of the Nyquist sampling theorem It has beenproven that using a rate which is lower than the Nyquistrate to sample the signal and reconstruct the data is feasibleData acquisitionmethods based on this theory use ldquosamplingand compressionrdquo technology and obtain the compressedsignal directly CS requires the number of samples to be farless than conventional signal processing and these samplescan recover the original signal exactly This reduces the costof the signalrsquos acquisition and transmission CS brings arevolutionary breakthrough for information acquisition andprocessing technology Currently CS is applied in manyfields especially image processing A number of results havedeveloped based on CS theory For example Rice Universityprofessor Duarte and his colleagues developed a single-pixel
Hindawi Publishing CorporationInternational Journal of Distributed Sensor NetworksVolume 2015 Article ID 901073 7 pageshttpdxdoiorg1011552015901073
2 International Journal of Distributed Sensor Networks
[7] compressive sensing camera which was based on CStheory This design included a new camera structure and isthe most famous proof of concept model for CS
Based on ldquosampling and compressionrdquo technology weacquire signal by projecting the high dimension signal to thelow dimension space using measurement matrix Measure-ment matrix is designed according to the characteristics ofthe signalThemeasurementmatrix and the sparse base mustsatisfy the restricted isometry property (RIP) conditionswhich can reconstruct the original signal Toeplitz matrixand circulant matrix have been applied in the channelestimation and synthetic aperture radar and so forth [2 8ndash12]Bajwa et al study some matrix elements that obey Bernoullidistribution system in the literature and follow-up studiesargue that the matrix element is bounded random matrixor distributed Gaussian random matrix in this case thesematrices can meet RIP conditions of the 2k-sparse vectorquantity on time domain [2 10]
As the apple is one ofChinarsquosmost important fruit speciesthe refined management of applersquos growing process is impor-tant to improve the yield which means that the efficiency ofapple image information system is of great importance Inthis paper we introduce the compressed sensing theory to theapple image acquisition process and improve both the speedand efficiency effectively In the process of applersquos growth a lotof apples images need to be acquired and transferred to mon-itor the growth status of apple pest control and the robotsmovementsThis paper researches on usingGaussian randommatrices Bernoulli random matrices Partial Orthogonalrandom matrices Partial Hadamard matrices and Toeplitzmatrix to measure the apple image With the same quantitycondition we select the best construction matrix as the appleimage measurement matrix The apple image measurementmatrix is applied to the applersquos manufacturing operationsprocess and overcomes the traditional method limit It canreduce the cost of data acquisition and transmission Thus ithas very important practical significance
The main contributions of this paper are as follows wegive the first trial to apply the image acquisition to theapple growing process We use ldquosampling and compressionrdquotechnology and obtain the compressed signal directly andit can enormously reduce the time and store space demandof acquisition data By conducting experiments on real lifeimages we figure out the best measurement matrix anddemonstrate the effectiveness and efficiency of our methods
2 Materials and Methods
21 Theory of Compressed Sensing Compressed sensing cansample and compress signals at the same time It can obtainthe compressed signal directly which includes three coreproblems sparse representation of a signal the measurementmatrix and the reconstruction algorithm [13]The theoreticalframework is shown in Figure 1
CS relies on two principles sparsity and incoherenceDuring signal conversation process signal is representedby several big coefficients and leave out the rest of thecoefficient which is not distortion of the signal The mostpractical signals (such as voice and video signal) can have
sparse representation in some transform domain A sparserepresentation of the signal is to change the original signalthrough a mathematical transformation after which thesignal is projected onto the orthogonal transformation basisIf most of the transform coefficients have an absolute value ofzero we call it a sparse signal after the transformation If thevalue of most of the transform coefficients is small and fewlarge coefficients can capture the main information about theobject we call it an approximation sparse signal
119883 = [1198831 1198832 1198833 119883
119899] is a one-dimensional discrete
signal of length 119873 which represents a sparse form of theoriginal signal in a transform basis that is
119883 = Ψ120572 (1)
In this expression the coefficient 119886119894
= [119909 120595119894] the
coefficient vector is 119886 = Ψ119879119883 the number of 120572rsquos nonzero
coefficients is 119896 The transfer matrix Ψ = [1205951 1205952 1205953 120595119899]is the basis of the orthogonal transformation It is moreflexible in choosing them according to the characteristicsof the original signal The typical frequently used sparsetransformation bases are Fourier basis DCT basis waveletbasis (wavelet domain) and so forth
After the sparse transformation of the original signal itis different from the traditional Nyquist sampling theoremcompressed sensing samples a signal through designingthe measurement matrix Φ Φ can make high-dimensionalsignals projected onto a low-dimensional space to realizethe compression of data Finally with the measurementvector in the low-dimensional projection domain we canreconstruct the original signal nearly lossless as a linearequation Consider
119910 = Φ119883 (2)
In the formula above 119910 represents linear measurementof the sparse signal 119883 119910 isin 119877
119872 is an 119872 dimensional vectorand 119883 isin 119877
119873 is the unknown signal which needs to bereconstructed Then plugging (1) into (2)
119910 = ΦΨ120572 (3)
Φ is called the measurement matrix Φ isin 119877119872times119873
(119872 ≪
119873) 119872 is the set of measurement times in the process ofcompressed sensing theory and 119873 is the length of thesignal Measured value of the unknown signal 119883 under themeasurement matrix Φ is 119910 The measured value 119910 is thecompressed representation of119883
The coherence between the measurement matrix Φ andthe representation basis Ψ is
120583 (ΦΨ) = radic119909 max1le119896 119895le119899
10038161003816100381610038161003816⟨120593119896 120595119895⟩10038161003816100381610038161003816 120583 (ΦΨ) isin [1 radic119899] (4)
The coherence measures the largest correlation betweenany two elements of Φ and Ψ which contain correlatedelements [14] the coherence 120583 is large Otherwise it is small
Signal reconstruction refers to the process inwhichwe geta sparse signal with length119873 or the original discrete signal119883again [15] through obtaining themeasurement vector 119910 by119872
International Journal of Distributed Sensor Networks 3
Measurementmatrix
observation
Thereconstructed
signal
Signalcharacteristics
Featureextraction
Signalreconstruction
Sparse samplingsignal
Figure 1 Signal processing framework based on compressed sensing
times The measurement matrix and sparse transformationbase satisfy the conditions of RIP or incoherence We canreconstruct the original signal119883 accurately or approximate itprecisely by solving the problemof optimal 119871
0norm119883meets
the type shown below
119883= argmin 1198830
st 119910 = Φ119883
(5)
Solving the above equation is an NP-hard problemFortunately prior work in compressed sensing has providedus with numerically feasible alternatives to this NP-hardproblem According to the theory of functional analysis andconvex optimization onemajor approach is that we can solvethe 1198710problem approximately through the 119871
1norm [16 17]
Common reconstruction algorithms include the interior-point method [18] Gradient Projection for Sparse Recon-struction (GPSR) [19] Homotopy algorithm [20] MP algo-rithm [21] Basis Pursuit (BP) algorithm [22] and Orthog-onal Matching Pursuit algorithm The interior-point algo-rithm is slow but generates very precise reconstructionsGPRS exhibits rapid computational speed and reconstructionresults are generally better Homotopy algorithm is morepractical for small-scale problems
MP algorithm reconstruction is fast but the reconstruc-tion result is bad BP algorithm suffers from slow recon-struction speed but the results are better OMP algorithmuses selection criteria in the matching pursuit In the rebuildprocess every iteration gets an atom that belongs to supportset of 119865 from 119909 By orthogonalization of the selected atomscollection we ensure the optimality of the iteration andreduce the number of iterations In this paper we have chosenthe OMP algorithm for apple image reconstruction
22 Selection on Measurement Matrix Another premiseof compressed sensing theory is incoherence Incoherencebelongs to the sensor model The measurement matrix Φ
and sparse base Ψ are multiplied to calculate the sensingmatrix In order to reconstruct the sparse signal Candes andTao proved that Φ must satisfy the isometry property (RIP)conditions For any119870-sparse signal and constant it satisfies
(1minus 120575119896) 119888
22 le
1003817100381710038171003817Φ1198791198881003817100381710038171003817
22 le (1+ 120575
119896) 119888
22 forall119888 isin 119877
|119879| (6)
119879 sub 1 119873 |119879| le 119870 Φ119879is a child of the matrix
composed of the related column indicated by the index119879withthe size of 119870 times |119879| The matrix Φ then satisfies the restrictedisometry property The limited isometric property defines animportant equidistant constant 120575
119896 it indicates that as long as
120575119896is not close to 1 it can approximately promise the Euclidean
distance of the constant 119878-order sparse signal This ensuresthe uniqueness of the reconstructed signal
Designing the measurement matrix should satisfy thefollowing conditions transformation matrix should satisfyRIP conditions or the measurement matrix and trans-formation matrix should meet incoherence The previousresearches prove that Gaussian random matrices partialrandom Fourier matrix Partial Hadamard matrix Toeplitzmatrix Partial Orthogonal random matrices Bernoullirandom matrix and so forth are feasible solutions Thispaper selects the Gaussian random matrices Bernoulli ran-dom matrices Partial Orthogonal random matrices PartialHadamardmatrices andToeplitzmatrices on the apple imagemeasurement
(1) Gaussian Random Measurement Matrix The probabilitydensity function of random variable 119883 is
119891 (119909) =1
radic2120587120590119890minus(119909minus120583)
221205902
minusinfin lt 119909 lt infin (7)
where 120583 120590 (120590 gt 0) are called parameters of normaldistribution or Gaussian distribution 119883 is denoted by 119883 sim
119873(120583 1205902) For Gaussian distribution we have 119864(119883) = 120583 and
119863(119883) = 1205902
The elements 119909119894119895
of Gaussian random matrices Φ areindependent random variables which obey the Gaussiandistribution with the mean of 0 and the square deviation of1119899 namely 119909
119894119895sim 119873(0 1119899)
(2) Bernoulli Random Measurement Matrices Bernoulli dis-tribution is a discrete probability distribution when theBernoulli trial succeeds it results in Bernoulli randomvariable as a value 119860 Otherwise it results in Bernoullirandomvariable as non-119860 Bernoulli randommatrices are theelements of thematrixΦwith the same probability with 1radic119899
or minus1radic119899
4 International Journal of Distributed Sensor Networks
Gaussian
Bernoulli
OrthogonalHadamard
Toeplitz
39540
40541
41542
42543
43544
445
342 344 346 348 35 352 354 356
Tim
e
PSNR
Figure 2 Reconstruction time and PSNR values of 5 differentmeasurement matrices with119872 = 150
Gaussian Bernoulli
Orthogonal
Hadamard
Toeplitz
44454647484950515253
345 35 355 36 365
Tim
e
PSNR
Figure 3 Reconstruction time and PSNR values of 5 differentmeasurement matrices with119872 = 160
(3) Partial Orthogonal RandomMatrices If119873 order square119860meets 119860119879119860 = 119864 119860 is called orthogonal matrix the sufficientand necessary conditions of the119873 order square matrix 119860 arethe fact that 119860 column vector is standard orthogonal vectorgroup
The method of constructing Partial Orthogonal mea-surement is illustrated as follows We first generate 119873 lowast 119873
dimension orthogonalmatrix119879 and thenwe randomly select119872 rows from 119879 construct the119872 lowast 119873matrix and then unitethe119873 columns to get the measurement matrix
(4) Partial Hadamard Matrices Hadamard matrix is orthog-onal square matrix composed by elements of +1 and 1The method of constructing Partial Hadamard matrix isillustrated as follows We first generate Hadamard matrix119873 lowast 119873 and then we from the generated matrix randomlyselect119872 row vector and generate119872lowast119873Hadamard randommeasurement matrix
(5) ToeplitzMatrix Toeplitzmatrix is also known as diagonal-constant matrix elements of line parallel to main diagonalline are the same The method of constructing Toeplitz ran-dommeasurement is illustrated as follows First we generatea random vector and then the random vector with relatedfunctions to generate the corresponding Toeplitz matricessecond we select the first119872 rows of the Toeplitz constructedas 119872 lowast 119873 Toeplitz random measurement matrix
Gaussian
Bernoulli
Orthogonal
Hadamard
Toeplitz
46474849505152535455
345 35 355 36 365 37
Tim
e
PSNR
Figure 4 Reconstruction time and PSNR values of 5 differentmeasurement matrices with119872 = 170
GaussianBernoulli
OrthogonalHadamard
Toeplitz
55555
56565
57575
58585
59
35 355 36 365 37 375 38
Tim
e
PSNR
Figure 5 Reconstruction time and PSNR values of 5 differentmeasurement matrices with119872 = 180
3 Results and Discussion
This paper chooses an apple image represented by 256 lowast
256RGB pixels We use sym5 wavelet bases to representthe original apple image sparsely with random Gaussianmeasurement matrix (119883 sim 119873(120583 120590
2)) Bernoulli random
matrices (119883 sim 119861(119899 119901)) Partial Orthogonal randommatricesand Partial Hadamard matrices and Toeplitz matrices arethen used for observational measuring of the signal Finallythe apple image is reconstructed by the OMP algorithmThe reconstructed PSNR values and runtime were used tocompare and contrast the simulation results The quality ofthe 5 reconstructed apple images is measured using PSNRFigures 2ndash5 show time and PSNR scatter diagrams for 5differentmeasurementmatrices used to reconstruct the appleimage when the sampling point119872 takes different values
In this experiment we vary the number of samplingpixels for example 119872 = 150 119872 = 160 119872 = 170119872 = 180 use sym5 wavelet base as sparse transformationbase sample with five different measurement matrices andreconstruct images throughOMP algorithmThe experimentverified that the colorized apple image constructed by thepartial orthogonal randommatrices with a sym5wavelet basehad a higher quality
Figures 6ndash9 compare the restored apple image by the CSreconstruction algorithm with the Toeplitz matrix and theOrthogonal matrix with the original apple image when thesampling point119872 assumes different values
International Journal of Distributed Sensor Networks 5
Original image Toeplitz matrix reconstruction image Orthogonal matrix reconstruction image
Figure 6 Original image Toeplitz matrix reconstruction image and partial orthogonal matrix reconstruction image with 119872 = 150
Original image Toeplitz matrix reconstruction image Orthogonal matrix reconstruction image
Figure 7 Original image Toeplitz matrix reconstruction image and partial orthogonal matrix reconstruction image with 119872 = 160
Original image Toeplitz matrix reconstruction image Orthogonal matrix reconstruction image
Figure 8 Original image Toeplitz matrix reconstruction image and partial orthogonal matrix reconstruction image with 119872 = 170
6 International Journal of Distributed Sensor Networks
Original image Toeplitz matrix reconstruction image Orthogonal matrix reconstruction image
Figure 9 Original image Toeplitz matrix reconstruction image and partial orthogonal matrix reconstruction image with 119872 = 180
The above results show the reconstruction precisionof the Orthogonal measurement matrix and the Toeplitzmeasurement matrix Under the condition of same samplingpoints the quality of reconstruction image with the Orthog-onal measurement matrix is much better than that of thereconstruction image with Toeplitz measurement matrix
In this paper we choose the sampling points from 150 to180 If we use too many points that is more than 180 pointsthe significance of using sampling methods is minimizedWhile if we use too few points that is less than 150 pointsthe effectiveness of our methods decline Hence betweenthe tradeoff of effectiveness and complexity we choose thesampling points from 150 and 180
Since PSNR is mostly commonly used to measure thequality of reconstruction images we choose the PSNR scoreas measurement metric SSIM is also a quality reconstructionmetric but it is more complicated to compute for whichthe formula involves a value for each pixel We conductexperiments using SSIM metric and it demonstrates thesuperiority of our method but it takes too much space in thepaper we do not list the details of the SSIM results
4 Conclusion
This paper uses sym5 wavelet base as apple image sparsetransformation base and then it uses Gaussian randommatrices Bernoulli random matrices Partial Orthogonalrandom matrices Partial Hadamard matrices and Toeplitzmatrices to measure apple image respectively Using thesame measure quantity we select the matrix that has bestreconstruction effect as the apple imagemeasurementmatrixAccording to the experiment results this paper selects PartialOrthogonal random matrices as apple image measurementmatrix This measurement matrix is used for the data pro-cessing of apple image it can better realize image datacompression and greatly reduced apple image sampling andtransport cost which has important application significancein the process of apple production operation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank Wanlin Gao for helpfulcomments andpointersThisworkwas supported byNationalKeyTechnologyRampDProgramofChina during the 12th Five-Year Plan Period (Grant no 2012BAD35B02)
References
[1] H Rauhut ldquoCirculant and Toeplitz matrices in compressedsensingrdquo httparxivorgabs09024394
[2] J Haupt W U Bajwa G Raz and R Nowak ldquoToeplitz com-pressed sensing matrices with applications to sparse channelestimationrdquo IEEE Transactions on Information Theory vol 56no 11 pp 5862ndash5875 2010
[3] J Romberg ldquoCompressive sensing by random convolutionrdquoSIAM Journal on Imaging Sciences vol 2 no 4 pp 1098ndash11282009
[4] BM Sanandaji T L Vincent andM BWakin ldquoConcentrationof measure inequalities for compressive Toeplitz matrices withapplications to detection and system identificationrdquo in Proceed-ings of the 49th IEEE Conference on Decision and Control (CDCrsquo10) pp 2922ndash2929 December 2010
[5] J Y Park H L Yap C J Rozell and M B Wakin ldquoConcentra-tion of measure for block diagonal matrices with applicationsto compressive signal processingrdquo IEEE Transactions on SignalProcessing vol 59 no 12 pp 5859ndash5875 2011
[6] M B Wakin B M Sanandaji and T L Vincent ldquoOn theobservability of linear systems from random compressivemeasurementsrdquo in Proceedings of the 49th IEEE Conference onDecision andControl (CDC rsquo10) pp 4447ndash4454December 2010
[7] M F Duarte M A Davenport D Takhar et al ldquoSingle-Pixel imaging via compressive samplingrdquo IEEE Signal ProcessingMagazine vol 25 no 2 pp 83ndash91 2008
International Journal of Distributed Sensor Networks 7
[8] H Rauhut ldquoCirculant and Toeplitz matrices in compressedsensingrdquo in Proceedings of the Signal Processing with AdaptiveSparse Structured Representations (SPARS rsquo09) Saint MaloFrance April 2009
[9] J Romberg ldquoCompressive sensing by random convolutionrdquoSIAM Journal on Imaging Sciences vol 2 no 4 pp 1098ndash11282009
[10] W U Bajwa J Haupt G Raz and R Nowak ldquoCompressedchannel sensingrdquo in Proceedings of the 42nd Annual Conferenceon Information Sciences and Systems (CISS rsquo08) pp 5ndash10 March2008
[11] J A Tropp M B Wakin M F Duarte D Baron and R GBaraniuk ldquoRandomfilters for compressive sampling and recon-structionrdquo in Proceedings of the IEEE International Conferenceon Acoustics Speech and Signal Processing (ICASSP rsquo06) vol 3pp 872ndash875 May 2006
[12] W U Bajwa J D Haupt G M Raz S J Wright and R DNowak ldquoToeplitz-structured compressed sensing matricesrdquo inProceedings of the IEEESP 14th Workshop on Statistical SignalProcessing (SSP rsquo07) pp 294ndash298 IEEE Madison Wis USAAugust 2007
[13] 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
[14] E J Candes and M B Wakin ldquoAn introduction to compressivesamplingrdquo IEEE Signal Processing Magazine vol 25 no 2 pp21ndash30 2008
[15] C Gao H Li and L Ma ldquoVoice signal compression andreconstruction method based on the theory of the compressivesensingrdquo in Proceedings of the 11th National Conference onMan-Machine Voice Communication 2011
[16] I Drori and D L Donoho ldquoSolution of l1minimization
problems by LARShomotopy methodsrdquo in Proceedings of theIEEE International Conference on Acoustics Speech and SignalProcessing (ICASSP rsquo06) vol 3 IEEE Toulouse France May2006
[17] P J Garrigues and L E Ghaoui ldquoAn homotopy algorithmfor Lasso with online observationsrdquo in Proceedings of theAnnual Conference on Neural Information Processing SystemsVancouver Canada December 2008
[18] S-J Kim K Koh M Lustig S Boyd and D GorinevskyldquoAn interior-point method for large-scale 119897
1-regularized least
squaresrdquo IEEE Journal of Selected Topics in Signal Processing vol1 no 4 pp 606ndash617 2007
[19] M A T Figueiredo R D Nowak and S J Wright ldquoGradientprojection for sparse reconstruction application to compressedsensing and other inverse problemsrdquo IEEE Journal on SelectedTopics in Signal Processing vol 1 no 4 pp 586ndash598 2007
[20] D L Donoho and Y Tsaig ldquoFast solution of ℓ1-normminimiza-
tion problems when the solution may be sparserdquo Tech RepDepartment of Statistics Stanford University Stanford CalifUSA 2008
[21] T T Do L Gan N Nguyen and T D Tran ldquoSparsity adaptivematching pursuit algorithm for practical compressed sensingrdquoin Proceedings of the 42nd Asilomar Conference on SignalsSystems and Computers pp 581ndash587 Pacific Grove Calif USAOctober 2008
[22] S S Chen D L Donoho and M A Saunders ldquoAtomicdecomposition by basis pursuitrdquo SIAM Journal on ScientificComputing vol 20 no 1 pp 33ndash61 1998
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DistributedSensor Networks
International Journal of
2 International Journal of Distributed Sensor Networks
[7] compressive sensing camera which was based on CStheory This design included a new camera structure and isthe most famous proof of concept model for CS
Based on ldquosampling and compressionrdquo technology weacquire signal by projecting the high dimension signal to thelow dimension space using measurement matrix Measure-ment matrix is designed according to the characteristics ofthe signalThemeasurementmatrix and the sparse base mustsatisfy the restricted isometry property (RIP) conditionswhich can reconstruct the original signal Toeplitz matrixand circulant matrix have been applied in the channelestimation and synthetic aperture radar and so forth [2 8ndash12]Bajwa et al study some matrix elements that obey Bernoullidistribution system in the literature and follow-up studiesargue that the matrix element is bounded random matrixor distributed Gaussian random matrix in this case thesematrices can meet RIP conditions of the 2k-sparse vectorquantity on time domain [2 10]
As the apple is one ofChinarsquosmost important fruit speciesthe refined management of applersquos growing process is impor-tant to improve the yield which means that the efficiency ofapple image information system is of great importance Inthis paper we introduce the compressed sensing theory to theapple image acquisition process and improve both the speedand efficiency effectively In the process of applersquos growth a lotof apples images need to be acquired and transferred to mon-itor the growth status of apple pest control and the robotsmovementsThis paper researches on usingGaussian randommatrices Bernoulli random matrices Partial Orthogonalrandom matrices Partial Hadamard matrices and Toeplitzmatrix to measure the apple image With the same quantitycondition we select the best construction matrix as the appleimage measurement matrix The apple image measurementmatrix is applied to the applersquos manufacturing operationsprocess and overcomes the traditional method limit It canreduce the cost of data acquisition and transmission Thus ithas very important practical significance
The main contributions of this paper are as follows wegive the first trial to apply the image acquisition to theapple growing process We use ldquosampling and compressionrdquotechnology and obtain the compressed signal directly andit can enormously reduce the time and store space demandof acquisition data By conducting experiments on real lifeimages we figure out the best measurement matrix anddemonstrate the effectiveness and efficiency of our methods
2 Materials and Methods
21 Theory of Compressed Sensing Compressed sensing cansample and compress signals at the same time It can obtainthe compressed signal directly which includes three coreproblems sparse representation of a signal the measurementmatrix and the reconstruction algorithm [13]The theoreticalframework is shown in Figure 1
CS relies on two principles sparsity and incoherenceDuring signal conversation process signal is representedby several big coefficients and leave out the rest of thecoefficient which is not distortion of the signal The mostpractical signals (such as voice and video signal) can have
sparse representation in some transform domain A sparserepresentation of the signal is to change the original signalthrough a mathematical transformation after which thesignal is projected onto the orthogonal transformation basisIf most of the transform coefficients have an absolute value ofzero we call it a sparse signal after the transformation If thevalue of most of the transform coefficients is small and fewlarge coefficients can capture the main information about theobject we call it an approximation sparse signal
119883 = [1198831 1198832 1198833 119883
119899] is a one-dimensional discrete
signal of length 119873 which represents a sparse form of theoriginal signal in a transform basis that is
119883 = Ψ120572 (1)
In this expression the coefficient 119886119894
= [119909 120595119894] the
coefficient vector is 119886 = Ψ119879119883 the number of 120572rsquos nonzero
coefficients is 119896 The transfer matrix Ψ = [1205951 1205952 1205953 120595119899]is the basis of the orthogonal transformation It is moreflexible in choosing them according to the characteristicsof the original signal The typical frequently used sparsetransformation bases are Fourier basis DCT basis waveletbasis (wavelet domain) and so forth
After the sparse transformation of the original signal itis different from the traditional Nyquist sampling theoremcompressed sensing samples a signal through designingthe measurement matrix Φ Φ can make high-dimensionalsignals projected onto a low-dimensional space to realizethe compression of data Finally with the measurementvector in the low-dimensional projection domain we canreconstruct the original signal nearly lossless as a linearequation Consider
119910 = Φ119883 (2)
In the formula above 119910 represents linear measurementof the sparse signal 119883 119910 isin 119877
119872 is an 119872 dimensional vectorand 119883 isin 119877
119873 is the unknown signal which needs to bereconstructed Then plugging (1) into (2)
119910 = ΦΨ120572 (3)
Φ is called the measurement matrix Φ isin 119877119872times119873
(119872 ≪
119873) 119872 is the set of measurement times in the process ofcompressed sensing theory and 119873 is the length of thesignal Measured value of the unknown signal 119883 under themeasurement matrix Φ is 119910 The measured value 119910 is thecompressed representation of119883
The coherence between the measurement matrix Φ andthe representation basis Ψ is
120583 (ΦΨ) = radic119909 max1le119896 119895le119899
10038161003816100381610038161003816⟨120593119896 120595119895⟩10038161003816100381610038161003816 120583 (ΦΨ) isin [1 radic119899] (4)
The coherence measures the largest correlation betweenany two elements of Φ and Ψ which contain correlatedelements [14] the coherence 120583 is large Otherwise it is small
Signal reconstruction refers to the process inwhichwe geta sparse signal with length119873 or the original discrete signal119883again [15] through obtaining themeasurement vector 119910 by119872
International Journal of Distributed Sensor Networks 3
Measurementmatrix
observation
Thereconstructed
signal
Signalcharacteristics
Featureextraction
Signalreconstruction
Sparse samplingsignal
Figure 1 Signal processing framework based on compressed sensing
times The measurement matrix and sparse transformationbase satisfy the conditions of RIP or incoherence We canreconstruct the original signal119883 accurately or approximate itprecisely by solving the problemof optimal 119871
0norm119883meets
the type shown below
119883= argmin 1198830
st 119910 = Φ119883
(5)
Solving the above equation is an NP-hard problemFortunately prior work in compressed sensing has providedus with numerically feasible alternatives to this NP-hardproblem According to the theory of functional analysis andconvex optimization onemajor approach is that we can solvethe 1198710problem approximately through the 119871
1norm [16 17]
Common reconstruction algorithms include the interior-point method [18] Gradient Projection for Sparse Recon-struction (GPSR) [19] Homotopy algorithm [20] MP algo-rithm [21] Basis Pursuit (BP) algorithm [22] and Orthog-onal Matching Pursuit algorithm The interior-point algo-rithm is slow but generates very precise reconstructionsGPRS exhibits rapid computational speed and reconstructionresults are generally better Homotopy algorithm is morepractical for small-scale problems
MP algorithm reconstruction is fast but the reconstruc-tion result is bad BP algorithm suffers from slow recon-struction speed but the results are better OMP algorithmuses selection criteria in the matching pursuit In the rebuildprocess every iteration gets an atom that belongs to supportset of 119865 from 119909 By orthogonalization of the selected atomscollection we ensure the optimality of the iteration andreduce the number of iterations In this paper we have chosenthe OMP algorithm for apple image reconstruction
22 Selection on Measurement Matrix Another premiseof compressed sensing theory is incoherence Incoherencebelongs to the sensor model The measurement matrix Φ
and sparse base Ψ are multiplied to calculate the sensingmatrix In order to reconstruct the sparse signal Candes andTao proved that Φ must satisfy the isometry property (RIP)conditions For any119870-sparse signal and constant it satisfies
(1minus 120575119896) 119888
22 le
1003817100381710038171003817Φ1198791198881003817100381710038171003817
22 le (1+ 120575
119896) 119888
22 forall119888 isin 119877
|119879| (6)
119879 sub 1 119873 |119879| le 119870 Φ119879is a child of the matrix
composed of the related column indicated by the index119879withthe size of 119870 times |119879| The matrix Φ then satisfies the restrictedisometry property The limited isometric property defines animportant equidistant constant 120575
119896 it indicates that as long as
120575119896is not close to 1 it can approximately promise the Euclidean
distance of the constant 119878-order sparse signal This ensuresthe uniqueness of the reconstructed signal
Designing the measurement matrix should satisfy thefollowing conditions transformation matrix should satisfyRIP conditions or the measurement matrix and trans-formation matrix should meet incoherence The previousresearches prove that Gaussian random matrices partialrandom Fourier matrix Partial Hadamard matrix Toeplitzmatrix Partial Orthogonal random matrices Bernoullirandom matrix and so forth are feasible solutions Thispaper selects the Gaussian random matrices Bernoulli ran-dom matrices Partial Orthogonal random matrices PartialHadamardmatrices andToeplitzmatrices on the apple imagemeasurement
(1) Gaussian Random Measurement Matrix The probabilitydensity function of random variable 119883 is
119891 (119909) =1
radic2120587120590119890minus(119909minus120583)
221205902
minusinfin lt 119909 lt infin (7)
where 120583 120590 (120590 gt 0) are called parameters of normaldistribution or Gaussian distribution 119883 is denoted by 119883 sim
119873(120583 1205902) For Gaussian distribution we have 119864(119883) = 120583 and
119863(119883) = 1205902
The elements 119909119894119895
of Gaussian random matrices Φ areindependent random variables which obey the Gaussiandistribution with the mean of 0 and the square deviation of1119899 namely 119909
119894119895sim 119873(0 1119899)
(2) Bernoulli Random Measurement Matrices Bernoulli dis-tribution is a discrete probability distribution when theBernoulli trial succeeds it results in Bernoulli randomvariable as a value 119860 Otherwise it results in Bernoullirandomvariable as non-119860 Bernoulli randommatrices are theelements of thematrixΦwith the same probability with 1radic119899
or minus1radic119899
4 International Journal of Distributed Sensor Networks
Gaussian
Bernoulli
OrthogonalHadamard
Toeplitz
39540
40541
41542
42543
43544
445
342 344 346 348 35 352 354 356
Tim
e
PSNR
Figure 2 Reconstruction time and PSNR values of 5 differentmeasurement matrices with119872 = 150
Gaussian Bernoulli
Orthogonal
Hadamard
Toeplitz
44454647484950515253
345 35 355 36 365
Tim
e
PSNR
Figure 3 Reconstruction time and PSNR values of 5 differentmeasurement matrices with119872 = 160
(3) Partial Orthogonal RandomMatrices If119873 order square119860meets 119860119879119860 = 119864 119860 is called orthogonal matrix the sufficientand necessary conditions of the119873 order square matrix 119860 arethe fact that 119860 column vector is standard orthogonal vectorgroup
The method of constructing Partial Orthogonal mea-surement is illustrated as follows We first generate 119873 lowast 119873
dimension orthogonalmatrix119879 and thenwe randomly select119872 rows from 119879 construct the119872 lowast 119873matrix and then unitethe119873 columns to get the measurement matrix
(4) Partial Hadamard Matrices Hadamard matrix is orthog-onal square matrix composed by elements of +1 and 1The method of constructing Partial Hadamard matrix isillustrated as follows We first generate Hadamard matrix119873 lowast 119873 and then we from the generated matrix randomlyselect119872 row vector and generate119872lowast119873Hadamard randommeasurement matrix
(5) ToeplitzMatrix Toeplitzmatrix is also known as diagonal-constant matrix elements of line parallel to main diagonalline are the same The method of constructing Toeplitz ran-dommeasurement is illustrated as follows First we generatea random vector and then the random vector with relatedfunctions to generate the corresponding Toeplitz matricessecond we select the first119872 rows of the Toeplitz constructedas 119872 lowast 119873 Toeplitz random measurement matrix
Gaussian
Bernoulli
Orthogonal
Hadamard
Toeplitz
46474849505152535455
345 35 355 36 365 37
Tim
e
PSNR
Figure 4 Reconstruction time and PSNR values of 5 differentmeasurement matrices with119872 = 170
GaussianBernoulli
OrthogonalHadamard
Toeplitz
55555
56565
57575
58585
59
35 355 36 365 37 375 38
Tim
e
PSNR
Figure 5 Reconstruction time and PSNR values of 5 differentmeasurement matrices with119872 = 180
3 Results and Discussion
This paper chooses an apple image represented by 256 lowast
256RGB pixels We use sym5 wavelet bases to representthe original apple image sparsely with random Gaussianmeasurement matrix (119883 sim 119873(120583 120590
2)) Bernoulli random
matrices (119883 sim 119861(119899 119901)) Partial Orthogonal randommatricesand Partial Hadamard matrices and Toeplitz matrices arethen used for observational measuring of the signal Finallythe apple image is reconstructed by the OMP algorithmThe reconstructed PSNR values and runtime were used tocompare and contrast the simulation results The quality ofthe 5 reconstructed apple images is measured using PSNRFigures 2ndash5 show time and PSNR scatter diagrams for 5differentmeasurementmatrices used to reconstruct the appleimage when the sampling point119872 takes different values
In this experiment we vary the number of samplingpixels for example 119872 = 150 119872 = 160 119872 = 170119872 = 180 use sym5 wavelet base as sparse transformationbase sample with five different measurement matrices andreconstruct images throughOMP algorithmThe experimentverified that the colorized apple image constructed by thepartial orthogonal randommatrices with a sym5wavelet basehad a higher quality
Figures 6ndash9 compare the restored apple image by the CSreconstruction algorithm with the Toeplitz matrix and theOrthogonal matrix with the original apple image when thesampling point119872 assumes different values
International Journal of Distributed Sensor Networks 5
Original image Toeplitz matrix reconstruction image Orthogonal matrix reconstruction image
Figure 6 Original image Toeplitz matrix reconstruction image and partial orthogonal matrix reconstruction image with 119872 = 150
Original image Toeplitz matrix reconstruction image Orthogonal matrix reconstruction image
Figure 7 Original image Toeplitz matrix reconstruction image and partial orthogonal matrix reconstruction image with 119872 = 160
Original image Toeplitz matrix reconstruction image Orthogonal matrix reconstruction image
Figure 8 Original image Toeplitz matrix reconstruction image and partial orthogonal matrix reconstruction image with 119872 = 170
6 International Journal of Distributed Sensor Networks
Original image Toeplitz matrix reconstruction image Orthogonal matrix reconstruction image
Figure 9 Original image Toeplitz matrix reconstruction image and partial orthogonal matrix reconstruction image with 119872 = 180
The above results show the reconstruction precisionof the Orthogonal measurement matrix and the Toeplitzmeasurement matrix Under the condition of same samplingpoints the quality of reconstruction image with the Orthog-onal measurement matrix is much better than that of thereconstruction image with Toeplitz measurement matrix
In this paper we choose the sampling points from 150 to180 If we use too many points that is more than 180 pointsthe significance of using sampling methods is minimizedWhile if we use too few points that is less than 150 pointsthe effectiveness of our methods decline Hence betweenthe tradeoff of effectiveness and complexity we choose thesampling points from 150 and 180
Since PSNR is mostly commonly used to measure thequality of reconstruction images we choose the PSNR scoreas measurement metric SSIM is also a quality reconstructionmetric but it is more complicated to compute for whichthe formula involves a value for each pixel We conductexperiments using SSIM metric and it demonstrates thesuperiority of our method but it takes too much space in thepaper we do not list the details of the SSIM results
4 Conclusion
This paper uses sym5 wavelet base as apple image sparsetransformation base and then it uses Gaussian randommatrices Bernoulli random matrices Partial Orthogonalrandom matrices Partial Hadamard matrices and Toeplitzmatrices to measure apple image respectively Using thesame measure quantity we select the matrix that has bestreconstruction effect as the apple imagemeasurementmatrixAccording to the experiment results this paper selects PartialOrthogonal random matrices as apple image measurementmatrix This measurement matrix is used for the data pro-cessing of apple image it can better realize image datacompression and greatly reduced apple image sampling andtransport cost which has important application significancein the process of apple production operation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank Wanlin Gao for helpfulcomments andpointersThisworkwas supported byNationalKeyTechnologyRampDProgramofChina during the 12th Five-Year Plan Period (Grant no 2012BAD35B02)
References
[1] H Rauhut ldquoCirculant and Toeplitz matrices in compressedsensingrdquo httparxivorgabs09024394
[2] J Haupt W U Bajwa G Raz and R Nowak ldquoToeplitz com-pressed sensing matrices with applications to sparse channelestimationrdquo IEEE Transactions on Information Theory vol 56no 11 pp 5862ndash5875 2010
[3] J Romberg ldquoCompressive sensing by random convolutionrdquoSIAM Journal on Imaging Sciences vol 2 no 4 pp 1098ndash11282009
[4] BM Sanandaji T L Vincent andM BWakin ldquoConcentrationof measure inequalities for compressive Toeplitz matrices withapplications to detection and system identificationrdquo in Proceed-ings of the 49th IEEE Conference on Decision and Control (CDCrsquo10) pp 2922ndash2929 December 2010
[5] J Y Park H L Yap C J Rozell and M B Wakin ldquoConcentra-tion of measure for block diagonal matrices with applicationsto compressive signal processingrdquo IEEE Transactions on SignalProcessing vol 59 no 12 pp 5859ndash5875 2011
[6] M B Wakin B M Sanandaji and T L Vincent ldquoOn theobservability of linear systems from random compressivemeasurementsrdquo in Proceedings of the 49th IEEE Conference onDecision andControl (CDC rsquo10) pp 4447ndash4454December 2010
[7] M F Duarte M A Davenport D Takhar et al ldquoSingle-Pixel imaging via compressive samplingrdquo IEEE Signal ProcessingMagazine vol 25 no 2 pp 83ndash91 2008
International Journal of Distributed Sensor Networks 7
[8] H Rauhut ldquoCirculant and Toeplitz matrices in compressedsensingrdquo in Proceedings of the Signal Processing with AdaptiveSparse Structured Representations (SPARS rsquo09) Saint MaloFrance April 2009
[9] J Romberg ldquoCompressive sensing by random convolutionrdquoSIAM Journal on Imaging Sciences vol 2 no 4 pp 1098ndash11282009
[10] W U Bajwa J Haupt G Raz and R Nowak ldquoCompressedchannel sensingrdquo in Proceedings of the 42nd Annual Conferenceon Information Sciences and Systems (CISS rsquo08) pp 5ndash10 March2008
[11] J A Tropp M B Wakin M F Duarte D Baron and R GBaraniuk ldquoRandomfilters for compressive sampling and recon-structionrdquo in Proceedings of the IEEE International Conferenceon Acoustics Speech and Signal Processing (ICASSP rsquo06) vol 3pp 872ndash875 May 2006
[12] W U Bajwa J D Haupt G M Raz S J Wright and R DNowak ldquoToeplitz-structured compressed sensing matricesrdquo inProceedings of the IEEESP 14th Workshop on Statistical SignalProcessing (SSP rsquo07) pp 294ndash298 IEEE Madison Wis USAAugust 2007
[13] 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
[14] E J Candes and M B Wakin ldquoAn introduction to compressivesamplingrdquo IEEE Signal Processing Magazine vol 25 no 2 pp21ndash30 2008
[15] C Gao H Li and L Ma ldquoVoice signal compression andreconstruction method based on the theory of the compressivesensingrdquo in Proceedings of the 11th National Conference onMan-Machine Voice Communication 2011
[16] I Drori and D L Donoho ldquoSolution of l1minimization
problems by LARShomotopy methodsrdquo in Proceedings of theIEEE International Conference on Acoustics Speech and SignalProcessing (ICASSP rsquo06) vol 3 IEEE Toulouse France May2006
[17] P J Garrigues and L E Ghaoui ldquoAn homotopy algorithmfor Lasso with online observationsrdquo in Proceedings of theAnnual Conference on Neural Information Processing SystemsVancouver Canada December 2008
[18] S-J Kim K Koh M Lustig S Boyd and D GorinevskyldquoAn interior-point method for large-scale 119897
1-regularized least
squaresrdquo IEEE Journal of Selected Topics in Signal Processing vol1 no 4 pp 606ndash617 2007
[19] M A T Figueiredo R D Nowak and S J Wright ldquoGradientprojection for sparse reconstruction application to compressedsensing and other inverse problemsrdquo IEEE Journal on SelectedTopics in Signal Processing vol 1 no 4 pp 586ndash598 2007
[20] D L Donoho and Y Tsaig ldquoFast solution of ℓ1-normminimiza-
tion problems when the solution may be sparserdquo Tech RepDepartment of Statistics Stanford University Stanford CalifUSA 2008
[21] T T Do L Gan N Nguyen and T D Tran ldquoSparsity adaptivematching pursuit algorithm for practical compressed sensingrdquoin Proceedings of the 42nd Asilomar Conference on SignalsSystems and Computers pp 581ndash587 Pacific Grove Calif USAOctober 2008
[22] S S Chen D L Donoho and M A Saunders ldquoAtomicdecomposition by basis pursuitrdquo SIAM Journal on ScientificComputing vol 20 no 1 pp 33ndash61 1998
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
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Shock and Vibration
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Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
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Electrical and Computer Engineering
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Advances inOptoElectronics
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Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
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Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of Distributed Sensor Networks 3
Measurementmatrix
observation
Thereconstructed
signal
Signalcharacteristics
Featureextraction
Signalreconstruction
Sparse samplingsignal
Figure 1 Signal processing framework based on compressed sensing
times The measurement matrix and sparse transformationbase satisfy the conditions of RIP or incoherence We canreconstruct the original signal119883 accurately or approximate itprecisely by solving the problemof optimal 119871
0norm119883meets
the type shown below
119883= argmin 1198830
st 119910 = Φ119883
(5)
Solving the above equation is an NP-hard problemFortunately prior work in compressed sensing has providedus with numerically feasible alternatives to this NP-hardproblem According to the theory of functional analysis andconvex optimization onemajor approach is that we can solvethe 1198710problem approximately through the 119871
1norm [16 17]
Common reconstruction algorithms include the interior-point method [18] Gradient Projection for Sparse Recon-struction (GPSR) [19] Homotopy algorithm [20] MP algo-rithm [21] Basis Pursuit (BP) algorithm [22] and Orthog-onal Matching Pursuit algorithm The interior-point algo-rithm is slow but generates very precise reconstructionsGPRS exhibits rapid computational speed and reconstructionresults are generally better Homotopy algorithm is morepractical for small-scale problems
MP algorithm reconstruction is fast but the reconstruc-tion result is bad BP algorithm suffers from slow recon-struction speed but the results are better OMP algorithmuses selection criteria in the matching pursuit In the rebuildprocess every iteration gets an atom that belongs to supportset of 119865 from 119909 By orthogonalization of the selected atomscollection we ensure the optimality of the iteration andreduce the number of iterations In this paper we have chosenthe OMP algorithm for apple image reconstruction
22 Selection on Measurement Matrix Another premiseof compressed sensing theory is incoherence Incoherencebelongs to the sensor model The measurement matrix Φ
and sparse base Ψ are multiplied to calculate the sensingmatrix In order to reconstruct the sparse signal Candes andTao proved that Φ must satisfy the isometry property (RIP)conditions For any119870-sparse signal and constant it satisfies
(1minus 120575119896) 119888
22 le
1003817100381710038171003817Φ1198791198881003817100381710038171003817
22 le (1+ 120575
119896) 119888
22 forall119888 isin 119877
|119879| (6)
119879 sub 1 119873 |119879| le 119870 Φ119879is a child of the matrix
composed of the related column indicated by the index119879withthe size of 119870 times |119879| The matrix Φ then satisfies the restrictedisometry property The limited isometric property defines animportant equidistant constant 120575
119896 it indicates that as long as
120575119896is not close to 1 it can approximately promise the Euclidean
distance of the constant 119878-order sparse signal This ensuresthe uniqueness of the reconstructed signal
Designing the measurement matrix should satisfy thefollowing conditions transformation matrix should satisfyRIP conditions or the measurement matrix and trans-formation matrix should meet incoherence The previousresearches prove that Gaussian random matrices partialrandom Fourier matrix Partial Hadamard matrix Toeplitzmatrix Partial Orthogonal random matrices Bernoullirandom matrix and so forth are feasible solutions Thispaper selects the Gaussian random matrices Bernoulli ran-dom matrices Partial Orthogonal random matrices PartialHadamardmatrices andToeplitzmatrices on the apple imagemeasurement
(1) Gaussian Random Measurement Matrix The probabilitydensity function of random variable 119883 is
119891 (119909) =1
radic2120587120590119890minus(119909minus120583)
221205902
minusinfin lt 119909 lt infin (7)
where 120583 120590 (120590 gt 0) are called parameters of normaldistribution or Gaussian distribution 119883 is denoted by 119883 sim
119873(120583 1205902) For Gaussian distribution we have 119864(119883) = 120583 and
119863(119883) = 1205902
The elements 119909119894119895
of Gaussian random matrices Φ areindependent random variables which obey the Gaussiandistribution with the mean of 0 and the square deviation of1119899 namely 119909
119894119895sim 119873(0 1119899)
(2) Bernoulli Random Measurement Matrices Bernoulli dis-tribution is a discrete probability distribution when theBernoulli trial succeeds it results in Bernoulli randomvariable as a value 119860 Otherwise it results in Bernoullirandomvariable as non-119860 Bernoulli randommatrices are theelements of thematrixΦwith the same probability with 1radic119899
or minus1radic119899
4 International Journal of Distributed Sensor Networks
Gaussian
Bernoulli
OrthogonalHadamard
Toeplitz
39540
40541
41542
42543
43544
445
342 344 346 348 35 352 354 356
Tim
e
PSNR
Figure 2 Reconstruction time and PSNR values of 5 differentmeasurement matrices with119872 = 150
Gaussian Bernoulli
Orthogonal
Hadamard
Toeplitz
44454647484950515253
345 35 355 36 365
Tim
e
PSNR
Figure 3 Reconstruction time and PSNR values of 5 differentmeasurement matrices with119872 = 160
(3) Partial Orthogonal RandomMatrices If119873 order square119860meets 119860119879119860 = 119864 119860 is called orthogonal matrix the sufficientand necessary conditions of the119873 order square matrix 119860 arethe fact that 119860 column vector is standard orthogonal vectorgroup
The method of constructing Partial Orthogonal mea-surement is illustrated as follows We first generate 119873 lowast 119873
dimension orthogonalmatrix119879 and thenwe randomly select119872 rows from 119879 construct the119872 lowast 119873matrix and then unitethe119873 columns to get the measurement matrix
(4) Partial Hadamard Matrices Hadamard matrix is orthog-onal square matrix composed by elements of +1 and 1The method of constructing Partial Hadamard matrix isillustrated as follows We first generate Hadamard matrix119873 lowast 119873 and then we from the generated matrix randomlyselect119872 row vector and generate119872lowast119873Hadamard randommeasurement matrix
(5) ToeplitzMatrix Toeplitzmatrix is also known as diagonal-constant matrix elements of line parallel to main diagonalline are the same The method of constructing Toeplitz ran-dommeasurement is illustrated as follows First we generatea random vector and then the random vector with relatedfunctions to generate the corresponding Toeplitz matricessecond we select the first119872 rows of the Toeplitz constructedas 119872 lowast 119873 Toeplitz random measurement matrix
Gaussian
Bernoulli
Orthogonal
Hadamard
Toeplitz
46474849505152535455
345 35 355 36 365 37
Tim
e
PSNR
Figure 4 Reconstruction time and PSNR values of 5 differentmeasurement matrices with119872 = 170
GaussianBernoulli
OrthogonalHadamard
Toeplitz
55555
56565
57575
58585
59
35 355 36 365 37 375 38
Tim
e
PSNR
Figure 5 Reconstruction time and PSNR values of 5 differentmeasurement matrices with119872 = 180
3 Results and Discussion
This paper chooses an apple image represented by 256 lowast
256RGB pixels We use sym5 wavelet bases to representthe original apple image sparsely with random Gaussianmeasurement matrix (119883 sim 119873(120583 120590
2)) Bernoulli random
matrices (119883 sim 119861(119899 119901)) Partial Orthogonal randommatricesand Partial Hadamard matrices and Toeplitz matrices arethen used for observational measuring of the signal Finallythe apple image is reconstructed by the OMP algorithmThe reconstructed PSNR values and runtime were used tocompare and contrast the simulation results The quality ofthe 5 reconstructed apple images is measured using PSNRFigures 2ndash5 show time and PSNR scatter diagrams for 5differentmeasurementmatrices used to reconstruct the appleimage when the sampling point119872 takes different values
In this experiment we vary the number of samplingpixels for example 119872 = 150 119872 = 160 119872 = 170119872 = 180 use sym5 wavelet base as sparse transformationbase sample with five different measurement matrices andreconstruct images throughOMP algorithmThe experimentverified that the colorized apple image constructed by thepartial orthogonal randommatrices with a sym5wavelet basehad a higher quality
Figures 6ndash9 compare the restored apple image by the CSreconstruction algorithm with the Toeplitz matrix and theOrthogonal matrix with the original apple image when thesampling point119872 assumes different values
International Journal of Distributed Sensor Networks 5
Original image Toeplitz matrix reconstruction image Orthogonal matrix reconstruction image
Figure 6 Original image Toeplitz matrix reconstruction image and partial orthogonal matrix reconstruction image with 119872 = 150
Original image Toeplitz matrix reconstruction image Orthogonal matrix reconstruction image
Figure 7 Original image Toeplitz matrix reconstruction image and partial orthogonal matrix reconstruction image with 119872 = 160
Original image Toeplitz matrix reconstruction image Orthogonal matrix reconstruction image
Figure 8 Original image Toeplitz matrix reconstruction image and partial orthogonal matrix reconstruction image with 119872 = 170
6 International Journal of Distributed Sensor Networks
Original image Toeplitz matrix reconstruction image Orthogonal matrix reconstruction image
Figure 9 Original image Toeplitz matrix reconstruction image and partial orthogonal matrix reconstruction image with 119872 = 180
The above results show the reconstruction precisionof the Orthogonal measurement matrix and the Toeplitzmeasurement matrix Under the condition of same samplingpoints the quality of reconstruction image with the Orthog-onal measurement matrix is much better than that of thereconstruction image with Toeplitz measurement matrix
In this paper we choose the sampling points from 150 to180 If we use too many points that is more than 180 pointsthe significance of using sampling methods is minimizedWhile if we use too few points that is less than 150 pointsthe effectiveness of our methods decline Hence betweenthe tradeoff of effectiveness and complexity we choose thesampling points from 150 and 180
Since PSNR is mostly commonly used to measure thequality of reconstruction images we choose the PSNR scoreas measurement metric SSIM is also a quality reconstructionmetric but it is more complicated to compute for whichthe formula involves a value for each pixel We conductexperiments using SSIM metric and it demonstrates thesuperiority of our method but it takes too much space in thepaper we do not list the details of the SSIM results
4 Conclusion
This paper uses sym5 wavelet base as apple image sparsetransformation base and then it uses Gaussian randommatrices Bernoulli random matrices Partial Orthogonalrandom matrices Partial Hadamard matrices and Toeplitzmatrices to measure apple image respectively Using thesame measure quantity we select the matrix that has bestreconstruction effect as the apple imagemeasurementmatrixAccording to the experiment results this paper selects PartialOrthogonal random matrices as apple image measurementmatrix This measurement matrix is used for the data pro-cessing of apple image it can better realize image datacompression and greatly reduced apple image sampling andtransport cost which has important application significancein the process of apple production operation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank Wanlin Gao for helpfulcomments andpointersThisworkwas supported byNationalKeyTechnologyRampDProgramofChina during the 12th Five-Year Plan Period (Grant no 2012BAD35B02)
References
[1] H Rauhut ldquoCirculant and Toeplitz matrices in compressedsensingrdquo httparxivorgabs09024394
[2] J Haupt W U Bajwa G Raz and R Nowak ldquoToeplitz com-pressed sensing matrices with applications to sparse channelestimationrdquo IEEE Transactions on Information Theory vol 56no 11 pp 5862ndash5875 2010
[3] J Romberg ldquoCompressive sensing by random convolutionrdquoSIAM Journal on Imaging Sciences vol 2 no 4 pp 1098ndash11282009
[4] BM Sanandaji T L Vincent andM BWakin ldquoConcentrationof measure inequalities for compressive Toeplitz matrices withapplications to detection and system identificationrdquo in Proceed-ings of the 49th IEEE Conference on Decision and Control (CDCrsquo10) pp 2922ndash2929 December 2010
[5] J Y Park H L Yap C J Rozell and M B Wakin ldquoConcentra-tion of measure for block diagonal matrices with applicationsto compressive signal processingrdquo IEEE Transactions on SignalProcessing vol 59 no 12 pp 5859ndash5875 2011
[6] M B Wakin B M Sanandaji and T L Vincent ldquoOn theobservability of linear systems from random compressivemeasurementsrdquo in Proceedings of the 49th IEEE Conference onDecision andControl (CDC rsquo10) pp 4447ndash4454December 2010
[7] M F Duarte M A Davenport D Takhar et al ldquoSingle-Pixel imaging via compressive samplingrdquo IEEE Signal ProcessingMagazine vol 25 no 2 pp 83ndash91 2008
International Journal of Distributed Sensor Networks 7
[8] H Rauhut ldquoCirculant and Toeplitz matrices in compressedsensingrdquo in Proceedings of the Signal Processing with AdaptiveSparse Structured Representations (SPARS rsquo09) Saint MaloFrance April 2009
[9] J Romberg ldquoCompressive sensing by random convolutionrdquoSIAM Journal on Imaging Sciences vol 2 no 4 pp 1098ndash11282009
[10] W U Bajwa J Haupt G Raz and R Nowak ldquoCompressedchannel sensingrdquo in Proceedings of the 42nd Annual Conferenceon Information Sciences and Systems (CISS rsquo08) pp 5ndash10 March2008
[11] J A Tropp M B Wakin M F Duarte D Baron and R GBaraniuk ldquoRandomfilters for compressive sampling and recon-structionrdquo in Proceedings of the IEEE International Conferenceon Acoustics Speech and Signal Processing (ICASSP rsquo06) vol 3pp 872ndash875 May 2006
[12] W U Bajwa J D Haupt G M Raz S J Wright and R DNowak ldquoToeplitz-structured compressed sensing matricesrdquo inProceedings of the IEEESP 14th Workshop on Statistical SignalProcessing (SSP rsquo07) pp 294ndash298 IEEE Madison Wis USAAugust 2007
[13] 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
[14] E J Candes and M B Wakin ldquoAn introduction to compressivesamplingrdquo IEEE Signal Processing Magazine vol 25 no 2 pp21ndash30 2008
[15] C Gao H Li and L Ma ldquoVoice signal compression andreconstruction method based on the theory of the compressivesensingrdquo in Proceedings of the 11th National Conference onMan-Machine Voice Communication 2011
[16] I Drori and D L Donoho ldquoSolution of l1minimization
problems by LARShomotopy methodsrdquo in Proceedings of theIEEE International Conference on Acoustics Speech and SignalProcessing (ICASSP rsquo06) vol 3 IEEE Toulouse France May2006
[17] P J Garrigues and L E Ghaoui ldquoAn homotopy algorithmfor Lasso with online observationsrdquo in Proceedings of theAnnual Conference on Neural Information Processing SystemsVancouver Canada December 2008
[18] S-J Kim K Koh M Lustig S Boyd and D GorinevskyldquoAn interior-point method for large-scale 119897
1-regularized least
squaresrdquo IEEE Journal of Selected Topics in Signal Processing vol1 no 4 pp 606ndash617 2007
[19] M A T Figueiredo R D Nowak and S J Wright ldquoGradientprojection for sparse reconstruction application to compressedsensing and other inverse problemsrdquo IEEE Journal on SelectedTopics in Signal Processing vol 1 no 4 pp 586ndash598 2007
[20] D L Donoho and Y Tsaig ldquoFast solution of ℓ1-normminimiza-
tion problems when the solution may be sparserdquo Tech RepDepartment of Statistics Stanford University Stanford CalifUSA 2008
[21] T T Do L Gan N Nguyen and T D Tran ldquoSparsity adaptivematching pursuit algorithm for practical compressed sensingrdquoin Proceedings of the 42nd Asilomar Conference on SignalsSystems and Computers pp 581ndash587 Pacific Grove Calif USAOctober 2008
[22] S S Chen D L Donoho and M A Saunders ldquoAtomicdecomposition by basis pursuitrdquo SIAM Journal on ScientificComputing vol 20 no 1 pp 33ndash61 1998
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
4 International Journal of Distributed Sensor Networks
Gaussian
Bernoulli
OrthogonalHadamard
Toeplitz
39540
40541
41542
42543
43544
445
342 344 346 348 35 352 354 356
Tim
e
PSNR
Figure 2 Reconstruction time and PSNR values of 5 differentmeasurement matrices with119872 = 150
Gaussian Bernoulli
Orthogonal
Hadamard
Toeplitz
44454647484950515253
345 35 355 36 365
Tim
e
PSNR
Figure 3 Reconstruction time and PSNR values of 5 differentmeasurement matrices with119872 = 160
(3) Partial Orthogonal RandomMatrices If119873 order square119860meets 119860119879119860 = 119864 119860 is called orthogonal matrix the sufficientand necessary conditions of the119873 order square matrix 119860 arethe fact that 119860 column vector is standard orthogonal vectorgroup
The method of constructing Partial Orthogonal mea-surement is illustrated as follows We first generate 119873 lowast 119873
dimension orthogonalmatrix119879 and thenwe randomly select119872 rows from 119879 construct the119872 lowast 119873matrix and then unitethe119873 columns to get the measurement matrix
(4) Partial Hadamard Matrices Hadamard matrix is orthog-onal square matrix composed by elements of +1 and 1The method of constructing Partial Hadamard matrix isillustrated as follows We first generate Hadamard matrix119873 lowast 119873 and then we from the generated matrix randomlyselect119872 row vector and generate119872lowast119873Hadamard randommeasurement matrix
(5) ToeplitzMatrix Toeplitzmatrix is also known as diagonal-constant matrix elements of line parallel to main diagonalline are the same The method of constructing Toeplitz ran-dommeasurement is illustrated as follows First we generatea random vector and then the random vector with relatedfunctions to generate the corresponding Toeplitz matricessecond we select the first119872 rows of the Toeplitz constructedas 119872 lowast 119873 Toeplitz random measurement matrix
Gaussian
Bernoulli
Orthogonal
Hadamard
Toeplitz
46474849505152535455
345 35 355 36 365 37
Tim
e
PSNR
Figure 4 Reconstruction time and PSNR values of 5 differentmeasurement matrices with119872 = 170
GaussianBernoulli
OrthogonalHadamard
Toeplitz
55555
56565
57575
58585
59
35 355 36 365 37 375 38
Tim
e
PSNR
Figure 5 Reconstruction time and PSNR values of 5 differentmeasurement matrices with119872 = 180
3 Results and Discussion
This paper chooses an apple image represented by 256 lowast
256RGB pixels We use sym5 wavelet bases to representthe original apple image sparsely with random Gaussianmeasurement matrix (119883 sim 119873(120583 120590
2)) Bernoulli random
matrices (119883 sim 119861(119899 119901)) Partial Orthogonal randommatricesand Partial Hadamard matrices and Toeplitz matrices arethen used for observational measuring of the signal Finallythe apple image is reconstructed by the OMP algorithmThe reconstructed PSNR values and runtime were used tocompare and contrast the simulation results The quality ofthe 5 reconstructed apple images is measured using PSNRFigures 2ndash5 show time and PSNR scatter diagrams for 5differentmeasurementmatrices used to reconstruct the appleimage when the sampling point119872 takes different values
In this experiment we vary the number of samplingpixels for example 119872 = 150 119872 = 160 119872 = 170119872 = 180 use sym5 wavelet base as sparse transformationbase sample with five different measurement matrices andreconstruct images throughOMP algorithmThe experimentverified that the colorized apple image constructed by thepartial orthogonal randommatrices with a sym5wavelet basehad a higher quality
Figures 6ndash9 compare the restored apple image by the CSreconstruction algorithm with the Toeplitz matrix and theOrthogonal matrix with the original apple image when thesampling point119872 assumes different values
International Journal of Distributed Sensor Networks 5
Original image Toeplitz matrix reconstruction image Orthogonal matrix reconstruction image
Figure 6 Original image Toeplitz matrix reconstruction image and partial orthogonal matrix reconstruction image with 119872 = 150
Original image Toeplitz matrix reconstruction image Orthogonal matrix reconstruction image
Figure 7 Original image Toeplitz matrix reconstruction image and partial orthogonal matrix reconstruction image with 119872 = 160
Original image Toeplitz matrix reconstruction image Orthogonal matrix reconstruction image
Figure 8 Original image Toeplitz matrix reconstruction image and partial orthogonal matrix reconstruction image with 119872 = 170
6 International Journal of Distributed Sensor Networks
Original image Toeplitz matrix reconstruction image Orthogonal matrix reconstruction image
Figure 9 Original image Toeplitz matrix reconstruction image and partial orthogonal matrix reconstruction image with 119872 = 180
The above results show the reconstruction precisionof the Orthogonal measurement matrix and the Toeplitzmeasurement matrix Under the condition of same samplingpoints the quality of reconstruction image with the Orthog-onal measurement matrix is much better than that of thereconstruction image with Toeplitz measurement matrix
In this paper we choose the sampling points from 150 to180 If we use too many points that is more than 180 pointsthe significance of using sampling methods is minimizedWhile if we use too few points that is less than 150 pointsthe effectiveness of our methods decline Hence betweenthe tradeoff of effectiveness and complexity we choose thesampling points from 150 and 180
Since PSNR is mostly commonly used to measure thequality of reconstruction images we choose the PSNR scoreas measurement metric SSIM is also a quality reconstructionmetric but it is more complicated to compute for whichthe formula involves a value for each pixel We conductexperiments using SSIM metric and it demonstrates thesuperiority of our method but it takes too much space in thepaper we do not list the details of the SSIM results
4 Conclusion
This paper uses sym5 wavelet base as apple image sparsetransformation base and then it uses Gaussian randommatrices Bernoulli random matrices Partial Orthogonalrandom matrices Partial Hadamard matrices and Toeplitzmatrices to measure apple image respectively Using thesame measure quantity we select the matrix that has bestreconstruction effect as the apple imagemeasurementmatrixAccording to the experiment results this paper selects PartialOrthogonal random matrices as apple image measurementmatrix This measurement matrix is used for the data pro-cessing of apple image it can better realize image datacompression and greatly reduced apple image sampling andtransport cost which has important application significancein the process of apple production operation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank Wanlin Gao for helpfulcomments andpointersThisworkwas supported byNationalKeyTechnologyRampDProgramofChina during the 12th Five-Year Plan Period (Grant no 2012BAD35B02)
References
[1] H Rauhut ldquoCirculant and Toeplitz matrices in compressedsensingrdquo httparxivorgabs09024394
[2] J Haupt W U Bajwa G Raz and R Nowak ldquoToeplitz com-pressed sensing matrices with applications to sparse channelestimationrdquo IEEE Transactions on Information Theory vol 56no 11 pp 5862ndash5875 2010
[3] J Romberg ldquoCompressive sensing by random convolutionrdquoSIAM Journal on Imaging Sciences vol 2 no 4 pp 1098ndash11282009
[4] BM Sanandaji T L Vincent andM BWakin ldquoConcentrationof measure inequalities for compressive Toeplitz matrices withapplications to detection and system identificationrdquo in Proceed-ings of the 49th IEEE Conference on Decision and Control (CDCrsquo10) pp 2922ndash2929 December 2010
[5] J Y Park H L Yap C J Rozell and M B Wakin ldquoConcentra-tion of measure for block diagonal matrices with applicationsto compressive signal processingrdquo IEEE Transactions on SignalProcessing vol 59 no 12 pp 5859ndash5875 2011
[6] M B Wakin B M Sanandaji and T L Vincent ldquoOn theobservability of linear systems from random compressivemeasurementsrdquo in Proceedings of the 49th IEEE Conference onDecision andControl (CDC rsquo10) pp 4447ndash4454December 2010
[7] M F Duarte M A Davenport D Takhar et al ldquoSingle-Pixel imaging via compressive samplingrdquo IEEE Signal ProcessingMagazine vol 25 no 2 pp 83ndash91 2008
International Journal of Distributed Sensor Networks 7
[8] H Rauhut ldquoCirculant and Toeplitz matrices in compressedsensingrdquo in Proceedings of the Signal Processing with AdaptiveSparse Structured Representations (SPARS rsquo09) Saint MaloFrance April 2009
[9] J Romberg ldquoCompressive sensing by random convolutionrdquoSIAM Journal on Imaging Sciences vol 2 no 4 pp 1098ndash11282009
[10] W U Bajwa J Haupt G Raz and R Nowak ldquoCompressedchannel sensingrdquo in Proceedings of the 42nd Annual Conferenceon Information Sciences and Systems (CISS rsquo08) pp 5ndash10 March2008
[11] J A Tropp M B Wakin M F Duarte D Baron and R GBaraniuk ldquoRandomfilters for compressive sampling and recon-structionrdquo in Proceedings of the IEEE International Conferenceon Acoustics Speech and Signal Processing (ICASSP rsquo06) vol 3pp 872ndash875 May 2006
[12] W U Bajwa J D Haupt G M Raz S J Wright and R DNowak ldquoToeplitz-structured compressed sensing matricesrdquo inProceedings of the IEEESP 14th Workshop on Statistical SignalProcessing (SSP rsquo07) pp 294ndash298 IEEE Madison Wis USAAugust 2007
[13] 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
[14] E J Candes and M B Wakin ldquoAn introduction to compressivesamplingrdquo IEEE Signal Processing Magazine vol 25 no 2 pp21ndash30 2008
[15] C Gao H Li and L Ma ldquoVoice signal compression andreconstruction method based on the theory of the compressivesensingrdquo in Proceedings of the 11th National Conference onMan-Machine Voice Communication 2011
[16] I Drori and D L Donoho ldquoSolution of l1minimization
problems by LARShomotopy methodsrdquo in Proceedings of theIEEE International Conference on Acoustics Speech and SignalProcessing (ICASSP rsquo06) vol 3 IEEE Toulouse France May2006
[17] P J Garrigues and L E Ghaoui ldquoAn homotopy algorithmfor Lasso with online observationsrdquo in Proceedings of theAnnual Conference on Neural Information Processing SystemsVancouver Canada December 2008
[18] S-J Kim K Koh M Lustig S Boyd and D GorinevskyldquoAn interior-point method for large-scale 119897
1-regularized least
squaresrdquo IEEE Journal of Selected Topics in Signal Processing vol1 no 4 pp 606ndash617 2007
[19] M A T Figueiredo R D Nowak and S J Wright ldquoGradientprojection for sparse reconstruction application to compressedsensing and other inverse problemsrdquo IEEE Journal on SelectedTopics in Signal Processing vol 1 no 4 pp 586ndash598 2007
[20] D L Donoho and Y Tsaig ldquoFast solution of ℓ1-normminimiza-
tion problems when the solution may be sparserdquo Tech RepDepartment of Statistics Stanford University Stanford CalifUSA 2008
[21] T T Do L Gan N Nguyen and T D Tran ldquoSparsity adaptivematching pursuit algorithm for practical compressed sensingrdquoin Proceedings of the 42nd Asilomar Conference on SignalsSystems and Computers pp 581ndash587 Pacific Grove Calif USAOctober 2008
[22] S S Chen D L Donoho and M A Saunders ldquoAtomicdecomposition by basis pursuitrdquo SIAM Journal on ScientificComputing vol 20 no 1 pp 33ndash61 1998
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of Distributed Sensor Networks 5
Original image Toeplitz matrix reconstruction image Orthogonal matrix reconstruction image
Figure 6 Original image Toeplitz matrix reconstruction image and partial orthogonal matrix reconstruction image with 119872 = 150
Original image Toeplitz matrix reconstruction image Orthogonal matrix reconstruction image
Figure 7 Original image Toeplitz matrix reconstruction image and partial orthogonal matrix reconstruction image with 119872 = 160
Original image Toeplitz matrix reconstruction image Orthogonal matrix reconstruction image
Figure 8 Original image Toeplitz matrix reconstruction image and partial orthogonal matrix reconstruction image with 119872 = 170
6 International Journal of Distributed Sensor Networks
Original image Toeplitz matrix reconstruction image Orthogonal matrix reconstruction image
Figure 9 Original image Toeplitz matrix reconstruction image and partial orthogonal matrix reconstruction image with 119872 = 180
The above results show the reconstruction precisionof the Orthogonal measurement matrix and the Toeplitzmeasurement matrix Under the condition of same samplingpoints the quality of reconstruction image with the Orthog-onal measurement matrix is much better than that of thereconstruction image with Toeplitz measurement matrix
In this paper we choose the sampling points from 150 to180 If we use too many points that is more than 180 pointsthe significance of using sampling methods is minimizedWhile if we use too few points that is less than 150 pointsthe effectiveness of our methods decline Hence betweenthe tradeoff of effectiveness and complexity we choose thesampling points from 150 and 180
Since PSNR is mostly commonly used to measure thequality of reconstruction images we choose the PSNR scoreas measurement metric SSIM is also a quality reconstructionmetric but it is more complicated to compute for whichthe formula involves a value for each pixel We conductexperiments using SSIM metric and it demonstrates thesuperiority of our method but it takes too much space in thepaper we do not list the details of the SSIM results
4 Conclusion
This paper uses sym5 wavelet base as apple image sparsetransformation base and then it uses Gaussian randommatrices Bernoulli random matrices Partial Orthogonalrandom matrices Partial Hadamard matrices and Toeplitzmatrices to measure apple image respectively Using thesame measure quantity we select the matrix that has bestreconstruction effect as the apple imagemeasurementmatrixAccording to the experiment results this paper selects PartialOrthogonal random matrices as apple image measurementmatrix This measurement matrix is used for the data pro-cessing of apple image it can better realize image datacompression and greatly reduced apple image sampling andtransport cost which has important application significancein the process of apple production operation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank Wanlin Gao for helpfulcomments andpointersThisworkwas supported byNationalKeyTechnologyRampDProgramofChina during the 12th Five-Year Plan Period (Grant no 2012BAD35B02)
References
[1] H Rauhut ldquoCirculant and Toeplitz matrices in compressedsensingrdquo httparxivorgabs09024394
[2] J Haupt W U Bajwa G Raz and R Nowak ldquoToeplitz com-pressed sensing matrices with applications to sparse channelestimationrdquo IEEE Transactions on Information Theory vol 56no 11 pp 5862ndash5875 2010
[3] J Romberg ldquoCompressive sensing by random convolutionrdquoSIAM Journal on Imaging Sciences vol 2 no 4 pp 1098ndash11282009
[4] BM Sanandaji T L Vincent andM BWakin ldquoConcentrationof measure inequalities for compressive Toeplitz matrices withapplications to detection and system identificationrdquo in Proceed-ings of the 49th IEEE Conference on Decision and Control (CDCrsquo10) pp 2922ndash2929 December 2010
[5] J Y Park H L Yap C J Rozell and M B Wakin ldquoConcentra-tion of measure for block diagonal matrices with applicationsto compressive signal processingrdquo IEEE Transactions on SignalProcessing vol 59 no 12 pp 5859ndash5875 2011
[6] M B Wakin B M Sanandaji and T L Vincent ldquoOn theobservability of linear systems from random compressivemeasurementsrdquo in Proceedings of the 49th IEEE Conference onDecision andControl (CDC rsquo10) pp 4447ndash4454December 2010
[7] M F Duarte M A Davenport D Takhar et al ldquoSingle-Pixel imaging via compressive samplingrdquo IEEE Signal ProcessingMagazine vol 25 no 2 pp 83ndash91 2008
International Journal of Distributed Sensor Networks 7
[8] H Rauhut ldquoCirculant and Toeplitz matrices in compressedsensingrdquo in Proceedings of the Signal Processing with AdaptiveSparse Structured Representations (SPARS rsquo09) Saint MaloFrance April 2009
[9] J Romberg ldquoCompressive sensing by random convolutionrdquoSIAM Journal on Imaging Sciences vol 2 no 4 pp 1098ndash11282009
[10] W U Bajwa J Haupt G Raz and R Nowak ldquoCompressedchannel sensingrdquo in Proceedings of the 42nd Annual Conferenceon Information Sciences and Systems (CISS rsquo08) pp 5ndash10 March2008
[11] J A Tropp M B Wakin M F Duarte D Baron and R GBaraniuk ldquoRandomfilters for compressive sampling and recon-structionrdquo in Proceedings of the IEEE International Conferenceon Acoustics Speech and Signal Processing (ICASSP rsquo06) vol 3pp 872ndash875 May 2006
[12] W U Bajwa J D Haupt G M Raz S J Wright and R DNowak ldquoToeplitz-structured compressed sensing matricesrdquo inProceedings of the IEEESP 14th Workshop on Statistical SignalProcessing (SSP rsquo07) pp 294ndash298 IEEE Madison Wis USAAugust 2007
[13] 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
[14] E J Candes and M B Wakin ldquoAn introduction to compressivesamplingrdquo IEEE Signal Processing Magazine vol 25 no 2 pp21ndash30 2008
[15] C Gao H Li and L Ma ldquoVoice signal compression andreconstruction method based on the theory of the compressivesensingrdquo in Proceedings of the 11th National Conference onMan-Machine Voice Communication 2011
[16] I Drori and D L Donoho ldquoSolution of l1minimization
problems by LARShomotopy methodsrdquo in Proceedings of theIEEE International Conference on Acoustics Speech and SignalProcessing (ICASSP rsquo06) vol 3 IEEE Toulouse France May2006
[17] P J Garrigues and L E Ghaoui ldquoAn homotopy algorithmfor Lasso with online observationsrdquo in Proceedings of theAnnual Conference on Neural Information Processing SystemsVancouver Canada December 2008
[18] S-J Kim K Koh M Lustig S Boyd and D GorinevskyldquoAn interior-point method for large-scale 119897
1-regularized least
squaresrdquo IEEE Journal of Selected Topics in Signal Processing vol1 no 4 pp 606ndash617 2007
[19] M A T Figueiredo R D Nowak and S J Wright ldquoGradientprojection for sparse reconstruction application to compressedsensing and other inverse problemsrdquo IEEE Journal on SelectedTopics in Signal Processing vol 1 no 4 pp 586ndash598 2007
[20] D L Donoho and Y Tsaig ldquoFast solution of ℓ1-normminimiza-
tion problems when the solution may be sparserdquo Tech RepDepartment of Statistics Stanford University Stanford CalifUSA 2008
[21] T T Do L Gan N Nguyen and T D Tran ldquoSparsity adaptivematching pursuit algorithm for practical compressed sensingrdquoin Proceedings of the 42nd Asilomar Conference on SignalsSystems and Computers pp 581ndash587 Pacific Grove Calif USAOctober 2008
[22] S S Chen D L Donoho and M A Saunders ldquoAtomicdecomposition by basis pursuitrdquo SIAM Journal on ScientificComputing vol 20 no 1 pp 33ndash61 1998
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
6 International Journal of Distributed Sensor Networks
Original image Toeplitz matrix reconstruction image Orthogonal matrix reconstruction image
Figure 9 Original image Toeplitz matrix reconstruction image and partial orthogonal matrix reconstruction image with 119872 = 180
The above results show the reconstruction precisionof the Orthogonal measurement matrix and the Toeplitzmeasurement matrix Under the condition of same samplingpoints the quality of reconstruction image with the Orthog-onal measurement matrix is much better than that of thereconstruction image with Toeplitz measurement matrix
In this paper we choose the sampling points from 150 to180 If we use too many points that is more than 180 pointsthe significance of using sampling methods is minimizedWhile if we use too few points that is less than 150 pointsthe effectiveness of our methods decline Hence betweenthe tradeoff of effectiveness and complexity we choose thesampling points from 150 and 180
Since PSNR is mostly commonly used to measure thequality of reconstruction images we choose the PSNR scoreas measurement metric SSIM is also a quality reconstructionmetric but it is more complicated to compute for whichthe formula involves a value for each pixel We conductexperiments using SSIM metric and it demonstrates thesuperiority of our method but it takes too much space in thepaper we do not list the details of the SSIM results
4 Conclusion
This paper uses sym5 wavelet base as apple image sparsetransformation base and then it uses Gaussian randommatrices Bernoulli random matrices Partial Orthogonalrandom matrices Partial Hadamard matrices and Toeplitzmatrices to measure apple image respectively Using thesame measure quantity we select the matrix that has bestreconstruction effect as the apple imagemeasurementmatrixAccording to the experiment results this paper selects PartialOrthogonal random matrices as apple image measurementmatrix This measurement matrix is used for the data pro-cessing of apple image it can better realize image datacompression and greatly reduced apple image sampling andtransport cost which has important application significancein the process of apple production operation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank Wanlin Gao for helpfulcomments andpointersThisworkwas supported byNationalKeyTechnologyRampDProgramofChina during the 12th Five-Year Plan Period (Grant no 2012BAD35B02)
References
[1] H Rauhut ldquoCirculant and Toeplitz matrices in compressedsensingrdquo httparxivorgabs09024394
[2] J Haupt W U Bajwa G Raz and R Nowak ldquoToeplitz com-pressed sensing matrices with applications to sparse channelestimationrdquo IEEE Transactions on Information Theory vol 56no 11 pp 5862ndash5875 2010
[3] J Romberg ldquoCompressive sensing by random convolutionrdquoSIAM Journal on Imaging Sciences vol 2 no 4 pp 1098ndash11282009
[4] BM Sanandaji T L Vincent andM BWakin ldquoConcentrationof measure inequalities for compressive Toeplitz matrices withapplications to detection and system identificationrdquo in Proceed-ings of the 49th IEEE Conference on Decision and Control (CDCrsquo10) pp 2922ndash2929 December 2010
[5] J Y Park H L Yap C J Rozell and M B Wakin ldquoConcentra-tion of measure for block diagonal matrices with applicationsto compressive signal processingrdquo IEEE Transactions on SignalProcessing vol 59 no 12 pp 5859ndash5875 2011
[6] M B Wakin B M Sanandaji and T L Vincent ldquoOn theobservability of linear systems from random compressivemeasurementsrdquo in Proceedings of the 49th IEEE Conference onDecision andControl (CDC rsquo10) pp 4447ndash4454December 2010
[7] M F Duarte M A Davenport D Takhar et al ldquoSingle-Pixel imaging via compressive samplingrdquo IEEE Signal ProcessingMagazine vol 25 no 2 pp 83ndash91 2008
International Journal of Distributed Sensor Networks 7
[8] H Rauhut ldquoCirculant and Toeplitz matrices in compressedsensingrdquo in Proceedings of the Signal Processing with AdaptiveSparse Structured Representations (SPARS rsquo09) Saint MaloFrance April 2009
[9] J Romberg ldquoCompressive sensing by random convolutionrdquoSIAM Journal on Imaging Sciences vol 2 no 4 pp 1098ndash11282009
[10] W U Bajwa J Haupt G Raz and R Nowak ldquoCompressedchannel sensingrdquo in Proceedings of the 42nd Annual Conferenceon Information Sciences and Systems (CISS rsquo08) pp 5ndash10 March2008
[11] J A Tropp M B Wakin M F Duarte D Baron and R GBaraniuk ldquoRandomfilters for compressive sampling and recon-structionrdquo in Proceedings of the IEEE International Conferenceon Acoustics Speech and Signal Processing (ICASSP rsquo06) vol 3pp 872ndash875 May 2006
[12] W U Bajwa J D Haupt G M Raz S J Wright and R DNowak ldquoToeplitz-structured compressed sensing matricesrdquo inProceedings of the IEEESP 14th Workshop on Statistical SignalProcessing (SSP rsquo07) pp 294ndash298 IEEE Madison Wis USAAugust 2007
[13] 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
[14] E J Candes and M B Wakin ldquoAn introduction to compressivesamplingrdquo IEEE Signal Processing Magazine vol 25 no 2 pp21ndash30 2008
[15] C Gao H Li and L Ma ldquoVoice signal compression andreconstruction method based on the theory of the compressivesensingrdquo in Proceedings of the 11th National Conference onMan-Machine Voice Communication 2011
[16] I Drori and D L Donoho ldquoSolution of l1minimization
problems by LARShomotopy methodsrdquo in Proceedings of theIEEE International Conference on Acoustics Speech and SignalProcessing (ICASSP rsquo06) vol 3 IEEE Toulouse France May2006
[17] P J Garrigues and L E Ghaoui ldquoAn homotopy algorithmfor Lasso with online observationsrdquo in Proceedings of theAnnual Conference on Neural Information Processing SystemsVancouver Canada December 2008
[18] S-J Kim K Koh M Lustig S Boyd and D GorinevskyldquoAn interior-point method for large-scale 119897
1-regularized least
squaresrdquo IEEE Journal of Selected Topics in Signal Processing vol1 no 4 pp 606ndash617 2007
[19] M A T Figueiredo R D Nowak and S J Wright ldquoGradientprojection for sparse reconstruction application to compressedsensing and other inverse problemsrdquo IEEE Journal on SelectedTopics in Signal Processing vol 1 no 4 pp 586ndash598 2007
[20] D L Donoho and Y Tsaig ldquoFast solution of ℓ1-normminimiza-
tion problems when the solution may be sparserdquo Tech RepDepartment of Statistics Stanford University Stanford CalifUSA 2008
[21] T T Do L Gan N Nguyen and T D Tran ldquoSparsity adaptivematching pursuit algorithm for practical compressed sensingrdquoin Proceedings of the 42nd Asilomar Conference on SignalsSystems and Computers pp 581ndash587 Pacific Grove Calif USAOctober 2008
[22] S S Chen D L Donoho and M A Saunders ldquoAtomicdecomposition by basis pursuitrdquo SIAM Journal on ScientificComputing vol 20 no 1 pp 33ndash61 1998
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of Distributed Sensor Networks 7
[8] H Rauhut ldquoCirculant and Toeplitz matrices in compressedsensingrdquo in Proceedings of the Signal Processing with AdaptiveSparse Structured Representations (SPARS rsquo09) Saint MaloFrance April 2009
[9] J Romberg ldquoCompressive sensing by random convolutionrdquoSIAM Journal on Imaging Sciences vol 2 no 4 pp 1098ndash11282009
[10] W U Bajwa J Haupt G Raz and R Nowak ldquoCompressedchannel sensingrdquo in Proceedings of the 42nd Annual Conferenceon Information Sciences and Systems (CISS rsquo08) pp 5ndash10 March2008
[11] J A Tropp M B Wakin M F Duarte D Baron and R GBaraniuk ldquoRandomfilters for compressive sampling and recon-structionrdquo in Proceedings of the IEEE International Conferenceon Acoustics Speech and Signal Processing (ICASSP rsquo06) vol 3pp 872ndash875 May 2006
[12] W U Bajwa J D Haupt G M Raz S J Wright and R DNowak ldquoToeplitz-structured compressed sensing matricesrdquo inProceedings of the IEEESP 14th Workshop on Statistical SignalProcessing (SSP rsquo07) pp 294ndash298 IEEE Madison Wis USAAugust 2007
[13] 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
[14] E J Candes and M B Wakin ldquoAn introduction to compressivesamplingrdquo IEEE Signal Processing Magazine vol 25 no 2 pp21ndash30 2008
[15] C Gao H Li and L Ma ldquoVoice signal compression andreconstruction method based on the theory of the compressivesensingrdquo in Proceedings of the 11th National Conference onMan-Machine Voice Communication 2011
[16] I Drori and D L Donoho ldquoSolution of l1minimization
problems by LARShomotopy methodsrdquo in Proceedings of theIEEE International Conference on Acoustics Speech and SignalProcessing (ICASSP rsquo06) vol 3 IEEE Toulouse France May2006
[17] P J Garrigues and L E Ghaoui ldquoAn homotopy algorithmfor Lasso with online observationsrdquo in Proceedings of theAnnual Conference on Neural Information Processing SystemsVancouver Canada December 2008
[18] S-J Kim K Koh M Lustig S Boyd and D GorinevskyldquoAn interior-point method for large-scale 119897
1-regularized least
squaresrdquo IEEE Journal of Selected Topics in Signal Processing vol1 no 4 pp 606ndash617 2007
[19] M A T Figueiredo R D Nowak and S J Wright ldquoGradientprojection for sparse reconstruction application to compressedsensing and other inverse problemsrdquo IEEE Journal on SelectedTopics in Signal Processing vol 1 no 4 pp 586ndash598 2007
[20] D L Donoho and Y Tsaig ldquoFast solution of ℓ1-normminimiza-
tion problems when the solution may be sparserdquo Tech RepDepartment of Statistics Stanford University Stanford CalifUSA 2008
[21] T T Do L Gan N Nguyen and T D Tran ldquoSparsity adaptivematching pursuit algorithm for practical compressed sensingrdquoin Proceedings of the 42nd Asilomar Conference on SignalsSystems and Computers pp 581ndash587 Pacific Grove Calif USAOctober 2008
[22] S S Chen D L Donoho and M A Saunders ldquoAtomicdecomposition by basis pursuitrdquo SIAM Journal on ScientificComputing vol 20 no 1 pp 33ndash61 1998
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of