research article compressed sensing based apple image...

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


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


(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


GaussianBernoulli

OrthogonalHadamard

Toeplitz

55555

56565

57575

58585

59

35 355 36 365 37 375 38

Tim

e

PSNR


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


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








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


[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


[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


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor 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


Measurementmatrix

observation

Thereconstructed

signal


Featureextraction





0norm119883meets


119883= argmin 1198830

st 119910 = Φ119883

(5)


1norm [16 17]





(1minus 120575119896) 119888

22 le

1003817100381710038171003817Φ1198791198881003817100381710038171003817

22 le (1+ 120575

119896) 119888

22 forall119888 isin 119877

|119879| (6)








119891 (119909) =1


221205902




119863(119883) = 1205902

The elements 119909119894119895


119894119895sim 119873(0 1119899)




Gaussian

Bernoulli

OrthogonalHadamard

Toeplitz

39540

40541

41542

42543

43544

445

342 344 346 348 35 352 354 356

Tim

e

PSNR


Gaussian Bernoulli

Orthogonal

Hadamard

Toeplitz

44454647484950515253

345 35 355 36 365

Tim

e

PSNR







Gaussian

Bernoulli

Orthogonal

Hadamard

Toeplitz

46474849505152535455

345 35 355 36 365 37

Tim

e

PSNR


GaussianBernoulli

OrthogonalHadamard

Toeplitz

55555

56565

57575

58585

59

35 355 36 365 37 375 38

Tim

e

PSNR






















4 Conclusion




Acknowledgments


References





















1-regularized least









RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Measurementmatrix

observation

Thereconstructed

signal


Featureextraction





0norm119883meets


119883= argmin 1198830

st 119910 = Φ119883

(5)


1norm [16 17]





(1minus 120575119896) 119888

22 le

1003817100381710038171003817Φ1198791198881003817100381710038171003817

22 le (1+ 120575

119896) 119888

22 forall119888 isin 119877

|119879| (6)








119891 (119909) =1


221205902




119863(119883) = 1205902

The elements 119909119894119895


119894119895sim 119873(0 1119899)




Gaussian

Bernoulli

OrthogonalHadamard

Toeplitz

39540

40541

41542

42543

43544

445

342 344 346 348 35 352 354 356

Tim

e

PSNR


Gaussian Bernoulli

Orthogonal

Hadamard

Toeplitz

44454647484950515253

345 35 355 36 365

Tim

e

PSNR







Gaussian

Bernoulli

Orthogonal

Hadamard

Toeplitz

46474849505152535455

345 35 355 36 365 37

Tim

e

PSNR


GaussianBernoulli

OrthogonalHadamard

Toeplitz

55555

56565

57575

58585

59

35 355 36 365 37 375 38

Tim

e

PSNR






















4 Conclusion




Acknowledgments


References





















1-regularized least









RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Gaussian

Bernoulli

OrthogonalHadamard

Toeplitz

39540

40541

41542

42543

43544

445

342 344 346 348 35 352 354 356

Tim

e

PSNR


Gaussian Bernoulli

Orthogonal

Hadamard

Toeplitz

44454647484950515253

345 35 355 36 365

Tim

e

PSNR







Gaussian

Bernoulli

Orthogonal

Hadamard

Toeplitz

46474849505152535455

345 35 355 36 365 37

Tim

e

PSNR


GaussianBernoulli

OrthogonalHadamard

Toeplitz

55555

56565

57575

58585

59

35 355 36 365 37 375 38

Tim

e

PSNR






















4 Conclusion




Acknowledgments


References





















1-regularized least









RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation






















4 Conclusion




Acknowledgments


References





















1-regularized least









RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation






















1-regularized least









RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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