Research ArticleA Novel Measurement Matrix Optimization Approach forHyperspectral Unmixing

Su Xu12 and Xiping He12

1College of Computer Science and Information Engineering Chongqing Technology and Business University Chongqing 400067 China2Chongqing Engineering Laboratory for Detection Control and Integrated System Chongqing Technology and Business UniversityChongqing 400067 China

Correspondence should be addressed to Su Xu xusu44163com

Received 25 January 2017 Revised 25 February 2017 Accepted 16 March 2017 Published 3 July 2017

Academic Editor Qiang Song

Copyright copy 2017 Su Xu and Xiping HeThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Each pixel in the hyperspectral unmixing process is modeled as a linear combination of endmembers which can be expressed inthe form of linear combinations of a number of pure spectral signatures that are known in advance However the limitation ofGaussian random variables on its computational complexity or sparsity affects the efficiency and accuracy This paper proposesa novel approach for the optimization of measurement matrix in compressive sensing (CS) theory for hyperspectral unmixingFirstly a new Toeplitz-structured chaotic measurement matrix (TSCMM) is formed by pseudo-random chaotic elements whichcan be implemented by a simple hardware secondly rank revealing QR factorization with eigenvalue decomposition is presentedto speed up the measurement time finally orthogonal gradient descent method for measurement matrix optimization is used toachieve optimal incoherence Experimental results demonstrate that the proposed approach can lead to better CS reconstructionperformance with low extra computational cost in hyperspectral unmixing

1 Introduction

Compressive sensing (CS) theory [1 2] is a new developedtheoretical framework on signal sampling and data compres-sion which indicates that if a signal is sparse or compressiblein a certain transform domain the transformed higher-dimensional signal can be projected onto a lower dimensionalspace by a measurement matrix It leads to nonadaptivemeasurement encoding on the signal at a rate far below theNyquist sampling rate converting from sampling the signalitself to sampling the information contained in the signalTherefore it has recently gained more and more attention invarious areas of applied mathematics computer science andelectrical engineering

Design of measurement matrix is a research hotspot inCS and measurement matrix optimization has become aninevitable trend to construct a new measurement matrix sys-tem In recent years scholars have yieldedmany optimizationmethods [3ndash22] to design measurement matrix to reducethe minimum coherence of Gram matrix These are typically

fallen into three categories iterative thresholdingmethod [3ndash14] gradient iteration process [15ndash19] and Tensor product[20ndash22]

Zhang et al [6] proposed that the Kronecker productmeasurementmatrix based on orthogonal basis canmaintainnonlinear correlation between columnsrsquo vector from high-dimensional data But Kronecker product [5 6] leads tolow sampling efficiency and poor computational complexitywhich limit the deep study for measurement matrix

By using iterative thresholding method Elad [7] itera-tively reduces the averagemutual coherence using a shrinkageoperation followed by singular value decomposition (SVD)step and shrinks the elements of Gram matrix Lustig et al[8] defined an incoherence criterion and proposed a MonteCarlo scheme for random incoherent sampling Abolghasemiet al [9] attempted a kind of nonuniform sampling bysegmenting the input signal and taking samples with differentrates from each segment Duarte-Carvajalino and Sapiro[10] take advantage of an eigenvalue decomposition process

HindawiJournal of Control Science and EngineeringVolume 2017 Article ID 8471024 13 pageshttpsdoiorg10115520178471024

2 Journal of Control Science and Engineering

followed by a KSVD-based algorithm to optimize measure-ment matrix and learn dictionary basis respectively Wanget al [11] propose to generate colored random projectionsusing an adaptive scheme Next they [12] propose a variabledensity sampling strategy by exploiting the prior informationabout the statistical distributions of natural images in thewavelet domain Although the algorithm is simple and easyto understand the research [13 14] found that iterativethresholding method has slow convergence speed and is easyto fall into local minimum problems in practical applicationMeanwhile it damages Restricted Isometric Property (RIP)and eventually may cause the collapse of the BP algorithm

By using gradient iteration process Xu et alrsquos algorithm[15] first shrinks and updates elements in Gram matrix withEquiangular Tight Frame (ETF) Li et al [18] constructsthe dimensional orthogonal matrix in SVD Abolghasemiet alrsquos algorithm [16] lies in the innovation of the gradientiteration process to obtainmeasurementmatrix Zhang et alrsquosalgorithm [17] adopts the spherical search steepest descentmethod Tian et alrsquos algorithm [19] shrinks Gram matrixwith the orthogonal gradient factor matrix to reduce themaximum and average mutual coherence of measurementmatrixThe appearance of different pursuit rates brings aboutthe saddle-point steady-state solution which only guaranteesa local minimum solution

Unfortunately most algorithms neglect intrasensor cor-relations between the samples of high-dimensional data like3D color image video and hyperspectral image it may affectthemultispectral features excessively and destroy the originalstructure of high-dimensional data and it ultimately affectsthe precision of hyperspectral unmixing

Toeplitz-structured chaoticmeasurementmatrix is one ofdeterministic measurement matrices in CS which requiresonly 119874(119872) independent variables and 119874(119872 log 2119872) opera-tions But it still has three attractiveweaknesses optimal inco-herence being unachieved reconstruction precision beinginsufficient and measurement time being unbearable

To eliminate the weaknesses we are inspired to workon optimizing the Toeplitz-structured chaotic measurementmatrix to obtain better results fromhigh-dimensional signalsMeanwhile the possibility of fusing these two attractive opti-mized methods is certain rank revealing QR factorizationwith eigenvalue decomposition from Xu et alrsquos algorithm[15] and orthogonal gradient descent approach from Tianet alrsquos algorithm [19] to obtain a new optimized method toovercome the computational complexity This is my intuitiveidea of this paper

The key contribution of this work can be elaborated asfollows

(1) In the domain of designing the measurement matrixit is crucial to achieve high quality with implement-ing effective hardware In this work some pseudo-random chaotic elements can approximate the ran-dom structure component which satisfies RIP prop-erty with overwhelming probability So these pseudo-random chaotic elements are applied to Toeplitz-structured chaotic measurement matrix which formsa new measurement matrix (TSCMM) Compared

with the others we attempt to prove that it satisfiesRIP conditions

(2) According to the properties of the separated contentsGram matrix is improved by a rank revealing QRfactorization with eigenvalue decomposition to speedup convergence rate The results of experiments showthat the proposed methods can greatly reduce com-putation complexity building on the convergence androbustness

(3) From a practical point of view high computationalcomplexity imposes restrictions on achieving optimalincoherence orthogonal gradient descent approach isproposed to acquire measurement matrix optimiza-tion The improved scheme can effectively reducethe reconstruction error and acquire satisfied imagequality compared to other conventional methods

The rest of this paper is organized as follows The mainpart of this paper starts with review of somemeasuring coher-ence criteria and develops to RIP in Section 2 Based on theprevious analysis the proposed approach for measurementmatrix optimization in Section 3 is developed Particularlymotivated by TSCMM QR factorization with eigenvaluedecomposition and improved gradient descent approach isthen suggested in the following three subsections startingwith Toeplitz-structured chaotic measurementmatrix in Sec-tion 31 followed byQR factorization algorithm in Section 32and an orthogonal gradient descent approach for measure-mentmatrix optimization in Section 33 Experimental resultsin Section 4 and conclusions in Section 5 are presented

2 Problem Formulation and Analysis

From the viewpoint of mathematics CS sample procession isapproximated by recovering 119909 from far incomplete measure-ments 119910 = Φ119909 = ΦΨ120579 (1)

where 120579 in some basis Ψ is sparse or compressible repre-sentation while Φ isin R119870times119873 is so-called CS measurementmatrix Because of 119870 ≪ 119873 the signal is measured throughthe projection by the measurement matrix Φ which leads tobeing highly underdetermined

If the D-dimensional sample signal 119883 = [1199091198791 1199091198792 119909119879119863]and independent measurements result 119884 = [1199101198791 1199101198792 119910119879119863]are unknown the Kronecker product measurement matrix[5] can be expressed as Φ = Φ1 otimes Φ2 otimes sdot sdot sdot otimes Φ119863 When eachsensor obtaining its independent measurements is the samemeasurementmatrixΦ119863 = Φ1015840 the jointmeasurementmatrixcan be expressed asΦ = 119868119863 otimesΦ1015840 where 119868119863 denotes the119863times119863identity matrix as shown in Figure 1

The constants 120575119870 for the matrix Φ are intrinsically tiedto the singular values of all column submatrices of a certainsize If Φ1 Φ2 Φ119863 are matrices with restricted isometryconstants (RIP) 120575119870(Φ1) 120575119870(Φ2) 120575119870(Φ119863) the structure of

Journal of Control Science and Engineering 3

[I] otimes [Φ120575] =[[[[[[[

[Φ120575][Φ120575][Φ120575][Φ120575]

]]]]]]]

S

S

S S S

Figure 1 The diagram of the joint measurement matrix in Duartersquosmethod

Kronecker product matrices yields simple bounds for theirRIP that can be expressed as

120575119870 (Φ1 otimes Φ2 otimes sdot sdot sdot otimes Φ119863) le 119863prod119889=1

(1 + 120575119870 (Φ119889)) minus 1 (2)

Considering the D-dimensional Kronecker sparsifyingbasisΨ = Ψ1otimesΨ2otimessdot sdot sdototimesΨ119863 and a globalmeasurement basis orframes obtained through a Kronecker product of individualmeasurement bases the definition of mutual coherence ispresented as120583 (Φ1 otimes Φ2 otimes sdot sdot sdot otimes Φ119863 Ψ1 otimes Ψ2 otimes sdot sdot sdot otimes Ψ119863)= 119863prod

119889=1

120583 (Φ119889 Ψ119889) (3)

High-dimensional Kronecker compressive sensing(HKCS) [6] proposed the optimal synthetic sensing matrixby taking Kronecker products of individual optimal sensingmatrix in each dimension The optimal sensing matrix thatminimizes the mutual coherence of the projection matrixcan be expressed as Φ1015840 = Φ119905 otimes Φ119904 (4)

With the same sampling rate matrices of HKCS haverelatively smaller mutual coherence It can be written as120583 (ΦΨ) le 120583 (Φ1015840 Ψ) (5)

It also indicates that the optimization process is dividablewhich preserves the block feature of Kronecker productmatrix and enables fast low-scale matrix computation Theoverall video acquisition is decomposed as shown in Figure 2

The high-dimensional Kronecker products measurementmatrix is our optimization goal as shown in Figure 3

If119860119888119904 = Ψ119905 otimesΦ119904 is defined as Grammatrix and minimumsquare error cost function is defined as 119864 the optimizationproblem can be written as119864 ≜ MSE = 100381710038171003817100381710038171003817(119860119888119904)119879119860119888119904 minus 1198681003817100381710038171003817100381710038172119865

st 119860119888119904 = Ψ119905 otimes Φ119904 (6)

Acquire in progressive

Dimensions and corresponding matrices

fashion

S

[ΦS] [ΨS][Φt] [Ψt]

t

t

t

t

Figure 2 The proposed multidimensional compressive sensing forvideo acquisition

SSSS

S

t

t

[Ψt] otimes [Φ120575] =[[[[[[[[[[[

[Ψt11

Φ120575

] [Ψt12

Φ120575

] [Ψt13

Φ120575

] [Ψt14

Φ120575

][Ψt21

Φ120575

] [Ψt22

Φ120575

] [Ψt23

Φ120575

] [Ψt24

Φ120575

][Ψt31

Φ120575

] [Ψt32

Φ120575

] [Ψt33

Φ120575

] [Ψt34

Φ120575

]

]]]]]]]]]]]

Figure 3 The diagram of the joint measurement matrix in HKCS

The goal of eliminating the correlation is to minimizethe difference between Gram matrix and identity matrix inthe form of Frobenius norm Considering Kronecker productproperties 119864 ≜ MSE = 10038171003817100381710038171003817(Ψ119879119905 Ψ119905) otimes (Φ119879119905 Φ119905) minus 119868100381710038171003817100381710038172119865 (7)

If the value Θ119879Θ = 119881119879Λ119881 can be replaced withits corresponding eigenvalue decomposition and Φ119879119905 Φ119905 =(8119898)sum119899119894=0[119909(119894)]119879119909(119899 minus 119894) (7) can then become as follows119864 ≜ MSE = 1003817100381710038171003817100381710038171003817100381710038171003817and minus and119881119879 8119898 119899sum

119894=0

[119909 (119894)]119879 119909 (119899 minus 119894) 119881and10038171003817100381710038171003817100381710038171003817100381710038172119865 (8)

Because and is real diagonal matrices and119881119879119881and = (119881and)119879119881andIf 119861 = 119881and then 119861 fl [|1198611198941198952|]

Here (8) can be reduced to119864 ≜ MSE = 1003817100381710038171003817100381710038171003817100381710038171003817and minus 119861119879 8119898 119899sum119894=0

[119909 (119894)]119879 119909 (119899 minus 119894) 11986110038171003817100381710038171003817100381710038171003817100381710038172119865 (9)

Supposing that 119861119894119895 is the elements in 119861 gradient decreaseiteration method is used to minimize mean square error(MSE) 119861119894119895 larr 119861119894119895 minus 120588nabla119864 where 120588 is step size and 120588 gt 0

4 Journal of Control Science and Engineering

Quasi-Toeplitzmatrix

QR factorizationwith eigenvaluedecomposition

Orthogonalgradient descent

method

Sparsifying

Randomly

columns

Discrete chaotic

sequence

Toeplitzmatrix

Toeplitzarrangement

Toeplitz-structured

chaoticmeasurement

measurementGrammatrix

The final

matrix

Randomly generated measurement matrix

Eigen-decomposition

Generate synthetic

measurementmatrix

matrix Φt

sampling Q

basis Ψt

Figure 4 The improved scheme by different methods

nabla119864 equiv 120597119864120597119861119894119895 is gradient value of 119864 then 119861(119894+1) = 119861119894minus120578119861119894(119861119894119879(8119898)sum119899119894=0[119909(119894)]119879119909(119899 minus 119894)119861119894) According to Ger-schgorin theorem the column coherence of 120593119894 can bededuced as follows120583 = max

1le119894119895le119873119894 =119895

10038161003816100381610038161003816⟨120593119894 Ψ119895⟩10038161003816100381610038161003816 (10)

If119873 le 119872(119872+1)2 the infimumof the column coherenceis called Welch bound120583 ge 120583119908 = 119873 minus 119872(119873 minus 1)119872 (11)

Equiangular Tight Frame (ETF) is derived from (11) if theconstraints are equality

3 The Proposed Approach

Based on previous conclusions the proposed algorithm aimsto optimize the Toeplitz-structured chaotic measurementmatrix to obtain better results from hyperspectral unmixingTherefore the research content in this paper mainly consistsof three parts designed TSCMM optimized Gram matrixand orthogonal gradient descent approach as shown inFigure 4

The study is given by taking the following methodsfirstly to obtain easy hardware implemented pseudo-randomchaotic elements are used to form a new Toeplitz-structuredchaotic measurement matrix (TSCMM) as discussed inSection 31 to overcome unbearable Cost Time Grammatrixis improved by a rank revealing QR factorization witheigenvalue decomposition as discussed in Section 32 toachieve optimal incoherence orthogonal gradient descentmethod for measurement matrix optimization is presentedin Section 33 Finally the improved scheme is presentedthrough explicit analysis and discussion

31 Xu-TSCMM Recently [23] is written by myself com-pletely and probed into its initial theory and researchDiscrete chaotic system function is proposed to generate aseries of pseudo-random numbers Based on those elementsToeplitz-structured chaotic measurement matrix (TSCMM)

is produced to guarantee the incoherence criterion To reducethe building time of TSCMM Circulantblock-diagonalsplitting structure is attached on TSCMM Although aboutone-third of matrix values are eliminated the measurementmatrix is proved to satisfy Johnson-Lindenstrauss (J-L)lemma and achieves the goal of satisfying RIP Logistic map[24] as the simplest dynamic systems evidencing chaoticbehavior is described as follows119909119899+1 = 120583119909119899 (1 minus 119909119899) 120583 isin [0 4] 119909119899 isin [0 1] sub 119877 (12)

where 120583 isin (35699 4] sub 119877 is the discrete state While para-meter 120583 is 4 the sequence 119909119899(119905) satisfies beta distributionwith 120572 = 05 and 120573 = 05 and the next probability densityfunction 119891(119909 05 05) = (1120587)(119909 minus 1199092)minus12 has been used forsimulation

Set 119911119894(119905) as the output sequence generated by (12) withinitial condition 119911119894(0) and let 119909119894(119905) denote the regularizationof 119911119894(119905) as the following form 119909119894(119905) = 119911119894(119905)minus05 119894 = 0 1 2

Approximately 119909119894(119905) can be considered as random vari-able and it satisfies the following distribution 119891(119909) =(1120587)(025 minus 1199092)minus12 Then by selecting 119898 different initialconditions 119911(0) isin [0 1]119898 sub 119877119898 one can obtain 119898 vectorswith dimension 119877 which enables us to construct the follow-ing matrixΦ scaled byradic8119898

Φ = radic 8119898 (1199090 (0) sdot sdot sdot 1199090 (119899 minus 1) d119909119898 (0) sdot sdot sdot 119909119898 (119899 minus 1)) (13)

Here (13) is called the beta-like matrixAccording to (13) set one initial condition 119911 isin (0) isin R

and generate a sequence 119909 isin R119899 in the chaotic system ThenToeplitz-structured matrix Φ = R119898times119899 is constructed in thefollowing form

Φ = radic 8119898 (119909(119899 minus 1) 119909 (119899 minus 2) sdot sdot sdot 119909 (0)119909 (0) 119909 (119899 minus 1) 119909 (1) d119909 (119898 minus 2) 119909 (119898 minus 3) sdot sdot sdot 119909 (119898 minus 1)) (14)

Journal of Control Science and Engineering 5

Hereradic8119898 is for normalization andΦ is called Toeplitz-structured chaotic measurement matrix (TSCMM) whichmeets J-L theorem

32 Duarte-ETF Method Minimum coherence property ofETF has been the main target to find a feasible solution Itis impossible to solve the problem exactly because of thecomplexity while the structure of Gram matrix has changedso much that the selection of new units in the following stepis very difficult

To minimize (14) an optimization approach is adoptedto reduce the maximum and average mutual coherence ofmeasurement matrix It shrinks Gram matrix based on ETFtheoryThemethod canminimize the globalmutual coherentcoefficient of TSCMM by adjusting the eigenvalues abovezero to the average value of the sumof these eigenvalues with-out changing the sum After performing alternating mini-mization the optimized measurement matrix can be con-structed from the output Gram matrix with a rank revealingQR factorization with eigenvalue decomposition

Theorem 1 Given a measurement matrix Φ = R119898times119871 andrepresenting matrix Θ = R119899times119898 there exists a matrix 119863 =ΘΦ and Gram matrix 119866 = 10067041198631198791006704119863 where 1006704119863 is the columnnormalization from119863 If the real positive definite matrix119866 has120582119896 gt 0 (119896 = 1 sim 119898) the following equality sum119898119896=1 120582119896 = 119899 andsum119898119896=1(120582119896)2 = sum119899119894119895=1(⟨119894 119895⟩)2 where 119894 (119894 = 1 sim 119899) is thecolumn of 1006704119863 which is obtained

Based on Theorem 1 to minimize the largest absolutevalues of the off-diagonals in the correspondingGrammatrixwe can determine the eigenvalues of Gram matrix by solvingthe following optimization problems

min sum119894 =119895

(119892119894119895)2 = 119898sum119896=1

(120582119896)2 minus 119899sum119894=1

(119892119894119895)2st 119898sum

119896=1

120582119896 = 119899 (15)

where 119892119894119895 is the element of Gram matrix The optimizationproblem (15) is to minimize the square sum of the elementof Gram matrix if the sum of the characteristic value 120582119896remains constant Though the eigenvalue decomposition ofGram matrix has 120582119896 gt 0 the value 119899119898 has been graduallyapproaching to minimize the square sum of sum119898119896=1(120582119896)2Because real symmetric matrix eigenvalue decomposition isorthogonal the square sum of nondiagonal elements fromGram matrix gradually decreases and further it attains theeffect as follows

min sum119894 =119895

(119892119894119895)2 = 119898sum119896=1

( 119899119898)2 = 119899 (16)

Finally 1006704119863 is obtained after several iterations which is alsothe optimal Gram matrix 119866best

33 The Orthogonal Gradient Descent Approach Accordingto the definition of Grammatrix Grammatrix is the productof measurement matrix and sparse matrix Therefore theoptimalmeasurementmatrix can be directly derived from theoptimal Gram matrix However as for overcomplete sparserepresentation based on redundant dictionary it is very hardto design an effective algorithm to construct measurementmatrix In order to solve this problem orthogonal gradientdescent method is employed to get the optimal measurementmatrix Θbest

If the optimal Gram matrix 119866best is obtained the optimalmeasurement matrix Θbest is as followsΘbest = argmin 10038171003817100381710038171003817119866best minus Θ119879Θ10038171003817100381710038171003817119865 ≜ 119865 (Θ) (17)

And then the complex problem from optimal Grammatrix119866best is transformed into simpleminimumof119865(Θ) Bydetermining the derivative of119865(Θ) a skew-symmetricmatrix119882 with the measurement matrix Θ and the gradient matrixnabla119865 is used to obtain revision factornabla119865 (Θ) = Θ (Θ119879Θ minus 119866best) 119882 (Θ nabla119865) = Θ119879 (nabla119865) minus (nabla119865)119879Θ (18)

Next if 119868 is identity matrix the orthogonal matrix 119878 canbe expressed through the Cayley transform to ensure thepositive definiteness of the revision factor119878 = (119868 minus 119882) (119868 + 119882)minus1 (19)

The orthogonal gradient factor matrix Δ can be obtainedto update the gradient directionΔ = nabla119865 + (nabla119865) 119878 = nabla119865 (119878 + 119868) (20)

Combining (19) and (20) (21) can be rewritten as followsΔ = nabla119865 (119868 + (119868 minus 119882) (119868 + 119882)minus1) (21)Δ = 2nabla119865 (119868 + 119882)minus1 (22)

Finally update measurement matrixΘ with the orthogo-nal gradient factor matrix ΔΘ larr997888 Θ minus 120578Δ (23)

Because 120578 is updating ratio 119865(Θ) gradually converges attheminimumvalueThen the optimalmeasurementmatrixΘis the goal of our pursuit According to the linearity propertyof Toeplitz measurement matrix [25]Θ satisfies J-L propertywith overwhelming probability FromTheorem 1 it has beenproven that J-L condition can replace RIP condition So Θalso satisfies RIP property with overwhelming probability

The flow diagram of the proposed method will be givenas shown in Algorithm 1

4 Experiments and Result Analysis

To illustrate the effectiveness of the proposed approach themost widely used hyperspectral images in unmixing such

6 Journal of Control Science and Engineering

Input number of measurements119872 dictionary Ψ119871times119873 threshold 120585 = radic(119871 minus 119873)119873(119871 minus 1)updating ratio 120578 number of iterations Iter1 Iter2Initialization Set Φ119872times119873 to be TSCMMUpdate (1) Set the initial value of iteration 1198961 = 0(2)Optimize Gram matrix

(a) Compute Gram matrix 119866 = Θ119879Θ = Ψ119879Φ1198791198961Φ1198961Ψ(b) Normalize 1006704119866 = diag(1radicdiag(119866)) lowast 119866 lowast diag(1radicdiag(119866))(c) Update the elements of Gram matrix 1006704119866(3) Optimize measurement matrix(a) Set the initial value of iteration 1198962 = 0(b) Compute the orthogonal gradient factor matrix Δ 1198962 (c) Update measurement matrix Θ1198962 Θ1198962 = Θ1198962 minus 120578Δ 1198962 (d) 1198962 = 1198962 + 1 if 1198962 = Iter2 stop else return to Step 32)(4)Compute measurement matrix Φ1198961 (5) 1198961 = 1198961 + 1 if 1198961 = Iter1 stop else return to Step (2)

Output Φbest is the optimal measurement matrixΦIter1minus1Further SNR and reconstructed signal

Algorithm 1 The flow diagram of the proposed method Optimization of TSCMM

as Cuprite Urban and Jasper Ridge were selected in thespectral range from 380 nm to 2500 nm each channel bandwidth is up to 946 nm All high-dimensional data is providedby the standard hyperspectral library of 224 bands whichcomes from [26] To reduce complexity there are only 128times 128 pixel blocks of original image which starts from the(0 0)th pixel Further only 8 channels (from 11 to 81 bandsevery 10 bands) were remained due to dense water vapor andatmospheric effects

In the course of the experiment the signal sparsitymethod is Fourier basis and reconstruction algorithm isStagewise Orthogonal Matching Pursuit (StOMP) [27] Var-ious kinds of measurement matrix (Circulant ToeplitzToeplitz-structured chaotic measurement matrix (TSCMM)[23] Elad-Optimization Method (EOM) [7] and Duarte-Carvajalino and Sapirorsquos Method (Duarte-ETF) [10]) areemployed to illustrate the effectiveness of proposed approachfor hyperspectral unmixing

To demonstrate the efficiency of these methods tradi-tional evaluation methods can generally be divided into twocategories (1) subjective assessment and (2) objective evalu-ation Mean Squared Error (MSE) and Peak-Signal-to-NoiseRatio (PSNR) as one of the most important indices fromobjective evaluation determine the quality of recovery imagewhile Cost Time (CT) verifies the efficiency of the proposedapproach They all testify experiment results of recoverysignals built on laptop with AthlonTM Processor 160G HZ1 GB RAM Matlab 70 andWindows XP operation platform

41 Cuprite To illustrate the use of the hyperspectral anal-ysis process a sample scene covers the Cuprite miningdistrict in western Nevada USA from NASArsquos AirborneVisibleInfrared Imaging Spectrometer (AVIRIS) is providedThe data provided here is one of the most widely used hyper-spectral images in unmixing studyThere are 210 wavelengthsranging from 400 nm to 2500 nm resulting in a spectralresolution of 10 nm

In Figure 5 the first image (Top) was taken in bluelight the second image (middle) was taken in red light andthe third image (bottom) was taken in near infrared lightcentered at a wavelength of 750 nanometers

Figure 5 presents the subjective evaluation by Circu-lant Toeplitz TSCMM EOM Duarte-ETF and proposedmethod Compared with other results the performance fromFigure 5(b) is the worst The reason is that the elementsfrom Circulant measurement matrix follow periodic rep-etition permutation which does not satisfy RIP propertywith overwhelming probability Figure 5(c) clearly demon-strates that Toeplitz measurement matrix can avoid theproblem well Because of the property of pseudo-randomof chaotic sequence the performance from TSCMM hasfurther improved as shown in Figure 5(d) The result fromFigures 5(e)ndash5(g) shows that there is a significant impacton different bands using different optimization methodsNear infrared image is less affected by dust and gas Visibleblue channel has strong capability to penetrate water andvisible red channel can reflect the health status of plantsTherefore the sorted off-diagonal entries of themeasurementmatrix from EOM are likely more sparse and diagonal entriesare more concentrated The results are clearly shown inFigure 5(g) that the proposed approach had a significantperformance compared to any others and closely resemblesoriginal image Furthermore the objective evaluations whichinclude Mean Squared Error (MSE) Peak-Signal-to-NoiseRatio (PSNR) and Cost Time (CT) can avoid artificial errorand draw compelling conclusion The results can be clearlyseen fromdifferentmethods on recoveryCuprite as shown inTable 1

Figure 5 andTable 1 report the recovery quality of the pro-posed method on recovery Cuprite The following observa-tions are summarized (1) of all evaluating indicators con-sidered here traditional Circulant had the worst perfor-mance in both subjective and objective evaluations (2) sincethe introduction of Toeplitz the performance gets major

Journal of Control Science and Engineering 7

(1) 11

(2) 41 bands (128 times 128 pixels)

bands (128 times 128 pixels)

(3) 81 bands (128 times 128 pixels)

(a1) original (b1) Circulant (c1) Toeplitz (d1) TSCMM (e1) EOM (f1) Duarte-ETF (g1) Proposed

(a2) original (b2) Circulant (c2) Toeplitz (d2) TSCMM (e2) EOM (f2) Duarte-ETF (g2) Proposed

(a3) original (b3) Circulant (c3) Toeplitz (d3) TSCMM (e3) EOM (f3) Duarte-ETF (g3) Proposed

Figure 5The subjective quality of different CSmeasurementmatrices from left to right (a) original (b) Circulant (c) Toeplitz (d) TSCMM(e) EOM (f) Duarte-ETF and (g) proposed method

Table 1 MSE and PSNR of different matrixes of recovery Cuprite

Algorithm 11 bands (64 times 64 pixels) 41 bands (64 times 64 pixels) 81 bands (64 times 64 pixels)PSNR MSE PSNR MSE PSNR MSE CT

Circulate 599634 00656 609680 00520 615821 00452 51070Toeplitz 599634 00656 619375 00416 633173 00303 35506TSCMM 600696 00640 619783 00412 633462 00301 17624EOM 602806 00610 622337 00389 633962 00297 2454345Duarte-ETF 606304 00562 623788 00376 634082 00297 1165994Proposed 643532 00239 638856 00266 637179 00276 516191

improvement on image quality while improved Toeplitz-structured matrix method (TSCMM) is slightly better thanclassical Toeplitz matrix method (3) EOM has significantperformance in image quality however the optimizationprocess is usually an iterative process which is also a verycomplicated and time-consuming process (4) Duarte-ETFhas better contrast and lower computational complexity (5)the proposed method takes advantage of improved Toeplitz-structured matrix to speed up the convergence speed andimprove traditional optimization method to better recoveryhigh-dimensional image Experimental results show that theproposed method has a better overall performance

42 Urban University of California Santa Barbara (UCSB)built an urban spectral library for the GoletaSanta BarbaraareaThe hyperspectral data of Urban were acquired betweenlate May and early June 2001 using an ASD full rangeinstrument on loan from the Jet Propulsion LaboratoryThesespectra of Urban are characterized by 499 roofs 179 roads

66 sidewalks 56 parking lots 40 road paints 37 types ofvegetation 47 types of nonphotosynthetic vegetation 88 baresoil and beach spectra 27 acquired from tennis courts and50 more from miscellaneous surfaces

Experiments on the hyperspectral data of Urban demon-strate that the proposed scheme substantially improves thereconstruction accuracy Clearly it comes to the same con-clusion from Figure 6 that compared with the previous fivemethods the effect of proposed approach is obviously sup-erior to any other methods and is the most similar to originalimage

To evaluate and compare the proposed method thefollowing performance indices such as Average Gradient(AG) Edge-Intensity (EI) Figure Definition (FD) GrayMean (GM) StandardDeviation (SD) Space Frequency (SF)Variance (VAR) and Structural Similarity (SSIM) were usedIt shows that the objective evaluation indices enhance theexperiment rigor and convincing The results are shown inTable 2

8 Journal of Control Science and Engineering

Table2Objectiv

eevaluationof

typicalm

etho

dsandprop

osed

metho

dfro

mrecovery

Urban

Band

sAlgorith

mTh

eobjectiv

eevaluationindices

AGEI

FDGM

SDMSE

PSNR

SFSSI

VAR

CT

11

Circulate

0070476

048360

012231

04106

0119

700929

584508

02034

098826

0119

7217

379

Toeplitz

0047687

034172

010125

032796

01137

00610

602804

018949

099361

011366

18706

TSCM

M0054635

03864

70114

14035344

01162

006

865976

58019866

099254

011624

15146

EOM

00566

08040

032

011519

035809

0117

200708

596279

020025

099226

0117

203051965

Duarte-ET

F0047687

034172

010125

032796

01137

00610

602804

018949

099361

011366

1366804

Prop

osed

004

6138

029

123

006

690

023194

01197

004

3461758

601195

6099

537

01198

0165975

41

Circulate

0072566

051888

012438

037371

01363

00631

6013

16020690

099349

013626

16382

Toeplitz

0058448

041868

012772

040

138

01390

00786

591767

022932

099207

013897

18069

TSCM

M0061306

043672

013128

039347

01362

00744

594172

022736

099259

013620

1538

1EO

M006

0340

04340

8012941

040222

01353

00771

59260

9022536

099215

013529

813160

Duarte-ET

F0062241

044

182

013123

042482

01402

00873

587190

023610

099089

014018

5974

19Prop

osed

005

0626

040

358

007

196

024

426

01307

004

52615816

012782

099

561

01306

8243532

81

Circulate

0082530

06279

013639

049357

01449

00635

6010

19022630

099397

014492

15829

Toeplitz

0077269

047993

014082

044

812

01390

00535

60844

2023904

099484

013905

18284

TSCM

M0070612

050737

014828

051106

01496

00716

595790

025291

099353

014958

15702

EOM

0070821

05100

6014940

050827

01503

00707

596361

02544

7099369

015030

9731

53Duarte-ET

F0074147

052535

014946

048963

01499

00652

599890

025959

099450

014986

590915

Prop

osed

006

8955

042

678

009

339

040

915

01571

004

84612814

01648

5099

580

01570

82874

26

Journal of Control Science and Engineering 9

(1) 11 bands (64 times 64 pixels)

(3) 81 bands (64 times 64 pixels)

(2) 41 bands (64 times 64 pixels)

(a1) original (b1) Circulant (c1) Toeplitz (d1) TSCMM (e1) EOM (f1) Duarte-ETF (g1) Proposed

(a2) original (b2) Circulant (c2) Toeplitz (d2) TSCMM (e2) EOM (f2) Duarte-ETF (g2) Proposed

(a3) original (b3) Circulant (c3) Toeplitz (d3) TSCMM (e3) EOM (f3) Duarte-ETF (g3) Proposed

Figure 6The subjective quality of different CSmeasurementmatrices from left to right (a) original (b) Circulant (c) Toeplitz (d) TSCMM(e) EOM (f) Duarte-ETF and (g) proposed method

The results of Table 2 show that the method has a higherperformance than traditional Toeplitz or Circulant matrixmethod Although improved Toeplitz-structured matrixmethod (TSCMM) is slightly better than classical Toeplitzboth classical optimization measurement matrix method(EOM) and proposed method have significant performancein image quality Furthermore the proposed method takesadvantage of improved Toeplitz-structured matrix to speedup the convergence speed and improve traditional opti-mization method to recover better high-dimensional imageExperimental results show that the proposed method has abetter overall performance

43 Jasper Ridge The hyperspectral image of Jasper Ridgewas obtained on June 2 September 4 and October 6 1992whichwas calibrated to surface reflectanceThe imagewas themost popular source to analyze with spectral mixture anal-ysis using library endmembers representing green foliagenonphotosynthetic vegetation and soils characteristic of thesite Field-based vegetation was obtained fromUSGeologicalService

From Figure 7 it is obvious that the worst effects fromtraditional Circulant have reached being almost intolerableWhile the results from TSCMM and Toeplitz are almostsimilar the former has only slight improvement comparedto the latter On the other hand the performance of theproposed approach was significantly improved compared tothat of EOM or Duarte-ETF The study concluded that theproposed approach had a significant performance comparedto that of others Furthermore the objective evaluations are

shown inTable 3The results can be clearly seen fromdifferentmethods on recovery Jasper Ridge

The results of Table 3 show that the proposed methodhas a higher performance than traditional Toeplitz or Circu-lant matrix method TSCMM takes advantage of improvedToeplitz-structuredmatrix to speedup the convergence speedand improve traditional Toeplitz or Circulant matrix methodto recover better high-dimensional data Although EOMhas the lower column coherence and faster convergence itweakens RIP condition and causes recovery performancedegradation While the absolute values from Duarte-ETFconcentrate around mutual coherence this can make theequivalent dictionary as close as possible to an ETF But thisalgorithm has high computational complexity Furthermoreexperimental results from the proposed method show thatthe proposed method has a better overall performance

44 Hyperspectral Unmixing The fourth experiment per-forms an experimental evaluation of the accuracy of thestandard hyperspectral unmixing districts known as Cuprite[28] To reduce complexity there are only 188 channels(3ndash103 114ndash147 and 168ndash220 bands) that were remained dueto dense water vapor and atmospheric effects In the courseof the experiment the signal sparsity method is Fourier basisand reconstruction algorithm is StOMP

The results are shown in Figure 8 for hyperspectralunmixing and closely resemble those obtained from hyper-spectral data Figure 8 compares the unmixing performanceof the proposed method with different endmembers and theabundance from different endmembers is totally dissimilar

10 Journal of Control Science and Engineering

Table 3 The objective evaluations of different matrixes of recovery Jasper Ridge

Algorithm 11 bands (64 times 64) 81 bands (64 times 64) 151 bands (64 times 64)SNR PSNR MSE SNR PSNR MSE SNR PSNR MSE CT

Circulate minus60698 573412 01199 116699 600402 00644 33299 603823 00595 15304Toeplitz 247566 707289 00055 294215 677497 00109 157386 657714 00172 15519TSCMM 153473 666425 00141 339793 697291 00069 200693 676521 00112 14869EOM 44853 619252 00417 350894 702112 00062 216750 683495 00095 788838Duarte-ETF 247566 707289 00055 333205 694430 00074 172452 664257 00148 665711Proposed 210504 691193 00080 549719 788460 848e minus 04 422277 772754 00012 345743

(1) 11 bands (64 times 64 pixels)

(2) 81 bands (64 times 64 pixels)

(3) 151 bands (64 times 64 pixels)

(a2) original (b2) Circulant (c2) Toeplitz (d2) TSCMM (e2) EOM (f2) Duarte-ETF (g2) Proposed

(a1) original (b1) Circulant (c1) Toeplitz (d1) TSCMM (e1) EOM (f1) DuarteminusETF (g1) Proposed

(a3) original (b3) Circulant (c3) Toeplitz (d3) TSCMM (e3) EOM (f3) Duarte-ETF (g3) Proposed

Figure 7The subjective quality of different CSmeasurementmatrices from left to right (a) original (b) Circulant (c) Toeplitz (d) TSCMM(e) EOM (f) Duarte-ETF and (g) proposed method

These features from the proposed method ensure clear andaccurate the abundances

By accessing information from USGS Digital Spec-tral Library [29] the unmixing performance has beenalmost correct Furthermore different endmembers fromUSGS 1995 Library [30] are used to verify that the predictionmodel of hyperspectral unmixing scheme is accurate inFigure 8

Apparently the proposed unmixing results (blue thinthread) have a strong correlationwith these endmembers (redthin thread) from USGS 1995 Library From Figure 9(a) asteep sloping line from Alunite suggests that the unmixingendmember (blue thin thread) has remarkable similarityThis same conclusion has been made in studies from Figures9(d)ndash9(h) On the other hand the unsatisfied results fromFigures 9(b) and 9(c) have been caused by smooth curveThe proposed method has good accuracy and is robust totraditional filtering compression cutting and noise attack

5 Conclusions

In this paper to overcome the limitation of Toeplitz-structured chaotic measurement matrix an improved mea-surement matrix has been carried out in the hyperspectralunmixing process to achieve multiple endmembers of hyper-spectral image And in theory it proves that this matrixhas retained the RIP property with overwhelming proba-bility Experimental results demonstrate that the proposedmethod to design of measurement matrix leads to better CSreconstruction performance with low extra computationalcost Compared with some traditional measurement matrixan improved method has highest technical feasibility lowestcomputational complexity and least computation time con-sumption in the same recovery qualityThe proposedmethodcan take the special advantage in hyperspectral unmixingprocess and explore the practical satellite system to remotesensing

Journal of Control Science and Engineering 11

(a) Alunite (b) Andradite (c) Pyrope

(d) Nontronite (e) Dumortierite (f) Kaolinite

(g) Chalcedony (h) Kaolinite (i) Buddingtonite

Figure 8 The results from different elements abundance in hyperspectral unmixing schemes

12 Journal of Control Science and Engineering

02

04

06

08

1Re

flect

ance

()

500 1000 1500 2000 2500 30000Wavelength (120583m)

(a) Alunite

0

01

02

03

04

Refle

ctan

ce (

)

500 1000 1500 2000 2500 30000Wavelength (120583m)

(b) Andradite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

1

Refle

ctan

ce (

)

(c) Pyrope

0

02

04

06

08

Refle

ctan

ce (

)

500 1000 1500 2000 2500 30000Wavelength (120583m)

(d) Nontronite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

1

Refle

ctan

ce (

)

(e) Dumortierite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

Refle

ctan

ce (

)

(f) Kaolinite

0

02

04

06

08

Refle

ctan

ce (

)

500 1000 1500 2000 2500 30000Wavelength (120583m)

(g) Chalcedony

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

Refle

ctan

ce (

)

(h) Kaolinite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

Refle

ctan

ce (

)

(i) Buddingtonite

Figure 9 The comparison map between actual unmixing effect and endmember library

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is supported by Chongqing Engineering Labo-ratory for Detection Control and Integrated System Theproject is also funded by Key Technology Research andIndustrialization of Fire Monitoring and Early Warn-ing Sensor Network for High Voltage Transmission Line(KJZH17124) This research is funded by Chongqing Edu-cation Commission Foundation (KJ1400612) This projectis also granted financial support from a CooperativeProject of Chongqing Technology and Business University(990516001)

References

[1] D L Donoho ldquoCompressed sensingrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol52 no 4 pp 1289ndash1306 2006

[2] E J Candes J Romberg and T Tao ldquoRobust uncertaintyprinciples exact signal reconstruction from highly incompletefrequency informationrdquo Institute of Electrical and ElectronicsEngineers Transactions on InformationTheory vol 52 no 2 pp489ndash509 2006

[3] T N Canh K D Quoc and B Jeon ldquoMulti-resolution kro-necker compressive sensingrdquo Transactions on Smart Processingamp Computing vol 3 no 1 pp 19ndash27 2014

[4] K Q Dinh H J Shim and B Jeon ldquoMeasurement coding forcompressive imaging using a structural measuremnet matrixrdquoin 2013 20th IEEE International Conference on Image ProcessingICIP 2013 pp 10ndash13 aus September 2013

[5] M F Duarte and R G Baraniuk ldquoKronecker compressive sens-ingrdquo IEEE Transactions on Image Processing vol 21 no 2 pp494ndash504 2012

[6] B Zhang X Tong W Wang and J Xie ldquoThe research ofKronecker product-based measurement matrix of compressivesensingrdquo EURASIP Journal on Wireless Communications andNetworking vol 2013 article 161 pp 1ndash5 2013

[7] M Elad ldquoOptimized projections for compressed sensingrdquo IEEETransactions on Signal Processing vol 55 no 12 pp 5695ndash57022007

Journal of Control Science and Engineering 13

[8] M Lustig D Donoho and J M Pauly ldquoSparse MRI the appli-cation of compressed sensing for rapid MR imagingrdquoMagneticResonance in Medicine vol 58 no 6 pp 1182ndash1195 2007

[9] V Abolghasemi S Sanei S Ferdowsi F Ghaderi and ABelcher ldquoSegmented compressive sensingrdquo in Proceedings of theIEEESP 15thWorkshop on Statistical Signal Processing (SSP rsquo09)pp 630ndash633 September 2009

[10] J M Duarte-Carvajalino and G Sapiro ldquoLearning to sensesparse signals simultaneous sensingmatrix and sparsifying dic-tionary optimizationrdquo IEEE Transactions on Image Processingvol 18 no 7 pp 1395ndash1408 2009

[11] ZWang G R Arce and J L Paredes ldquoColored randomprojec-tions for compressed sensingrdquo in 2007 IEEE International Con-ference on Acoustics Speech and Signal Processing ICASSP rsquo07pp III873ndashIII876 usa April 2007

[12] Z Wang and G R Arce ldquoVariable density compressed imagesamplingrdquo IEEE Transactions on Image Processing vol 19 no 1pp 264ndash270 2010

[13] M Elad and M Aharon ldquoImage denoising via learned dic-tionaries and sparse representationrdquo in 2006 IEEE ComputerSociety Conference on Computer Vision and Pattern RecognitionCVPR 2006 pp 895ndash900 usa June 2006

[14] M Aharon M Elad and A Bruckstein ldquoK-SVD an algorithmfor designing overcomplete dictionaries for sparse representa-tionrdquo IEEE Transactions on Signal Processing vol 54 no 11 pp4311ndash4322 2006

[15] J Xu Y Pi and Z Cao ldquoOptimized projection matrix for com-pressive sensingrdquo EURASIP Journal on Advances in SignalProcessing vol 2010 Article ID 560349 2010

[16] V Abolghasemi S Ferdowsi and S Sanei ldquoA gradient-basedalternating minimization approach for optimization of themeasurementmatrix in compressive sensingrdquo Signal Processingvol 92 no 4 pp 999ndash1009 2012

[17] Q Zhang Y Fu H Li and R Rong ldquoOptimized projectionmatrix for compressed sensingrdquo Circuits Systems and SignalProcessing vol 33 no 5 pp 1627ndash1636 2014

[18] G Li Z Zhu D Yang L Chang and H Bai ldquoOn projectionmatrix optimization for compressive sensing systemsrdquo IEEETransactions on Signal Processing vol 61 no 11 pp 2887ndash28982013

[19] S Tian X Fan and L I Zhetao ldquoOrthogonal-gradient mea-surement matrix construction algorithmrdquo Chinese Journal ofElectronics vol 25 no 1 pp 81ndash87 2016

[20] V Abolghasemi S Ferdowsi and BMakkiabadi ldquoOn optimiza-tion of the measurement matrix for compressive sensingrdquo inSignal Processing Conference European IEEE Ed pp 427ndash431August 2010

[21] Q Li D Schonfeld and S Friedland ldquoGeneralized tensorcompressive sensingrdquo in 2013 IEEE International Conference onMultimedia and Expo ICME 2013 usa July 2013

[22] S Friedland Q Li and D Schonfeld ldquoCompressive sensing ofsparse tensorsrdquo IEEE Transactions on Image Processing vol 23no 10 pp 4438ndash4447 2014

[23] S u Xu H Yin C Yi Y Xiong and T Xue ldquoAn ImprovedToeplitz Measurement Matrix for Compressive Sensingrdquo Inter-national Journal of Distributed Sensor Networks vol 8 pp 1ndash82014

[24] R L Devaney in Practical Numerical Algorithms for ChaoticSystems T S Parker and L O Chua Eds vol 32 pp 501ndash503Siam Review 3 edition 2006

[25] R Baraniuk M Davenport R DeVore and M Wakin ldquoAsimple proof of the restricted isometry property for randommatricesrdquoConstructiveApproximation An International Journalfor Approximations and Expansions vol 28 no 3 pp 253ndash2632008

[26] Available httpwwwesciencecnpeoplefeiyunZHUDatasetGThtml

[27] D L Donoho Y Tsaig I Drori and J-L Starck ldquoSparse solu-tion of underdetermined systems of linear equations by stage-wise orthogonal matching pursuitrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol58 no 2 pp 1094ndash1121 2012

[28] K Lang ldquoNewsWeeder Learning to Filter Netnewsrdquo in Inter-national Machine Learning Conference vol 1995 pp 331ndash339

[29] Available httpfeatureselectionasuedudatasetsphp[30] Available httpclopinetcomisabelleProjectsNIPS

Journal of Control Science and Engineering 3

[I] otimes [Φ120575] =[[[[[[[

[Φ120575][Φ120575][Φ120575][Φ120575]

]]]]]]]

S

S

S S S

Figure 1 The diagram of the joint measurement matrix in Duartersquosmethod

Kronecker product matrices yields simple bounds for theirRIP that can be expressed as

120575119870 (Φ1 otimes Φ2 otimes sdot sdot sdot otimes Φ119863) le 119863prod119889=1

(1 + 120575119870 (Φ119889)) minus 1 (2)

Considering the D-dimensional Kronecker sparsifyingbasisΨ = Ψ1otimesΨ2otimessdot sdot sdototimesΨ119863 and a globalmeasurement basis orframes obtained through a Kronecker product of individualmeasurement bases the definition of mutual coherence ispresented as120583 (Φ1 otimes Φ2 otimes sdot sdot sdot otimes Φ119863 Ψ1 otimes Ψ2 otimes sdot sdot sdot otimes Ψ119863)= 119863prod

119889=1

120583 (Φ119889 Ψ119889) (3)

High-dimensional Kronecker compressive sensing(HKCS) [6] proposed the optimal synthetic sensing matrixby taking Kronecker products of individual optimal sensingmatrix in each dimension The optimal sensing matrix thatminimizes the mutual coherence of the projection matrixcan be expressed as Φ1015840 = Φ119905 otimes Φ119904 (4)

With the same sampling rate matrices of HKCS haverelatively smaller mutual coherence It can be written as120583 (ΦΨ) le 120583 (Φ1015840 Ψ) (5)

It also indicates that the optimization process is dividablewhich preserves the block feature of Kronecker productmatrix and enables fast low-scale matrix computation Theoverall video acquisition is decomposed as shown in Figure 2

The high-dimensional Kronecker products measurementmatrix is our optimization goal as shown in Figure 3

If119860119888119904 = Ψ119905 otimesΦ119904 is defined as Grammatrix and minimumsquare error cost function is defined as 119864 the optimizationproblem can be written as119864 ≜ MSE = 100381710038171003817100381710038171003817(119860119888119904)119879119860119888119904 minus 1198681003817100381710038171003817100381710038172119865

st 119860119888119904 = Ψ119905 otimes Φ119904 (6)

Acquire in progressive

Dimensions and corresponding matrices

fashion

S

[ΦS] [ΨS][Φt] [Ψt]

t

t

t

t

Figure 2 The proposed multidimensional compressive sensing forvideo acquisition

SSSS

S

t

t

[Ψt] otimes [Φ120575] =[[[[[[[[[[[

[Ψt11

Φ120575

] [Ψt12

Φ120575

] [Ψt13

Φ120575

] [Ψt14

Φ120575

][Ψt21

Φ120575

] [Ψt22

Φ120575

] [Ψt23

Φ120575

] [Ψt24

Φ120575

][Ψt31

Φ120575

] [Ψt32

Φ120575

] [Ψt33

Φ120575

] [Ψt34

Φ120575

]

]]]]]]]]]]]

Figure 3 The diagram of the joint measurement matrix in HKCS

The goal of eliminating the correlation is to minimizethe difference between Gram matrix and identity matrix inthe form of Frobenius norm Considering Kronecker productproperties 119864 ≜ MSE = 10038171003817100381710038171003817(Ψ119879119905 Ψ119905) otimes (Φ119879119905 Φ119905) minus 119868100381710038171003817100381710038172119865 (7)

If the value Θ119879Θ = 119881119879Λ119881 can be replaced withits corresponding eigenvalue decomposition and Φ119879119905 Φ119905 =(8119898)sum119899119894=0[119909(119894)]119879119909(119899 minus 119894) (7) can then become as follows119864 ≜ MSE = 1003817100381710038171003817100381710038171003817100381710038171003817and minus and119881119879 8119898 119899sum

119894=0

[119909 (119894)]119879 119909 (119899 minus 119894) 119881and10038171003817100381710038171003817100381710038171003817100381710038172119865 (8)

Because and is real diagonal matrices and119881119879119881and = (119881and)119879119881andIf 119861 = 119881and then 119861 fl [|1198611198941198952|]

Here (8) can be reduced to119864 ≜ MSE = 1003817100381710038171003817100381710038171003817100381710038171003817and minus 119861119879 8119898 119899sum119894=0

[119909 (119894)]119879 119909 (119899 minus 119894) 11986110038171003817100381710038171003817100381710038171003817100381710038172119865 (9)

Supposing that 119861119894119895 is the elements in 119861 gradient decreaseiteration method is used to minimize mean square error(MSE) 119861119894119895 larr 119861119894119895 minus 120588nabla119864 where 120588 is step size and 120588 gt 0

4 Journal of Control Science and Engineering

Quasi-Toeplitzmatrix

QR factorizationwith eigenvaluedecomposition

Orthogonalgradient descent

method

Sparsifying

Randomly

columns

Discrete chaotic

sequence

Toeplitzmatrix

Toeplitzarrangement

Toeplitz-structured

chaoticmeasurement

measurementGrammatrix

The final

matrix

Randomly generated measurement matrix

Eigen-decomposition

Generate synthetic

measurementmatrix

matrix Φt

sampling Q

basis Ψt

Figure 4 The improved scheme by different methods

nabla119864 equiv 120597119864120597119861119894119895 is gradient value of 119864 then 119861(119894+1) = 119861119894minus120578119861119894(119861119894119879(8119898)sum119899119894=0[119909(119894)]119879119909(119899 minus 119894)119861119894) According to Ger-schgorin theorem the column coherence of 120593119894 can bededuced as follows120583 = max

1le119894119895le119873119894 =119895

10038161003816100381610038161003816⟨120593119894 Ψ119895⟩10038161003816100381610038161003816 (10)

If119873 le 119872(119872+1)2 the infimumof the column coherenceis called Welch bound120583 ge 120583119908 = 119873 minus 119872(119873 minus 1)119872 (11)

Equiangular Tight Frame (ETF) is derived from (11) if theconstraints are equality

3 The Proposed Approach

Based on previous conclusions the proposed algorithm aimsto optimize the Toeplitz-structured chaotic measurementmatrix to obtain better results from hyperspectral unmixingTherefore the research content in this paper mainly consistsof three parts designed TSCMM optimized Gram matrixand orthogonal gradient descent approach as shown inFigure 4

The study is given by taking the following methodsfirstly to obtain easy hardware implemented pseudo-randomchaotic elements are used to form a new Toeplitz-structuredchaotic measurement matrix (TSCMM) as discussed inSection 31 to overcome unbearable Cost Time Grammatrixis improved by a rank revealing QR factorization witheigenvalue decomposition as discussed in Section 32 toachieve optimal incoherence orthogonal gradient descentmethod for measurement matrix optimization is presentedin Section 33 Finally the improved scheme is presentedthrough explicit analysis and discussion

31 Xu-TSCMM Recently [23] is written by myself com-pletely and probed into its initial theory and researchDiscrete chaotic system function is proposed to generate aseries of pseudo-random numbers Based on those elementsToeplitz-structured chaotic measurement matrix (TSCMM)

is produced to guarantee the incoherence criterion To reducethe building time of TSCMM Circulantblock-diagonalsplitting structure is attached on TSCMM Although aboutone-third of matrix values are eliminated the measurementmatrix is proved to satisfy Johnson-Lindenstrauss (J-L)lemma and achieves the goal of satisfying RIP Logistic map[24] as the simplest dynamic systems evidencing chaoticbehavior is described as follows119909119899+1 = 120583119909119899 (1 minus 119909119899) 120583 isin [0 4] 119909119899 isin [0 1] sub 119877 (12)

where 120583 isin (35699 4] sub 119877 is the discrete state While para-meter 120583 is 4 the sequence 119909119899(119905) satisfies beta distributionwith 120572 = 05 and 120573 = 05 and the next probability densityfunction 119891(119909 05 05) = (1120587)(119909 minus 1199092)minus12 has been used forsimulation

Set 119911119894(119905) as the output sequence generated by (12) withinitial condition 119911119894(0) and let 119909119894(119905) denote the regularizationof 119911119894(119905) as the following form 119909119894(119905) = 119911119894(119905)minus05 119894 = 0 1 2

Approximately 119909119894(119905) can be considered as random vari-able and it satisfies the following distribution 119891(119909) =(1120587)(025 minus 1199092)minus12 Then by selecting 119898 different initialconditions 119911(0) isin [0 1]119898 sub 119877119898 one can obtain 119898 vectorswith dimension 119877 which enables us to construct the follow-ing matrixΦ scaled byradic8119898

Φ = radic 8119898 (1199090 (0) sdot sdot sdot 1199090 (119899 minus 1) d119909119898 (0) sdot sdot sdot 119909119898 (119899 minus 1)) (13)

Here (13) is called the beta-like matrixAccording to (13) set one initial condition 119911 isin (0) isin R

and generate a sequence 119909 isin R119899 in the chaotic system ThenToeplitz-structured matrix Φ = R119898times119899 is constructed in thefollowing form

Φ = radic 8119898 (119909(119899 minus 1) 119909 (119899 minus 2) sdot sdot sdot 119909 (0)119909 (0) 119909 (119899 minus 1) 119909 (1) d119909 (119898 minus 2) 119909 (119898 minus 3) sdot sdot sdot 119909 (119898 minus 1)) (14)

Journal of Control Science and Engineering 5

Hereradic8119898 is for normalization andΦ is called Toeplitz-structured chaotic measurement matrix (TSCMM) whichmeets J-L theorem

32 Duarte-ETF Method Minimum coherence property ofETF has been the main target to find a feasible solution Itis impossible to solve the problem exactly because of thecomplexity while the structure of Gram matrix has changedso much that the selection of new units in the following stepis very difficult

To minimize (14) an optimization approach is adoptedto reduce the maximum and average mutual coherence ofmeasurement matrix It shrinks Gram matrix based on ETFtheoryThemethod canminimize the globalmutual coherentcoefficient of TSCMM by adjusting the eigenvalues abovezero to the average value of the sumof these eigenvalues with-out changing the sum After performing alternating mini-mization the optimized measurement matrix can be con-structed from the output Gram matrix with a rank revealingQR factorization with eigenvalue decomposition

Theorem 1 Given a measurement matrix Φ = R119898times119871 andrepresenting matrix Θ = R119899times119898 there exists a matrix 119863 =ΘΦ and Gram matrix 119866 = 10067041198631198791006704119863 where 1006704119863 is the columnnormalization from119863 If the real positive definite matrix119866 has120582119896 gt 0 (119896 = 1 sim 119898) the following equality sum119898119896=1 120582119896 = 119899 andsum119898119896=1(120582119896)2 = sum119899119894119895=1(⟨119894 119895⟩)2 where 119894 (119894 = 1 sim 119899) is thecolumn of 1006704119863 which is obtained

Based on Theorem 1 to minimize the largest absolutevalues of the off-diagonals in the correspondingGrammatrixwe can determine the eigenvalues of Gram matrix by solvingthe following optimization problems

min sum119894 =119895

(119892119894119895)2 = 119898sum119896=1

(120582119896)2 minus 119899sum119894=1

(119892119894119895)2st 119898sum

119896=1

120582119896 = 119899 (15)

where 119892119894119895 is the element of Gram matrix The optimizationproblem (15) is to minimize the square sum of the elementof Gram matrix if the sum of the characteristic value 120582119896remains constant Though the eigenvalue decomposition ofGram matrix has 120582119896 gt 0 the value 119899119898 has been graduallyapproaching to minimize the square sum of sum119898119896=1(120582119896)2Because real symmetric matrix eigenvalue decomposition isorthogonal the square sum of nondiagonal elements fromGram matrix gradually decreases and further it attains theeffect as follows

min sum119894 =119895

(119892119894119895)2 = 119898sum119896=1

( 119899119898)2 = 119899 (16)

Finally 1006704119863 is obtained after several iterations which is alsothe optimal Gram matrix 119866best

33 The Orthogonal Gradient Descent Approach Accordingto the definition of Grammatrix Grammatrix is the productof measurement matrix and sparse matrix Therefore theoptimalmeasurementmatrix can be directly derived from theoptimal Gram matrix However as for overcomplete sparserepresentation based on redundant dictionary it is very hardto design an effective algorithm to construct measurementmatrix In order to solve this problem orthogonal gradientdescent method is employed to get the optimal measurementmatrix Θbest

If the optimal Gram matrix 119866best is obtained the optimalmeasurement matrix Θbest is as followsΘbest = argmin 10038171003817100381710038171003817119866best minus Θ119879Θ10038171003817100381710038171003817119865 ≜ 119865 (Θ) (17)

And then the complex problem from optimal Grammatrix119866best is transformed into simpleminimumof119865(Θ) Bydetermining the derivative of119865(Θ) a skew-symmetricmatrix119882 with the measurement matrix Θ and the gradient matrixnabla119865 is used to obtain revision factornabla119865 (Θ) = Θ (Θ119879Θ minus 119866best) 119882 (Θ nabla119865) = Θ119879 (nabla119865) minus (nabla119865)119879Θ (18)

Next if 119868 is identity matrix the orthogonal matrix 119878 canbe expressed through the Cayley transform to ensure thepositive definiteness of the revision factor119878 = (119868 minus 119882) (119868 + 119882)minus1 (19)

The orthogonal gradient factor matrix Δ can be obtainedto update the gradient directionΔ = nabla119865 + (nabla119865) 119878 = nabla119865 (119878 + 119868) (20)

Combining (19) and (20) (21) can be rewritten as followsΔ = nabla119865 (119868 + (119868 minus 119882) (119868 + 119882)minus1) (21)Δ = 2nabla119865 (119868 + 119882)minus1 (22)

Finally update measurement matrixΘ with the orthogo-nal gradient factor matrix ΔΘ larr997888 Θ minus 120578Δ (23)

Because 120578 is updating ratio 119865(Θ) gradually converges attheminimumvalueThen the optimalmeasurementmatrixΘis the goal of our pursuit According to the linearity propertyof Toeplitz measurement matrix [25]Θ satisfies J-L propertywith overwhelming probability FromTheorem 1 it has beenproven that J-L condition can replace RIP condition So Θalso satisfies RIP property with overwhelming probability

The flow diagram of the proposed method will be givenas shown in Algorithm 1

4 Experiments and Result Analysis

To illustrate the effectiveness of the proposed approach themost widely used hyperspectral images in unmixing such

6 Journal of Control Science and Engineering

Input number of measurements119872 dictionary Ψ119871times119873 threshold 120585 = radic(119871 minus 119873)119873(119871 minus 1)updating ratio 120578 number of iterations Iter1 Iter2Initialization Set Φ119872times119873 to be TSCMMUpdate (1) Set the initial value of iteration 1198961 = 0(2)Optimize Gram matrix

(a) Compute Gram matrix 119866 = Θ119879Θ = Ψ119879Φ1198791198961Φ1198961Ψ(b) Normalize 1006704119866 = diag(1radicdiag(119866)) lowast 119866 lowast diag(1radicdiag(119866))(c) Update the elements of Gram matrix 1006704119866(3) Optimize measurement matrix(a) Set the initial value of iteration 1198962 = 0(b) Compute the orthogonal gradient factor matrix Δ 1198962 (c) Update measurement matrix Θ1198962 Θ1198962 = Θ1198962 minus 120578Δ 1198962 (d) 1198962 = 1198962 + 1 if 1198962 = Iter2 stop else return to Step 32)(4)Compute measurement matrix Φ1198961 (5) 1198961 = 1198961 + 1 if 1198961 = Iter1 stop else return to Step (2)

Output Φbest is the optimal measurement matrixΦIter1minus1Further SNR and reconstructed signal

Algorithm 1 The flow diagram of the proposed method Optimization of TSCMM

as Cuprite Urban and Jasper Ridge were selected in thespectral range from 380 nm to 2500 nm each channel bandwidth is up to 946 nm All high-dimensional data is providedby the standard hyperspectral library of 224 bands whichcomes from [26] To reduce complexity there are only 128times 128 pixel blocks of original image which starts from the(0 0)th pixel Further only 8 channels (from 11 to 81 bandsevery 10 bands) were remained due to dense water vapor andatmospheric effects

In the course of the experiment the signal sparsitymethod is Fourier basis and reconstruction algorithm isStagewise Orthogonal Matching Pursuit (StOMP) [27] Var-ious kinds of measurement matrix (Circulant ToeplitzToeplitz-structured chaotic measurement matrix (TSCMM)[23] Elad-Optimization Method (EOM) [7] and Duarte-Carvajalino and Sapirorsquos Method (Duarte-ETF) [10]) areemployed to illustrate the effectiveness of proposed approachfor hyperspectral unmixing

To demonstrate the efficiency of these methods tradi-tional evaluation methods can generally be divided into twocategories (1) subjective assessment and (2) objective evalu-ation Mean Squared Error (MSE) and Peak-Signal-to-NoiseRatio (PSNR) as one of the most important indices fromobjective evaluation determine the quality of recovery imagewhile Cost Time (CT) verifies the efficiency of the proposedapproach They all testify experiment results of recoverysignals built on laptop with AthlonTM Processor 160G HZ1 GB RAM Matlab 70 andWindows XP operation platform

41 Cuprite To illustrate the use of the hyperspectral anal-ysis process a sample scene covers the Cuprite miningdistrict in western Nevada USA from NASArsquos AirborneVisibleInfrared Imaging Spectrometer (AVIRIS) is providedThe data provided here is one of the most widely used hyper-spectral images in unmixing studyThere are 210 wavelengthsranging from 400 nm to 2500 nm resulting in a spectralresolution of 10 nm

In Figure 5 the first image (Top) was taken in bluelight the second image (middle) was taken in red light andthe third image (bottom) was taken in near infrared lightcentered at a wavelength of 750 nanometers

Figure 5 presents the subjective evaluation by Circu-lant Toeplitz TSCMM EOM Duarte-ETF and proposedmethod Compared with other results the performance fromFigure 5(b) is the worst The reason is that the elementsfrom Circulant measurement matrix follow periodic rep-etition permutation which does not satisfy RIP propertywith overwhelming probability Figure 5(c) clearly demon-strates that Toeplitz measurement matrix can avoid theproblem well Because of the property of pseudo-randomof chaotic sequence the performance from TSCMM hasfurther improved as shown in Figure 5(d) The result fromFigures 5(e)ndash5(g) shows that there is a significant impacton different bands using different optimization methodsNear infrared image is less affected by dust and gas Visibleblue channel has strong capability to penetrate water andvisible red channel can reflect the health status of plantsTherefore the sorted off-diagonal entries of themeasurementmatrix from EOM are likely more sparse and diagonal entriesare more concentrated The results are clearly shown inFigure 5(g) that the proposed approach had a significantperformance compared to any others and closely resemblesoriginal image Furthermore the objective evaluations whichinclude Mean Squared Error (MSE) Peak-Signal-to-NoiseRatio (PSNR) and Cost Time (CT) can avoid artificial errorand draw compelling conclusion The results can be clearlyseen fromdifferentmethods on recoveryCuprite as shown inTable 1

Figure 5 andTable 1 report the recovery quality of the pro-posed method on recovery Cuprite The following observa-tions are summarized (1) of all evaluating indicators con-sidered here traditional Circulant had the worst perfor-mance in both subjective and objective evaluations (2) sincethe introduction of Toeplitz the performance gets major

Journal of Control Science and Engineering 7

(1) 11

(2) 41 bands (128 times 128 pixels)

bands (128 times 128 pixels)

(3) 81 bands (128 times 128 pixels)

(a1) original (b1) Circulant (c1) Toeplitz (d1) TSCMM (e1) EOM (f1) Duarte-ETF (g1) Proposed

(a2) original (b2) Circulant (c2) Toeplitz (d2) TSCMM (e2) EOM (f2) Duarte-ETF (g2) Proposed

(a3) original (b3) Circulant (c3) Toeplitz (d3) TSCMM (e3) EOM (f3) Duarte-ETF (g3) Proposed

Figure 5The subjective quality of different CSmeasurementmatrices from left to right (a) original (b) Circulant (c) Toeplitz (d) TSCMM(e) EOM (f) Duarte-ETF and (g) proposed method

Table 1 MSE and PSNR of different matrixes of recovery Cuprite

Algorithm 11 bands (64 times 64 pixels) 41 bands (64 times 64 pixels) 81 bands (64 times 64 pixels)PSNR MSE PSNR MSE PSNR MSE CT

Circulate 599634 00656 609680 00520 615821 00452 51070Toeplitz 599634 00656 619375 00416 633173 00303 35506TSCMM 600696 00640 619783 00412 633462 00301 17624EOM 602806 00610 622337 00389 633962 00297 2454345Duarte-ETF 606304 00562 623788 00376 634082 00297 1165994Proposed 643532 00239 638856 00266 637179 00276 516191

improvement on image quality while improved Toeplitz-structured matrix method (TSCMM) is slightly better thanclassical Toeplitz matrix method (3) EOM has significantperformance in image quality however the optimizationprocess is usually an iterative process which is also a verycomplicated and time-consuming process (4) Duarte-ETFhas better contrast and lower computational complexity (5)the proposed method takes advantage of improved Toeplitz-structured matrix to speed up the convergence speed andimprove traditional optimization method to better recoveryhigh-dimensional image Experimental results show that theproposed method has a better overall performance

42 Urban University of California Santa Barbara (UCSB)built an urban spectral library for the GoletaSanta BarbaraareaThe hyperspectral data of Urban were acquired betweenlate May and early June 2001 using an ASD full rangeinstrument on loan from the Jet Propulsion LaboratoryThesespectra of Urban are characterized by 499 roofs 179 roads

66 sidewalks 56 parking lots 40 road paints 37 types ofvegetation 47 types of nonphotosynthetic vegetation 88 baresoil and beach spectra 27 acquired from tennis courts and50 more from miscellaneous surfaces

Experiments on the hyperspectral data of Urban demon-strate that the proposed scheme substantially improves thereconstruction accuracy Clearly it comes to the same con-clusion from Figure 6 that compared with the previous fivemethods the effect of proposed approach is obviously sup-erior to any other methods and is the most similar to originalimage

To evaluate and compare the proposed method thefollowing performance indices such as Average Gradient(AG) Edge-Intensity (EI) Figure Definition (FD) GrayMean (GM) StandardDeviation (SD) Space Frequency (SF)Variance (VAR) and Structural Similarity (SSIM) were usedIt shows that the objective evaluation indices enhance theexperiment rigor and convincing The results are shown inTable 2

8 Journal of Control Science and Engineering

Table2Objectiv

eevaluationof

typicalm

etho

dsandprop

osed

metho

dfro

mrecovery

Urban

Band

sAlgorith

mTh

eobjectiv

eevaluationindices

AGEI

FDGM

SDMSE

PSNR

SFSSI

VAR

CT

11

Circulate

0070476

048360

012231

04106

0119

700929

584508

02034

098826

0119

7217

379

Toeplitz

0047687

034172

010125

032796

01137

00610

602804

018949

099361

011366

18706

TSCM

M0054635

03864

70114

14035344

01162

006

865976

58019866

099254

011624

15146

EOM

00566

08040

032

011519

035809

0117

200708

596279

020025

099226

0117

203051965

Duarte-ET

F0047687

034172

010125

032796

01137

00610

602804

018949

099361

011366

1366804

Prop

osed

004

6138

029

123

006

690

023194

01197

004

3461758

601195

6099

537

01198

0165975

41

Circulate

0072566

051888

012438

037371

01363

00631

6013

16020690

099349

013626

16382

Toeplitz

0058448

041868

012772

040

138

01390

00786

591767

022932

099207

013897

18069

TSCM

M0061306

043672

013128

039347

01362

00744

594172

022736

099259

013620

1538

1EO

M006

0340

04340

8012941

040222

01353

00771

59260

9022536

099215

013529

813160

Duarte-ET

F0062241

044

182

013123

042482

01402

00873

587190

023610

099089

014018

5974

19Prop

osed

005

0626

040

358

007

196

024

426

01307

004

52615816

012782

099

561

01306

8243532

81

Circulate

0082530

06279

013639

049357

01449

00635

6010

19022630

099397

014492

15829

Toeplitz

0077269

047993

014082

044

812

01390

00535

60844

2023904

099484

013905

18284

TSCM

M0070612

050737

014828

051106

01496

00716

595790

025291

099353

014958

15702

EOM

0070821

05100

6014940

050827

01503

00707

596361

02544

7099369

015030

9731

53Duarte-ET

F0074147

052535

014946

048963

01499

00652

599890

025959

099450

014986

590915

Prop

osed

006

8955

042

678

009

339

040

915

01571

004

84612814

01648

5099

580

01570

82874

26

Journal of Control Science and Engineering 9

(1) 11 bands (64 times 64 pixels)

(3) 81 bands (64 times 64 pixels)

(2) 41 bands (64 times 64 pixels)

(a1) original (b1) Circulant (c1) Toeplitz (d1) TSCMM (e1) EOM (f1) Duarte-ETF (g1) Proposed

(a2) original (b2) Circulant (c2) Toeplitz (d2) TSCMM (e2) EOM (f2) Duarte-ETF (g2) Proposed

(a3) original (b3) Circulant (c3) Toeplitz (d3) TSCMM (e3) EOM (f3) Duarte-ETF (g3) Proposed

Figure 6The subjective quality of different CSmeasurementmatrices from left to right (a) original (b) Circulant (c) Toeplitz (d) TSCMM(e) EOM (f) Duarte-ETF and (g) proposed method

The results of Table 2 show that the method has a higherperformance than traditional Toeplitz or Circulant matrixmethod Although improved Toeplitz-structured matrixmethod (TSCMM) is slightly better than classical Toeplitzboth classical optimization measurement matrix method(EOM) and proposed method have significant performancein image quality Furthermore the proposed method takesadvantage of improved Toeplitz-structured matrix to speedup the convergence speed and improve traditional opti-mization method to recover better high-dimensional imageExperimental results show that the proposed method has abetter overall performance

43 Jasper Ridge The hyperspectral image of Jasper Ridgewas obtained on June 2 September 4 and October 6 1992whichwas calibrated to surface reflectanceThe imagewas themost popular source to analyze with spectral mixture anal-ysis using library endmembers representing green foliagenonphotosynthetic vegetation and soils characteristic of thesite Field-based vegetation was obtained fromUSGeologicalService

From Figure 7 it is obvious that the worst effects fromtraditional Circulant have reached being almost intolerableWhile the results from TSCMM and Toeplitz are almostsimilar the former has only slight improvement comparedto the latter On the other hand the performance of theproposed approach was significantly improved compared tothat of EOM or Duarte-ETF The study concluded that theproposed approach had a significant performance comparedto that of others Furthermore the objective evaluations are

shown inTable 3The results can be clearly seen fromdifferentmethods on recovery Jasper Ridge

The results of Table 3 show that the proposed methodhas a higher performance than traditional Toeplitz or Circu-lant matrix method TSCMM takes advantage of improvedToeplitz-structuredmatrix to speedup the convergence speedand improve traditional Toeplitz or Circulant matrix methodto recover better high-dimensional data Although EOMhas the lower column coherence and faster convergence itweakens RIP condition and causes recovery performancedegradation While the absolute values from Duarte-ETFconcentrate around mutual coherence this can make theequivalent dictionary as close as possible to an ETF But thisalgorithm has high computational complexity Furthermoreexperimental results from the proposed method show thatthe proposed method has a better overall performance

44 Hyperspectral Unmixing The fourth experiment per-forms an experimental evaluation of the accuracy of thestandard hyperspectral unmixing districts known as Cuprite[28] To reduce complexity there are only 188 channels(3ndash103 114ndash147 and 168ndash220 bands) that were remained dueto dense water vapor and atmospheric effects In the courseof the experiment the signal sparsity method is Fourier basisand reconstruction algorithm is StOMP

The results are shown in Figure 8 for hyperspectralunmixing and closely resemble those obtained from hyper-spectral data Figure 8 compares the unmixing performanceof the proposed method with different endmembers and theabundance from different endmembers is totally dissimilar

10 Journal of Control Science and Engineering

Table 3 The objective evaluations of different matrixes of recovery Jasper Ridge

Algorithm 11 bands (64 times 64) 81 bands (64 times 64) 151 bands (64 times 64)SNR PSNR MSE SNR PSNR MSE SNR PSNR MSE CT

Circulate minus60698 573412 01199 116699 600402 00644 33299 603823 00595 15304Toeplitz 247566 707289 00055 294215 677497 00109 157386 657714 00172 15519TSCMM 153473 666425 00141 339793 697291 00069 200693 676521 00112 14869EOM 44853 619252 00417 350894 702112 00062 216750 683495 00095 788838Duarte-ETF 247566 707289 00055 333205 694430 00074 172452 664257 00148 665711Proposed 210504 691193 00080 549719 788460 848e minus 04 422277 772754 00012 345743

(1) 11 bands (64 times 64 pixels)

(2) 81 bands (64 times 64 pixels)

(3) 151 bands (64 times 64 pixels)

(a2) original (b2) Circulant (c2) Toeplitz (d2) TSCMM (e2) EOM (f2) Duarte-ETF (g2) Proposed

(a1) original (b1) Circulant (c1) Toeplitz (d1) TSCMM (e1) EOM (f1) DuarteminusETF (g1) Proposed

(a3) original (b3) Circulant (c3) Toeplitz (d3) TSCMM (e3) EOM (f3) Duarte-ETF (g3) Proposed

Figure 7The subjective quality of different CSmeasurementmatrices from left to right (a) original (b) Circulant (c) Toeplitz (d) TSCMM(e) EOM (f) Duarte-ETF and (g) proposed method

These features from the proposed method ensure clear andaccurate the abundances

By accessing information from USGS Digital Spec-tral Library [29] the unmixing performance has beenalmost correct Furthermore different endmembers fromUSGS 1995 Library [30] are used to verify that the predictionmodel of hyperspectral unmixing scheme is accurate inFigure 8

Apparently the proposed unmixing results (blue thinthread) have a strong correlationwith these endmembers (redthin thread) from USGS 1995 Library From Figure 9(a) asteep sloping line from Alunite suggests that the unmixingendmember (blue thin thread) has remarkable similarityThis same conclusion has been made in studies from Figures9(d)ndash9(h) On the other hand the unsatisfied results fromFigures 9(b) and 9(c) have been caused by smooth curveThe proposed method has good accuracy and is robust totraditional filtering compression cutting and noise attack

5 Conclusions

In this paper to overcome the limitation of Toeplitz-structured chaotic measurement matrix an improved mea-surement matrix has been carried out in the hyperspectralunmixing process to achieve multiple endmembers of hyper-spectral image And in theory it proves that this matrixhas retained the RIP property with overwhelming proba-bility Experimental results demonstrate that the proposedmethod to design of measurement matrix leads to better CSreconstruction performance with low extra computationalcost Compared with some traditional measurement matrixan improved method has highest technical feasibility lowestcomputational complexity and least computation time con-sumption in the same recovery qualityThe proposedmethodcan take the special advantage in hyperspectral unmixingprocess and explore the practical satellite system to remotesensing

Journal of Control Science and Engineering 11

(a) Alunite (b) Andradite (c) Pyrope

(d) Nontronite (e) Dumortierite (f) Kaolinite

(g) Chalcedony (h) Kaolinite (i) Buddingtonite

Figure 8 The results from different elements abundance in hyperspectral unmixing schemes

12 Journal of Control Science and Engineering

02

04

06

08

1Re

flect

ance

()

500 1000 1500 2000 2500 30000Wavelength (120583m)

(a) Alunite

0

01

02

03

04

Refle

ctan

ce (

)

500 1000 1500 2000 2500 30000Wavelength (120583m)

(b) Andradite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

1

Refle

ctan

ce (

)

(c) Pyrope

0

02

04

06

08

Refle

ctan

ce (

)

500 1000 1500 2000 2500 30000Wavelength (120583m)

(d) Nontronite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

1

Refle

ctan

ce (

)

(e) Dumortierite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

Refle

ctan

ce (

)

(f) Kaolinite

0

02

04

06

08

Refle

ctan

ce (

)

500 1000 1500 2000 2500 30000Wavelength (120583m)

(g) Chalcedony

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

Refle

ctan

ce (

)

(h) Kaolinite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

Refle

ctan

ce (

)

(i) Buddingtonite

Figure 9 The comparison map between actual unmixing effect and endmember library

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is supported by Chongqing Engineering Labo-ratory for Detection Control and Integrated System Theproject is also funded by Key Technology Research andIndustrialization of Fire Monitoring and Early Warn-ing Sensor Network for High Voltage Transmission Line(KJZH17124) This research is funded by Chongqing Edu-cation Commission Foundation (KJ1400612) This projectis also granted financial support from a CooperativeProject of Chongqing Technology and Business University(990516001)

References

[1] D L Donoho ldquoCompressed sensingrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol52 no 4 pp 1289ndash1306 2006

[2] E J Candes J Romberg and T Tao ldquoRobust uncertaintyprinciples exact signal reconstruction from highly incompletefrequency informationrdquo Institute of Electrical and ElectronicsEngineers Transactions on InformationTheory vol 52 no 2 pp489ndash509 2006

[3] T N Canh K D Quoc and B Jeon ldquoMulti-resolution kro-necker compressive sensingrdquo Transactions on Smart Processingamp Computing vol 3 no 1 pp 19ndash27 2014

[4] K Q Dinh H J Shim and B Jeon ldquoMeasurement coding forcompressive imaging using a structural measuremnet matrixrdquoin 2013 20th IEEE International Conference on Image ProcessingICIP 2013 pp 10ndash13 aus September 2013

[5] M F Duarte and R G Baraniuk ldquoKronecker compressive sens-ingrdquo IEEE Transactions on Image Processing vol 21 no 2 pp494ndash504 2012

[6] B Zhang X Tong W Wang and J Xie ldquoThe research ofKronecker product-based measurement matrix of compressivesensingrdquo EURASIP Journal on Wireless Communications andNetworking vol 2013 article 161 pp 1ndash5 2013

[7] M Elad ldquoOptimized projections for compressed sensingrdquo IEEETransactions on Signal Processing vol 55 no 12 pp 5695ndash57022007

Journal of Control Science and Engineering 13

[8] M Lustig D Donoho and J M Pauly ldquoSparse MRI the appli-cation of compressed sensing for rapid MR imagingrdquoMagneticResonance in Medicine vol 58 no 6 pp 1182ndash1195 2007

[9] V Abolghasemi S Sanei S Ferdowsi F Ghaderi and ABelcher ldquoSegmented compressive sensingrdquo in Proceedings of theIEEESP 15thWorkshop on Statistical Signal Processing (SSP rsquo09)pp 630ndash633 September 2009

[10] J M Duarte-Carvajalino and G Sapiro ldquoLearning to sensesparse signals simultaneous sensingmatrix and sparsifying dic-tionary optimizationrdquo IEEE Transactions on Image Processingvol 18 no 7 pp 1395ndash1408 2009

[11] ZWang G R Arce and J L Paredes ldquoColored randomprojec-tions for compressed sensingrdquo in 2007 IEEE International Con-ference on Acoustics Speech and Signal Processing ICASSP rsquo07pp III873ndashIII876 usa April 2007

[12] Z Wang and G R Arce ldquoVariable density compressed imagesamplingrdquo IEEE Transactions on Image Processing vol 19 no 1pp 264ndash270 2010

[13] M Elad and M Aharon ldquoImage denoising via learned dic-tionaries and sparse representationrdquo in 2006 IEEE ComputerSociety Conference on Computer Vision and Pattern RecognitionCVPR 2006 pp 895ndash900 usa June 2006

[14] M Aharon M Elad and A Bruckstein ldquoK-SVD an algorithmfor designing overcomplete dictionaries for sparse representa-tionrdquo IEEE Transactions on Signal Processing vol 54 no 11 pp4311ndash4322 2006

[15] J Xu Y Pi and Z Cao ldquoOptimized projection matrix for com-pressive sensingrdquo EURASIP Journal on Advances in SignalProcessing vol 2010 Article ID 560349 2010

[16] V Abolghasemi S Ferdowsi and S Sanei ldquoA gradient-basedalternating minimization approach for optimization of themeasurementmatrix in compressive sensingrdquo Signal Processingvol 92 no 4 pp 999ndash1009 2012

[17] Q Zhang Y Fu H Li and R Rong ldquoOptimized projectionmatrix for compressed sensingrdquo Circuits Systems and SignalProcessing vol 33 no 5 pp 1627ndash1636 2014

[18] G Li Z Zhu D Yang L Chang and H Bai ldquoOn projectionmatrix optimization for compressive sensing systemsrdquo IEEETransactions on Signal Processing vol 61 no 11 pp 2887ndash28982013

[19] S Tian X Fan and L I Zhetao ldquoOrthogonal-gradient mea-surement matrix construction algorithmrdquo Chinese Journal ofElectronics vol 25 no 1 pp 81ndash87 2016

[20] V Abolghasemi S Ferdowsi and BMakkiabadi ldquoOn optimiza-tion of the measurement matrix for compressive sensingrdquo inSignal Processing Conference European IEEE Ed pp 427ndash431August 2010

[21] Q Li D Schonfeld and S Friedland ldquoGeneralized tensorcompressive sensingrdquo in 2013 IEEE International Conference onMultimedia and Expo ICME 2013 usa July 2013

[22] S Friedland Q Li and D Schonfeld ldquoCompressive sensing ofsparse tensorsrdquo IEEE Transactions on Image Processing vol 23no 10 pp 4438ndash4447 2014

[23] S u Xu H Yin C Yi Y Xiong and T Xue ldquoAn ImprovedToeplitz Measurement Matrix for Compressive Sensingrdquo Inter-national Journal of Distributed Sensor Networks vol 8 pp 1ndash82014

[24] R L Devaney in Practical Numerical Algorithms for ChaoticSystems T S Parker and L O Chua Eds vol 32 pp 501ndash503Siam Review 3 edition 2006

[25] R Baraniuk M Davenport R DeVore and M Wakin ldquoAsimple proof of the restricted isometry property for randommatricesrdquoConstructiveApproximation An International Journalfor Approximations and Expansions vol 28 no 3 pp 253ndash2632008

[26] Available httpwwwesciencecnpeoplefeiyunZHUDatasetGThtml

[27] D L Donoho Y Tsaig I Drori and J-L Starck ldquoSparse solu-tion of underdetermined systems of linear equations by stage-wise orthogonal matching pursuitrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol58 no 2 pp 1094ndash1121 2012

[28] K Lang ldquoNewsWeeder Learning to Filter Netnewsrdquo in Inter-national Machine Learning Conference vol 1995 pp 331ndash339

[29] Available httpfeatureselectionasuedudatasetsphp[30] Available httpclopinetcomisabelleProjectsNIPS

4 Journal of Control Science and Engineering

Quasi-Toeplitzmatrix

QR factorizationwith eigenvaluedecomposition

Orthogonalgradient descent

method

Sparsifying

Randomly

columns

Discrete chaotic

sequence

Toeplitzmatrix

Toeplitzarrangement

Toeplitz-structured

chaoticmeasurement

measurementGrammatrix

The final

matrix

Randomly generated measurement matrix

Eigen-decomposition

Generate synthetic

measurementmatrix

matrix Φt

sampling Q

basis Ψt

Figure 4 The improved scheme by different methods

nabla119864 equiv 120597119864120597119861119894119895 is gradient value of 119864 then 119861(119894+1) = 119861119894minus120578119861119894(119861119894119879(8119898)sum119899119894=0[119909(119894)]119879119909(119899 minus 119894)119861119894) According to Ger-schgorin theorem the column coherence of 120593119894 can bededuced as follows120583 = max

1le119894119895le119873119894 =119895

10038161003816100381610038161003816⟨120593119894 Ψ119895⟩10038161003816100381610038161003816 (10)

If119873 le 119872(119872+1)2 the infimumof the column coherenceis called Welch bound120583 ge 120583119908 = 119873 minus 119872(119873 minus 1)119872 (11)

Equiangular Tight Frame (ETF) is derived from (11) if theconstraints are equality

3 The Proposed Approach

Based on previous conclusions the proposed algorithm aimsto optimize the Toeplitz-structured chaotic measurementmatrix to obtain better results from hyperspectral unmixingTherefore the research content in this paper mainly consistsof three parts designed TSCMM optimized Gram matrixand orthogonal gradient descent approach as shown inFigure 4

The study is given by taking the following methodsfirstly to obtain easy hardware implemented pseudo-randomchaotic elements are used to form a new Toeplitz-structuredchaotic measurement matrix (TSCMM) as discussed inSection 31 to overcome unbearable Cost Time Grammatrixis improved by a rank revealing QR factorization witheigenvalue decomposition as discussed in Section 32 toachieve optimal incoherence orthogonal gradient descentmethod for measurement matrix optimization is presentedin Section 33 Finally the improved scheme is presentedthrough explicit analysis and discussion

31 Xu-TSCMM Recently [23] is written by myself com-pletely and probed into its initial theory and researchDiscrete chaotic system function is proposed to generate aseries of pseudo-random numbers Based on those elementsToeplitz-structured chaotic measurement matrix (TSCMM)

is produced to guarantee the incoherence criterion To reducethe building time of TSCMM Circulantblock-diagonalsplitting structure is attached on TSCMM Although aboutone-third of matrix values are eliminated the measurementmatrix is proved to satisfy Johnson-Lindenstrauss (J-L)lemma and achieves the goal of satisfying RIP Logistic map[24] as the simplest dynamic systems evidencing chaoticbehavior is described as follows119909119899+1 = 120583119909119899 (1 minus 119909119899) 120583 isin [0 4] 119909119899 isin [0 1] sub 119877 (12)

where 120583 isin (35699 4] sub 119877 is the discrete state While para-meter 120583 is 4 the sequence 119909119899(119905) satisfies beta distributionwith 120572 = 05 and 120573 = 05 and the next probability densityfunction 119891(119909 05 05) = (1120587)(119909 minus 1199092)minus12 has been used forsimulation

Set 119911119894(119905) as the output sequence generated by (12) withinitial condition 119911119894(0) and let 119909119894(119905) denote the regularizationof 119911119894(119905) as the following form 119909119894(119905) = 119911119894(119905)minus05 119894 = 0 1 2

Approximately 119909119894(119905) can be considered as random vari-able and it satisfies the following distribution 119891(119909) =(1120587)(025 minus 1199092)minus12 Then by selecting 119898 different initialconditions 119911(0) isin [0 1]119898 sub 119877119898 one can obtain 119898 vectorswith dimension 119877 which enables us to construct the follow-ing matrixΦ scaled byradic8119898

Φ = radic 8119898 (1199090 (0) sdot sdot sdot 1199090 (119899 minus 1) d119909119898 (0) sdot sdot sdot 119909119898 (119899 minus 1)) (13)

Here (13) is called the beta-like matrixAccording to (13) set one initial condition 119911 isin (0) isin R

and generate a sequence 119909 isin R119899 in the chaotic system ThenToeplitz-structured matrix Φ = R119898times119899 is constructed in thefollowing form

Φ = radic 8119898 (119909(119899 minus 1) 119909 (119899 minus 2) sdot sdot sdot 119909 (0)119909 (0) 119909 (119899 minus 1) 119909 (1) d119909 (119898 minus 2) 119909 (119898 minus 3) sdot sdot sdot 119909 (119898 minus 1)) (14)

Journal of Control Science and Engineering 5

Hereradic8119898 is for normalization andΦ is called Toeplitz-structured chaotic measurement matrix (TSCMM) whichmeets J-L theorem

32 Duarte-ETF Method Minimum coherence property ofETF has been the main target to find a feasible solution Itis impossible to solve the problem exactly because of thecomplexity while the structure of Gram matrix has changedso much that the selection of new units in the following stepis very difficult

To minimize (14) an optimization approach is adoptedto reduce the maximum and average mutual coherence ofmeasurement matrix It shrinks Gram matrix based on ETFtheoryThemethod canminimize the globalmutual coherentcoefficient of TSCMM by adjusting the eigenvalues abovezero to the average value of the sumof these eigenvalues with-out changing the sum After performing alternating mini-mization the optimized measurement matrix can be con-structed from the output Gram matrix with a rank revealingQR factorization with eigenvalue decomposition

Theorem 1 Given a measurement matrix Φ = R119898times119871 andrepresenting matrix Θ = R119899times119898 there exists a matrix 119863 =ΘΦ and Gram matrix 119866 = 10067041198631198791006704119863 where 1006704119863 is the columnnormalization from119863 If the real positive definite matrix119866 has120582119896 gt 0 (119896 = 1 sim 119898) the following equality sum119898119896=1 120582119896 = 119899 andsum119898119896=1(120582119896)2 = sum119899119894119895=1(⟨119894 119895⟩)2 where 119894 (119894 = 1 sim 119899) is thecolumn of 1006704119863 which is obtained

Based on Theorem 1 to minimize the largest absolutevalues of the off-diagonals in the correspondingGrammatrixwe can determine the eigenvalues of Gram matrix by solvingthe following optimization problems

min sum119894 =119895

(119892119894119895)2 = 119898sum119896=1

(120582119896)2 minus 119899sum119894=1

(119892119894119895)2st 119898sum

119896=1

120582119896 = 119899 (15)

where 119892119894119895 is the element of Gram matrix The optimizationproblem (15) is to minimize the square sum of the elementof Gram matrix if the sum of the characteristic value 120582119896remains constant Though the eigenvalue decomposition ofGram matrix has 120582119896 gt 0 the value 119899119898 has been graduallyapproaching to minimize the square sum of sum119898119896=1(120582119896)2Because real symmetric matrix eigenvalue decomposition isorthogonal the square sum of nondiagonal elements fromGram matrix gradually decreases and further it attains theeffect as follows

min sum119894 =119895

(119892119894119895)2 = 119898sum119896=1

( 119899119898)2 = 119899 (16)

Finally 1006704119863 is obtained after several iterations which is alsothe optimal Gram matrix 119866best

33 The Orthogonal Gradient Descent Approach Accordingto the definition of Grammatrix Grammatrix is the productof measurement matrix and sparse matrix Therefore theoptimalmeasurementmatrix can be directly derived from theoptimal Gram matrix However as for overcomplete sparserepresentation based on redundant dictionary it is very hardto design an effective algorithm to construct measurementmatrix In order to solve this problem orthogonal gradientdescent method is employed to get the optimal measurementmatrix Θbest

If the optimal Gram matrix 119866best is obtained the optimalmeasurement matrix Θbest is as followsΘbest = argmin 10038171003817100381710038171003817119866best minus Θ119879Θ10038171003817100381710038171003817119865 ≜ 119865 (Θ) (17)

And then the complex problem from optimal Grammatrix119866best is transformed into simpleminimumof119865(Θ) Bydetermining the derivative of119865(Θ) a skew-symmetricmatrix119882 with the measurement matrix Θ and the gradient matrixnabla119865 is used to obtain revision factornabla119865 (Θ) = Θ (Θ119879Θ minus 119866best) 119882 (Θ nabla119865) = Θ119879 (nabla119865) minus (nabla119865)119879Θ (18)

Next if 119868 is identity matrix the orthogonal matrix 119878 canbe expressed through the Cayley transform to ensure thepositive definiteness of the revision factor119878 = (119868 minus 119882) (119868 + 119882)minus1 (19)

The orthogonal gradient factor matrix Δ can be obtainedto update the gradient directionΔ = nabla119865 + (nabla119865) 119878 = nabla119865 (119878 + 119868) (20)

Combining (19) and (20) (21) can be rewritten as followsΔ = nabla119865 (119868 + (119868 minus 119882) (119868 + 119882)minus1) (21)Δ = 2nabla119865 (119868 + 119882)minus1 (22)

Finally update measurement matrixΘ with the orthogo-nal gradient factor matrix ΔΘ larr997888 Θ minus 120578Δ (23)

Because 120578 is updating ratio 119865(Θ) gradually converges attheminimumvalueThen the optimalmeasurementmatrixΘis the goal of our pursuit According to the linearity propertyof Toeplitz measurement matrix [25]Θ satisfies J-L propertywith overwhelming probability FromTheorem 1 it has beenproven that J-L condition can replace RIP condition So Θalso satisfies RIP property with overwhelming probability

The flow diagram of the proposed method will be givenas shown in Algorithm 1

4 Experiments and Result Analysis

To illustrate the effectiveness of the proposed approach themost widely used hyperspectral images in unmixing such

6 Journal of Control Science and Engineering

Input number of measurements119872 dictionary Ψ119871times119873 threshold 120585 = radic(119871 minus 119873)119873(119871 minus 1)updating ratio 120578 number of iterations Iter1 Iter2Initialization Set Φ119872times119873 to be TSCMMUpdate (1) Set the initial value of iteration 1198961 = 0(2)Optimize Gram matrix

(a) Compute Gram matrix 119866 = Θ119879Θ = Ψ119879Φ1198791198961Φ1198961Ψ(b) Normalize 1006704119866 = diag(1radicdiag(119866)) lowast 119866 lowast diag(1radicdiag(119866))(c) Update the elements of Gram matrix 1006704119866(3) Optimize measurement matrix(a) Set the initial value of iteration 1198962 = 0(b) Compute the orthogonal gradient factor matrix Δ 1198962 (c) Update measurement matrix Θ1198962 Θ1198962 = Θ1198962 minus 120578Δ 1198962 (d) 1198962 = 1198962 + 1 if 1198962 = Iter2 stop else return to Step 32)(4)Compute measurement matrix Φ1198961 (5) 1198961 = 1198961 + 1 if 1198961 = Iter1 stop else return to Step (2)

Output Φbest is the optimal measurement matrixΦIter1minus1Further SNR and reconstructed signal

Algorithm 1 The flow diagram of the proposed method Optimization of TSCMM

as Cuprite Urban and Jasper Ridge were selected in thespectral range from 380 nm to 2500 nm each channel bandwidth is up to 946 nm All high-dimensional data is providedby the standard hyperspectral library of 224 bands whichcomes from [26] To reduce complexity there are only 128times 128 pixel blocks of original image which starts from the(0 0)th pixel Further only 8 channels (from 11 to 81 bandsevery 10 bands) were remained due to dense water vapor andatmospheric effects

In the course of the experiment the signal sparsitymethod is Fourier basis and reconstruction algorithm isStagewise Orthogonal Matching Pursuit (StOMP) [27] Var-ious kinds of measurement matrix (Circulant ToeplitzToeplitz-structured chaotic measurement matrix (TSCMM)[23] Elad-Optimization Method (EOM) [7] and Duarte-Carvajalino and Sapirorsquos Method (Duarte-ETF) [10]) areemployed to illustrate the effectiveness of proposed approachfor hyperspectral unmixing

To demonstrate the efficiency of these methods tradi-tional evaluation methods can generally be divided into twocategories (1) subjective assessment and (2) objective evalu-ation Mean Squared Error (MSE) and Peak-Signal-to-NoiseRatio (PSNR) as one of the most important indices fromobjective evaluation determine the quality of recovery imagewhile Cost Time (CT) verifies the efficiency of the proposedapproach They all testify experiment results of recoverysignals built on laptop with AthlonTM Processor 160G HZ1 GB RAM Matlab 70 andWindows XP operation platform

41 Cuprite To illustrate the use of the hyperspectral anal-ysis process a sample scene covers the Cuprite miningdistrict in western Nevada USA from NASArsquos AirborneVisibleInfrared Imaging Spectrometer (AVIRIS) is providedThe data provided here is one of the most widely used hyper-spectral images in unmixing studyThere are 210 wavelengthsranging from 400 nm to 2500 nm resulting in a spectralresolution of 10 nm

In Figure 5 the first image (Top) was taken in bluelight the second image (middle) was taken in red light andthe third image (bottom) was taken in near infrared lightcentered at a wavelength of 750 nanometers

Figure 5 presents the subjective evaluation by Circu-lant Toeplitz TSCMM EOM Duarte-ETF and proposedmethod Compared with other results the performance fromFigure 5(b) is the worst The reason is that the elementsfrom Circulant measurement matrix follow periodic rep-etition permutation which does not satisfy RIP propertywith overwhelming probability Figure 5(c) clearly demon-strates that Toeplitz measurement matrix can avoid theproblem well Because of the property of pseudo-randomof chaotic sequence the performance from TSCMM hasfurther improved as shown in Figure 5(d) The result fromFigures 5(e)ndash5(g) shows that there is a significant impacton different bands using different optimization methodsNear infrared image is less affected by dust and gas Visibleblue channel has strong capability to penetrate water andvisible red channel can reflect the health status of plantsTherefore the sorted off-diagonal entries of themeasurementmatrix from EOM are likely more sparse and diagonal entriesare more concentrated The results are clearly shown inFigure 5(g) that the proposed approach had a significantperformance compared to any others and closely resemblesoriginal image Furthermore the objective evaluations whichinclude Mean Squared Error (MSE) Peak-Signal-to-NoiseRatio (PSNR) and Cost Time (CT) can avoid artificial errorand draw compelling conclusion The results can be clearlyseen fromdifferentmethods on recoveryCuprite as shown inTable 1

Figure 5 andTable 1 report the recovery quality of the pro-posed method on recovery Cuprite The following observa-tions are summarized (1) of all evaluating indicators con-sidered here traditional Circulant had the worst perfor-mance in both subjective and objective evaluations (2) sincethe introduction of Toeplitz the performance gets major

Journal of Control Science and Engineering 7

(1) 11

(2) 41 bands (128 times 128 pixels)

bands (128 times 128 pixels)

(3) 81 bands (128 times 128 pixels)

(a1) original (b1) Circulant (c1) Toeplitz (d1) TSCMM (e1) EOM (f1) Duarte-ETF (g1) Proposed

(a2) original (b2) Circulant (c2) Toeplitz (d2) TSCMM (e2) EOM (f2) Duarte-ETF (g2) Proposed

(a3) original (b3) Circulant (c3) Toeplitz (d3) TSCMM (e3) EOM (f3) Duarte-ETF (g3) Proposed

Figure 5The subjective quality of different CSmeasurementmatrices from left to right (a) original (b) Circulant (c) Toeplitz (d) TSCMM(e) EOM (f) Duarte-ETF and (g) proposed method

Table 1 MSE and PSNR of different matrixes of recovery Cuprite

Algorithm 11 bands (64 times 64 pixels) 41 bands (64 times 64 pixels) 81 bands (64 times 64 pixels)PSNR MSE PSNR MSE PSNR MSE CT

Circulate 599634 00656 609680 00520 615821 00452 51070Toeplitz 599634 00656 619375 00416 633173 00303 35506TSCMM 600696 00640 619783 00412 633462 00301 17624EOM 602806 00610 622337 00389 633962 00297 2454345Duarte-ETF 606304 00562 623788 00376 634082 00297 1165994Proposed 643532 00239 638856 00266 637179 00276 516191

improvement on image quality while improved Toeplitz-structured matrix method (TSCMM) is slightly better thanclassical Toeplitz matrix method (3) EOM has significantperformance in image quality however the optimizationprocess is usually an iterative process which is also a verycomplicated and time-consuming process (4) Duarte-ETFhas better contrast and lower computational complexity (5)the proposed method takes advantage of improved Toeplitz-structured matrix to speed up the convergence speed andimprove traditional optimization method to better recoveryhigh-dimensional image Experimental results show that theproposed method has a better overall performance

42 Urban University of California Santa Barbara (UCSB)built an urban spectral library for the GoletaSanta BarbaraareaThe hyperspectral data of Urban were acquired betweenlate May and early June 2001 using an ASD full rangeinstrument on loan from the Jet Propulsion LaboratoryThesespectra of Urban are characterized by 499 roofs 179 roads

66 sidewalks 56 parking lots 40 road paints 37 types ofvegetation 47 types of nonphotosynthetic vegetation 88 baresoil and beach spectra 27 acquired from tennis courts and50 more from miscellaneous surfaces

Experiments on the hyperspectral data of Urban demon-strate that the proposed scheme substantially improves thereconstruction accuracy Clearly it comes to the same con-clusion from Figure 6 that compared with the previous fivemethods the effect of proposed approach is obviously sup-erior to any other methods and is the most similar to originalimage

To evaluate and compare the proposed method thefollowing performance indices such as Average Gradient(AG) Edge-Intensity (EI) Figure Definition (FD) GrayMean (GM) StandardDeviation (SD) Space Frequency (SF)Variance (VAR) and Structural Similarity (SSIM) were usedIt shows that the objective evaluation indices enhance theexperiment rigor and convincing The results are shown inTable 2

8 Journal of Control Science and Engineering

Table2Objectiv

eevaluationof

typicalm

etho

dsandprop

osed

metho

dfro

mrecovery

Urban

Band

sAlgorith

mTh

eobjectiv

eevaluationindices

AGEI

FDGM

SDMSE

PSNR

SFSSI

VAR

CT

11

Circulate

0070476

048360

012231

04106

0119

700929

584508

02034

098826

0119

7217

379

Toeplitz

0047687

034172

010125

032796

01137

00610

602804

018949

099361

011366

18706

TSCM

M0054635

03864

70114

14035344

01162

006

865976

58019866

099254

011624

15146

EOM

00566

08040

032

011519

035809

0117

200708

596279

020025

099226

0117

203051965

Duarte-ET

F0047687

034172

010125

032796

01137

00610

602804

018949

099361

011366

1366804

Prop

osed

004

6138

029

123

006

690

023194

01197

004

3461758

601195

6099

537

01198

0165975

41

Circulate

0072566

051888

012438

037371

01363

00631

6013

16020690

099349

013626

16382

Toeplitz

0058448

041868

012772

040

138

01390

00786

591767

022932

099207

013897

18069

TSCM

M0061306

043672

013128

039347

01362

00744

594172

022736

099259

013620

1538

1EO

M006

0340

04340

8012941

040222

01353

00771

59260

9022536

099215

013529

813160

Duarte-ET

F0062241

044

182

013123

042482

01402

00873

587190

023610

099089

014018

5974

19Prop

osed

005

0626

040

358

007

196

024

426

01307

004

52615816

012782

099

561

01306

8243532

81

Circulate

0082530

06279

013639

049357

01449

00635

6010

19022630

099397

014492

15829

Toeplitz

0077269

047993

014082

044

812

01390

00535

60844

2023904

099484

013905

18284

TSCM

M0070612

050737

014828

051106

01496

00716

595790

025291

099353

014958

15702

EOM

0070821

05100

6014940

050827

01503

00707

596361

02544

7099369

015030

9731

53Duarte-ET

F0074147

052535

014946

048963

01499

00652

599890

025959

099450

014986

590915

Prop

osed

006

8955

042

678

009

339

040

915

01571

004

84612814

01648

5099

580

01570

82874

26

Journal of Control Science and Engineering 9

(1) 11 bands (64 times 64 pixels)

(3) 81 bands (64 times 64 pixels)

(2) 41 bands (64 times 64 pixels)

(a1) original (b1) Circulant (c1) Toeplitz (d1) TSCMM (e1) EOM (f1) Duarte-ETF (g1) Proposed

(a2) original (b2) Circulant (c2) Toeplitz (d2) TSCMM (e2) EOM (f2) Duarte-ETF (g2) Proposed

(a3) original (b3) Circulant (c3) Toeplitz (d3) TSCMM (e3) EOM (f3) Duarte-ETF (g3) Proposed

Figure 6The subjective quality of different CSmeasurementmatrices from left to right (a) original (b) Circulant (c) Toeplitz (d) TSCMM(e) EOM (f) Duarte-ETF and (g) proposed method

The results of Table 2 show that the method has a higherperformance than traditional Toeplitz or Circulant matrixmethod Although improved Toeplitz-structured matrixmethod (TSCMM) is slightly better than classical Toeplitzboth classical optimization measurement matrix method(EOM) and proposed method have significant performancein image quality Furthermore the proposed method takesadvantage of improved Toeplitz-structured matrix to speedup the convergence speed and improve traditional opti-mization method to recover better high-dimensional imageExperimental results show that the proposed method has abetter overall performance

43 Jasper Ridge The hyperspectral image of Jasper Ridgewas obtained on June 2 September 4 and October 6 1992whichwas calibrated to surface reflectanceThe imagewas themost popular source to analyze with spectral mixture anal-ysis using library endmembers representing green foliagenonphotosynthetic vegetation and soils characteristic of thesite Field-based vegetation was obtained fromUSGeologicalService

From Figure 7 it is obvious that the worst effects fromtraditional Circulant have reached being almost intolerableWhile the results from TSCMM and Toeplitz are almostsimilar the former has only slight improvement comparedto the latter On the other hand the performance of theproposed approach was significantly improved compared tothat of EOM or Duarte-ETF The study concluded that theproposed approach had a significant performance comparedto that of others Furthermore the objective evaluations are

shown inTable 3The results can be clearly seen fromdifferentmethods on recovery Jasper Ridge

The results of Table 3 show that the proposed methodhas a higher performance than traditional Toeplitz or Circu-lant matrix method TSCMM takes advantage of improvedToeplitz-structuredmatrix to speedup the convergence speedand improve traditional Toeplitz or Circulant matrix methodto recover better high-dimensional data Although EOMhas the lower column coherence and faster convergence itweakens RIP condition and causes recovery performancedegradation While the absolute values from Duarte-ETFconcentrate around mutual coherence this can make theequivalent dictionary as close as possible to an ETF But thisalgorithm has high computational complexity Furthermoreexperimental results from the proposed method show thatthe proposed method has a better overall performance

44 Hyperspectral Unmixing The fourth experiment per-forms an experimental evaluation of the accuracy of thestandard hyperspectral unmixing districts known as Cuprite[28] To reduce complexity there are only 188 channels(3ndash103 114ndash147 and 168ndash220 bands) that were remained dueto dense water vapor and atmospheric effects In the courseof the experiment the signal sparsity method is Fourier basisand reconstruction algorithm is StOMP

The results are shown in Figure 8 for hyperspectralunmixing and closely resemble those obtained from hyper-spectral data Figure 8 compares the unmixing performanceof the proposed method with different endmembers and theabundance from different endmembers is totally dissimilar

10 Journal of Control Science and Engineering

Table 3 The objective evaluations of different matrixes of recovery Jasper Ridge

Algorithm 11 bands (64 times 64) 81 bands (64 times 64) 151 bands (64 times 64)SNR PSNR MSE SNR PSNR MSE SNR PSNR MSE CT

Circulate minus60698 573412 01199 116699 600402 00644 33299 603823 00595 15304Toeplitz 247566 707289 00055 294215 677497 00109 157386 657714 00172 15519TSCMM 153473 666425 00141 339793 697291 00069 200693 676521 00112 14869EOM 44853 619252 00417 350894 702112 00062 216750 683495 00095 788838Duarte-ETF 247566 707289 00055 333205 694430 00074 172452 664257 00148 665711Proposed 210504 691193 00080 549719 788460 848e minus 04 422277 772754 00012 345743

(1) 11 bands (64 times 64 pixels)

(2) 81 bands (64 times 64 pixels)

(3) 151 bands (64 times 64 pixels)

(a2) original (b2) Circulant (c2) Toeplitz (d2) TSCMM (e2) EOM (f2) Duarte-ETF (g2) Proposed

(a1) original (b1) Circulant (c1) Toeplitz (d1) TSCMM (e1) EOM (f1) DuarteminusETF (g1) Proposed

(a3) original (b3) Circulant (c3) Toeplitz (d3) TSCMM (e3) EOM (f3) Duarte-ETF (g3) Proposed

Figure 7The subjective quality of different CSmeasurementmatrices from left to right (a) original (b) Circulant (c) Toeplitz (d) TSCMM(e) EOM (f) Duarte-ETF and (g) proposed method

These features from the proposed method ensure clear andaccurate the abundances

By accessing information from USGS Digital Spec-tral Library [29] the unmixing performance has beenalmost correct Furthermore different endmembers fromUSGS 1995 Library [30] are used to verify that the predictionmodel of hyperspectral unmixing scheme is accurate inFigure 8

Apparently the proposed unmixing results (blue thinthread) have a strong correlationwith these endmembers (redthin thread) from USGS 1995 Library From Figure 9(a) asteep sloping line from Alunite suggests that the unmixingendmember (blue thin thread) has remarkable similarityThis same conclusion has been made in studies from Figures9(d)ndash9(h) On the other hand the unsatisfied results fromFigures 9(b) and 9(c) have been caused by smooth curveThe proposed method has good accuracy and is robust totraditional filtering compression cutting and noise attack

5 Conclusions

In this paper to overcome the limitation of Toeplitz-structured chaotic measurement matrix an improved mea-surement matrix has been carried out in the hyperspectralunmixing process to achieve multiple endmembers of hyper-spectral image And in theory it proves that this matrixhas retained the RIP property with overwhelming proba-bility Experimental results demonstrate that the proposedmethod to design of measurement matrix leads to better CSreconstruction performance with low extra computationalcost Compared with some traditional measurement matrixan improved method has highest technical feasibility lowestcomputational complexity and least computation time con-sumption in the same recovery qualityThe proposedmethodcan take the special advantage in hyperspectral unmixingprocess and explore the practical satellite system to remotesensing

Journal of Control Science and Engineering 11

(a) Alunite (b) Andradite (c) Pyrope

(d) Nontronite (e) Dumortierite (f) Kaolinite

(g) Chalcedony (h) Kaolinite (i) Buddingtonite

Figure 8 The results from different elements abundance in hyperspectral unmixing schemes

12 Journal of Control Science and Engineering

02

04

06

08

1Re

flect

ance

()

500 1000 1500 2000 2500 30000Wavelength (120583m)

(a) Alunite

0

01

02

03

04

Refle

ctan

ce (

)

500 1000 1500 2000 2500 30000Wavelength (120583m)

(b) Andradite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

1

Refle

ctan

ce (

)

(c) Pyrope

0

02

04

06

08

Refle

ctan

ce (

)

500 1000 1500 2000 2500 30000Wavelength (120583m)

(d) Nontronite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

1

Refle

ctan

ce (

)

(e) Dumortierite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

Refle

ctan

ce (

)

(f) Kaolinite

0

02

04

06

08

Refle

ctan

ce (

)

500 1000 1500 2000 2500 30000Wavelength (120583m)

(g) Chalcedony

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

Refle

ctan

ce (

)

(h) Kaolinite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

Refle

ctan

ce (

)

(i) Buddingtonite

Figure 9 The comparison map between actual unmixing effect and endmember library

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is supported by Chongqing Engineering Labo-ratory for Detection Control and Integrated System Theproject is also funded by Key Technology Research andIndustrialization of Fire Monitoring and Early Warn-ing Sensor Network for High Voltage Transmission Line(KJZH17124) This research is funded by Chongqing Edu-cation Commission Foundation (KJ1400612) This projectis also granted financial support from a CooperativeProject of Chongqing Technology and Business University(990516001)

References

[1] D L Donoho ldquoCompressed sensingrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol52 no 4 pp 1289ndash1306 2006

[2] E J Candes J Romberg and T Tao ldquoRobust uncertaintyprinciples exact signal reconstruction from highly incompletefrequency informationrdquo Institute of Electrical and ElectronicsEngineers Transactions on InformationTheory vol 52 no 2 pp489ndash509 2006

[3] T N Canh K D Quoc and B Jeon ldquoMulti-resolution kro-necker compressive sensingrdquo Transactions on Smart Processingamp Computing vol 3 no 1 pp 19ndash27 2014

[4] K Q Dinh H J Shim and B Jeon ldquoMeasurement coding forcompressive imaging using a structural measuremnet matrixrdquoin 2013 20th IEEE International Conference on Image ProcessingICIP 2013 pp 10ndash13 aus September 2013

[5] M F Duarte and R G Baraniuk ldquoKronecker compressive sens-ingrdquo IEEE Transactions on Image Processing vol 21 no 2 pp494ndash504 2012

[6] B Zhang X Tong W Wang and J Xie ldquoThe research ofKronecker product-based measurement matrix of compressivesensingrdquo EURASIP Journal on Wireless Communications andNetworking vol 2013 article 161 pp 1ndash5 2013

[7] M Elad ldquoOptimized projections for compressed sensingrdquo IEEETransactions on Signal Processing vol 55 no 12 pp 5695ndash57022007

Journal of Control Science and Engineering 13

[8] M Lustig D Donoho and J M Pauly ldquoSparse MRI the appli-cation of compressed sensing for rapid MR imagingrdquoMagneticResonance in Medicine vol 58 no 6 pp 1182ndash1195 2007

[9] V Abolghasemi S Sanei S Ferdowsi F Ghaderi and ABelcher ldquoSegmented compressive sensingrdquo in Proceedings of theIEEESP 15thWorkshop on Statistical Signal Processing (SSP rsquo09)pp 630ndash633 September 2009

[10] J M Duarte-Carvajalino and G Sapiro ldquoLearning to sensesparse signals simultaneous sensingmatrix and sparsifying dic-tionary optimizationrdquo IEEE Transactions on Image Processingvol 18 no 7 pp 1395ndash1408 2009

[11] ZWang G R Arce and J L Paredes ldquoColored randomprojec-tions for compressed sensingrdquo in 2007 IEEE International Con-ference on Acoustics Speech and Signal Processing ICASSP rsquo07pp III873ndashIII876 usa April 2007

[12] Z Wang and G R Arce ldquoVariable density compressed imagesamplingrdquo IEEE Transactions on Image Processing vol 19 no 1pp 264ndash270 2010

[13] M Elad and M Aharon ldquoImage denoising via learned dic-tionaries and sparse representationrdquo in 2006 IEEE ComputerSociety Conference on Computer Vision and Pattern RecognitionCVPR 2006 pp 895ndash900 usa June 2006

[14] M Aharon M Elad and A Bruckstein ldquoK-SVD an algorithmfor designing overcomplete dictionaries for sparse representa-tionrdquo IEEE Transactions on Signal Processing vol 54 no 11 pp4311ndash4322 2006

[15] J Xu Y Pi and Z Cao ldquoOptimized projection matrix for com-pressive sensingrdquo EURASIP Journal on Advances in SignalProcessing vol 2010 Article ID 560349 2010

[16] V Abolghasemi S Ferdowsi and S Sanei ldquoA gradient-basedalternating minimization approach for optimization of themeasurementmatrix in compressive sensingrdquo Signal Processingvol 92 no 4 pp 999ndash1009 2012

[17] Q Zhang Y Fu H Li and R Rong ldquoOptimized projectionmatrix for compressed sensingrdquo Circuits Systems and SignalProcessing vol 33 no 5 pp 1627ndash1636 2014

[18] G Li Z Zhu D Yang L Chang and H Bai ldquoOn projectionmatrix optimization for compressive sensing systemsrdquo IEEETransactions on Signal Processing vol 61 no 11 pp 2887ndash28982013

[19] S Tian X Fan and L I Zhetao ldquoOrthogonal-gradient mea-surement matrix construction algorithmrdquo Chinese Journal ofElectronics vol 25 no 1 pp 81ndash87 2016

[20] V Abolghasemi S Ferdowsi and BMakkiabadi ldquoOn optimiza-tion of the measurement matrix for compressive sensingrdquo inSignal Processing Conference European IEEE Ed pp 427ndash431August 2010

[21] Q Li D Schonfeld and S Friedland ldquoGeneralized tensorcompressive sensingrdquo in 2013 IEEE International Conference onMultimedia and Expo ICME 2013 usa July 2013

[22] S Friedland Q Li and D Schonfeld ldquoCompressive sensing ofsparse tensorsrdquo IEEE Transactions on Image Processing vol 23no 10 pp 4438ndash4447 2014

[23] S u Xu H Yin C Yi Y Xiong and T Xue ldquoAn ImprovedToeplitz Measurement Matrix for Compressive Sensingrdquo Inter-national Journal of Distributed Sensor Networks vol 8 pp 1ndash82014

[24] R L Devaney in Practical Numerical Algorithms for ChaoticSystems T S Parker and L O Chua Eds vol 32 pp 501ndash503Siam Review 3 edition 2006

[25] R Baraniuk M Davenport R DeVore and M Wakin ldquoAsimple proof of the restricted isometry property for randommatricesrdquoConstructiveApproximation An International Journalfor Approximations and Expansions vol 28 no 3 pp 253ndash2632008

[26] Available httpwwwesciencecnpeoplefeiyunZHUDatasetGThtml

[27] D L Donoho Y Tsaig I Drori and J-L Starck ldquoSparse solu-tion of underdetermined systems of linear equations by stage-wise orthogonal matching pursuitrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol58 no 2 pp 1094ndash1121 2012

[28] K Lang ldquoNewsWeeder Learning to Filter Netnewsrdquo in Inter-national Machine Learning Conference vol 1995 pp 331ndash339

[29] Available httpfeatureselectionasuedudatasetsphp[30] Available httpclopinetcomisabelleProjectsNIPS

Journal of Control Science and Engineering 5

Hereradic8119898 is for normalization andΦ is called Toeplitz-structured chaotic measurement matrix (TSCMM) whichmeets J-L theorem

32 Duarte-ETF Method Minimum coherence property ofETF has been the main target to find a feasible solution Itis impossible to solve the problem exactly because of thecomplexity while the structure of Gram matrix has changedso much that the selection of new units in the following stepis very difficult

To minimize (14) an optimization approach is adoptedto reduce the maximum and average mutual coherence ofmeasurement matrix It shrinks Gram matrix based on ETFtheoryThemethod canminimize the globalmutual coherentcoefficient of TSCMM by adjusting the eigenvalues abovezero to the average value of the sumof these eigenvalues with-out changing the sum After performing alternating mini-mization the optimized measurement matrix can be con-structed from the output Gram matrix with a rank revealingQR factorization with eigenvalue decomposition

Theorem 1 Given a measurement matrix Φ = R119898times119871 andrepresenting matrix Θ = R119899times119898 there exists a matrix 119863 =ΘΦ and Gram matrix 119866 = 10067041198631198791006704119863 where 1006704119863 is the columnnormalization from119863 If the real positive definite matrix119866 has120582119896 gt 0 (119896 = 1 sim 119898) the following equality sum119898119896=1 120582119896 = 119899 andsum119898119896=1(120582119896)2 = sum119899119894119895=1(⟨119894 119895⟩)2 where 119894 (119894 = 1 sim 119899) is thecolumn of 1006704119863 which is obtained

Based on Theorem 1 to minimize the largest absolutevalues of the off-diagonals in the correspondingGrammatrixwe can determine the eigenvalues of Gram matrix by solvingthe following optimization problems

min sum119894 =119895

(119892119894119895)2 = 119898sum119896=1

(120582119896)2 minus 119899sum119894=1

(119892119894119895)2st 119898sum

119896=1

120582119896 = 119899 (15)

where 119892119894119895 is the element of Gram matrix The optimizationproblem (15) is to minimize the square sum of the elementof Gram matrix if the sum of the characteristic value 120582119896remains constant Though the eigenvalue decomposition ofGram matrix has 120582119896 gt 0 the value 119899119898 has been graduallyapproaching to minimize the square sum of sum119898119896=1(120582119896)2Because real symmetric matrix eigenvalue decomposition isorthogonal the square sum of nondiagonal elements fromGram matrix gradually decreases and further it attains theeffect as follows

min sum119894 =119895

(119892119894119895)2 = 119898sum119896=1

( 119899119898)2 = 119899 (16)

Finally 1006704119863 is obtained after several iterations which is alsothe optimal Gram matrix 119866best

33 The Orthogonal Gradient Descent Approach Accordingto the definition of Grammatrix Grammatrix is the productof measurement matrix and sparse matrix Therefore theoptimalmeasurementmatrix can be directly derived from theoptimal Gram matrix However as for overcomplete sparserepresentation based on redundant dictionary it is very hardto design an effective algorithm to construct measurementmatrix In order to solve this problem orthogonal gradientdescent method is employed to get the optimal measurementmatrix Θbest

If the optimal Gram matrix 119866best is obtained the optimalmeasurement matrix Θbest is as followsΘbest = argmin 10038171003817100381710038171003817119866best minus Θ119879Θ10038171003817100381710038171003817119865 ≜ 119865 (Θ) (17)

And then the complex problem from optimal Grammatrix119866best is transformed into simpleminimumof119865(Θ) Bydetermining the derivative of119865(Θ) a skew-symmetricmatrix119882 with the measurement matrix Θ and the gradient matrixnabla119865 is used to obtain revision factornabla119865 (Θ) = Θ (Θ119879Θ minus 119866best) 119882 (Θ nabla119865) = Θ119879 (nabla119865) minus (nabla119865)119879Θ (18)

Next if 119868 is identity matrix the orthogonal matrix 119878 canbe expressed through the Cayley transform to ensure thepositive definiteness of the revision factor119878 = (119868 minus 119882) (119868 + 119882)minus1 (19)

The orthogonal gradient factor matrix Δ can be obtainedto update the gradient directionΔ = nabla119865 + (nabla119865) 119878 = nabla119865 (119878 + 119868) (20)

Combining (19) and (20) (21) can be rewritten as followsΔ = nabla119865 (119868 + (119868 minus 119882) (119868 + 119882)minus1) (21)Δ = 2nabla119865 (119868 + 119882)minus1 (22)

Finally update measurement matrixΘ with the orthogo-nal gradient factor matrix ΔΘ larr997888 Θ minus 120578Δ (23)

Because 120578 is updating ratio 119865(Θ) gradually converges attheminimumvalueThen the optimalmeasurementmatrixΘis the goal of our pursuit According to the linearity propertyof Toeplitz measurement matrix [25]Θ satisfies J-L propertywith overwhelming probability FromTheorem 1 it has beenproven that J-L condition can replace RIP condition So Θalso satisfies RIP property with overwhelming probability

The flow diagram of the proposed method will be givenas shown in Algorithm 1

4 Experiments and Result Analysis

To illustrate the effectiveness of the proposed approach themost widely used hyperspectral images in unmixing such

6 Journal of Control Science and Engineering

Input number of measurements119872 dictionary Ψ119871times119873 threshold 120585 = radic(119871 minus 119873)119873(119871 minus 1)updating ratio 120578 number of iterations Iter1 Iter2Initialization Set Φ119872times119873 to be TSCMMUpdate (1) Set the initial value of iteration 1198961 = 0(2)Optimize Gram matrix

(a) Compute Gram matrix 119866 = Θ119879Θ = Ψ119879Φ1198791198961Φ1198961Ψ(b) Normalize 1006704119866 = diag(1radicdiag(119866)) lowast 119866 lowast diag(1radicdiag(119866))(c) Update the elements of Gram matrix 1006704119866(3) Optimize measurement matrix(a) Set the initial value of iteration 1198962 = 0(b) Compute the orthogonal gradient factor matrix Δ 1198962 (c) Update measurement matrix Θ1198962 Θ1198962 = Θ1198962 minus 120578Δ 1198962 (d) 1198962 = 1198962 + 1 if 1198962 = Iter2 stop else return to Step 32)(4)Compute measurement matrix Φ1198961 (5) 1198961 = 1198961 + 1 if 1198961 = Iter1 stop else return to Step (2)

Output Φbest is the optimal measurement matrixΦIter1minus1Further SNR and reconstructed signal

Algorithm 1 The flow diagram of the proposed method Optimization of TSCMM

as Cuprite Urban and Jasper Ridge were selected in thespectral range from 380 nm to 2500 nm each channel bandwidth is up to 946 nm All high-dimensional data is providedby the standard hyperspectral library of 224 bands whichcomes from [26] To reduce complexity there are only 128times 128 pixel blocks of original image which starts from the(0 0)th pixel Further only 8 channels (from 11 to 81 bandsevery 10 bands) were remained due to dense water vapor andatmospheric effects

In the course of the experiment the signal sparsitymethod is Fourier basis and reconstruction algorithm isStagewise Orthogonal Matching Pursuit (StOMP) [27] Var-ious kinds of measurement matrix (Circulant ToeplitzToeplitz-structured chaotic measurement matrix (TSCMM)[23] Elad-Optimization Method (EOM) [7] and Duarte-Carvajalino and Sapirorsquos Method (Duarte-ETF) [10]) areemployed to illustrate the effectiveness of proposed approachfor hyperspectral unmixing

To demonstrate the efficiency of these methods tradi-tional evaluation methods can generally be divided into twocategories (1) subjective assessment and (2) objective evalu-ation Mean Squared Error (MSE) and Peak-Signal-to-NoiseRatio (PSNR) as one of the most important indices fromobjective evaluation determine the quality of recovery imagewhile Cost Time (CT) verifies the efficiency of the proposedapproach They all testify experiment results of recoverysignals built on laptop with AthlonTM Processor 160G HZ1 GB RAM Matlab 70 andWindows XP operation platform

41 Cuprite To illustrate the use of the hyperspectral anal-ysis process a sample scene covers the Cuprite miningdistrict in western Nevada USA from NASArsquos AirborneVisibleInfrared Imaging Spectrometer (AVIRIS) is providedThe data provided here is one of the most widely used hyper-spectral images in unmixing studyThere are 210 wavelengthsranging from 400 nm to 2500 nm resulting in a spectralresolution of 10 nm

In Figure 5 the first image (Top) was taken in bluelight the second image (middle) was taken in red light andthe third image (bottom) was taken in near infrared lightcentered at a wavelength of 750 nanometers

Figure 5 presents the subjective evaluation by Circu-lant Toeplitz TSCMM EOM Duarte-ETF and proposedmethod Compared with other results the performance fromFigure 5(b) is the worst The reason is that the elementsfrom Circulant measurement matrix follow periodic rep-etition permutation which does not satisfy RIP propertywith overwhelming probability Figure 5(c) clearly demon-strates that Toeplitz measurement matrix can avoid theproblem well Because of the property of pseudo-randomof chaotic sequence the performance from TSCMM hasfurther improved as shown in Figure 5(d) The result fromFigures 5(e)ndash5(g) shows that there is a significant impacton different bands using different optimization methodsNear infrared image is less affected by dust and gas Visibleblue channel has strong capability to penetrate water andvisible red channel can reflect the health status of plantsTherefore the sorted off-diagonal entries of themeasurementmatrix from EOM are likely more sparse and diagonal entriesare more concentrated The results are clearly shown inFigure 5(g) that the proposed approach had a significantperformance compared to any others and closely resemblesoriginal image Furthermore the objective evaluations whichinclude Mean Squared Error (MSE) Peak-Signal-to-NoiseRatio (PSNR) and Cost Time (CT) can avoid artificial errorand draw compelling conclusion The results can be clearlyseen fromdifferentmethods on recoveryCuprite as shown inTable 1

Figure 5 andTable 1 report the recovery quality of the pro-posed method on recovery Cuprite The following observa-tions are summarized (1) of all evaluating indicators con-sidered here traditional Circulant had the worst perfor-mance in both subjective and objective evaluations (2) sincethe introduction of Toeplitz the performance gets major

Journal of Control Science and Engineering 7

(1) 11

(2) 41 bands (128 times 128 pixels)

bands (128 times 128 pixels)

(3) 81 bands (128 times 128 pixels)

(a1) original (b1) Circulant (c1) Toeplitz (d1) TSCMM (e1) EOM (f1) Duarte-ETF (g1) Proposed

(a2) original (b2) Circulant (c2) Toeplitz (d2) TSCMM (e2) EOM (f2) Duarte-ETF (g2) Proposed

(a3) original (b3) Circulant (c3) Toeplitz (d3) TSCMM (e3) EOM (f3) Duarte-ETF (g3) Proposed

Figure 5The subjective quality of different CSmeasurementmatrices from left to right (a) original (b) Circulant (c) Toeplitz (d) TSCMM(e) EOM (f) Duarte-ETF and (g) proposed method

Table 1 MSE and PSNR of different matrixes of recovery Cuprite

Algorithm 11 bands (64 times 64 pixels) 41 bands (64 times 64 pixels) 81 bands (64 times 64 pixels)PSNR MSE PSNR MSE PSNR MSE CT

Circulate 599634 00656 609680 00520 615821 00452 51070Toeplitz 599634 00656 619375 00416 633173 00303 35506TSCMM 600696 00640 619783 00412 633462 00301 17624EOM 602806 00610 622337 00389 633962 00297 2454345Duarte-ETF 606304 00562 623788 00376 634082 00297 1165994Proposed 643532 00239 638856 00266 637179 00276 516191

improvement on image quality while improved Toeplitz-structured matrix method (TSCMM) is slightly better thanclassical Toeplitz matrix method (3) EOM has significantperformance in image quality however the optimizationprocess is usually an iterative process which is also a verycomplicated and time-consuming process (4) Duarte-ETFhas better contrast and lower computational complexity (5)the proposed method takes advantage of improved Toeplitz-structured matrix to speed up the convergence speed andimprove traditional optimization method to better recoveryhigh-dimensional image Experimental results show that theproposed method has a better overall performance

42 Urban University of California Santa Barbara (UCSB)built an urban spectral library for the GoletaSanta BarbaraareaThe hyperspectral data of Urban were acquired betweenlate May and early June 2001 using an ASD full rangeinstrument on loan from the Jet Propulsion LaboratoryThesespectra of Urban are characterized by 499 roofs 179 roads

66 sidewalks 56 parking lots 40 road paints 37 types ofvegetation 47 types of nonphotosynthetic vegetation 88 baresoil and beach spectra 27 acquired from tennis courts and50 more from miscellaneous surfaces

Experiments on the hyperspectral data of Urban demon-strate that the proposed scheme substantially improves thereconstruction accuracy Clearly it comes to the same con-clusion from Figure 6 that compared with the previous fivemethods the effect of proposed approach is obviously sup-erior to any other methods and is the most similar to originalimage

To evaluate and compare the proposed method thefollowing performance indices such as Average Gradient(AG) Edge-Intensity (EI) Figure Definition (FD) GrayMean (GM) StandardDeviation (SD) Space Frequency (SF)Variance (VAR) and Structural Similarity (SSIM) were usedIt shows that the objective evaluation indices enhance theexperiment rigor and convincing The results are shown inTable 2

8 Journal of Control Science and Engineering

Table2Objectiv

eevaluationof

typicalm

etho

dsandprop

osed

metho

dfro

mrecovery

Urban

Band

sAlgorith

mTh

eobjectiv

eevaluationindices

AGEI

FDGM

SDMSE

PSNR

SFSSI

VAR

CT

11

Circulate

0070476

048360

012231

04106

0119

700929

584508

02034

098826

0119

7217

379

Toeplitz

0047687

034172

010125

032796

01137

00610

602804

018949

099361

011366

18706

TSCM

M0054635

03864

70114

14035344

01162

006

865976

58019866

099254

011624

15146

EOM

00566

08040

032

011519

035809

0117

200708

596279

020025

099226

0117

203051965

Duarte-ET

F0047687

034172

010125

032796

01137

00610

602804

018949

099361

011366

1366804

Prop

osed

004

6138

029

123

006

690

023194

01197

004

3461758

601195

6099

537

01198

0165975

41

Circulate

0072566

051888

012438

037371

01363

00631

6013

16020690

099349

013626

16382

Toeplitz

0058448

041868

012772

040

138

01390

00786

591767

022932

099207

013897

18069

TSCM

M0061306

043672

013128

039347

01362

00744

594172

022736

099259

013620

1538

1EO

M006

0340

04340

8012941

040222

01353

00771

59260

9022536

099215

013529

813160

Duarte-ET

F0062241

044

182

013123

042482

01402

00873

587190

023610

099089

014018

5974

19Prop

osed

005

0626

040

358

007

196

024

426

01307

004

52615816

012782

099

561

01306

8243532

81

Circulate

0082530

06279

013639

049357

01449

00635

6010

19022630

099397

014492

15829

Toeplitz

0077269

047993

014082

044

812

01390

00535

60844

2023904

099484

013905

18284

TSCM

M0070612

050737

014828

051106

01496

00716

595790

025291

099353

014958

15702

EOM

0070821

05100

6014940

050827

01503

00707

596361

02544

7099369

015030

9731

53Duarte-ET

F0074147

052535

014946

048963

01499

00652

599890

025959

099450

014986

590915

Prop

osed

006

8955

042

678

009

339

040

915

01571

004

84612814

01648

5099

580

01570

82874

26

Journal of Control Science and Engineering 9

(1) 11 bands (64 times 64 pixels)

(3) 81 bands (64 times 64 pixels)

(2) 41 bands (64 times 64 pixels)

(a1) original (b1) Circulant (c1) Toeplitz (d1) TSCMM (e1) EOM (f1) Duarte-ETF (g1) Proposed

(a2) original (b2) Circulant (c2) Toeplitz (d2) TSCMM (e2) EOM (f2) Duarte-ETF (g2) Proposed

(a3) original (b3) Circulant (c3) Toeplitz (d3) TSCMM (e3) EOM (f3) Duarte-ETF (g3) Proposed

Figure 6The subjective quality of different CSmeasurementmatrices from left to right (a) original (b) Circulant (c) Toeplitz (d) TSCMM(e) EOM (f) Duarte-ETF and (g) proposed method

The results of Table 2 show that the method has a higherperformance than traditional Toeplitz or Circulant matrixmethod Although improved Toeplitz-structured matrixmethod (TSCMM) is slightly better than classical Toeplitzboth classical optimization measurement matrix method(EOM) and proposed method have significant performancein image quality Furthermore the proposed method takesadvantage of improved Toeplitz-structured matrix to speedup the convergence speed and improve traditional opti-mization method to recover better high-dimensional imageExperimental results show that the proposed method has abetter overall performance

43 Jasper Ridge The hyperspectral image of Jasper Ridgewas obtained on June 2 September 4 and October 6 1992whichwas calibrated to surface reflectanceThe imagewas themost popular source to analyze with spectral mixture anal-ysis using library endmembers representing green foliagenonphotosynthetic vegetation and soils characteristic of thesite Field-based vegetation was obtained fromUSGeologicalService

From Figure 7 it is obvious that the worst effects fromtraditional Circulant have reached being almost intolerableWhile the results from TSCMM and Toeplitz are almostsimilar the former has only slight improvement comparedto the latter On the other hand the performance of theproposed approach was significantly improved compared tothat of EOM or Duarte-ETF The study concluded that theproposed approach had a significant performance comparedto that of others Furthermore the objective evaluations are

shown inTable 3The results can be clearly seen fromdifferentmethods on recovery Jasper Ridge

The results of Table 3 show that the proposed methodhas a higher performance than traditional Toeplitz or Circu-lant matrix method TSCMM takes advantage of improvedToeplitz-structuredmatrix to speedup the convergence speedand improve traditional Toeplitz or Circulant matrix methodto recover better high-dimensional data Although EOMhas the lower column coherence and faster convergence itweakens RIP condition and causes recovery performancedegradation While the absolute values from Duarte-ETFconcentrate around mutual coherence this can make theequivalent dictionary as close as possible to an ETF But thisalgorithm has high computational complexity Furthermoreexperimental results from the proposed method show thatthe proposed method has a better overall performance

44 Hyperspectral Unmixing The fourth experiment per-forms an experimental evaluation of the accuracy of thestandard hyperspectral unmixing districts known as Cuprite[28] To reduce complexity there are only 188 channels(3ndash103 114ndash147 and 168ndash220 bands) that were remained dueto dense water vapor and atmospheric effects In the courseof the experiment the signal sparsity method is Fourier basisand reconstruction algorithm is StOMP

The results are shown in Figure 8 for hyperspectralunmixing and closely resemble those obtained from hyper-spectral data Figure 8 compares the unmixing performanceof the proposed method with different endmembers and theabundance from different endmembers is totally dissimilar

10 Journal of Control Science and Engineering

Table 3 The objective evaluations of different matrixes of recovery Jasper Ridge

Algorithm 11 bands (64 times 64) 81 bands (64 times 64) 151 bands (64 times 64)SNR PSNR MSE SNR PSNR MSE SNR PSNR MSE CT

Circulate minus60698 573412 01199 116699 600402 00644 33299 603823 00595 15304Toeplitz 247566 707289 00055 294215 677497 00109 157386 657714 00172 15519TSCMM 153473 666425 00141 339793 697291 00069 200693 676521 00112 14869EOM 44853 619252 00417 350894 702112 00062 216750 683495 00095 788838Duarte-ETF 247566 707289 00055 333205 694430 00074 172452 664257 00148 665711Proposed 210504 691193 00080 549719 788460 848e minus 04 422277 772754 00012 345743

(1) 11 bands (64 times 64 pixels)

(2) 81 bands (64 times 64 pixels)

(3) 151 bands (64 times 64 pixels)

(a2) original (b2) Circulant (c2) Toeplitz (d2) TSCMM (e2) EOM (f2) Duarte-ETF (g2) Proposed

(a1) original (b1) Circulant (c1) Toeplitz (d1) TSCMM (e1) EOM (f1) DuarteminusETF (g1) Proposed

(a3) original (b3) Circulant (c3) Toeplitz (d3) TSCMM (e3) EOM (f3) Duarte-ETF (g3) Proposed

Figure 7The subjective quality of different CSmeasurementmatrices from left to right (a) original (b) Circulant (c) Toeplitz (d) TSCMM(e) EOM (f) Duarte-ETF and (g) proposed method

These features from the proposed method ensure clear andaccurate the abundances

By accessing information from USGS Digital Spec-tral Library [29] the unmixing performance has beenalmost correct Furthermore different endmembers fromUSGS 1995 Library [30] are used to verify that the predictionmodel of hyperspectral unmixing scheme is accurate inFigure 8

Apparently the proposed unmixing results (blue thinthread) have a strong correlationwith these endmembers (redthin thread) from USGS 1995 Library From Figure 9(a) asteep sloping line from Alunite suggests that the unmixingendmember (blue thin thread) has remarkable similarityThis same conclusion has been made in studies from Figures9(d)ndash9(h) On the other hand the unsatisfied results fromFigures 9(b) and 9(c) have been caused by smooth curveThe proposed method has good accuracy and is robust totraditional filtering compression cutting and noise attack

5 Conclusions

In this paper to overcome the limitation of Toeplitz-structured chaotic measurement matrix an improved mea-surement matrix has been carried out in the hyperspectralunmixing process to achieve multiple endmembers of hyper-spectral image And in theory it proves that this matrixhas retained the RIP property with overwhelming proba-bility Experimental results demonstrate that the proposedmethod to design of measurement matrix leads to better CSreconstruction performance with low extra computationalcost Compared with some traditional measurement matrixan improved method has highest technical feasibility lowestcomputational complexity and least computation time con-sumption in the same recovery qualityThe proposedmethodcan take the special advantage in hyperspectral unmixingprocess and explore the practical satellite system to remotesensing

Journal of Control Science and Engineering 11

(a) Alunite (b) Andradite (c) Pyrope

(d) Nontronite (e) Dumortierite (f) Kaolinite

(g) Chalcedony (h) Kaolinite (i) Buddingtonite

Figure 8 The results from different elements abundance in hyperspectral unmixing schemes

12 Journal of Control Science and Engineering

02

04

06

08

1Re

flect

ance

()

500 1000 1500 2000 2500 30000Wavelength (120583m)

(a) Alunite

0

01

02

03

04

Refle

ctan

ce (

)

500 1000 1500 2000 2500 30000Wavelength (120583m)

(b) Andradite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

1

Refle

ctan

ce (

)

(c) Pyrope

0

02

04

06

08

Refle

ctan

ce (

)

500 1000 1500 2000 2500 30000Wavelength (120583m)

(d) Nontronite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

1

Refle

ctan

ce (

)

(e) Dumortierite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

Refle

ctan

ce (

)

(f) Kaolinite

0

02

04

06

08

Refle

ctan

ce (

)

500 1000 1500 2000 2500 30000Wavelength (120583m)

(g) Chalcedony

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

Refle

ctan

ce (

)

(h) Kaolinite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

Refle

ctan

ce (

)

(i) Buddingtonite

Figure 9 The comparison map between actual unmixing effect and endmember library

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is supported by Chongqing Engineering Labo-ratory for Detection Control and Integrated System Theproject is also funded by Key Technology Research andIndustrialization of Fire Monitoring and Early Warn-ing Sensor Network for High Voltage Transmission Line(KJZH17124) This research is funded by Chongqing Edu-cation Commission Foundation (KJ1400612) This projectis also granted financial support from a CooperativeProject of Chongqing Technology and Business University(990516001)

References

[1] D L Donoho ldquoCompressed sensingrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol52 no 4 pp 1289ndash1306 2006

[2] E J Candes J Romberg and T Tao ldquoRobust uncertaintyprinciples exact signal reconstruction from highly incompletefrequency informationrdquo Institute of Electrical and ElectronicsEngineers Transactions on InformationTheory vol 52 no 2 pp489ndash509 2006

[3] T N Canh K D Quoc and B Jeon ldquoMulti-resolution kro-necker compressive sensingrdquo Transactions on Smart Processingamp Computing vol 3 no 1 pp 19ndash27 2014

[4] K Q Dinh H J Shim and B Jeon ldquoMeasurement coding forcompressive imaging using a structural measuremnet matrixrdquoin 2013 20th IEEE International Conference on Image ProcessingICIP 2013 pp 10ndash13 aus September 2013

[5] M F Duarte and R G Baraniuk ldquoKronecker compressive sens-ingrdquo IEEE Transactions on Image Processing vol 21 no 2 pp494ndash504 2012

[6] B Zhang X Tong W Wang and J Xie ldquoThe research ofKronecker product-based measurement matrix of compressivesensingrdquo EURASIP Journal on Wireless Communications andNetworking vol 2013 article 161 pp 1ndash5 2013

[7] M Elad ldquoOptimized projections for compressed sensingrdquo IEEETransactions on Signal Processing vol 55 no 12 pp 5695ndash57022007

Journal of Control Science and Engineering 13

[8] M Lustig D Donoho and J M Pauly ldquoSparse MRI the appli-cation of compressed sensing for rapid MR imagingrdquoMagneticResonance in Medicine vol 58 no 6 pp 1182ndash1195 2007

[9] V Abolghasemi S Sanei S Ferdowsi F Ghaderi and ABelcher ldquoSegmented compressive sensingrdquo in Proceedings of theIEEESP 15thWorkshop on Statistical Signal Processing (SSP rsquo09)pp 630ndash633 September 2009

[10] J M Duarte-Carvajalino and G Sapiro ldquoLearning to sensesparse signals simultaneous sensingmatrix and sparsifying dic-tionary optimizationrdquo IEEE Transactions on Image Processingvol 18 no 7 pp 1395ndash1408 2009

[11] ZWang G R Arce and J L Paredes ldquoColored randomprojec-tions for compressed sensingrdquo in 2007 IEEE International Con-ference on Acoustics Speech and Signal Processing ICASSP rsquo07pp III873ndashIII876 usa April 2007

[12] Z Wang and G R Arce ldquoVariable density compressed imagesamplingrdquo IEEE Transactions on Image Processing vol 19 no 1pp 264ndash270 2010

[13] M Elad and M Aharon ldquoImage denoising via learned dic-tionaries and sparse representationrdquo in 2006 IEEE ComputerSociety Conference on Computer Vision and Pattern RecognitionCVPR 2006 pp 895ndash900 usa June 2006

[14] M Aharon M Elad and A Bruckstein ldquoK-SVD an algorithmfor designing overcomplete dictionaries for sparse representa-tionrdquo IEEE Transactions on Signal Processing vol 54 no 11 pp4311ndash4322 2006

[15] J Xu Y Pi and Z Cao ldquoOptimized projection matrix for com-pressive sensingrdquo EURASIP Journal on Advances in SignalProcessing vol 2010 Article ID 560349 2010

[16] V Abolghasemi S Ferdowsi and S Sanei ldquoA gradient-basedalternating minimization approach for optimization of themeasurementmatrix in compressive sensingrdquo Signal Processingvol 92 no 4 pp 999ndash1009 2012

[17] Q Zhang Y Fu H Li and R Rong ldquoOptimized projectionmatrix for compressed sensingrdquo Circuits Systems and SignalProcessing vol 33 no 5 pp 1627ndash1636 2014

[18] G Li Z Zhu D Yang L Chang and H Bai ldquoOn projectionmatrix optimization for compressive sensing systemsrdquo IEEETransactions on Signal Processing vol 61 no 11 pp 2887ndash28982013

[19] S Tian X Fan and L I Zhetao ldquoOrthogonal-gradient mea-surement matrix construction algorithmrdquo Chinese Journal ofElectronics vol 25 no 1 pp 81ndash87 2016

[20] V Abolghasemi S Ferdowsi and BMakkiabadi ldquoOn optimiza-tion of the measurement matrix for compressive sensingrdquo inSignal Processing Conference European IEEE Ed pp 427ndash431August 2010

[21] Q Li D Schonfeld and S Friedland ldquoGeneralized tensorcompressive sensingrdquo in 2013 IEEE International Conference onMultimedia and Expo ICME 2013 usa July 2013

[22] S Friedland Q Li and D Schonfeld ldquoCompressive sensing ofsparse tensorsrdquo IEEE Transactions on Image Processing vol 23no 10 pp 4438ndash4447 2014

[23] S u Xu H Yin C Yi Y Xiong and T Xue ldquoAn ImprovedToeplitz Measurement Matrix for Compressive Sensingrdquo Inter-national Journal of Distributed Sensor Networks vol 8 pp 1ndash82014

[24] R L Devaney in Practical Numerical Algorithms for ChaoticSystems T S Parker and L O Chua Eds vol 32 pp 501ndash503Siam Review 3 edition 2006

[25] R Baraniuk M Davenport R DeVore and M Wakin ldquoAsimple proof of the restricted isometry property for randommatricesrdquoConstructiveApproximation An International Journalfor Approximations and Expansions vol 28 no 3 pp 253ndash2632008

[26] Available httpwwwesciencecnpeoplefeiyunZHUDatasetGThtml

[27] D L Donoho Y Tsaig I Drori and J-L Starck ldquoSparse solu-tion of underdetermined systems of linear equations by stage-wise orthogonal matching pursuitrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol58 no 2 pp 1094ndash1121 2012

[28] K Lang ldquoNewsWeeder Learning to Filter Netnewsrdquo in Inter-national Machine Learning Conference vol 1995 pp 331ndash339

[29] Available httpfeatureselectionasuedudatasetsphp[30] Available httpclopinetcomisabelleProjectsNIPS

6 Journal of Control Science and Engineering

Input number of measurements119872 dictionary Ψ119871times119873 threshold 120585 = radic(119871 minus 119873)119873(119871 minus 1)updating ratio 120578 number of iterations Iter1 Iter2Initialization Set Φ119872times119873 to be TSCMMUpdate (1) Set the initial value of iteration 1198961 = 0(2)Optimize Gram matrix

(a) Compute Gram matrix 119866 = Θ119879Θ = Ψ119879Φ1198791198961Φ1198961Ψ(b) Normalize 1006704119866 = diag(1radicdiag(119866)) lowast 119866 lowast diag(1radicdiag(119866))(c) Update the elements of Gram matrix 1006704119866(3) Optimize measurement matrix(a) Set the initial value of iteration 1198962 = 0(b) Compute the orthogonal gradient factor matrix Δ 1198962 (c) Update measurement matrix Θ1198962 Θ1198962 = Θ1198962 minus 120578Δ 1198962 (d) 1198962 = 1198962 + 1 if 1198962 = Iter2 stop else return to Step 32)(4)Compute measurement matrix Φ1198961 (5) 1198961 = 1198961 + 1 if 1198961 = Iter1 stop else return to Step (2)

Output Φbest is the optimal measurement matrixΦIter1minus1Further SNR and reconstructed signal

Algorithm 1 The flow diagram of the proposed method Optimization of TSCMM

as Cuprite Urban and Jasper Ridge were selected in thespectral range from 380 nm to 2500 nm each channel bandwidth is up to 946 nm All high-dimensional data is providedby the standard hyperspectral library of 224 bands whichcomes from [26] To reduce complexity there are only 128times 128 pixel blocks of original image which starts from the(0 0)th pixel Further only 8 channels (from 11 to 81 bandsevery 10 bands) were remained due to dense water vapor andatmospheric effects

In the course of the experiment the signal sparsitymethod is Fourier basis and reconstruction algorithm isStagewise Orthogonal Matching Pursuit (StOMP) [27] Var-ious kinds of measurement matrix (Circulant ToeplitzToeplitz-structured chaotic measurement matrix (TSCMM)[23] Elad-Optimization Method (EOM) [7] and Duarte-Carvajalino and Sapirorsquos Method (Duarte-ETF) [10]) areemployed to illustrate the effectiveness of proposed approachfor hyperspectral unmixing

To demonstrate the efficiency of these methods tradi-tional evaluation methods can generally be divided into twocategories (1) subjective assessment and (2) objective evalu-ation Mean Squared Error (MSE) and Peak-Signal-to-NoiseRatio (PSNR) as one of the most important indices fromobjective evaluation determine the quality of recovery imagewhile Cost Time (CT) verifies the efficiency of the proposedapproach They all testify experiment results of recoverysignals built on laptop with AthlonTM Processor 160G HZ1 GB RAM Matlab 70 andWindows XP operation platform

41 Cuprite To illustrate the use of the hyperspectral anal-ysis process a sample scene covers the Cuprite miningdistrict in western Nevada USA from NASArsquos AirborneVisibleInfrared Imaging Spectrometer (AVIRIS) is providedThe data provided here is one of the most widely used hyper-spectral images in unmixing studyThere are 210 wavelengthsranging from 400 nm to 2500 nm resulting in a spectralresolution of 10 nm

In Figure 5 the first image (Top) was taken in bluelight the second image (middle) was taken in red light andthe third image (bottom) was taken in near infrared lightcentered at a wavelength of 750 nanometers

Figure 5 presents the subjective evaluation by Circu-lant Toeplitz TSCMM EOM Duarte-ETF and proposedmethod Compared with other results the performance fromFigure 5(b) is the worst The reason is that the elementsfrom Circulant measurement matrix follow periodic rep-etition permutation which does not satisfy RIP propertywith overwhelming probability Figure 5(c) clearly demon-strates that Toeplitz measurement matrix can avoid theproblem well Because of the property of pseudo-randomof chaotic sequence the performance from TSCMM hasfurther improved as shown in Figure 5(d) The result fromFigures 5(e)ndash5(g) shows that there is a significant impacton different bands using different optimization methodsNear infrared image is less affected by dust and gas Visibleblue channel has strong capability to penetrate water andvisible red channel can reflect the health status of plantsTherefore the sorted off-diagonal entries of themeasurementmatrix from EOM are likely more sparse and diagonal entriesare more concentrated The results are clearly shown inFigure 5(g) that the proposed approach had a significantperformance compared to any others and closely resemblesoriginal image Furthermore the objective evaluations whichinclude Mean Squared Error (MSE) Peak-Signal-to-NoiseRatio (PSNR) and Cost Time (CT) can avoid artificial errorand draw compelling conclusion The results can be clearlyseen fromdifferentmethods on recoveryCuprite as shown inTable 1

Figure 5 andTable 1 report the recovery quality of the pro-posed method on recovery Cuprite The following observa-tions are summarized (1) of all evaluating indicators con-sidered here traditional Circulant had the worst perfor-mance in both subjective and objective evaluations (2) sincethe introduction of Toeplitz the performance gets major

Journal of Control Science and Engineering 7

(1) 11

(2) 41 bands (128 times 128 pixels)

bands (128 times 128 pixels)

(3) 81 bands (128 times 128 pixels)

(a1) original (b1) Circulant (c1) Toeplitz (d1) TSCMM (e1) EOM (f1) Duarte-ETF (g1) Proposed

(a2) original (b2) Circulant (c2) Toeplitz (d2) TSCMM (e2) EOM (f2) Duarte-ETF (g2) Proposed

(a3) original (b3) Circulant (c3) Toeplitz (d3) TSCMM (e3) EOM (f3) Duarte-ETF (g3) Proposed

Figure 5The subjective quality of different CSmeasurementmatrices from left to right (a) original (b) Circulant (c) Toeplitz (d) TSCMM(e) EOM (f) Duarte-ETF and (g) proposed method

Table 1 MSE and PSNR of different matrixes of recovery Cuprite

Algorithm 11 bands (64 times 64 pixels) 41 bands (64 times 64 pixels) 81 bands (64 times 64 pixels)PSNR MSE PSNR MSE PSNR MSE CT

Circulate 599634 00656 609680 00520 615821 00452 51070Toeplitz 599634 00656 619375 00416 633173 00303 35506TSCMM 600696 00640 619783 00412 633462 00301 17624EOM 602806 00610 622337 00389 633962 00297 2454345Duarte-ETF 606304 00562 623788 00376 634082 00297 1165994Proposed 643532 00239 638856 00266 637179 00276 516191

improvement on image quality while improved Toeplitz-structured matrix method (TSCMM) is slightly better thanclassical Toeplitz matrix method (3) EOM has significantperformance in image quality however the optimizationprocess is usually an iterative process which is also a verycomplicated and time-consuming process (4) Duarte-ETFhas better contrast and lower computational complexity (5)the proposed method takes advantage of improved Toeplitz-structured matrix to speed up the convergence speed andimprove traditional optimization method to better recoveryhigh-dimensional image Experimental results show that theproposed method has a better overall performance

42 Urban University of California Santa Barbara (UCSB)built an urban spectral library for the GoletaSanta BarbaraareaThe hyperspectral data of Urban were acquired betweenlate May and early June 2001 using an ASD full rangeinstrument on loan from the Jet Propulsion LaboratoryThesespectra of Urban are characterized by 499 roofs 179 roads

66 sidewalks 56 parking lots 40 road paints 37 types ofvegetation 47 types of nonphotosynthetic vegetation 88 baresoil and beach spectra 27 acquired from tennis courts and50 more from miscellaneous surfaces

Experiments on the hyperspectral data of Urban demon-strate that the proposed scheme substantially improves thereconstruction accuracy Clearly it comes to the same con-clusion from Figure 6 that compared with the previous fivemethods the effect of proposed approach is obviously sup-erior to any other methods and is the most similar to originalimage

To evaluate and compare the proposed method thefollowing performance indices such as Average Gradient(AG) Edge-Intensity (EI) Figure Definition (FD) GrayMean (GM) StandardDeviation (SD) Space Frequency (SF)Variance (VAR) and Structural Similarity (SSIM) were usedIt shows that the objective evaluation indices enhance theexperiment rigor and convincing The results are shown inTable 2

8 Journal of Control Science and Engineering

Table2Objectiv

eevaluationof

typicalm

etho

dsandprop

osed

metho

dfro

mrecovery

Urban

Band

sAlgorith

mTh

eobjectiv

eevaluationindices

AGEI

FDGM

SDMSE

PSNR

SFSSI

VAR

CT

11

Circulate

0070476

048360

012231

04106

0119

700929

584508

02034

098826

0119

7217

379

Toeplitz

0047687

034172

010125

032796

01137

00610

602804

018949

099361

011366

18706

TSCM

M0054635

03864

70114

14035344

01162

006

865976

58019866

099254

011624

15146

EOM

00566

08040

032

011519

035809

0117

200708

596279

020025

099226

0117

203051965

Duarte-ET

F0047687

034172

010125

032796

01137

00610

602804

018949

099361

011366

1366804

Prop

osed

004

6138

029

123

006

690

023194

01197

004

3461758

601195

6099

537

01198

0165975

41

Circulate

0072566

051888

012438

037371

01363

00631

6013

16020690

099349

013626

16382

Toeplitz

0058448

041868

012772

040

138

01390

00786

591767

022932

099207

013897

18069

TSCM

M0061306

043672

013128

039347

01362

00744

594172

022736

099259

013620

1538

1EO

M006

0340

04340

8012941

040222

01353

00771

59260

9022536

099215

013529

813160

Duarte-ET

F0062241

044

182

013123

042482

01402

00873

587190

023610

099089

014018

5974

19Prop

osed

005

0626

040

358

007

196

024

426

01307

004

52615816

012782

099

561

01306

8243532

81

Circulate

0082530

06279

013639

049357

01449

00635

6010

19022630

099397

014492

15829

Toeplitz

0077269

047993

014082

044

812

01390

00535

60844

2023904

099484

013905

18284

TSCM

M0070612

050737

014828

051106

01496

00716

595790

025291

099353

014958

15702

EOM

0070821

05100

6014940

050827

01503

00707

596361

02544

7099369

015030

9731

53Duarte-ET

F0074147

052535

014946

048963

01499

00652

599890

025959

099450

014986

590915

Prop

osed

006

8955

042

678

009

339

040

915

01571

004

84612814

01648

5099

580

01570

82874

26

Journal of Control Science and Engineering 9

(1) 11 bands (64 times 64 pixels)

(3) 81 bands (64 times 64 pixels)

(2) 41 bands (64 times 64 pixels)

(a1) original (b1) Circulant (c1) Toeplitz (d1) TSCMM (e1) EOM (f1) Duarte-ETF (g1) Proposed

(a2) original (b2) Circulant (c2) Toeplitz (d2) TSCMM (e2) EOM (f2) Duarte-ETF (g2) Proposed

(a3) original (b3) Circulant (c3) Toeplitz (d3) TSCMM (e3) EOM (f3) Duarte-ETF (g3) Proposed

Figure 6The subjective quality of different CSmeasurementmatrices from left to right (a) original (b) Circulant (c) Toeplitz (d) TSCMM(e) EOM (f) Duarte-ETF and (g) proposed method

The results of Table 2 show that the method has a higherperformance than traditional Toeplitz or Circulant matrixmethod Although improved Toeplitz-structured matrixmethod (TSCMM) is slightly better than classical Toeplitzboth classical optimization measurement matrix method(EOM) and proposed method have significant performancein image quality Furthermore the proposed method takesadvantage of improved Toeplitz-structured matrix to speedup the convergence speed and improve traditional opti-mization method to recover better high-dimensional imageExperimental results show that the proposed method has abetter overall performance

43 Jasper Ridge The hyperspectral image of Jasper Ridgewas obtained on June 2 September 4 and October 6 1992whichwas calibrated to surface reflectanceThe imagewas themost popular source to analyze with spectral mixture anal-ysis using library endmembers representing green foliagenonphotosynthetic vegetation and soils characteristic of thesite Field-based vegetation was obtained fromUSGeologicalService

From Figure 7 it is obvious that the worst effects fromtraditional Circulant have reached being almost intolerableWhile the results from TSCMM and Toeplitz are almostsimilar the former has only slight improvement comparedto the latter On the other hand the performance of theproposed approach was significantly improved compared tothat of EOM or Duarte-ETF The study concluded that theproposed approach had a significant performance comparedto that of others Furthermore the objective evaluations are

shown inTable 3The results can be clearly seen fromdifferentmethods on recovery Jasper Ridge

The results of Table 3 show that the proposed methodhas a higher performance than traditional Toeplitz or Circu-lant matrix method TSCMM takes advantage of improvedToeplitz-structuredmatrix to speedup the convergence speedand improve traditional Toeplitz or Circulant matrix methodto recover better high-dimensional data Although EOMhas the lower column coherence and faster convergence itweakens RIP condition and causes recovery performancedegradation While the absolute values from Duarte-ETFconcentrate around mutual coherence this can make theequivalent dictionary as close as possible to an ETF But thisalgorithm has high computational complexity Furthermoreexperimental results from the proposed method show thatthe proposed method has a better overall performance

44 Hyperspectral Unmixing The fourth experiment per-forms an experimental evaluation of the accuracy of thestandard hyperspectral unmixing districts known as Cuprite[28] To reduce complexity there are only 188 channels(3ndash103 114ndash147 and 168ndash220 bands) that were remained dueto dense water vapor and atmospheric effects In the courseof the experiment the signal sparsity method is Fourier basisand reconstruction algorithm is StOMP

The results are shown in Figure 8 for hyperspectralunmixing and closely resemble those obtained from hyper-spectral data Figure 8 compares the unmixing performanceof the proposed method with different endmembers and theabundance from different endmembers is totally dissimilar

10 Journal of Control Science and Engineering

Table 3 The objective evaluations of different matrixes of recovery Jasper Ridge

Algorithm 11 bands (64 times 64) 81 bands (64 times 64) 151 bands (64 times 64)SNR PSNR MSE SNR PSNR MSE SNR PSNR MSE CT

Circulate minus60698 573412 01199 116699 600402 00644 33299 603823 00595 15304Toeplitz 247566 707289 00055 294215 677497 00109 157386 657714 00172 15519TSCMM 153473 666425 00141 339793 697291 00069 200693 676521 00112 14869EOM 44853 619252 00417 350894 702112 00062 216750 683495 00095 788838Duarte-ETF 247566 707289 00055 333205 694430 00074 172452 664257 00148 665711Proposed 210504 691193 00080 549719 788460 848e minus 04 422277 772754 00012 345743

(1) 11 bands (64 times 64 pixels)

(2) 81 bands (64 times 64 pixels)

(3) 151 bands (64 times 64 pixels)

(a2) original (b2) Circulant (c2) Toeplitz (d2) TSCMM (e2) EOM (f2) Duarte-ETF (g2) Proposed

(a1) original (b1) Circulant (c1) Toeplitz (d1) TSCMM (e1) EOM (f1) DuarteminusETF (g1) Proposed

(a3) original (b3) Circulant (c3) Toeplitz (d3) TSCMM (e3) EOM (f3) Duarte-ETF (g3) Proposed

Figure 7The subjective quality of different CSmeasurementmatrices from left to right (a) original (b) Circulant (c) Toeplitz (d) TSCMM(e) EOM (f) Duarte-ETF and (g) proposed method

These features from the proposed method ensure clear andaccurate the abundances

By accessing information from USGS Digital Spec-tral Library [29] the unmixing performance has beenalmost correct Furthermore different endmembers fromUSGS 1995 Library [30] are used to verify that the predictionmodel of hyperspectral unmixing scheme is accurate inFigure 8

Apparently the proposed unmixing results (blue thinthread) have a strong correlationwith these endmembers (redthin thread) from USGS 1995 Library From Figure 9(a) asteep sloping line from Alunite suggests that the unmixingendmember (blue thin thread) has remarkable similarityThis same conclusion has been made in studies from Figures9(d)ndash9(h) On the other hand the unsatisfied results fromFigures 9(b) and 9(c) have been caused by smooth curveThe proposed method has good accuracy and is robust totraditional filtering compression cutting and noise attack

5 Conclusions

In this paper to overcome the limitation of Toeplitz-structured chaotic measurement matrix an improved mea-surement matrix has been carried out in the hyperspectralunmixing process to achieve multiple endmembers of hyper-spectral image And in theory it proves that this matrixhas retained the RIP property with overwhelming proba-bility Experimental results demonstrate that the proposedmethod to design of measurement matrix leads to better CSreconstruction performance with low extra computationalcost Compared with some traditional measurement matrixan improved method has highest technical feasibility lowestcomputational complexity and least computation time con-sumption in the same recovery qualityThe proposedmethodcan take the special advantage in hyperspectral unmixingprocess and explore the practical satellite system to remotesensing

Journal of Control Science and Engineering 11

(a) Alunite (b) Andradite (c) Pyrope

(d) Nontronite (e) Dumortierite (f) Kaolinite

(g) Chalcedony (h) Kaolinite (i) Buddingtonite

Figure 8 The results from different elements abundance in hyperspectral unmixing schemes

12 Journal of Control Science and Engineering

02

04

06

08

1Re

flect

ance

()

500 1000 1500 2000 2500 30000Wavelength (120583m)

(a) Alunite

0

01

02

03

04

Refle

ctan

ce (

)

500 1000 1500 2000 2500 30000Wavelength (120583m)

(b) Andradite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

1

Refle

ctan

ce (

)

(c) Pyrope

0

02

04

06

08

Refle

ctan

ce (

)

500 1000 1500 2000 2500 30000Wavelength (120583m)

(d) Nontronite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

1

Refle

ctan

ce (

)

(e) Dumortierite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

Refle

ctan

ce (

)

(f) Kaolinite

0

02

04

06

08

Refle

ctan

ce (

)

500 1000 1500 2000 2500 30000Wavelength (120583m)

(g) Chalcedony

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

Refle

ctan

ce (

)

(h) Kaolinite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

Refle

ctan

ce (

)

(i) Buddingtonite

Figure 9 The comparison map between actual unmixing effect and endmember library

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is supported by Chongqing Engineering Labo-ratory for Detection Control and Integrated System Theproject is also funded by Key Technology Research andIndustrialization of Fire Monitoring and Early Warn-ing Sensor Network for High Voltage Transmission Line(KJZH17124) This research is funded by Chongqing Edu-cation Commission Foundation (KJ1400612) This projectis also granted financial support from a CooperativeProject of Chongqing Technology and Business University(990516001)

References

[1] D L Donoho ldquoCompressed sensingrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol52 no 4 pp 1289ndash1306 2006

[2] E J Candes J Romberg and T Tao ldquoRobust uncertaintyprinciples exact signal reconstruction from highly incompletefrequency informationrdquo Institute of Electrical and ElectronicsEngineers Transactions on InformationTheory vol 52 no 2 pp489ndash509 2006

[3] T N Canh K D Quoc and B Jeon ldquoMulti-resolution kro-necker compressive sensingrdquo Transactions on Smart Processingamp Computing vol 3 no 1 pp 19ndash27 2014

[4] K Q Dinh H J Shim and B Jeon ldquoMeasurement coding forcompressive imaging using a structural measuremnet matrixrdquoin 2013 20th IEEE International Conference on Image ProcessingICIP 2013 pp 10ndash13 aus September 2013

[5] M F Duarte and R G Baraniuk ldquoKronecker compressive sens-ingrdquo IEEE Transactions on Image Processing vol 21 no 2 pp494ndash504 2012

[6] B Zhang X Tong W Wang and J Xie ldquoThe research ofKronecker product-based measurement matrix of compressivesensingrdquo EURASIP Journal on Wireless Communications andNetworking vol 2013 article 161 pp 1ndash5 2013

[7] M Elad ldquoOptimized projections for compressed sensingrdquo IEEETransactions on Signal Processing vol 55 no 12 pp 5695ndash57022007

Journal of Control Science and Engineering 13

[8] M Lustig D Donoho and J M Pauly ldquoSparse MRI the appli-cation of compressed sensing for rapid MR imagingrdquoMagneticResonance in Medicine vol 58 no 6 pp 1182ndash1195 2007

[9] V Abolghasemi S Sanei S Ferdowsi F Ghaderi and ABelcher ldquoSegmented compressive sensingrdquo in Proceedings of theIEEESP 15thWorkshop on Statistical Signal Processing (SSP rsquo09)pp 630ndash633 September 2009

[10] J M Duarte-Carvajalino and G Sapiro ldquoLearning to sensesparse signals simultaneous sensingmatrix and sparsifying dic-tionary optimizationrdquo IEEE Transactions on Image Processingvol 18 no 7 pp 1395ndash1408 2009

[11] ZWang G R Arce and J L Paredes ldquoColored randomprojec-tions for compressed sensingrdquo in 2007 IEEE International Con-ference on Acoustics Speech and Signal Processing ICASSP rsquo07pp III873ndashIII876 usa April 2007

[12] Z Wang and G R Arce ldquoVariable density compressed imagesamplingrdquo IEEE Transactions on Image Processing vol 19 no 1pp 264ndash270 2010

[13] M Elad and M Aharon ldquoImage denoising via learned dic-tionaries and sparse representationrdquo in 2006 IEEE ComputerSociety Conference on Computer Vision and Pattern RecognitionCVPR 2006 pp 895ndash900 usa June 2006

[14] M Aharon M Elad and A Bruckstein ldquoK-SVD an algorithmfor designing overcomplete dictionaries for sparse representa-tionrdquo IEEE Transactions on Signal Processing vol 54 no 11 pp4311ndash4322 2006

[15] J Xu Y Pi and Z Cao ldquoOptimized projection matrix for com-pressive sensingrdquo EURASIP Journal on Advances in SignalProcessing vol 2010 Article ID 560349 2010

[16] V Abolghasemi S Ferdowsi and S Sanei ldquoA gradient-basedalternating minimization approach for optimization of themeasurementmatrix in compressive sensingrdquo Signal Processingvol 92 no 4 pp 999ndash1009 2012

[17] Q Zhang Y Fu H Li and R Rong ldquoOptimized projectionmatrix for compressed sensingrdquo Circuits Systems and SignalProcessing vol 33 no 5 pp 1627ndash1636 2014

[18] G Li Z Zhu D Yang L Chang and H Bai ldquoOn projectionmatrix optimization for compressive sensing systemsrdquo IEEETransactions on Signal Processing vol 61 no 11 pp 2887ndash28982013

[19] S Tian X Fan and L I Zhetao ldquoOrthogonal-gradient mea-surement matrix construction algorithmrdquo Chinese Journal ofElectronics vol 25 no 1 pp 81ndash87 2016

[20] V Abolghasemi S Ferdowsi and BMakkiabadi ldquoOn optimiza-tion of the measurement matrix for compressive sensingrdquo inSignal Processing Conference European IEEE Ed pp 427ndash431August 2010

[21] Q Li D Schonfeld and S Friedland ldquoGeneralized tensorcompressive sensingrdquo in 2013 IEEE International Conference onMultimedia and Expo ICME 2013 usa July 2013

[22] S Friedland Q Li and D Schonfeld ldquoCompressive sensing ofsparse tensorsrdquo IEEE Transactions on Image Processing vol 23no 10 pp 4438ndash4447 2014

[23] S u Xu H Yin C Yi Y Xiong and T Xue ldquoAn ImprovedToeplitz Measurement Matrix for Compressive Sensingrdquo Inter-national Journal of Distributed Sensor Networks vol 8 pp 1ndash82014

[24] R L Devaney in Practical Numerical Algorithms for ChaoticSystems T S Parker and L O Chua Eds vol 32 pp 501ndash503Siam Review 3 edition 2006

[25] R Baraniuk M Davenport R DeVore and M Wakin ldquoAsimple proof of the restricted isometry property for randommatricesrdquoConstructiveApproximation An International Journalfor Approximations and Expansions vol 28 no 3 pp 253ndash2632008

[26] Available httpwwwesciencecnpeoplefeiyunZHUDatasetGThtml

[27] D L Donoho Y Tsaig I Drori and J-L Starck ldquoSparse solu-tion of underdetermined systems of linear equations by stage-wise orthogonal matching pursuitrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol58 no 2 pp 1094ndash1121 2012

[28] K Lang ldquoNewsWeeder Learning to Filter Netnewsrdquo in Inter-national Machine Learning Conference vol 1995 pp 331ndash339

[29] Available httpfeatureselectionasuedudatasetsphp[30] Available httpclopinetcomisabelleProjectsNIPS

Journal of Control Science and Engineering 7

(1) 11

(2) 41 bands (128 times 128 pixels)

bands (128 times 128 pixels)

(3) 81 bands (128 times 128 pixels)

(a1) original (b1) Circulant (c1) Toeplitz (d1) TSCMM (e1) EOM (f1) Duarte-ETF (g1) Proposed

(a2) original (b2) Circulant (c2) Toeplitz (d2) TSCMM (e2) EOM (f2) Duarte-ETF (g2) Proposed

(a3) original (b3) Circulant (c3) Toeplitz (d3) TSCMM (e3) EOM (f3) Duarte-ETF (g3) Proposed

Figure 5The subjective quality of different CSmeasurementmatrices from left to right (a) original (b) Circulant (c) Toeplitz (d) TSCMM(e) EOM (f) Duarte-ETF and (g) proposed method

Table 1 MSE and PSNR of different matrixes of recovery Cuprite

Algorithm 11 bands (64 times 64 pixels) 41 bands (64 times 64 pixels) 81 bands (64 times 64 pixels)PSNR MSE PSNR MSE PSNR MSE CT

Circulate 599634 00656 609680 00520 615821 00452 51070Toeplitz 599634 00656 619375 00416 633173 00303 35506TSCMM 600696 00640 619783 00412 633462 00301 17624EOM 602806 00610 622337 00389 633962 00297 2454345Duarte-ETF 606304 00562 623788 00376 634082 00297 1165994Proposed 643532 00239 638856 00266 637179 00276 516191

improvement on image quality while improved Toeplitz-structured matrix method (TSCMM) is slightly better thanclassical Toeplitz matrix method (3) EOM has significantperformance in image quality however the optimizationprocess is usually an iterative process which is also a verycomplicated and time-consuming process (4) Duarte-ETFhas better contrast and lower computational complexity (5)the proposed method takes advantage of improved Toeplitz-structured matrix to speed up the convergence speed andimprove traditional optimization method to better recoveryhigh-dimensional image Experimental results show that theproposed method has a better overall performance

42 Urban University of California Santa Barbara (UCSB)built an urban spectral library for the GoletaSanta BarbaraareaThe hyperspectral data of Urban were acquired betweenlate May and early June 2001 using an ASD full rangeinstrument on loan from the Jet Propulsion LaboratoryThesespectra of Urban are characterized by 499 roofs 179 roads

66 sidewalks 56 parking lots 40 road paints 37 types ofvegetation 47 types of nonphotosynthetic vegetation 88 baresoil and beach spectra 27 acquired from tennis courts and50 more from miscellaneous surfaces

Experiments on the hyperspectral data of Urban demon-strate that the proposed scheme substantially improves thereconstruction accuracy Clearly it comes to the same con-clusion from Figure 6 that compared with the previous fivemethods the effect of proposed approach is obviously sup-erior to any other methods and is the most similar to originalimage

To evaluate and compare the proposed method thefollowing performance indices such as Average Gradient(AG) Edge-Intensity (EI) Figure Definition (FD) GrayMean (GM) StandardDeviation (SD) Space Frequency (SF)Variance (VAR) and Structural Similarity (SSIM) were usedIt shows that the objective evaluation indices enhance theexperiment rigor and convincing The results are shown inTable 2

8 Journal of Control Science and Engineering

Table2Objectiv

eevaluationof

typicalm

etho

dsandprop

osed

metho

dfro

mrecovery

Urban

Band

sAlgorith

mTh

eobjectiv

eevaluationindices

AGEI

FDGM

SDMSE

PSNR

SFSSI

VAR

CT

11

Circulate

0070476

048360

012231

04106

0119

700929

584508

02034

098826

0119

7217

379

Toeplitz

0047687

034172

010125

032796

01137

00610

602804

018949

099361

011366

18706

TSCM

M0054635

03864

70114

14035344

01162

006

865976

58019866

099254

011624

15146

EOM

00566

08040

032

011519

035809

0117

200708

596279

020025

099226

0117

203051965

Duarte-ET

F0047687

034172

010125

032796

01137

00610

602804

018949

099361

011366

1366804

Prop

osed

004

6138

029

123

006

690

023194

01197

004

3461758

601195

6099

537

01198

0165975

41

Circulate

0072566

051888

012438

037371

01363

00631

6013

16020690

099349

013626

16382

Toeplitz

0058448

041868

012772

040

138

01390

00786

591767

022932

099207

013897

18069

TSCM

M0061306

043672

013128

039347

01362

00744

594172

022736

099259

013620

1538

1EO

M006

0340

04340

8012941

040222

01353

00771

59260

9022536

099215

013529

813160

Duarte-ET

F0062241

044

182

013123

042482

01402

00873

587190

023610

099089

014018

5974

19Prop

osed

005

0626

040

358

007

196

024

426

01307

004

52615816

012782

099

561

01306

8243532

81

Circulate

0082530

06279

013639

049357

01449

00635

6010

19022630

099397

014492

15829

Toeplitz

0077269

047993

014082

044

812

01390

00535

60844

2023904

099484

013905

18284

TSCM

M0070612

050737

014828

051106

01496

00716

595790

025291

099353

014958

15702

EOM

0070821

05100

6014940

050827

01503

00707

596361

02544

7099369

015030

9731

53Duarte-ET

F0074147

052535

014946

048963

01499

00652

599890

025959

099450

014986

590915

Prop

osed

006

8955

042

678

009

339

040

915

01571

004

84612814

01648

5099

580

01570

82874

26

Journal of Control Science and Engineering 9

(1) 11 bands (64 times 64 pixels)

(3) 81 bands (64 times 64 pixels)

(2) 41 bands (64 times 64 pixels)

(a1) original (b1) Circulant (c1) Toeplitz (d1) TSCMM (e1) EOM (f1) Duarte-ETF (g1) Proposed

(a2) original (b2) Circulant (c2) Toeplitz (d2) TSCMM (e2) EOM (f2) Duarte-ETF (g2) Proposed

(a3) original (b3) Circulant (c3) Toeplitz (d3) TSCMM (e3) EOM (f3) Duarte-ETF (g3) Proposed

Figure 6The subjective quality of different CSmeasurementmatrices from left to right (a) original (b) Circulant (c) Toeplitz (d) TSCMM(e) EOM (f) Duarte-ETF and (g) proposed method

The results of Table 2 show that the method has a higherperformance than traditional Toeplitz or Circulant matrixmethod Although improved Toeplitz-structured matrixmethod (TSCMM) is slightly better than classical Toeplitzboth classical optimization measurement matrix method(EOM) and proposed method have significant performancein image quality Furthermore the proposed method takesadvantage of improved Toeplitz-structured matrix to speedup the convergence speed and improve traditional opti-mization method to recover better high-dimensional imageExperimental results show that the proposed method has abetter overall performance

43 Jasper Ridge The hyperspectral image of Jasper Ridgewas obtained on June 2 September 4 and October 6 1992whichwas calibrated to surface reflectanceThe imagewas themost popular source to analyze with spectral mixture anal-ysis using library endmembers representing green foliagenonphotosynthetic vegetation and soils characteristic of thesite Field-based vegetation was obtained fromUSGeologicalService

From Figure 7 it is obvious that the worst effects fromtraditional Circulant have reached being almost intolerableWhile the results from TSCMM and Toeplitz are almostsimilar the former has only slight improvement comparedto the latter On the other hand the performance of theproposed approach was significantly improved compared tothat of EOM or Duarte-ETF The study concluded that theproposed approach had a significant performance comparedto that of others Furthermore the objective evaluations are

shown inTable 3The results can be clearly seen fromdifferentmethods on recovery Jasper Ridge

The results of Table 3 show that the proposed methodhas a higher performance than traditional Toeplitz or Circu-lant matrix method TSCMM takes advantage of improvedToeplitz-structuredmatrix to speedup the convergence speedand improve traditional Toeplitz or Circulant matrix methodto recover better high-dimensional data Although EOMhas the lower column coherence and faster convergence itweakens RIP condition and causes recovery performancedegradation While the absolute values from Duarte-ETFconcentrate around mutual coherence this can make theequivalent dictionary as close as possible to an ETF But thisalgorithm has high computational complexity Furthermoreexperimental results from the proposed method show thatthe proposed method has a better overall performance

44 Hyperspectral Unmixing The fourth experiment per-forms an experimental evaluation of the accuracy of thestandard hyperspectral unmixing districts known as Cuprite[28] To reduce complexity there are only 188 channels(3ndash103 114ndash147 and 168ndash220 bands) that were remained dueto dense water vapor and atmospheric effects In the courseof the experiment the signal sparsity method is Fourier basisand reconstruction algorithm is StOMP

The results are shown in Figure 8 for hyperspectralunmixing and closely resemble those obtained from hyper-spectral data Figure 8 compares the unmixing performanceof the proposed method with different endmembers and theabundance from different endmembers is totally dissimilar

10 Journal of Control Science and Engineering

Table 3 The objective evaluations of different matrixes of recovery Jasper Ridge

Algorithm 11 bands (64 times 64) 81 bands (64 times 64) 151 bands (64 times 64)SNR PSNR MSE SNR PSNR MSE SNR PSNR MSE CT

Circulate minus60698 573412 01199 116699 600402 00644 33299 603823 00595 15304Toeplitz 247566 707289 00055 294215 677497 00109 157386 657714 00172 15519TSCMM 153473 666425 00141 339793 697291 00069 200693 676521 00112 14869EOM 44853 619252 00417 350894 702112 00062 216750 683495 00095 788838Duarte-ETF 247566 707289 00055 333205 694430 00074 172452 664257 00148 665711Proposed 210504 691193 00080 549719 788460 848e minus 04 422277 772754 00012 345743

(1) 11 bands (64 times 64 pixels)

(2) 81 bands (64 times 64 pixels)

(3) 151 bands (64 times 64 pixels)

(a2) original (b2) Circulant (c2) Toeplitz (d2) TSCMM (e2) EOM (f2) Duarte-ETF (g2) Proposed

(a1) original (b1) Circulant (c1) Toeplitz (d1) TSCMM (e1) EOM (f1) DuarteminusETF (g1) Proposed

(a3) original (b3) Circulant (c3) Toeplitz (d3) TSCMM (e3) EOM (f3) Duarte-ETF (g3) Proposed

Figure 7The subjective quality of different CSmeasurementmatrices from left to right (a) original (b) Circulant (c) Toeplitz (d) TSCMM(e) EOM (f) Duarte-ETF and (g) proposed method

These features from the proposed method ensure clear andaccurate the abundances

By accessing information from USGS Digital Spec-tral Library [29] the unmixing performance has beenalmost correct Furthermore different endmembers fromUSGS 1995 Library [30] are used to verify that the predictionmodel of hyperspectral unmixing scheme is accurate inFigure 8

Apparently the proposed unmixing results (blue thinthread) have a strong correlationwith these endmembers (redthin thread) from USGS 1995 Library From Figure 9(a) asteep sloping line from Alunite suggests that the unmixingendmember (blue thin thread) has remarkable similarityThis same conclusion has been made in studies from Figures9(d)ndash9(h) On the other hand the unsatisfied results fromFigures 9(b) and 9(c) have been caused by smooth curveThe proposed method has good accuracy and is robust totraditional filtering compression cutting and noise attack

5 Conclusions

In this paper to overcome the limitation of Toeplitz-structured chaotic measurement matrix an improved mea-surement matrix has been carried out in the hyperspectralunmixing process to achieve multiple endmembers of hyper-spectral image And in theory it proves that this matrixhas retained the RIP property with overwhelming proba-bility Experimental results demonstrate that the proposedmethod to design of measurement matrix leads to better CSreconstruction performance with low extra computationalcost Compared with some traditional measurement matrixan improved method has highest technical feasibility lowestcomputational complexity and least computation time con-sumption in the same recovery qualityThe proposedmethodcan take the special advantage in hyperspectral unmixingprocess and explore the practical satellite system to remotesensing

Journal of Control Science and Engineering 11

(a) Alunite (b) Andradite (c) Pyrope

(d) Nontronite (e) Dumortierite (f) Kaolinite

(g) Chalcedony (h) Kaolinite (i) Buddingtonite

Figure 8 The results from different elements abundance in hyperspectral unmixing schemes

12 Journal of Control Science and Engineering

02

04

06

08

1Re

flect

ance

()

500 1000 1500 2000 2500 30000Wavelength (120583m)

(a) Alunite

0

01

02

03

04

Refle

ctan

ce (

)

500 1000 1500 2000 2500 30000Wavelength (120583m)

(b) Andradite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

1

Refle

ctan

ce (

)

(c) Pyrope

0

02

04

06

08

Refle

ctan

ce (

)

500 1000 1500 2000 2500 30000Wavelength (120583m)

(d) Nontronite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

1

Refle

ctan

ce (

)

(e) Dumortierite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

Refle

ctan

ce (

)

(f) Kaolinite

0

02

04

06

08

Refle

ctan

ce (

)

500 1000 1500 2000 2500 30000Wavelength (120583m)

(g) Chalcedony

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

Refle

ctan

ce (

)

(h) Kaolinite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

Refle

ctan

ce (

)

(i) Buddingtonite

Figure 9 The comparison map between actual unmixing effect and endmember library

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is supported by Chongqing Engineering Labo-ratory for Detection Control and Integrated System Theproject is also funded by Key Technology Research andIndustrialization of Fire Monitoring and Early Warn-ing Sensor Network for High Voltage Transmission Line(KJZH17124) This research is funded by Chongqing Edu-cation Commission Foundation (KJ1400612) This projectis also granted financial support from a CooperativeProject of Chongqing Technology and Business University(990516001)

References

[1] D L Donoho ldquoCompressed sensingrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol52 no 4 pp 1289ndash1306 2006

[2] E J Candes J Romberg and T Tao ldquoRobust uncertaintyprinciples exact signal reconstruction from highly incompletefrequency informationrdquo Institute of Electrical and ElectronicsEngineers Transactions on InformationTheory vol 52 no 2 pp489ndash509 2006

[3] T N Canh K D Quoc and B Jeon ldquoMulti-resolution kro-necker compressive sensingrdquo Transactions on Smart Processingamp Computing vol 3 no 1 pp 19ndash27 2014

[4] K Q Dinh H J Shim and B Jeon ldquoMeasurement coding forcompressive imaging using a structural measuremnet matrixrdquoin 2013 20th IEEE International Conference on Image ProcessingICIP 2013 pp 10ndash13 aus September 2013

[5] M F Duarte and R G Baraniuk ldquoKronecker compressive sens-ingrdquo IEEE Transactions on Image Processing vol 21 no 2 pp494ndash504 2012

[6] B Zhang X Tong W Wang and J Xie ldquoThe research ofKronecker product-based measurement matrix of compressivesensingrdquo EURASIP Journal on Wireless Communications andNetworking vol 2013 article 161 pp 1ndash5 2013

[7] M Elad ldquoOptimized projections for compressed sensingrdquo IEEETransactions on Signal Processing vol 55 no 12 pp 5695ndash57022007

Journal of Control Science and Engineering 13

[8] M Lustig D Donoho and J M Pauly ldquoSparse MRI the appli-cation of compressed sensing for rapid MR imagingrdquoMagneticResonance in Medicine vol 58 no 6 pp 1182ndash1195 2007

[9] V Abolghasemi S Sanei S Ferdowsi F Ghaderi and ABelcher ldquoSegmented compressive sensingrdquo in Proceedings of theIEEESP 15thWorkshop on Statistical Signal Processing (SSP rsquo09)pp 630ndash633 September 2009

[10] J M Duarte-Carvajalino and G Sapiro ldquoLearning to sensesparse signals simultaneous sensingmatrix and sparsifying dic-tionary optimizationrdquo IEEE Transactions on Image Processingvol 18 no 7 pp 1395ndash1408 2009

[11] ZWang G R Arce and J L Paredes ldquoColored randomprojec-tions for compressed sensingrdquo in 2007 IEEE International Con-ference on Acoustics Speech and Signal Processing ICASSP rsquo07pp III873ndashIII876 usa April 2007

[12] Z Wang and G R Arce ldquoVariable density compressed imagesamplingrdquo IEEE Transactions on Image Processing vol 19 no 1pp 264ndash270 2010

[13] M Elad and M Aharon ldquoImage denoising via learned dic-tionaries and sparse representationrdquo in 2006 IEEE ComputerSociety Conference on Computer Vision and Pattern RecognitionCVPR 2006 pp 895ndash900 usa June 2006

[14] M Aharon M Elad and A Bruckstein ldquoK-SVD an algorithmfor designing overcomplete dictionaries for sparse representa-tionrdquo IEEE Transactions on Signal Processing vol 54 no 11 pp4311ndash4322 2006

[15] J Xu Y Pi and Z Cao ldquoOptimized projection matrix for com-pressive sensingrdquo EURASIP Journal on Advances in SignalProcessing vol 2010 Article ID 560349 2010

[16] V Abolghasemi S Ferdowsi and S Sanei ldquoA gradient-basedalternating minimization approach for optimization of themeasurementmatrix in compressive sensingrdquo Signal Processingvol 92 no 4 pp 999ndash1009 2012

[17] Q Zhang Y Fu H Li and R Rong ldquoOptimized projectionmatrix for compressed sensingrdquo Circuits Systems and SignalProcessing vol 33 no 5 pp 1627ndash1636 2014

[18] G Li Z Zhu D Yang L Chang and H Bai ldquoOn projectionmatrix optimization for compressive sensing systemsrdquo IEEETransactions on Signal Processing vol 61 no 11 pp 2887ndash28982013

[19] S Tian X Fan and L I Zhetao ldquoOrthogonal-gradient mea-surement matrix construction algorithmrdquo Chinese Journal ofElectronics vol 25 no 1 pp 81ndash87 2016

[20] V Abolghasemi S Ferdowsi and BMakkiabadi ldquoOn optimiza-tion of the measurement matrix for compressive sensingrdquo inSignal Processing Conference European IEEE Ed pp 427ndash431August 2010

[21] Q Li D Schonfeld and S Friedland ldquoGeneralized tensorcompressive sensingrdquo in 2013 IEEE International Conference onMultimedia and Expo ICME 2013 usa July 2013

[22] S Friedland Q Li and D Schonfeld ldquoCompressive sensing ofsparse tensorsrdquo IEEE Transactions on Image Processing vol 23no 10 pp 4438ndash4447 2014

[23] S u Xu H Yin C Yi Y Xiong and T Xue ldquoAn ImprovedToeplitz Measurement Matrix for Compressive Sensingrdquo Inter-national Journal of Distributed Sensor Networks vol 8 pp 1ndash82014

[24] R L Devaney in Practical Numerical Algorithms for ChaoticSystems T S Parker and L O Chua Eds vol 32 pp 501ndash503Siam Review 3 edition 2006

[25] R Baraniuk M Davenport R DeVore and M Wakin ldquoAsimple proof of the restricted isometry property for randommatricesrdquoConstructiveApproximation An International Journalfor Approximations and Expansions vol 28 no 3 pp 253ndash2632008

[26] Available httpwwwesciencecnpeoplefeiyunZHUDatasetGThtml

[27] D L Donoho Y Tsaig I Drori and J-L Starck ldquoSparse solu-tion of underdetermined systems of linear equations by stage-wise orthogonal matching pursuitrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol58 no 2 pp 1094ndash1121 2012

[28] K Lang ldquoNewsWeeder Learning to Filter Netnewsrdquo in Inter-national Machine Learning Conference vol 1995 pp 331ndash339

[29] Available httpfeatureselectionasuedudatasetsphp[30] Available httpclopinetcomisabelleProjectsNIPS

8 Journal of Control Science and Engineering

Table2Objectiv

eevaluationof

typicalm

etho

dsandprop

osed

metho

dfro

mrecovery

Urban

Band

sAlgorith

mTh

eobjectiv

eevaluationindices

AGEI

FDGM

SDMSE

PSNR

SFSSI

VAR

CT

11

Circulate

0070476

048360

012231

04106

0119

700929

584508

02034

098826

0119

7217

379

Toeplitz

0047687

034172

010125

032796

01137

00610

602804

018949

099361

011366

18706

TSCM

M0054635

03864

70114

14035344

01162

006

865976

58019866

099254

011624

15146

EOM

00566

08040

032

011519

035809

0117

200708

596279

020025

099226

0117

203051965

Duarte-ET

F0047687

034172

010125

032796

01137

00610

602804

018949

099361

011366

1366804

Prop

osed

004

6138

029

123

006

690

023194

01197

004

3461758

601195

6099

537

01198

0165975

41

Circulate

0072566

051888

012438

037371

01363

00631

6013

16020690

099349

013626

16382

Toeplitz

0058448

041868

012772

040

138

01390

00786

591767

022932

099207

013897

18069

TSCM

M0061306

043672

013128

039347

01362

00744

594172

022736

099259

013620

1538

1EO

M006

0340

04340

8012941

040222

01353

00771

59260

9022536

099215

013529

813160

Duarte-ET

F0062241

044

182

013123

042482

01402

00873

587190

023610

099089

014018

5974

19Prop

osed

005

0626

040

358

007

196

024

426

01307

004

52615816

012782

099

561

01306

8243532

81

Circulate

0082530

06279

013639

049357

01449

00635

6010

19022630

099397

014492

15829

Toeplitz

0077269

047993

014082

044

812

01390

00535

60844

2023904

099484

013905

18284

TSCM

M0070612

050737

014828

051106

01496

00716

595790

025291

099353

014958

15702

EOM

0070821

05100

6014940

050827

01503

00707

596361

02544

7099369

015030

9731

53Duarte-ET

F0074147

052535

014946

048963

01499

00652

599890

025959

099450

014986

590915

Prop

osed

006

8955

042

678

009

339

040

915

01571

004

84612814

01648

5099

580

01570

82874

26

Journal of Control Science and Engineering 9

(1) 11 bands (64 times 64 pixels)

(3) 81 bands (64 times 64 pixels)

(2) 41 bands (64 times 64 pixels)

(a1) original (b1) Circulant (c1) Toeplitz (d1) TSCMM (e1) EOM (f1) Duarte-ETF (g1) Proposed

(a2) original (b2) Circulant (c2) Toeplitz (d2) TSCMM (e2) EOM (f2) Duarte-ETF (g2) Proposed

(a3) original (b3) Circulant (c3) Toeplitz (d3) TSCMM (e3) EOM (f3) Duarte-ETF (g3) Proposed

Figure 6The subjective quality of different CSmeasurementmatrices from left to right (a) original (b) Circulant (c) Toeplitz (d) TSCMM(e) EOM (f) Duarte-ETF and (g) proposed method

The results of Table 2 show that the method has a higherperformance than traditional Toeplitz or Circulant matrixmethod Although improved Toeplitz-structured matrixmethod (TSCMM) is slightly better than classical Toeplitzboth classical optimization measurement matrix method(EOM) and proposed method have significant performancein image quality Furthermore the proposed method takesadvantage of improved Toeplitz-structured matrix to speedup the convergence speed and improve traditional opti-mization method to recover better high-dimensional imageExperimental results show that the proposed method has abetter overall performance

43 Jasper Ridge The hyperspectral image of Jasper Ridgewas obtained on June 2 September 4 and October 6 1992whichwas calibrated to surface reflectanceThe imagewas themost popular source to analyze with spectral mixture anal-ysis using library endmembers representing green foliagenonphotosynthetic vegetation and soils characteristic of thesite Field-based vegetation was obtained fromUSGeologicalService

From Figure 7 it is obvious that the worst effects fromtraditional Circulant have reached being almost intolerableWhile the results from TSCMM and Toeplitz are almostsimilar the former has only slight improvement comparedto the latter On the other hand the performance of theproposed approach was significantly improved compared tothat of EOM or Duarte-ETF The study concluded that theproposed approach had a significant performance comparedto that of others Furthermore the objective evaluations are

shown inTable 3The results can be clearly seen fromdifferentmethods on recovery Jasper Ridge

The results of Table 3 show that the proposed methodhas a higher performance than traditional Toeplitz or Circu-lant matrix method TSCMM takes advantage of improvedToeplitz-structuredmatrix to speedup the convergence speedand improve traditional Toeplitz or Circulant matrix methodto recover better high-dimensional data Although EOMhas the lower column coherence and faster convergence itweakens RIP condition and causes recovery performancedegradation While the absolute values from Duarte-ETFconcentrate around mutual coherence this can make theequivalent dictionary as close as possible to an ETF But thisalgorithm has high computational complexity Furthermoreexperimental results from the proposed method show thatthe proposed method has a better overall performance

44 Hyperspectral Unmixing The fourth experiment per-forms an experimental evaluation of the accuracy of thestandard hyperspectral unmixing districts known as Cuprite[28] To reduce complexity there are only 188 channels(3ndash103 114ndash147 and 168ndash220 bands) that were remained dueto dense water vapor and atmospheric effects In the courseof the experiment the signal sparsity method is Fourier basisand reconstruction algorithm is StOMP

The results are shown in Figure 8 for hyperspectralunmixing and closely resemble those obtained from hyper-spectral data Figure 8 compares the unmixing performanceof the proposed method with different endmembers and theabundance from different endmembers is totally dissimilar

10 Journal of Control Science and Engineering

Table 3 The objective evaluations of different matrixes of recovery Jasper Ridge

Algorithm 11 bands (64 times 64) 81 bands (64 times 64) 151 bands (64 times 64)SNR PSNR MSE SNR PSNR MSE SNR PSNR MSE CT

Circulate minus60698 573412 01199 116699 600402 00644 33299 603823 00595 15304Toeplitz 247566 707289 00055 294215 677497 00109 157386 657714 00172 15519TSCMM 153473 666425 00141 339793 697291 00069 200693 676521 00112 14869EOM 44853 619252 00417 350894 702112 00062 216750 683495 00095 788838Duarte-ETF 247566 707289 00055 333205 694430 00074 172452 664257 00148 665711Proposed 210504 691193 00080 549719 788460 848e minus 04 422277 772754 00012 345743

(1) 11 bands (64 times 64 pixels)

(2) 81 bands (64 times 64 pixels)

(3) 151 bands (64 times 64 pixels)

(a2) original (b2) Circulant (c2) Toeplitz (d2) TSCMM (e2) EOM (f2) Duarte-ETF (g2) Proposed

(a1) original (b1) Circulant (c1) Toeplitz (d1) TSCMM (e1) EOM (f1) DuarteminusETF (g1) Proposed

(a3) original (b3) Circulant (c3) Toeplitz (d3) TSCMM (e3) EOM (f3) Duarte-ETF (g3) Proposed

Figure 7The subjective quality of different CSmeasurementmatrices from left to right (a) original (b) Circulant (c) Toeplitz (d) TSCMM(e) EOM (f) Duarte-ETF and (g) proposed method

These features from the proposed method ensure clear andaccurate the abundances

By accessing information from USGS Digital Spec-tral Library [29] the unmixing performance has beenalmost correct Furthermore different endmembers fromUSGS 1995 Library [30] are used to verify that the predictionmodel of hyperspectral unmixing scheme is accurate inFigure 8

Apparently the proposed unmixing results (blue thinthread) have a strong correlationwith these endmembers (redthin thread) from USGS 1995 Library From Figure 9(a) asteep sloping line from Alunite suggests that the unmixingendmember (blue thin thread) has remarkable similarityThis same conclusion has been made in studies from Figures9(d)ndash9(h) On the other hand the unsatisfied results fromFigures 9(b) and 9(c) have been caused by smooth curveThe proposed method has good accuracy and is robust totraditional filtering compression cutting and noise attack

5 Conclusions

In this paper to overcome the limitation of Toeplitz-structured chaotic measurement matrix an improved mea-surement matrix has been carried out in the hyperspectralunmixing process to achieve multiple endmembers of hyper-spectral image And in theory it proves that this matrixhas retained the RIP property with overwhelming proba-bility Experimental results demonstrate that the proposedmethod to design of measurement matrix leads to better CSreconstruction performance with low extra computationalcost Compared with some traditional measurement matrixan improved method has highest technical feasibility lowestcomputational complexity and least computation time con-sumption in the same recovery qualityThe proposedmethodcan take the special advantage in hyperspectral unmixingprocess and explore the practical satellite system to remotesensing

Journal of Control Science and Engineering 11

(a) Alunite (b) Andradite (c) Pyrope

(d) Nontronite (e) Dumortierite (f) Kaolinite

(g) Chalcedony (h) Kaolinite (i) Buddingtonite

Figure 8 The results from different elements abundance in hyperspectral unmixing schemes

12 Journal of Control Science and Engineering

02

04

06

08

1Re

flect

ance

()

500 1000 1500 2000 2500 30000Wavelength (120583m)

(a) Alunite

0

01

02

03

04

Refle

ctan

ce (

)

500 1000 1500 2000 2500 30000Wavelength (120583m)

(b) Andradite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

1

Refle

ctan

ce (

)

(c) Pyrope

0

02

04

06

08

Refle

ctan

ce (

)

500 1000 1500 2000 2500 30000Wavelength (120583m)

(d) Nontronite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

1

Refle

ctan

ce (

)

(e) Dumortierite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

Refle

ctan

ce (

)

(f) Kaolinite

0

02

04

06

08

Refle

ctan

ce (

)

500 1000 1500 2000 2500 30000Wavelength (120583m)

(g) Chalcedony

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

Refle

ctan

ce (

)

(h) Kaolinite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

Refle

ctan

ce (

)

(i) Buddingtonite

Figure 9 The comparison map between actual unmixing effect and endmember library

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is supported by Chongqing Engineering Labo-ratory for Detection Control and Integrated System Theproject is also funded by Key Technology Research andIndustrialization of Fire Monitoring and Early Warn-ing Sensor Network for High Voltage Transmission Line(KJZH17124) This research is funded by Chongqing Edu-cation Commission Foundation (KJ1400612) This projectis also granted financial support from a CooperativeProject of Chongqing Technology and Business University(990516001)

References

[1] D L Donoho ldquoCompressed sensingrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol52 no 4 pp 1289ndash1306 2006

[2] E J Candes J Romberg and T Tao ldquoRobust uncertaintyprinciples exact signal reconstruction from highly incompletefrequency informationrdquo Institute of Electrical and ElectronicsEngineers Transactions on InformationTheory vol 52 no 2 pp489ndash509 2006

[3] T N Canh K D Quoc and B Jeon ldquoMulti-resolution kro-necker compressive sensingrdquo Transactions on Smart Processingamp Computing vol 3 no 1 pp 19ndash27 2014

[4] K Q Dinh H J Shim and B Jeon ldquoMeasurement coding forcompressive imaging using a structural measuremnet matrixrdquoin 2013 20th IEEE International Conference on Image ProcessingICIP 2013 pp 10ndash13 aus September 2013

[5] M F Duarte and R G Baraniuk ldquoKronecker compressive sens-ingrdquo IEEE Transactions on Image Processing vol 21 no 2 pp494ndash504 2012

[6] B Zhang X Tong W Wang and J Xie ldquoThe research ofKronecker product-based measurement matrix of compressivesensingrdquo EURASIP Journal on Wireless Communications andNetworking vol 2013 article 161 pp 1ndash5 2013

[7] M Elad ldquoOptimized projections for compressed sensingrdquo IEEETransactions on Signal Processing vol 55 no 12 pp 5695ndash57022007

Journal of Control Science and Engineering 13

[8] M Lustig D Donoho and J M Pauly ldquoSparse MRI the appli-cation of compressed sensing for rapid MR imagingrdquoMagneticResonance in Medicine vol 58 no 6 pp 1182ndash1195 2007

[9] V Abolghasemi S Sanei S Ferdowsi F Ghaderi and ABelcher ldquoSegmented compressive sensingrdquo in Proceedings of theIEEESP 15thWorkshop on Statistical Signal Processing (SSP rsquo09)pp 630ndash633 September 2009

[10] J M Duarte-Carvajalino and G Sapiro ldquoLearning to sensesparse signals simultaneous sensingmatrix and sparsifying dic-tionary optimizationrdquo IEEE Transactions on Image Processingvol 18 no 7 pp 1395ndash1408 2009

[11] ZWang G R Arce and J L Paredes ldquoColored randomprojec-tions for compressed sensingrdquo in 2007 IEEE International Con-ference on Acoustics Speech and Signal Processing ICASSP rsquo07pp III873ndashIII876 usa April 2007

[12] Z Wang and G R Arce ldquoVariable density compressed imagesamplingrdquo IEEE Transactions on Image Processing vol 19 no 1pp 264ndash270 2010

[13] M Elad and M Aharon ldquoImage denoising via learned dic-tionaries and sparse representationrdquo in 2006 IEEE ComputerSociety Conference on Computer Vision and Pattern RecognitionCVPR 2006 pp 895ndash900 usa June 2006

[14] M Aharon M Elad and A Bruckstein ldquoK-SVD an algorithmfor designing overcomplete dictionaries for sparse representa-tionrdquo IEEE Transactions on Signal Processing vol 54 no 11 pp4311ndash4322 2006

[15] J Xu Y Pi and Z Cao ldquoOptimized projection matrix for com-pressive sensingrdquo EURASIP Journal on Advances in SignalProcessing vol 2010 Article ID 560349 2010

[16] V Abolghasemi S Ferdowsi and S Sanei ldquoA gradient-basedalternating minimization approach for optimization of themeasurementmatrix in compressive sensingrdquo Signal Processingvol 92 no 4 pp 999ndash1009 2012

[17] Q Zhang Y Fu H Li and R Rong ldquoOptimized projectionmatrix for compressed sensingrdquo Circuits Systems and SignalProcessing vol 33 no 5 pp 1627ndash1636 2014

[18] G Li Z Zhu D Yang L Chang and H Bai ldquoOn projectionmatrix optimization for compressive sensing systemsrdquo IEEETransactions on Signal Processing vol 61 no 11 pp 2887ndash28982013

[19] S Tian X Fan and L I Zhetao ldquoOrthogonal-gradient mea-surement matrix construction algorithmrdquo Chinese Journal ofElectronics vol 25 no 1 pp 81ndash87 2016

[20] V Abolghasemi S Ferdowsi and BMakkiabadi ldquoOn optimiza-tion of the measurement matrix for compressive sensingrdquo inSignal Processing Conference European IEEE Ed pp 427ndash431August 2010

[21] Q Li D Schonfeld and S Friedland ldquoGeneralized tensorcompressive sensingrdquo in 2013 IEEE International Conference onMultimedia and Expo ICME 2013 usa July 2013

[22] S Friedland Q Li and D Schonfeld ldquoCompressive sensing ofsparse tensorsrdquo IEEE Transactions on Image Processing vol 23no 10 pp 4438ndash4447 2014

[23] S u Xu H Yin C Yi Y Xiong and T Xue ldquoAn ImprovedToeplitz Measurement Matrix for Compressive Sensingrdquo Inter-national Journal of Distributed Sensor Networks vol 8 pp 1ndash82014

[24] R L Devaney in Practical Numerical Algorithms for ChaoticSystems T S Parker and L O Chua Eds vol 32 pp 501ndash503Siam Review 3 edition 2006

[25] R Baraniuk M Davenport R DeVore and M Wakin ldquoAsimple proof of the restricted isometry property for randommatricesrdquoConstructiveApproximation An International Journalfor Approximations and Expansions vol 28 no 3 pp 253ndash2632008

[26] Available httpwwwesciencecnpeoplefeiyunZHUDatasetGThtml

[27] D L Donoho Y Tsaig I Drori and J-L Starck ldquoSparse solu-tion of underdetermined systems of linear equations by stage-wise orthogonal matching pursuitrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol58 no 2 pp 1094ndash1121 2012

[28] K Lang ldquoNewsWeeder Learning to Filter Netnewsrdquo in Inter-national Machine Learning Conference vol 1995 pp 331ndash339

[29] Available httpfeatureselectionasuedudatasetsphp[30] Available httpclopinetcomisabelleProjectsNIPS

Journal of Control Science and Engineering 9

(1) 11 bands (64 times 64 pixels)

(3) 81 bands (64 times 64 pixels)

(2) 41 bands (64 times 64 pixels)

(a1) original (b1) Circulant (c1) Toeplitz (d1) TSCMM (e1) EOM (f1) Duarte-ETF (g1) Proposed

(a2) original (b2) Circulant (c2) Toeplitz (d2) TSCMM (e2) EOM (f2) Duarte-ETF (g2) Proposed

(a3) original (b3) Circulant (c3) Toeplitz (d3) TSCMM (e3) EOM (f3) Duarte-ETF (g3) Proposed

Figure 6The subjective quality of different CSmeasurementmatrices from left to right (a) original (b) Circulant (c) Toeplitz (d) TSCMM(e) EOM (f) Duarte-ETF and (g) proposed method

The results of Table 2 show that the method has a higherperformance than traditional Toeplitz or Circulant matrixmethod Although improved Toeplitz-structured matrixmethod (TSCMM) is slightly better than classical Toeplitzboth classical optimization measurement matrix method(EOM) and proposed method have significant performancein image quality Furthermore the proposed method takesadvantage of improved Toeplitz-structured matrix to speedup the convergence speed and improve traditional opti-mization method to recover better high-dimensional imageExperimental results show that the proposed method has abetter overall performance

43 Jasper Ridge The hyperspectral image of Jasper Ridgewas obtained on June 2 September 4 and October 6 1992whichwas calibrated to surface reflectanceThe imagewas themost popular source to analyze with spectral mixture anal-ysis using library endmembers representing green foliagenonphotosynthetic vegetation and soils characteristic of thesite Field-based vegetation was obtained fromUSGeologicalService

From Figure 7 it is obvious that the worst effects fromtraditional Circulant have reached being almost intolerableWhile the results from TSCMM and Toeplitz are almostsimilar the former has only slight improvement comparedto the latter On the other hand the performance of theproposed approach was significantly improved compared tothat of EOM or Duarte-ETF The study concluded that theproposed approach had a significant performance comparedto that of others Furthermore the objective evaluations are

shown inTable 3The results can be clearly seen fromdifferentmethods on recovery Jasper Ridge

The results of Table 3 show that the proposed methodhas a higher performance than traditional Toeplitz or Circu-lant matrix method TSCMM takes advantage of improvedToeplitz-structuredmatrix to speedup the convergence speedand improve traditional Toeplitz or Circulant matrix methodto recover better high-dimensional data Although EOMhas the lower column coherence and faster convergence itweakens RIP condition and causes recovery performancedegradation While the absolute values from Duarte-ETFconcentrate around mutual coherence this can make theequivalent dictionary as close as possible to an ETF But thisalgorithm has high computational complexity Furthermoreexperimental results from the proposed method show thatthe proposed method has a better overall performance

44 Hyperspectral Unmixing The fourth experiment per-forms an experimental evaluation of the accuracy of thestandard hyperspectral unmixing districts known as Cuprite[28] To reduce complexity there are only 188 channels(3ndash103 114ndash147 and 168ndash220 bands) that were remained dueto dense water vapor and atmospheric effects In the courseof the experiment the signal sparsity method is Fourier basisand reconstruction algorithm is StOMP

The results are shown in Figure 8 for hyperspectralunmixing and closely resemble those obtained from hyper-spectral data Figure 8 compares the unmixing performanceof the proposed method with different endmembers and theabundance from different endmembers is totally dissimilar

10 Journal of Control Science and Engineering

Table 3 The objective evaluations of different matrixes of recovery Jasper Ridge

Algorithm 11 bands (64 times 64) 81 bands (64 times 64) 151 bands (64 times 64)SNR PSNR MSE SNR PSNR MSE SNR PSNR MSE CT

Circulate minus60698 573412 01199 116699 600402 00644 33299 603823 00595 15304Toeplitz 247566 707289 00055 294215 677497 00109 157386 657714 00172 15519TSCMM 153473 666425 00141 339793 697291 00069 200693 676521 00112 14869EOM 44853 619252 00417 350894 702112 00062 216750 683495 00095 788838Duarte-ETF 247566 707289 00055 333205 694430 00074 172452 664257 00148 665711Proposed 210504 691193 00080 549719 788460 848e minus 04 422277 772754 00012 345743

(1) 11 bands (64 times 64 pixels)

(2) 81 bands (64 times 64 pixels)

(3) 151 bands (64 times 64 pixels)

(a2) original (b2) Circulant (c2) Toeplitz (d2) TSCMM (e2) EOM (f2) Duarte-ETF (g2) Proposed

(a1) original (b1) Circulant (c1) Toeplitz (d1) TSCMM (e1) EOM (f1) DuarteminusETF (g1) Proposed

(a3) original (b3) Circulant (c3) Toeplitz (d3) TSCMM (e3) EOM (f3) Duarte-ETF (g3) Proposed

Figure 7The subjective quality of different CSmeasurementmatrices from left to right (a) original (b) Circulant (c) Toeplitz (d) TSCMM(e) EOM (f) Duarte-ETF and (g) proposed method

These features from the proposed method ensure clear andaccurate the abundances

By accessing information from USGS Digital Spec-tral Library [29] the unmixing performance has beenalmost correct Furthermore different endmembers fromUSGS 1995 Library [30] are used to verify that the predictionmodel of hyperspectral unmixing scheme is accurate inFigure 8

Apparently the proposed unmixing results (blue thinthread) have a strong correlationwith these endmembers (redthin thread) from USGS 1995 Library From Figure 9(a) asteep sloping line from Alunite suggests that the unmixingendmember (blue thin thread) has remarkable similarityThis same conclusion has been made in studies from Figures9(d)ndash9(h) On the other hand the unsatisfied results fromFigures 9(b) and 9(c) have been caused by smooth curveThe proposed method has good accuracy and is robust totraditional filtering compression cutting and noise attack

5 Conclusions

In this paper to overcome the limitation of Toeplitz-structured chaotic measurement matrix an improved mea-surement matrix has been carried out in the hyperspectralunmixing process to achieve multiple endmembers of hyper-spectral image And in theory it proves that this matrixhas retained the RIP property with overwhelming proba-bility Experimental results demonstrate that the proposedmethod to design of measurement matrix leads to better CSreconstruction performance with low extra computationalcost Compared with some traditional measurement matrixan improved method has highest technical feasibility lowestcomputational complexity and least computation time con-sumption in the same recovery qualityThe proposedmethodcan take the special advantage in hyperspectral unmixingprocess and explore the practical satellite system to remotesensing

Journal of Control Science and Engineering 11

(a) Alunite (b) Andradite (c) Pyrope

(d) Nontronite (e) Dumortierite (f) Kaolinite

(g) Chalcedony (h) Kaolinite (i) Buddingtonite

Figure 8 The results from different elements abundance in hyperspectral unmixing schemes

12 Journal of Control Science and Engineering

02

04

06

08

1Re

flect

ance

()

500 1000 1500 2000 2500 30000Wavelength (120583m)

(a) Alunite

0

01

02

03

04

Refle

ctan

ce (

)

500 1000 1500 2000 2500 30000Wavelength (120583m)

(b) Andradite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

1

Refle

ctan

ce (

)

(c) Pyrope

0

02

04

06

08

Refle

ctan

ce (

)

500 1000 1500 2000 2500 30000Wavelength (120583m)

(d) Nontronite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

1

Refle

ctan

ce (

)

(e) Dumortierite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

Refle

ctan

ce (

)

(f) Kaolinite

0

02

04

06

08

Refle

ctan

ce (

)

500 1000 1500 2000 2500 30000Wavelength (120583m)

(g) Chalcedony

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

Refle

ctan

ce (

)

(h) Kaolinite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

Refle

ctan

ce (

)

(i) Buddingtonite

Figure 9 The comparison map between actual unmixing effect and endmember library

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is supported by Chongqing Engineering Labo-ratory for Detection Control and Integrated System Theproject is also funded by Key Technology Research andIndustrialization of Fire Monitoring and Early Warn-ing Sensor Network for High Voltage Transmission Line(KJZH17124) This research is funded by Chongqing Edu-cation Commission Foundation (KJ1400612) This projectis also granted financial support from a CooperativeProject of Chongqing Technology and Business University(990516001)

References

[1] D L Donoho ldquoCompressed sensingrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol52 no 4 pp 1289ndash1306 2006

[2] E J Candes J Romberg and T Tao ldquoRobust uncertaintyprinciples exact signal reconstruction from highly incompletefrequency informationrdquo Institute of Electrical and ElectronicsEngineers Transactions on InformationTheory vol 52 no 2 pp489ndash509 2006

[3] T N Canh K D Quoc and B Jeon ldquoMulti-resolution kro-necker compressive sensingrdquo Transactions on Smart Processingamp Computing vol 3 no 1 pp 19ndash27 2014

[4] K Q Dinh H J Shim and B Jeon ldquoMeasurement coding forcompressive imaging using a structural measuremnet matrixrdquoin 2013 20th IEEE International Conference on Image ProcessingICIP 2013 pp 10ndash13 aus September 2013

[5] M F Duarte and R G Baraniuk ldquoKronecker compressive sens-ingrdquo IEEE Transactions on Image Processing vol 21 no 2 pp494ndash504 2012

[6] B Zhang X Tong W Wang and J Xie ldquoThe research ofKronecker product-based measurement matrix of compressivesensingrdquo EURASIP Journal on Wireless Communications andNetworking vol 2013 article 161 pp 1ndash5 2013

[7] M Elad ldquoOptimized projections for compressed sensingrdquo IEEETransactions on Signal Processing vol 55 no 12 pp 5695ndash57022007

Journal of Control Science and Engineering 13

[8] M Lustig D Donoho and J M Pauly ldquoSparse MRI the appli-cation of compressed sensing for rapid MR imagingrdquoMagneticResonance in Medicine vol 58 no 6 pp 1182ndash1195 2007

[9] V Abolghasemi S Sanei S Ferdowsi F Ghaderi and ABelcher ldquoSegmented compressive sensingrdquo in Proceedings of theIEEESP 15thWorkshop on Statistical Signal Processing (SSP rsquo09)pp 630ndash633 September 2009

[10] J M Duarte-Carvajalino and G Sapiro ldquoLearning to sensesparse signals simultaneous sensingmatrix and sparsifying dic-tionary optimizationrdquo IEEE Transactions on Image Processingvol 18 no 7 pp 1395ndash1408 2009

[11] ZWang G R Arce and J L Paredes ldquoColored randomprojec-tions for compressed sensingrdquo in 2007 IEEE International Con-ference on Acoustics Speech and Signal Processing ICASSP rsquo07pp III873ndashIII876 usa April 2007

[12] Z Wang and G R Arce ldquoVariable density compressed imagesamplingrdquo IEEE Transactions on Image Processing vol 19 no 1pp 264ndash270 2010

[13] M Elad and M Aharon ldquoImage denoising via learned dic-tionaries and sparse representationrdquo in 2006 IEEE ComputerSociety Conference on Computer Vision and Pattern RecognitionCVPR 2006 pp 895ndash900 usa June 2006

[14] M Aharon M Elad and A Bruckstein ldquoK-SVD an algorithmfor designing overcomplete dictionaries for sparse representa-tionrdquo IEEE Transactions on Signal Processing vol 54 no 11 pp4311ndash4322 2006

[15] J Xu Y Pi and Z Cao ldquoOptimized projection matrix for com-pressive sensingrdquo EURASIP Journal on Advances in SignalProcessing vol 2010 Article ID 560349 2010

[16] V Abolghasemi S Ferdowsi and S Sanei ldquoA gradient-basedalternating minimization approach for optimization of themeasurementmatrix in compressive sensingrdquo Signal Processingvol 92 no 4 pp 999ndash1009 2012

[17] Q Zhang Y Fu H Li and R Rong ldquoOptimized projectionmatrix for compressed sensingrdquo Circuits Systems and SignalProcessing vol 33 no 5 pp 1627ndash1636 2014

[18] G Li Z Zhu D Yang L Chang and H Bai ldquoOn projectionmatrix optimization for compressive sensing systemsrdquo IEEETransactions on Signal Processing vol 61 no 11 pp 2887ndash28982013

[19] S Tian X Fan and L I Zhetao ldquoOrthogonal-gradient mea-surement matrix construction algorithmrdquo Chinese Journal ofElectronics vol 25 no 1 pp 81ndash87 2016

[20] V Abolghasemi S Ferdowsi and BMakkiabadi ldquoOn optimiza-tion of the measurement matrix for compressive sensingrdquo inSignal Processing Conference European IEEE Ed pp 427ndash431August 2010

[21] Q Li D Schonfeld and S Friedland ldquoGeneralized tensorcompressive sensingrdquo in 2013 IEEE International Conference onMultimedia and Expo ICME 2013 usa July 2013

[22] S Friedland Q Li and D Schonfeld ldquoCompressive sensing ofsparse tensorsrdquo IEEE Transactions on Image Processing vol 23no 10 pp 4438ndash4447 2014

[23] S u Xu H Yin C Yi Y Xiong and T Xue ldquoAn ImprovedToeplitz Measurement Matrix for Compressive Sensingrdquo Inter-national Journal of Distributed Sensor Networks vol 8 pp 1ndash82014

[24] R L Devaney in Practical Numerical Algorithms for ChaoticSystems T S Parker and L O Chua Eds vol 32 pp 501ndash503Siam Review 3 edition 2006

[25] R Baraniuk M Davenport R DeVore and M Wakin ldquoAsimple proof of the restricted isometry property for randommatricesrdquoConstructiveApproximation An International Journalfor Approximations and Expansions vol 28 no 3 pp 253ndash2632008

[26] Available httpwwwesciencecnpeoplefeiyunZHUDatasetGThtml

[27] D L Donoho Y Tsaig I Drori and J-L Starck ldquoSparse solu-tion of underdetermined systems of linear equations by stage-wise orthogonal matching pursuitrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol58 no 2 pp 1094ndash1121 2012

[28] K Lang ldquoNewsWeeder Learning to Filter Netnewsrdquo in Inter-national Machine Learning Conference vol 1995 pp 331ndash339

[29] Available httpfeatureselectionasuedudatasetsphp[30] Available httpclopinetcomisabelleProjectsNIPS

10 Journal of Control Science and Engineering

Table 3 The objective evaluations of different matrixes of recovery Jasper Ridge

Algorithm 11 bands (64 times 64) 81 bands (64 times 64) 151 bands (64 times 64)SNR PSNR MSE SNR PSNR MSE SNR PSNR MSE CT

Circulate minus60698 573412 01199 116699 600402 00644 33299 603823 00595 15304Toeplitz 247566 707289 00055 294215 677497 00109 157386 657714 00172 15519TSCMM 153473 666425 00141 339793 697291 00069 200693 676521 00112 14869EOM 44853 619252 00417 350894 702112 00062 216750 683495 00095 788838Duarte-ETF 247566 707289 00055 333205 694430 00074 172452 664257 00148 665711Proposed 210504 691193 00080 549719 788460 848e minus 04 422277 772754 00012 345743

(1) 11 bands (64 times 64 pixels)

(2) 81 bands (64 times 64 pixels)

(3) 151 bands (64 times 64 pixels)

(a2) original (b2) Circulant (c2) Toeplitz (d2) TSCMM (e2) EOM (f2) Duarte-ETF (g2) Proposed

(a1) original (b1) Circulant (c1) Toeplitz (d1) TSCMM (e1) EOM (f1) DuarteminusETF (g1) Proposed

(a3) original (b3) Circulant (c3) Toeplitz (d3) TSCMM (e3) EOM (f3) Duarte-ETF (g3) Proposed

Figure 7The subjective quality of different CSmeasurementmatrices from left to right (a) original (b) Circulant (c) Toeplitz (d) TSCMM(e) EOM (f) Duarte-ETF and (g) proposed method

These features from the proposed method ensure clear andaccurate the abundances

By accessing information from USGS Digital Spec-tral Library [29] the unmixing performance has beenalmost correct Furthermore different endmembers fromUSGS 1995 Library [30] are used to verify that the predictionmodel of hyperspectral unmixing scheme is accurate inFigure 8

Apparently the proposed unmixing results (blue thinthread) have a strong correlationwith these endmembers (redthin thread) from USGS 1995 Library From Figure 9(a) asteep sloping line from Alunite suggests that the unmixingendmember (blue thin thread) has remarkable similarityThis same conclusion has been made in studies from Figures9(d)ndash9(h) On the other hand the unsatisfied results fromFigures 9(b) and 9(c) have been caused by smooth curveThe proposed method has good accuracy and is robust totraditional filtering compression cutting and noise attack

5 Conclusions

In this paper to overcome the limitation of Toeplitz-structured chaotic measurement matrix an improved mea-surement matrix has been carried out in the hyperspectralunmixing process to achieve multiple endmembers of hyper-spectral image And in theory it proves that this matrixhas retained the RIP property with overwhelming proba-bility Experimental results demonstrate that the proposedmethod to design of measurement matrix leads to better CSreconstruction performance with low extra computationalcost Compared with some traditional measurement matrixan improved method has highest technical feasibility lowestcomputational complexity and least computation time con-sumption in the same recovery qualityThe proposedmethodcan take the special advantage in hyperspectral unmixingprocess and explore the practical satellite system to remotesensing

Journal of Control Science and Engineering 11

(a) Alunite (b) Andradite (c) Pyrope

(d) Nontronite (e) Dumortierite (f) Kaolinite

(g) Chalcedony (h) Kaolinite (i) Buddingtonite

Figure 8 The results from different elements abundance in hyperspectral unmixing schemes

12 Journal of Control Science and Engineering

02

04

06

08

1Re

flect

ance

()

500 1000 1500 2000 2500 30000Wavelength (120583m)

(a) Alunite

0

01

02

03

04

Refle

ctan

ce (

)

500 1000 1500 2000 2500 30000Wavelength (120583m)

(b) Andradite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

1

Refle

ctan

ce (

)

(c) Pyrope

0

02

04

06

08

Refle

ctan

ce (

)

500 1000 1500 2000 2500 30000Wavelength (120583m)

(d) Nontronite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

1

Refle

ctan

ce (

)

(e) Dumortierite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

Refle

ctan

ce (

)

(f) Kaolinite

0

02

04

06

08

Refle

ctan

ce (

)

500 1000 1500 2000 2500 30000Wavelength (120583m)

(g) Chalcedony

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

Refle

ctan

ce (

)

(h) Kaolinite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

Refle

ctan

ce (

)

(i) Buddingtonite

Figure 9 The comparison map between actual unmixing effect and endmember library

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is supported by Chongqing Engineering Labo-ratory for Detection Control and Integrated System Theproject is also funded by Key Technology Research andIndustrialization of Fire Monitoring and Early Warn-ing Sensor Network for High Voltage Transmission Line(KJZH17124) This research is funded by Chongqing Edu-cation Commission Foundation (KJ1400612) This projectis also granted financial support from a CooperativeProject of Chongqing Technology and Business University(990516001)

References

[1] D L Donoho ldquoCompressed sensingrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol52 no 4 pp 1289ndash1306 2006

[2] E J Candes J Romberg and T Tao ldquoRobust uncertaintyprinciples exact signal reconstruction from highly incompletefrequency informationrdquo Institute of Electrical and ElectronicsEngineers Transactions on InformationTheory vol 52 no 2 pp489ndash509 2006

[3] T N Canh K D Quoc and B Jeon ldquoMulti-resolution kro-necker compressive sensingrdquo Transactions on Smart Processingamp Computing vol 3 no 1 pp 19ndash27 2014

[4] K Q Dinh H J Shim and B Jeon ldquoMeasurement coding forcompressive imaging using a structural measuremnet matrixrdquoin 2013 20th IEEE International Conference on Image ProcessingICIP 2013 pp 10ndash13 aus September 2013

[5] M F Duarte and R G Baraniuk ldquoKronecker compressive sens-ingrdquo IEEE Transactions on Image Processing vol 21 no 2 pp494ndash504 2012

[6] B Zhang X Tong W Wang and J Xie ldquoThe research ofKronecker product-based measurement matrix of compressivesensingrdquo EURASIP Journal on Wireless Communications andNetworking vol 2013 article 161 pp 1ndash5 2013

[7] M Elad ldquoOptimized projections for compressed sensingrdquo IEEETransactions on Signal Processing vol 55 no 12 pp 5695ndash57022007

Journal of Control Science and Engineering 13

[8] M Lustig D Donoho and J M Pauly ldquoSparse MRI the appli-cation of compressed sensing for rapid MR imagingrdquoMagneticResonance in Medicine vol 58 no 6 pp 1182ndash1195 2007

[9] V Abolghasemi S Sanei S Ferdowsi F Ghaderi and ABelcher ldquoSegmented compressive sensingrdquo in Proceedings of theIEEESP 15thWorkshop on Statistical Signal Processing (SSP rsquo09)pp 630ndash633 September 2009

[10] J M Duarte-Carvajalino and G Sapiro ldquoLearning to sensesparse signals simultaneous sensingmatrix and sparsifying dic-tionary optimizationrdquo IEEE Transactions on Image Processingvol 18 no 7 pp 1395ndash1408 2009

[11] ZWang G R Arce and J L Paredes ldquoColored randomprojec-tions for compressed sensingrdquo in 2007 IEEE International Con-ference on Acoustics Speech and Signal Processing ICASSP rsquo07pp III873ndashIII876 usa April 2007

[12] Z Wang and G R Arce ldquoVariable density compressed imagesamplingrdquo IEEE Transactions on Image Processing vol 19 no 1pp 264ndash270 2010

[13] M Elad and M Aharon ldquoImage denoising via learned dic-tionaries and sparse representationrdquo in 2006 IEEE ComputerSociety Conference on Computer Vision and Pattern RecognitionCVPR 2006 pp 895ndash900 usa June 2006

[14] M Aharon M Elad and A Bruckstein ldquoK-SVD an algorithmfor designing overcomplete dictionaries for sparse representa-tionrdquo IEEE Transactions on Signal Processing vol 54 no 11 pp4311ndash4322 2006

[15] J Xu Y Pi and Z Cao ldquoOptimized projection matrix for com-pressive sensingrdquo EURASIP Journal on Advances in SignalProcessing vol 2010 Article ID 560349 2010

[16] V Abolghasemi S Ferdowsi and S Sanei ldquoA gradient-basedalternating minimization approach for optimization of themeasurementmatrix in compressive sensingrdquo Signal Processingvol 92 no 4 pp 999ndash1009 2012

[17] Q Zhang Y Fu H Li and R Rong ldquoOptimized projectionmatrix for compressed sensingrdquo Circuits Systems and SignalProcessing vol 33 no 5 pp 1627ndash1636 2014

[18] G Li Z Zhu D Yang L Chang and H Bai ldquoOn projectionmatrix optimization for compressive sensing systemsrdquo IEEETransactions on Signal Processing vol 61 no 11 pp 2887ndash28982013

[19] S Tian X Fan and L I Zhetao ldquoOrthogonal-gradient mea-surement matrix construction algorithmrdquo Chinese Journal ofElectronics vol 25 no 1 pp 81ndash87 2016

[20] V Abolghasemi S Ferdowsi and BMakkiabadi ldquoOn optimiza-tion of the measurement matrix for compressive sensingrdquo inSignal Processing Conference European IEEE Ed pp 427ndash431August 2010

[21] Q Li D Schonfeld and S Friedland ldquoGeneralized tensorcompressive sensingrdquo in 2013 IEEE International Conference onMultimedia and Expo ICME 2013 usa July 2013

[22] S Friedland Q Li and D Schonfeld ldquoCompressive sensing ofsparse tensorsrdquo IEEE Transactions on Image Processing vol 23no 10 pp 4438ndash4447 2014

[23] S u Xu H Yin C Yi Y Xiong and T Xue ldquoAn ImprovedToeplitz Measurement Matrix for Compressive Sensingrdquo Inter-national Journal of Distributed Sensor Networks vol 8 pp 1ndash82014

[24] R L Devaney in Practical Numerical Algorithms for ChaoticSystems T S Parker and L O Chua Eds vol 32 pp 501ndash503Siam Review 3 edition 2006

[25] R Baraniuk M Davenport R DeVore and M Wakin ldquoAsimple proof of the restricted isometry property for randommatricesrdquoConstructiveApproximation An International Journalfor Approximations and Expansions vol 28 no 3 pp 253ndash2632008

[26] Available httpwwwesciencecnpeoplefeiyunZHUDatasetGThtml

[27] D L Donoho Y Tsaig I Drori and J-L Starck ldquoSparse solu-tion of underdetermined systems of linear equations by stage-wise orthogonal matching pursuitrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol58 no 2 pp 1094ndash1121 2012

[28] K Lang ldquoNewsWeeder Learning to Filter Netnewsrdquo in Inter-national Machine Learning Conference vol 1995 pp 331ndash339

[29] Available httpfeatureselectionasuedudatasetsphp[30] Available httpclopinetcomisabelleProjectsNIPS

Journal of Control Science and Engineering 11

(a) Alunite (b) Andradite (c) Pyrope

(d) Nontronite (e) Dumortierite (f) Kaolinite

(g) Chalcedony (h) Kaolinite (i) Buddingtonite

Figure 8 The results from different elements abundance in hyperspectral unmixing schemes

12 Journal of Control Science and Engineering

02

04

06

08

1Re

flect

ance

()

500 1000 1500 2000 2500 30000Wavelength (120583m)

(a) Alunite

0

01

02

03

04

Refle

ctan

ce (

)

500 1000 1500 2000 2500 30000Wavelength (120583m)

(b) Andradite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

1

Refle

ctan

ce (

)

(c) Pyrope

0

02

04

06

08

Refle

ctan

ce (

)

500 1000 1500 2000 2500 30000Wavelength (120583m)

(d) Nontronite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

1

Refle

ctan

ce (

)

(e) Dumortierite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

Refle

ctan

ce (

)

(f) Kaolinite

0

02

04

06

08

Refle

ctan

ce (

)

500 1000 1500 2000 2500 30000Wavelength (120583m)

(g) Chalcedony

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

Refle

ctan

ce (

)

(h) Kaolinite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

Refle

ctan

ce (

)

(i) Buddingtonite

Figure 9 The comparison map between actual unmixing effect and endmember library

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is supported by Chongqing Engineering Labo-ratory for Detection Control and Integrated System Theproject is also funded by Key Technology Research andIndustrialization of Fire Monitoring and Early Warn-ing Sensor Network for High Voltage Transmission Line(KJZH17124) This research is funded by Chongqing Edu-cation Commission Foundation (KJ1400612) This projectis also granted financial support from a CooperativeProject of Chongqing Technology and Business University(990516001)

References

[1] D L Donoho ldquoCompressed sensingrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol52 no 4 pp 1289ndash1306 2006

[2] E J Candes J Romberg and T Tao ldquoRobust uncertaintyprinciples exact signal reconstruction from highly incompletefrequency informationrdquo Institute of Electrical and ElectronicsEngineers Transactions on InformationTheory vol 52 no 2 pp489ndash509 2006

[3] T N Canh K D Quoc and B Jeon ldquoMulti-resolution kro-necker compressive sensingrdquo Transactions on Smart Processingamp Computing vol 3 no 1 pp 19ndash27 2014

[4] K Q Dinh H J Shim and B Jeon ldquoMeasurement coding forcompressive imaging using a structural measuremnet matrixrdquoin 2013 20th IEEE International Conference on Image ProcessingICIP 2013 pp 10ndash13 aus September 2013

[5] M F Duarte and R G Baraniuk ldquoKronecker compressive sens-ingrdquo IEEE Transactions on Image Processing vol 21 no 2 pp494ndash504 2012

[6] B Zhang X Tong W Wang and J Xie ldquoThe research ofKronecker product-based measurement matrix of compressivesensingrdquo EURASIP Journal on Wireless Communications andNetworking vol 2013 article 161 pp 1ndash5 2013

[7] M Elad ldquoOptimized projections for compressed sensingrdquo IEEETransactions on Signal Processing vol 55 no 12 pp 5695ndash57022007

Journal of Control Science and Engineering 13

[8] M Lustig D Donoho and J M Pauly ldquoSparse MRI the appli-cation of compressed sensing for rapid MR imagingrdquoMagneticResonance in Medicine vol 58 no 6 pp 1182ndash1195 2007

[9] V Abolghasemi S Sanei S Ferdowsi F Ghaderi and ABelcher ldquoSegmented compressive sensingrdquo in Proceedings of theIEEESP 15thWorkshop on Statistical Signal Processing (SSP rsquo09)pp 630ndash633 September 2009

[10] J M Duarte-Carvajalino and G Sapiro ldquoLearning to sensesparse signals simultaneous sensingmatrix and sparsifying dic-tionary optimizationrdquo IEEE Transactions on Image Processingvol 18 no 7 pp 1395ndash1408 2009

[11] ZWang G R Arce and J L Paredes ldquoColored randomprojec-tions for compressed sensingrdquo in 2007 IEEE International Con-ference on Acoustics Speech and Signal Processing ICASSP rsquo07pp III873ndashIII876 usa April 2007

[12] Z Wang and G R Arce ldquoVariable density compressed imagesamplingrdquo IEEE Transactions on Image Processing vol 19 no 1pp 264ndash270 2010

[13] M Elad and M Aharon ldquoImage denoising via learned dic-tionaries and sparse representationrdquo in 2006 IEEE ComputerSociety Conference on Computer Vision and Pattern RecognitionCVPR 2006 pp 895ndash900 usa June 2006

[14] M Aharon M Elad and A Bruckstein ldquoK-SVD an algorithmfor designing overcomplete dictionaries for sparse representa-tionrdquo IEEE Transactions on Signal Processing vol 54 no 11 pp4311ndash4322 2006

[15] J Xu Y Pi and Z Cao ldquoOptimized projection matrix for com-pressive sensingrdquo EURASIP Journal on Advances in SignalProcessing vol 2010 Article ID 560349 2010

[16] V Abolghasemi S Ferdowsi and S Sanei ldquoA gradient-basedalternating minimization approach for optimization of themeasurementmatrix in compressive sensingrdquo Signal Processingvol 92 no 4 pp 999ndash1009 2012

[17] Q Zhang Y Fu H Li and R Rong ldquoOptimized projectionmatrix for compressed sensingrdquo Circuits Systems and SignalProcessing vol 33 no 5 pp 1627ndash1636 2014

[18] G Li Z Zhu D Yang L Chang and H Bai ldquoOn projectionmatrix optimization for compressive sensing systemsrdquo IEEETransactions on Signal Processing vol 61 no 11 pp 2887ndash28982013

[19] S Tian X Fan and L I Zhetao ldquoOrthogonal-gradient mea-surement matrix construction algorithmrdquo Chinese Journal ofElectronics vol 25 no 1 pp 81ndash87 2016

[20] V Abolghasemi S Ferdowsi and BMakkiabadi ldquoOn optimiza-tion of the measurement matrix for compressive sensingrdquo inSignal Processing Conference European IEEE Ed pp 427ndash431August 2010

[21] Q Li D Schonfeld and S Friedland ldquoGeneralized tensorcompressive sensingrdquo in 2013 IEEE International Conference onMultimedia and Expo ICME 2013 usa July 2013

[22] S Friedland Q Li and D Schonfeld ldquoCompressive sensing ofsparse tensorsrdquo IEEE Transactions on Image Processing vol 23no 10 pp 4438ndash4447 2014

[23] S u Xu H Yin C Yi Y Xiong and T Xue ldquoAn ImprovedToeplitz Measurement Matrix for Compressive Sensingrdquo Inter-national Journal of Distributed Sensor Networks vol 8 pp 1ndash82014

[24] R L Devaney in Practical Numerical Algorithms for ChaoticSystems T S Parker and L O Chua Eds vol 32 pp 501ndash503Siam Review 3 edition 2006

[25] R Baraniuk M Davenport R DeVore and M Wakin ldquoAsimple proof of the restricted isometry property for randommatricesrdquoConstructiveApproximation An International Journalfor Approximations and Expansions vol 28 no 3 pp 253ndash2632008

[26] Available httpwwwesciencecnpeoplefeiyunZHUDatasetGThtml

[27] D L Donoho Y Tsaig I Drori and J-L Starck ldquoSparse solu-tion of underdetermined systems of linear equations by stage-wise orthogonal matching pursuitrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol58 no 2 pp 1094ndash1121 2012

[28] K Lang ldquoNewsWeeder Learning to Filter Netnewsrdquo in Inter-national Machine Learning Conference vol 1995 pp 331ndash339

[29] Available httpfeatureselectionasuedudatasetsphp[30] Available httpclopinetcomisabelleProjectsNIPS

12 Journal of Control Science and Engineering

02

04

06

08

1Re

flect

ance

()

500 1000 1500 2000 2500 30000Wavelength (120583m)

(a) Alunite

0

01

02

03

04

Refle

ctan

ce (

)

500 1000 1500 2000 2500 30000Wavelength (120583m)

(b) Andradite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

1

Refle

ctan

ce (

)

(c) Pyrope

0

02

04

06

08

Refle

ctan

ce (

)

500 1000 1500 2000 2500 30000Wavelength (120583m)

(d) Nontronite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

1

Refle

ctan

ce (

)

(e) Dumortierite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

Refle

ctan

ce (

)

(f) Kaolinite

0

02

04

06

08

Refle

ctan

ce (

)

500 1000 1500 2000 2500 30000Wavelength (120583m)

(g) Chalcedony

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

Refle

ctan

ce (

)

(h) Kaolinite

500 1000 1500 2000 2500 30000Wavelength (120583m)

0

02

04

06

08

Refle

ctan

ce (

)

(i) Buddingtonite

Figure 9 The comparison map between actual unmixing effect and endmember library

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is supported by Chongqing Engineering Labo-ratory for Detection Control and Integrated System Theproject is also funded by Key Technology Research andIndustrialization of Fire Monitoring and Early Warn-ing Sensor Network for High Voltage Transmission Line(KJZH17124) This research is funded by Chongqing Edu-cation Commission Foundation (KJ1400612) This projectis also granted financial support from a CooperativeProject of Chongqing Technology and Business University(990516001)

References

[1] D L Donoho ldquoCompressed sensingrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol52 no 4 pp 1289ndash1306 2006

[2] E J Candes J Romberg and T Tao ldquoRobust uncertaintyprinciples exact signal reconstruction from highly incompletefrequency informationrdquo Institute of Electrical and ElectronicsEngineers Transactions on InformationTheory vol 52 no 2 pp489ndash509 2006

[3] T N Canh K D Quoc and B Jeon ldquoMulti-resolution kro-necker compressive sensingrdquo Transactions on Smart Processingamp Computing vol 3 no 1 pp 19ndash27 2014

[4] K Q Dinh H J Shim and B Jeon ldquoMeasurement coding forcompressive imaging using a structural measuremnet matrixrdquoin 2013 20th IEEE International Conference on Image ProcessingICIP 2013 pp 10ndash13 aus September 2013

[5] M F Duarte and R G Baraniuk ldquoKronecker compressive sens-ingrdquo IEEE Transactions on Image Processing vol 21 no 2 pp494ndash504 2012

[6] B Zhang X Tong W Wang and J Xie ldquoThe research ofKronecker product-based measurement matrix of compressivesensingrdquo EURASIP Journal on Wireless Communications andNetworking vol 2013 article 161 pp 1ndash5 2013

[7] M Elad ldquoOptimized projections for compressed sensingrdquo IEEETransactions on Signal Processing vol 55 no 12 pp 5695ndash57022007

Journal of Control Science and Engineering 13

[8] M Lustig D Donoho and J M Pauly ldquoSparse MRI the appli-cation of compressed sensing for rapid MR imagingrdquoMagneticResonance in Medicine vol 58 no 6 pp 1182ndash1195 2007

[9] V Abolghasemi S Sanei S Ferdowsi F Ghaderi and ABelcher ldquoSegmented compressive sensingrdquo in Proceedings of theIEEESP 15thWorkshop on Statistical Signal Processing (SSP rsquo09)pp 630ndash633 September 2009

[10] J M Duarte-Carvajalino and G Sapiro ldquoLearning to sensesparse signals simultaneous sensingmatrix and sparsifying dic-tionary optimizationrdquo IEEE Transactions on Image Processingvol 18 no 7 pp 1395ndash1408 2009

[11] ZWang G R Arce and J L Paredes ldquoColored randomprojec-tions for compressed sensingrdquo in 2007 IEEE International Con-ference on Acoustics Speech and Signal Processing ICASSP rsquo07pp III873ndashIII876 usa April 2007

[12] Z Wang and G R Arce ldquoVariable density compressed imagesamplingrdquo IEEE Transactions on Image Processing vol 19 no 1pp 264ndash270 2010

[13] M Elad and M Aharon ldquoImage denoising via learned dic-tionaries and sparse representationrdquo in 2006 IEEE ComputerSociety Conference on Computer Vision and Pattern RecognitionCVPR 2006 pp 895ndash900 usa June 2006

[14] M Aharon M Elad and A Bruckstein ldquoK-SVD an algorithmfor designing overcomplete dictionaries for sparse representa-tionrdquo IEEE Transactions on Signal Processing vol 54 no 11 pp4311ndash4322 2006

[15] J Xu Y Pi and Z Cao ldquoOptimized projection matrix for com-pressive sensingrdquo EURASIP Journal on Advances in SignalProcessing vol 2010 Article ID 560349 2010

[16] V Abolghasemi S Ferdowsi and S Sanei ldquoA gradient-basedalternating minimization approach for optimization of themeasurementmatrix in compressive sensingrdquo Signal Processingvol 92 no 4 pp 999ndash1009 2012

[17] Q Zhang Y Fu H Li and R Rong ldquoOptimized projectionmatrix for compressed sensingrdquo Circuits Systems and SignalProcessing vol 33 no 5 pp 1627ndash1636 2014

[18] G Li Z Zhu D Yang L Chang and H Bai ldquoOn projectionmatrix optimization for compressive sensing systemsrdquo IEEETransactions on Signal Processing vol 61 no 11 pp 2887ndash28982013

[19] S Tian X Fan and L I Zhetao ldquoOrthogonal-gradient mea-surement matrix construction algorithmrdquo Chinese Journal ofElectronics vol 25 no 1 pp 81ndash87 2016

[20] V Abolghasemi S Ferdowsi and BMakkiabadi ldquoOn optimiza-tion of the measurement matrix for compressive sensingrdquo inSignal Processing Conference European IEEE Ed pp 427ndash431August 2010

[21] Q Li D Schonfeld and S Friedland ldquoGeneralized tensorcompressive sensingrdquo in 2013 IEEE International Conference onMultimedia and Expo ICME 2013 usa July 2013

[22] S Friedland Q Li and D Schonfeld ldquoCompressive sensing ofsparse tensorsrdquo IEEE Transactions on Image Processing vol 23no 10 pp 4438ndash4447 2014

[23] S u Xu H Yin C Yi Y Xiong and T Xue ldquoAn ImprovedToeplitz Measurement Matrix for Compressive Sensingrdquo Inter-national Journal of Distributed Sensor Networks vol 8 pp 1ndash82014

[24] R L Devaney in Practical Numerical Algorithms for ChaoticSystems T S Parker and L O Chua Eds vol 32 pp 501ndash503Siam Review 3 edition 2006

[25] R Baraniuk M Davenport R DeVore and M Wakin ldquoAsimple proof of the restricted isometry property for randommatricesrdquoConstructiveApproximation An International Journalfor Approximations and Expansions vol 28 no 3 pp 253ndash2632008

[26] Available httpwwwesciencecnpeoplefeiyunZHUDatasetGThtml

[27] D L Donoho Y Tsaig I Drori and J-L Starck ldquoSparse solu-tion of underdetermined systems of linear equations by stage-wise orthogonal matching pursuitrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol58 no 2 pp 1094ndash1121 2012

[28] K Lang ldquoNewsWeeder Learning to Filter Netnewsrdquo in Inter-national Machine Learning Conference vol 1995 pp 331ndash339

[29] Available httpfeatureselectionasuedudatasetsphp[30] Available httpclopinetcomisabelleProjectsNIPS

Journal of Control Science and Engineering 13

[8] M Lustig D Donoho and J M Pauly ldquoSparse MRI the appli-cation of compressed sensing for rapid MR imagingrdquoMagneticResonance in Medicine vol 58 no 6 pp 1182ndash1195 2007

[9] V Abolghasemi S Sanei S Ferdowsi F Ghaderi and ABelcher ldquoSegmented compressive sensingrdquo in Proceedings of theIEEESP 15thWorkshop on Statistical Signal Processing (SSP rsquo09)pp 630ndash633 September 2009

[10] J M Duarte-Carvajalino and G Sapiro ldquoLearning to sensesparse signals simultaneous sensingmatrix and sparsifying dic-tionary optimizationrdquo IEEE Transactions on Image Processingvol 18 no 7 pp 1395ndash1408 2009

[11] ZWang G R Arce and J L Paredes ldquoColored randomprojec-tions for compressed sensingrdquo in 2007 IEEE International Con-ference on Acoustics Speech and Signal Processing ICASSP rsquo07pp III873ndashIII876 usa April 2007

[12] Z Wang and G R Arce ldquoVariable density compressed imagesamplingrdquo IEEE Transactions on Image Processing vol 19 no 1pp 264ndash270 2010

[13] M Elad and M Aharon ldquoImage denoising via learned dic-tionaries and sparse representationrdquo in 2006 IEEE ComputerSociety Conference on Computer Vision and Pattern RecognitionCVPR 2006 pp 895ndash900 usa June 2006

[14] M Aharon M Elad and A Bruckstein ldquoK-SVD an algorithmfor designing overcomplete dictionaries for sparse representa-tionrdquo IEEE Transactions on Signal Processing vol 54 no 11 pp4311ndash4322 2006

[15] J Xu Y Pi and Z Cao ldquoOptimized projection matrix for com-pressive sensingrdquo EURASIP Journal on Advances in SignalProcessing vol 2010 Article ID 560349 2010

[16] V Abolghasemi S Ferdowsi and S Sanei ldquoA gradient-basedalternating minimization approach for optimization of themeasurementmatrix in compressive sensingrdquo Signal Processingvol 92 no 4 pp 999ndash1009 2012

[17] Q Zhang Y Fu H Li and R Rong ldquoOptimized projectionmatrix for compressed sensingrdquo Circuits Systems and SignalProcessing vol 33 no 5 pp 1627ndash1636 2014

[18] G Li Z Zhu D Yang L Chang and H Bai ldquoOn projectionmatrix optimization for compressive sensing systemsrdquo IEEETransactions on Signal Processing vol 61 no 11 pp 2887ndash28982013

[19] S Tian X Fan and L I Zhetao ldquoOrthogonal-gradient mea-surement matrix construction algorithmrdquo Chinese Journal ofElectronics vol 25 no 1 pp 81ndash87 2016

[20] V Abolghasemi S Ferdowsi and BMakkiabadi ldquoOn optimiza-tion of the measurement matrix for compressive sensingrdquo inSignal Processing Conference European IEEE Ed pp 427ndash431August 2010

[21] Q Li D Schonfeld and S Friedland ldquoGeneralized tensorcompressive sensingrdquo in 2013 IEEE International Conference onMultimedia and Expo ICME 2013 usa July 2013

[22] S Friedland Q Li and D Schonfeld ldquoCompressive sensing ofsparse tensorsrdquo IEEE Transactions on Image Processing vol 23no 10 pp 4438ndash4447 2014

[23] S u Xu H Yin C Yi Y Xiong and T Xue ldquoAn ImprovedToeplitz Measurement Matrix for Compressive Sensingrdquo Inter-national Journal of Distributed Sensor Networks vol 8 pp 1ndash82014

[24] R L Devaney in Practical Numerical Algorithms for ChaoticSystems T S Parker and L O Chua Eds vol 32 pp 501ndash503Siam Review 3 edition 2006

[25] R Baraniuk M Davenport R DeVore and M Wakin ldquoAsimple proof of the restricted isometry property for randommatricesrdquoConstructiveApproximation An International Journalfor Approximations and Expansions vol 28 no 3 pp 253ndash2632008

[26] Available httpwwwesciencecnpeoplefeiyunZHUDatasetGThtml

[27] D L Donoho Y Tsaig I Drori and J-L Starck ldquoSparse solu-tion of underdetermined systems of linear equations by stage-wise orthogonal matching pursuitrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol58 no 2 pp 1094ndash1121 2012

[28] K Lang ldquoNewsWeeder Learning to Filter Netnewsrdquo in Inter-national Machine Learning Conference vol 1995 pp 331ndash339

[29] Available httpfeatureselectionasuedudatasetsphp[30] Available httpclopinetcomisabelleProjectsNIPS

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