Hypergraph-Regularized Sparse NMF

Hypergraph-Regularized Sparse NMF for Hyperspectral Unmixing Wenhong Wang,Yuntao Qian,Member,IEEE,and Yuan Yan Tang,Fellow,IEEEAbstract—Hyperspectral image(HSI)unmixing has attracted increasing research interests in recent decades.The major diffi-culty of it lies in that the endmembers and the associated abun-dances need to be separated from highly mixed observation data with few a priori information.Recently,sparsity-constrained non-negative matrix factorization(NMF)algorithms have been proved effective for hyperspectral unmixing(HU)since they can suffi-ciently utilize the sparsity property of HSIs.In order to improve the performance of NMF-based unmixing approaches,spectral and spatial constrains have been added into the unmixing model, but spectral–spatial joint structure is required to be more accu-rately estimated.To exploit the property that similar pixels within a small spatial neighborhood have higher possibility to share sim-ilar abundances,hypergraph structure is employed to capture the similarity relationship among the spatial nearby pixels.In the con-struction of a hypergraph,each pixel is taken as a vertex of the hypergraph,and each vertex with its k nearest spatial neighbor-ing pixels form a ing the hypergraph,the pixels with similar abundances can be accurately found,which enables the unmixing algorithm to obtain promising results.Experiments on synthetic data and real HSIs are conducted to investigate the performance of the proposed algorithm.The superiority of the proposed algorithm is demonstrated by comparing it with some state-of-the-art methods.Index Terms—Hypergraph learning,hyperspectral unmixing (HU),nonnegative matrix factorization(NMF),sparse coding.I.I NTRODUCTIONH YPERSPECTRAL remote sensing image generally con-tains hundreds of spectral bands about the scene,which is effective for detecting and recognizing the land-cover materials. Due to the limited spatial resolution of hyperspectral sensors, different types of materials may simultaneously exist in a sin-gle pixel,called“mixed pixel,”and the corresponding spectrum Manuscript received August12,2015;revised October24,2015;accepted December01,2015.Date of publication January05,2016;date of current version February09,2016.This work was supported in part by the National Natural Science Foundation of China under Grant61571393and Grant 61273244,in part by the Research Grants of the University of Macau under Grant MYRG2015-00049-FST and Grant MYRG2015-00050-FST,in part by the Science and Technology Development Fund(FDCT)of Macau under Grant 100-2012-A3and Grant026-2013-A,and in part by the Macau-China join Project008-2014-AMJ.(Corresponding author:Yuntao Qian.)W.Wang is with the Institute of Artificial Intelligence,College of Computer Science,Zhejiang University,Hangzhou310027,China,and also with the College of Computer Science,Liaocheng University,Liaocheng252059, China.Y.Qian is with the Institute of Artificial Intelligence,College of Computer Science,Zhejiang University,Hangzhou310027,China(e-mail: ytqian@).Y.Y.Tang is with the Faculty of Science and Technology,University of Macau,Macau999078,China.Color versions of one or more of thefigures in this paper are available online at .Digital Object Identifier10.1109/JSTARS.2015.2508448is a mixture of several pure spectrums(i.e.,endmembers).Theoccurrence of mixed pixels has become a challenging problemfor analyzing hyperspectral data quantitatively[1].Currently,hyperspectral unmixing(HU)is an effective technology totackle this problem[2],which aims to identify a collection ofendmembers corresponding to the pure materials present in thescene,and estimate the proportions(i.e.,abundances)of theendmembers in each pixel.Under the linear mixture model(LMM)of hyperspectralimage(HSI)[1],a number of HU methods have been proposedin recent decades.Among them,nonnegative matrix factoriza-tion(NMF)[3]and its variants attract increasing attentionsdue to the good unmixing performance for highly mixed HSIs[4]–[8].NMF can learn a parts-based representation of thedata by factorizing a nonnegative matrix into two nonnegativefactor matrices.When it is applied to HU,NMF can esti-mate the endmembers and abundances simultaneously withoutthe pure-pixel assumption.However,NMF has a large num-ber of local minima owning to the nonconvexity of its costfunction,which leads to unstable unmixing results.To dealwith this problem,an effective scheme is to introduce addi-tional structure constraints into NMF model.For example,theconstraint of sparsity has been introduced into NMF-basedunmixing methods,so that the inherent sparseness property ofHSI is sufficiently exploited[4],[5],[9],[10].Recently,a novelsparsity-constrained NMF(L1/2-NMF)unmixing method[4]has been widely used in HU[2],[8],[11].In L1/2-NMF,L1/2regularizer,instead of L1regularizer,is imposed on the abundances to enforce the sparse estimations.L1/2-NMF hasbeen demonstrated to be superior compared with some ofthe state-of-the-art approaches.However,like other sparsity-constrained NMF,the unmixing performance of L1/2-NMF isstill susceptible to the initial setting and the noise disturbance.Besides the constraint of sparsity,other constrains based onthe characteristics of HSI have been proposed to improve theNMF-based unmixing methods.Some constraints are depen-dent on spectral similarity between pixels,which aim toexploit the similarity relations of pixels in the spectral space.For example,a graph-regularized L1/2-NMF unmixing algo-rithm(GLNMF)[8]is proposed by incorporating graph-basedregularization into sparse NMF,which tries to make moreaccurate abundance estimations by capturing the intrinsic man-ifold structure of the data.Different from the constraint ofspectral similarity,some methods adopt the spatial similar-ity constraints,by which the structure consistency of HSIamong the spatially nearby pixels are considered.In[6],avariant of NMF,named abundance separation and smoothness-constrained NMF(ASSNMF),is proposed by integrating an1939-1404©2015IEEE.Personal use is permitted,but republication/redistribution requires IEEE permission.See /publications_standards/publications/rights/index.html for more information.abundance separation constraint with a spatial similarity con-straint,so that the abundances of a pixel are promoted to be consistent with the average abundances of its surrounding blocks.In[12],a weighted NMF unmixing algorithm(WNMF) is presented,where a local neighborhood weight is designed to enforce the abundance similarity among spatial neighbor-ing pixels.To utilize both spectral and spatial information,a few NMF variants incorporate the spectral–spatial similarity constrains.In[13],a neighborhood-preserving regularization, which assumes that each pixel can be linearly reconstructed by its spatial nearby pixels,is added into NMF to keep the local geometric structure of the data.In[11],a dual graph-regularized sparse NMF(DGNMF)is proposed by integrating the data consistency constraint in the spatial space with the intrinsic manifold constraint in the spectral space.Among these constraints,spectral–spatial joint constraint has shown con-siderable potentials for HU applications because it can make use of the spatial and spectral information simultaneously. Nevertheless,how to effectively model the spectral–spatial joint relationship remains a challenging problem.In order to make full use of the spatial–spectral joint infor-mation in the sparse unmixing model,in this paper,a novel unmixing algorithm,named hypergraph-regularized L1/2-NMF (HG L1/2-NMF),is proposed.According to the basic spatial–spectral correlation property of HSI,the similar pixels within a small spatial neighborhood have higher possibility to share similar abundances.Thus,the spectral similarity of the spa-tial nearby pixels can be used to promote the consistency of the abundances.In our method,this spectral–spatial joint struc-ture is modeled by a hypergraph rather than a simple graph because hypergraph learning has some attractive advantages. Currently,hypergraph learning has been successfully applied in manyfields such as image classification[14]and cluster-ing[15],information retrieval[16],sparse coding[17],and HSI analysis[18]–[20].Compared with the simple graph that just represents the pairwise relation between two pixels,the hypergraph can model the high-order relation among multiple pixels by including more than two vertices in its hyperedges. This enables us to model the spatial and spectral similarity relationships more accurately.In a hypergraph model,each vertex represents a pixel of HSI,and each hyperedge rep-resents the similarity relation among a group of pixels with higher similarity in a spatial block.By hypergraph learning, the spatial–spectral joint structure as a regularization term is incorporated into the sparse unmixing model to enforce the consistency of abundances among the pixels in the same hyper-edge.Moreover,with its hyperedge structure and the associated weights,the hypergraph can provide a robust representation of the similarity relationship among the spatial neighboring pixels,which alleviates the noise disturbance to the unmixing results.The performance of the proposed unmixing method is investigated by conducting a series of experiments on synthetic data sets and real-world HSIs.Our method is compared with some of the state-of-the-art unmixing algorithms to validate its effectiveness.This paper is organized as follows.In Section II,we give a brief review of LMM and NMF.In Section III,we present the HG L1/2-NMF unmixing algorithm,its update rules,and some implementation issues.Extensive experimental results on syn-thetic data and real HSIs are given in Section IV.We conclude this paper in Section V.II.P RELIMINARYA.Linear Spectrum Mixture ModelAccording to the LMM,an observed pixel spectrum is assumed to be a linear combination of a set of endmember signatures weighted by the associated abundance fractions[1]. Given that a hypespectral image with L bands and N pixels is denoted by Y∈R L×N,each pixel of Y can be formalized asy=As+e(1) where y∈R L×1is a vector denoting the target pixel,A∈R L×P is the endmember matrix consisting of P endmembersignatures a j,for j=1,...,P,s∈R P×1is the abundance vector including the proportions of corresponding endmem-bers in the pixel y,and e∈R L×1represents a measurement error vector.In general,the abundance vector s always satisfies the abundance nonnegativity constraint(ANC)and the abun-dance sum-to-one constraint(ASC),which are respectively represented byANC:s j≥0,j=1,...,P(2)ASC:Pj=1s j=1.(3)Using the matrix operations,(1)can be re-expressed by a compact formY=AS(4)where S=[s1,s2,...,s N]∈R P×N denotes the abundance matrix,and the measure error term is removed from the model by merging it into the endmember matrix[7].Under LMM,HU can be regarded as a blind source sep-aration problem,in which the endmember matrix A and the abundance matrix S need to be estimated simultaneously.B.Nonnegative Matrix FactorizationNMF is a powerful tool for data analysis,which provides a parts-based representation for nonnegative data.At present, NMF has been demonstrated effective for HU analysis owning to its inherent advantages.Given a nonnegative matrix Y∈R L×N and a prespecified positive integer P<min(L,N),NMF aims to factorize Y as a product of two nonnegative matrix factors A∈R L×P and S∈R P×N such that the error between Y and its reconstruction AS is minimized.Generally,NMF is formalized as the following optimization problem:minA,SF(A,S)= Y−AS 2F;s.t.A,S 0(5)where“ ”represents the element-wise great-or-equal relation, and · F denotes the Frobenious norm.W ANG et al.:HYPERGRAPH-REGULARIZED SPARSE NMF FOR HU683 NMF wasfirst introduced by Paatero and Tapper underthe name of positive matrix factorization[21].It became tobe widely used since Lee and Seung proposed the followingmultiplicative update rules(MURs)[22]:A←A.∗(Y S T./ASS T)(6)S←S.∗(A T Y./A T AS)(7)where(·)T is the matrix transpose operator,and“.∗”and“./”stand for element-wise matrix multiplication and division,respectively.Thereafter,many effective optimization methodsfor NMF have been proposed,such as active set method[23],projected gradient method[24],projected nonnegative leastsquares[25],and optimal gradient method[26].Among them,MUR is the most used approach due to its simplicity andeasiness to implement.Although(5)is convex with respect to either A or S,it is non-convex for both of them simultaneously.As a result,there area large amount of local minima in the solution space of NMF.This will lead to instable unmixing results when NMF is appliedto HU.To alleviate this problem,an effective method is tointroduce additional constraints into NMF,so that the feasiblesolutions of the algorithm are confined.Recently,sparsity-constrained NMF has received increasingattentions because sparseness is an important property of HSI[4],[5],[9],[10].In L1/2-NMF unmixing method[4],L1/2regularizer as sparseness constraint is incorporated into NMFto pursue sparser abundances.Since L1/2-NMF can effectivelyexploit the inherent sparseness of the data,it shows com-parative superiority compared with the basic NMF and othersparse NMF approaches.The objective function of L1/2-NMFis formulated asF(A,S)=12Y−AS 2F+λ S 1/2(8)where S 1/2= P,Np,n=1(s pn)1/2is the L1/2regularizationandλ∈R+is the regularization parameter.However,the unmixing performance of L1/2-NMF is still not robust to the initial setting of the endmember and abundance matrices. In addition,the separation results yielded by L1/2-NMF are often influenced by noise interference.In order to get superior unmixing solutions,more constraints concerning other a priori structures of HSI should be added into NMF model.III.HG NMF U NMIXING A LGORITHMIn this section,wefirst introduce the concepts about hyper-graph learning.Then,the proposed HG L1/2-NMF algorithm is described.In addition,the MURs for the optimization proce-dure and the implementation issues are addressed.A.Hypergraph LearningHypergraph learning[15],[27]is extended from the learn-ing technologies based on simple graph.A simple graph model only considers the pair-wise relationship between data samples. This will lead to the loss of information about some com-plex relationships that are crucial to particular applications.For Fig.1.Illustration of the hypergraph˜G with its incidence matrix. example,it is easy tofind two close data samples by the pair-wise similarity,while it is hard to infer whether there is a group of(more than two)similar samples.Unlike the simple graph, a hypergraph can capture the high-order relationships among multiple data samples,which is expected to effectively model the spectral–spatial joint structure of HSI.Now,some concepts about the hypergraph are introduced.Definition1:Let G=(V,E),where V={v1,v2,...,v N} is afinite set of elements,and E={e1,e2,...,e M}is a collection of subsets of V.G is a hypergraph defined on V if:1)e j=∅,for j=1,2,...,M;2)e1∪e2,...,∪e M=V.An element of V is called a vertex,and each e j∈E is known as a hyperedge of G.Each hyperedge e can be assigned with a positive value w(e),which is referred to a hyperedge weight. Commonly,an incidence matrix H∈R|V|×|E|is used to rep-resent the incidence relation between a vertex and a hyperedge, which is formalized asH(v,e)=1,if v∈e0,otherwise.(9)Fig.1gives an illustration of a hypergraph˜G=(˜V,˜E)with its incidence matrix˜H,where˜V={v1,v2,v3,v4,v5,v6,v7,v8} and˜E={e1={v1,v2,v3},e2={v3,v4,v5,v6},e3={v6, v7,v8}}.In Fig.1,the solid nodes stand for the vertices and the node sets marked by the ellipses denote the hyperedges. The corresponding incidence matrix˜H is shown in the right of the hypergraph˜G.The degree of each vertex v∈V is defined as the sum of weights associated with the hyperedges that v belongs to,i.e., d(v)={e∈E|v∈e}w(e).The hyperedge degree of each e∈E is defined as the number of vertices included in e,i.e.,δ(e)= |e|.Specifically,d(v)andδ(e)can be calculated,respectively, byd(v)=Mj=1w(e j)H(v i,e j)(10)δ(e)=Nj=1H(v j,e).(11)Let D v stands for the diagonal matrix of vertex degree whose entry(D v)ij=d(v i)if i=j,and(D v)ij=0other-wise.Similarly,D e and W individually denote the matrices of684IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERV ATIONS AND REMOTE SENSING,VOL.9,NO.2,FEBRUARY2016 Fig.2.Generation procedure of the hypergraph.Atfirst,each pixel of the given HSI is regarded as a centroid node to determine a spatial neighborhood.Next,the KNNs of the centroid pixel(marked by dashed line square),which are chosen within the corresponding spatial neighborhood,are used to generate a hyperedge. At last,all the obtained hyperedges form a hypergraph as a whole.hyperedge degree and hyperedge weight,which consist ofδ(e j) and w(e j),for j=1,2,...,M,respectively,in their diagonal entries.Then,the unnormalized hypergraph Laplacian matrix can be formalized asL Hyper=D v−B(12) where B=HW(D e)−1H T.Generally,a hyperedge is allowed to include an arbitrary number of vertices according to its definition.For the sake of simplicity,in this paper,the uniform hypergraph which contains the same number of nodes in each hyperedge is employed.B.HG L1/2-NMFAs aforementioned,NMF-based unmixing methods should benefit from both spatial and spectral information of HSI. Therefore,to improve the unmixing performance of the sparse NMF,spatial and spectral joint information available in the data needs to be fully exploited in the unmixing process.According to the basic property of HSI,the spatial-nearby pixels(but they are not requested to be spatial-adjacent)generally have highly consistent mixture structure such that they have great probability to share similar abundances.At the same time,the similarity between two pixels in the spectral space is also help-ful to identify the pixels with similar abundances.Therefore, the similarity relationships between pixels in the spectral space and in spatial neighborhood both play important roles in HU. In order to accuratelyfind the similar pixels within local spatial region,in this paper,a hypergraph G s=(V s,E s,W s) is constructed based on the overlapped spatial neighborhoods of a given HSI.Fig.2gives an illustration of the generation procedure of the hypergraph.Atfirst,each pixel spectrum y i in the HSI Y=[y1,y2,...,y N]is taken as an element of the vertex set V s.Then,each vertex v i∈V s acts as the“centroid”node of the located spatial region,and the corresponding hyper-edge e i∈E s is constructed by including v i and its K-nearest neighbors(KNNs)that are chosen from the same spatial neigh-borhood as v i.Therefore,the hypergraph G s consists of N hyperedges that are constructed on different spatial regions. Based on these hyperedges,the incidence matrix H ∈R N×N is obtained by following(9).Next,each hyperedge e i∈E s is assigned a positive weight,which models the spectral similar-ity of the pixels included in the hyperedge e i.The weight can be calculated as follows[18]:w(e i)=y j∈e iexp−y j−y i 22(13)whereδ=1KNNi=1{y j∈e i}y i−y j denotes the mean distance among all the spatial neighboring ing this weight estimation method,a hyperedge that consists of pixels with higher similarity will be associated with a larger weight. According to the above steps,each vertex is connected with at least one hyperedge and each hyperedge is associated with a weight.Then,the diagonal weight matrix W s∈R N×N can be given by(W s)ij=w(e i),if i=j0,otherwise.(14)With the obtained weight matrix W s,we can enforce the consistency of abundances corresponding to the pixels in each hyperedge by minimizing the following function:C=e∈E s{i,j}∈ew(e)δ(e)s i−s j 2(15) which can be reexpressed asC=Tr(SL Hypers S)(16)W ANG et al.:HYPERGRAPH-REGULARIZED SPARSE NMF FOR HU685 where Tr(·)stands for the trace operation of a matrix andL Hypers is the hypergraph Laplacian matrix of G s and definedbyL Hypers=D v−H W s(D e)−1(H )T.(17)Note that D v∈R N×N and D e∈R N×N in(17),respectively,denote the matrices of vertex degree and hyperedge degree ofhypergraph G s.By introducing the hypergraph-based regular-ization term(16)into the objective function of the originalL1/2-NMF,we obtain the HG L1/2-NMF as follows:F(A,S)= Y−AS 2F+α S 12+βTr(SL Hypers ST)(18)whereα∈R+andβ∈R+are the regularization parameters.C.Update RulesThe objective function F of HG L1/2-NMF in(18)is noncon-vex for both A and S simultaneously.Therefore,it is unrealistic to expect an optimization procedure to produce the global opti-mal solution.In this section,we deduce the MURs using the method of Lagrange multipliers[28],[29]tofind the local minima via an alternating optimization procedure.The objective function F in(18)can be reexpressed asF=Tr((Y−AS)(Y−AS)T)+α S 12+βTr(SL Hypers ST)=Tr(Y Y T)−2Tr(Y S T A T)+Tr(ASS T A T)+α A 12+βTr(SL Hypers ST).(19)Considering that the objective function is subject to the con-straints A lp≥0and S pn≥0,the corresponding Lagrange multipliers are defined asΥlp≥0andΛpn≥0,respectively. Then,the Lagrange functionΨis given byΨ=Tr(Y Y T)−2Tr(Y S T A T)+Tr(ASS T A T)+α S 12+βTr(SL Hypers ST)+Tr(ΥA T)+Tr(ΛS T).(20)Taking the partial derivatives ofΨwith respect to A and S, respectively,we get∂Ψ∂A =−2Y S T+2ASS T+Υ(21)∂Ψ∂S =−2A T Y+2A T AS+α2S−12+2βSL Hypers+Λ.(22)Using the Karush–Kuhn–Tucker(KKT)conditionsΥlp A lp=0 andΛpn S pn=0,the following formulas for A lp and S pn are obtained(ASS T)lp A lp−(Y S T)lp A lp=0(23)(A T AS)pn S pn+(α4S−12)pn S pn+(βSD v)pn S pn =(A T Y)pn S pn+β(SH W s(D e)−1(H )T)pn S pn.(24)Fig.3.Six endmembers used to create synthetic data sets.From the above equations,we can deduce the following MURs:A lp←A lp(Y S T)lp(ASS T)lp(25)S pn←S pn(A T Y+βSH W s(D e)−1(H )T)pn(A T AS+(α4S−12)+βSD v)pn.(26)The convergences of these two update rules are guaranteed by the Theorem1.Theorem1:The objective function F in(18)is nonincreas-ing under the update rules in(25)and(26).The detail proof for Theorem1is presented in the Appendix. The proof procedure is similar to the convergence proof of the standard NMF[22].D.Algorithm ImplementationIn this section,several implementation issues of the pro-posed unmixing algorithm are addressed.Thefirst issue is about the initialization of the matrices A and S.Since the objective function in(18)is nonconvex with respect to A and S simul-taneously,we can only expect to achieve local minima with the aforementioned update rules.Thus,it will yield different unmixing outputs when different initial settings are provided for these two matrices.In this paper,we adopt two initialization methods:1)random initialization and2)VCA-FCLS initial-ization.VCA-FCLS initialization provides initial endmembers and abundances via vertex component analysis(VCA)[30] and fully constrained least squares(FCLSs)[31]algorithms, respectively.In the experiments on synthetic HSI data,ran-dom initialization is employed,which generates the initial value for each entry of A and S randomly in interval[0,1],whereas VCA-FCLS initialization is used in the experiments on real HSI data.The second issue involves the ASC constraint.As the method suggested in[4]and[31],the observed HSI data matrix Y and the endmember matrix A are,respectively,augmented as686IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERV ATIONS AND REMOTE SENSING,VOL.9,NO.2,FEBRUARY2016 Fig.4.Performance of HG L1/2-NMF with respect to parameters(a)βand(b)K.Fig.5.Results on synthetic data with different SNRs:(a)SAD and(b)RMSE.Y a←Yδ1T N(27)A a←Aδ1T P(28)whereδcontrols the impact of ASC on the cost function,whichis set to15in the experiments as suggested in[32].1T N and1T Pare one row vectors with the dimensions of P and N,respec-tively.Accordingly,in each update step of S,the matrices Yand A are replaced by Y a and A a,respectively,in the updaterule(26).The third issue concerns the L1/2sparsity operator in theupdate rule(26).Since the operation(s pn)−12is invalid if s pnequals zero,effective measures should be taken to keep therobustness of the algorithm.Here,we follow the method used in[4],in which those entries of S with the values less than a pre-defined threshold10−4will be neglected by the correspondingsparsity operator.The last issue is about the selection of the parameters ofHG L1/2-NMF algorithm.There are four critical parameters:the size of neighborhood,the regularization parametersα,β,and the number of neighbors K.In the experiments,the size ofneighborhood is defined as5×5.The regularization parameterαcan impact the effect of sparsity constraint,which has beeninvestigated in[4].A proper value forαis determined byα=1√LLl=1√N− [Y]l,: 1/ [Y]l,: 2√N−1(29)where[Y]l,:denotes the l th band of the observed HSI Y.As forhow to select the suitable values forβand K,it will be analyzedin the experiments.Based on these implementation techniques,the proposedHG L1/2-NMF algorithm for unmixing is given in Algorithm1.Its computational cost primarily concerns the update steps ofmatrices A and S included in the iteration body.For the pur-pose of accurately analyzing the computational complexity,we count thefloating-point arithmetic operations involved inthe algorithm.To be specific,the number offloating-pointoperations needed by the step6is2N(LP+P2)+P2(2L−1);whereas,in step8,it is N2(P2+2P)+N(2LP+P2+6P)+P2(2L−1).Given that the algorithm terminates afterm iterations,the overall computational cost of the proposedalgorithm is O(mN2(P2+2P)).Algorithm1.HG L1/2-NMF Algorithm For HyperspectralUnmixingInput:The hyperspectral image Y≡[y1,y2,...,y N]∈R L×N;The algorithm regularization parametersαandβ;The number of nearest neighbors K;Output:The abundance matrix S;The endmember matrix A;1:Estimate the number of endmember P;2:Initialize endmember matrix A and abundance matrix S;3:Augment Y to generate Y a by(27);4:Construct hypergraph G s and compute L Hypers by(17);5:repeat6:Update A according to(25);7:Augment A to obtain A a following(28);8:Update S by(26);9:until the stopping condition of the algorithm is satisfied.W ANG et al.:HYPERGRAPH-REGULARIZED SPARSE NMF FOR HU687 Fig.6.Results on synthetic data with different degrees of mixing:(a)SAD and(b)RMSE.Fig.7.Results on synthetic data with different numbers of pixels:(a)SAD;(b)RMSE;and(c)Runningtime.Fig.8.Convergence curves of all thealgorithms.Fig.9.False-color image of the HYDICE Washington DC Mall data.IV.E XPERIMENTS R ESULTSIn this section,a series of experiments are conducted on syn-thetic and real-world HSI data to investigate the performanceof our algorithm.In the experiments on synthetic data,wecompare the proposed HG L1/2-NMF algorithm with threerepresentative algorithms:1)L1/2-NMF;2)GLNMF;and3)VCA-FCLS.In the experiments on real data,the proposedHG L1/2-NMF algorithm is compared with six representativemethods:1)ASSNMF;2)L1/2-NMF;3)GLNMF;4)DGNMF;5)minimum-volume-constrained NMF(MVCNMF)[7];and6)VCA.In these algorithms,HG L1/2-NMF,ASSNMF,andDGNMF consider the spatial information in unmixing.A.Performance CriteriaTo measure the unmixing accuracy of the algorithms,twowidely used criteria named spectral angle distance(SAD)androot-mean-square error(RMSE)are adopted in our experi-ments.For the p th endmember,the SAD between the estimatedendmemberˆa p and the corresponding reference signature a p isdefined asSAD p=cos−1a T pˆa pa p ˆa p(30)where · stands for the Euclidean norm of vector.In order tocompare the similarity between the reference abundances andthe estimated ones,the RMSE criteria for the p th endmember isdefined asRMSE p=1NNj=1(s pj−ˆs pj)2(31)where s pj denotes the reference abundance for the p th endmem-ber at the j th pixel,andˆs pj represents the estimated one.The。