Low-Rank Subspace Representation for Estimating

Low-Rank Subspace Representation for Estimating the Number of Signal Subspaces in

Hyperspectral Imagery

Alex Sumarsono,Member,IEEE,and Qian Du,Senior Member,IEEE

Abstract—In this paper,we consider signal subspace estimation based on low-rank representation for hyperspectral imagery.It is often assumed that major signal sources occupy a low-rank subspace.Due to the mixed nature of hyperspectral remote sensing data,the underlying data structure may include multiple sub-spaces instead of a single subspace.Therefore,in this paper,we propose the use of low-rank subspace representation to estimate the number of subspaces in hyperspectral imagery.In particular, we develop simple estimation approaches without user-defined parameters because these parameters can befixed as constants. Both real data experiments and computer simulations demon-strate excellent performance of the proposed approaches over those currently in the literature.

Index Terms—Data dimensionality,hyperspectral imagery, low-rank representation(LRR),low-rank subspace representation (LRSR),rank estimation,signal subspace estimation.

I.I NTRODUCTION

E STIMATION of data intrinsic dimensionality(ID)is a

very challenging problem[1].By definition,ID is referred to as the minimum number of parameters required to account for the observed properties of a data set[2].ID is often much less than data dimensionality.For remotely sensed hyperspec-tral data that includes hundreds of spectral bands,data dimen-sionality is very high,which is equal to the number of spectral channels.However,its ID can be a very small value.For in-stance,in the linear mixture model,a pixel in a remotely sensed image with relatively rough spatial resolution is often consid-ered as linear mixture of pure materials,called endmembers, present in the image scenes.Obviously,based on the definition of ID,the number of true endmembers that construct an entire image data may be closely related to the ID.In[3],the concept of virtual dimensionality(VD)was proposed,which is the min-imum number of distinct signal sources.The value of VD may be larger than the ID,but it is relatively easier to be estimated. ID estimation can be approached by rank estimation tech-niques,which are often based on information criteria,such as the Akaike information criterion[4]and the minimum

Manuscript received January28,2015;revised April1,2015and April25, 2015;accepted May23,2015.

The authors are with the Department of Electrical and Computer Engineer-ing,Mississippi State University,Mississippi State,MS39762USA.

Color versions of one or more of thefigures in this paper are available online at .

Digital Object Identifier10.1109/TGRS.2015.2438079description length[5].More discussion can be found in[6]. Unfortunately,it has been shown in[3]that those methods do not perform well for hyperspectral image data when noise is often not white.

Another frequently used approach is principal component analysis(PCA)by searching for the most significant eigenval-ues of the data covariance matrix.Due to the complexity of real data,the significance level is difficult to be predetermined; thus,some thresholding methods,such as the one using an energy percentage,Malinowsk’s method in[7],etc.,can be applied.More sophisticated generalized PCA(GPCA)was also proposed[8].However,threshold selection in these methods is critical but unfortunately empirical.

To statistically gauge the significance of eigenvalues,several methods,such as Harsanyi-Farrand-Chang(HFC),the noise-whitened HFC(NWHFC),and the noise subspace projection (NSP),were proposed in[3].The HFC was modified to be parameter-free in[9],which also examines the difference be-tween eigenvalue pairs of covariance and correlation matrices of original data as in[3].

It is well known that major data information tends to be distributed in a low-dimensional subspace,although hyperspec-tral imagery has very high data dimensionality.An interesting method called hyperspectral signal identification by minimum error(HySime)was proposed in[10],which searches for sig-nificant eigenvectors to construct the low-rank signal subspace. Considering the fact that data may contain rare components, such as outliers,the maximum orthogonal complement analysis algorithm(MOCA,[11])and the robust signal subspace estima-tion algorithm(RSSE,[12])were ter,the compu-tationally efficient MOCA,called modified MOCA(MMOCA) [13],and robust signal subspace identification in the presence of dependent noise(RSSI-SD)method[14]were also developed. Other work considering outliers during rank estimation can be found in[15].Nearest neighbor distance and random matrix theory are also investigated for ID or VD estimation[16]–[18]. In this paper,the recently popular low-rank and sparse matrix decomposition technique is considered to estimate the signal subspace and a sparse matrix accommodating outliers or rare components.The original idea was proposed as robust prin-cipal component analysis(RPCA)[19].If data recovery can be achieved by low-rank representation(LRR),more accurate estimation of signal subspaces may be possible by working on the low-rank matrix.Unfortunately,due to the complexity of real data,such as hyperspectral image data,the resulting

0196-2892©2015IEEE.Personal use is permitted,but republication/redistribution requires IEEE permission.

See /publications_standards/publications/rights/index.html for more information.