1946 ieeetransactionsonsignalprocessing,vol.64,no.8,april15… · 2020-03-22 · 1950...

1946 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 64, NO. 8, APRIL 15, 2016

A Fast Hyperplane-Based Minimum-VolumeEnclosing Simplex Algorithm for Blind

Hyperspectral UnmixingChia-Hsiang Lin, Chong-Yung Chi, Senior Member, IEEE, Yu-Hsiang Wang, and Tsung-Han Chan

Abstract—Hyperspectral unmixing (HU) is a crucial signalprocessing procedure to identify the underlying materials (or end-members) and their corresponding proportions (or abundances)from an observed hyperspectral scene. A well-known blind HUcriterion, advocated by Craig during the early 1990s, considersthe vertices of the minimum-volume enclosing simplex of the datacloud as good endmember estimates, and it has been empiricallyand theoretically found effective even in the scenario of no purepixels. However, such kinds of algorithms may suffer from heavysimplex volume computations in numerical optimization, etc. Inthis paper, without involving any simplex volume computations,by exploiting a convex geometry fact that a simplest simplex ofvertices can be defined by associated hyperplanes, we proposea fast blind HU algorithm, for which each of the hyperplanesassociated with the Craig’s simplex of vertices is constructedfrom affinely independent data pixels, together with anendmember identifiability analysis for its performance support.Without resorting to numerical optimization, the devised algo-rithm searches for the active data pixels via simplelinear algebraic computations, accounting for its computationalefficiency. Monte Carlo simulations and real data experiments areprovided to demonstrate its superior efficacy over some bench-mark Craig-criterion-based algorithms in both computationalefficiency and estimation accuracy.

Index Terms—Hyperspectral unmixing, Craig’s criterion,convex geometry, minimum-volume enclosing simplex, hyper-plane.

I. INTRODUCTION

H YPERSPECTRAL remote sensing (HRS) [2]–[4], alsoknown as imaging spectroscopy, is a crucial technology

to the identification of material substances (or endmembers) as

Manuscript received April 01, 2015; revised October 02, 2015; accepted De-cember 01, 2015. Date of publication December 17, 2015; date of current ver-sion February 26, 2016. The associate editor coordinating the review of thismanuscript and approving it for publication was Prof. Andre de Almeida. Thiswork is supported partly by the National Science Council, R.O.C., under GrantNSC 102-2221-E-007-035-MY2, and partly by Ministry of Science and Tech-nology, R.O.C., under Grant MOST 104-2221-E-007-069-MY3. Part of thiswork was presented at IEEE ICASSP 2015 [1]. (Corresponding author:Chia-Hsiang Lin.)C.-H. Lin, C.-Y. Chi, and Y.-H. Wang are with the Institute of

Communications Engineering and the Department of Electrical En-gineering, National Tsing Hua University, Hsinchu, Taiwan 30013,R.O.C. (e-mail: [email protected]; [email protected];[email protected]).T.-H. Chan is with MediaTek Inc., National Science Park, Hsinchu, Taiwan

30013, R.O.C. (e-mail: [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TSP.2015.2508778

well as their corresponding fractions (or abundances) presentin a scene of interest from observed hyperspectral data, havingvarious applications such as planetary exploration, land map-ping and classification, environmental monitoring, and mineralidentification and quantification [5]–[7]. The observed pixels inthe hyperspectral imaging data cube are often spectral mixturesof multiple substances, the so-called mixed pixel phenomenon[8], owing to the limited spatial resolution of the hyperspectralsensor (usually equipped on board the satellite or aircraft) uti-lized for recording the electromagnetic scattering patterns of theunderlying materials in the observed hyperspectal scene overabout several hundreds of narrowly spaced (typically, 5–10 nm)wavelengths that contiguously range from visible to near-in-frared bands. Occasionally, the mixed pixel phenomenon canalso be present in hyperspectral scenes where the underlyingmaterials are intimately mixed with each other [9] even thoughthe spatial resolution is high enough. Hyperspectral unmixing(HU) [8], [10], an essential procedure of extracting individualspectral signatures of the underlying materials in the capturedscene from these measured spectral mixtures, is therefore ofparamount importance in the HRS context.Blind HU, or unsupervised HU, involves two core stages,

namely endmember extraction and abundance estimation,without (or with very limited) prior knowledge about the end-members’ nature or the mixing mechanism. Some endmemberextraction algorithms (EEAs), such as alternating projectedsubgradients (APS) [11], joint Bayesian approach (JBA) [12],and iterated constrained endmembers (ICE) [13] (also thesparsity promoting ICE (SPICE) [14]), can simultaneouslydetermine the associated abundance fractions while extractingthe endmember signatures. Nevertheless, some EEAs performendmember estimation, followed by abundance estimationusing such as the fully constrained least squares (FCLS) [15] tocomplete the entire HU processing.The pure-pixel assumption has been exploited in devising

fast blind HU algorithms to search for the purest pixels overthe data set as the endmember estimates, and such searchingprocedure can always be carried out through simple linear al-gebraic formulations; see, e.g., pixel purity index (PPI) [16]and vertex component analysis (VCA) [17]. An important blindHU criterion, called Winter’s criterion [18], also based on thepure-pixel assumption, is to identify the vertices of the max-imum-volume simplex inscribed in the observed data cloud asendmember estimates. HU algorithms in this category include

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LIN et al.: A FAST HYPERPLANE-BASED MINIMUM-VOLUME ENCLOSING SIMPLEX ALGORITHM FOR BLIND HU 1947

N-finder (N-FINDR) [18], simplex growing algorithm (SGA)[19] (also the real-time implemented SGA [20]), and worst casealternating volume maximization (WAVMAX) [21], to name afew. However, the pure-pixel assumption could be seriously in-fringed in practical scenarios especially when the pixels are in-timately mixed, for instance, the hyperspectral imaging data forretinal analysis in the ophthalmology [9]. In these scenarios, HUalgorithms in this category could completely fail; actually, it isproven that perfect endmember identifiability is impossible forWinter-criterion-based algorithms if the pure-pixel assumptionis violated [21].Without relying on the existence of pure pixels, another

promising blind HU approach, advocated by Craig in early1990’s [22], exploits the simplex structure of hyperspectraldata, and believes that the vertices of the minimum-volumedata-enclosing simplex can yield good endmember estimates,and algorithms developed accordingly include such as min-imum-volume transform (MVT) [22], minimum-volume con-strained nonnegative matrix factorization (MVC-NMF) [23],and minimum-volume-based elimination strategy (MINVEST)[24]. Moreover, some linearization-based methods have alsobeen reported to practically identify Craig’s minimum-volumesimplex, e.g., the iterative linear approximation in min-imum-volume simplex analysis (MVSA) [25] (also its fastimplementation using the interior-point method [26], termedas ipMVSA [27]), and the alternating linear programming inminimum-volume enclosing simplex (MVES) [28]. Empiricalevidences do well support that this minimum-volume approachis resistant to lack of pure pixels, and can recover ground truthendmembers quite accurately even when the observed pixelsare heavily mixed. Very recently, the validity of this empiricalbelief has been theoretically justified; specifically, we showthat, as long as a key measure concerning the pixels’ mixinglevel is above a certain (small) threshold, Craig’s simplexcan perfectly identify the true endmembers in the noiselessscenario [29]. However, without the guidance of the pure-pixelassumption, this more sophisticated criterion would generallylead to more computationally expensive HU algorithms. To thebest of our knowledge, the ipMVSA algorithm [27] and thesimplex identification via split augmented Lagrangian (SISAL)algorithm [30] are the two state-of-the-art Craig-criterion-basedalgorithms in terms of computational efficiency. Nevertheless,in view of not only the NP-hardness of the Craig-simplex-iden-tification (CSI) problem [31] but also heavy simplex volumecomputations, all the above mentioned HU algorithms are yetto be much more computationally efficient. Moreover, theirperformances may not be very reliable owing to the sensitivityto regularization parameter tuning, non-deterministic (i.e.,non-reproducible) endmember estimates caused by random ini-tializations, and, most seriously, lack of rigorous identifiabilityanalysis.In this work, we break the deadlock on the trade-off between

a simple fast algorithmic scheme and the estimation accuracy inthe no pure-pixel case. We have observed that when the pure-pixel assumption holds true, the effectiveness of a simple fastHU algorithmic scheme could be attributed to that the desiredsolutions (i.e., pure pixels) already exist in the data set. Inspired

by this observation, we naturally raise a question: Can Craig’sminimum-volume simplex be identified by simply searching fora specific set of pixels in the data set regardless of the existenceof pure pixels? The answer is affirmative and will be given inthis paper.Based on the convex geometry fact that a simplest simplex

of vertices can be characterized by the associated hy-perplanes, this paper proposes an efficient and effective unsu-pervised Craig-criterion-based HU algorithm, together with anendmember identifiability analysis. Each hyperplane, parame-terized by a normal vector and an inner product constant [26],can then be estimated from affinely independent pixelsin the data set via simple linear algebraic formulations. The re-sulting hyperplane-based CSI (HyperCSI) algorithm, based onthe above pixel search scheme, can withstand the no pure-pixelscenario, and can yield deterministic, non-negative, and, mostimportantly, accurate endmember estimates. After endmemberestimation, a closed-form expression in terms of the identifiedhyperplanes’ parameters is derived for abundance estimation.Then some Monte Carlo numerical simulations and real hyper-spectral data experiments are presented to demonstrate the su-perior efficacy of the proposed HyperCSI algorithm over somebenchmark Craig-criterion-based HU algorithms in both esti-mation accuracy and computational efficiency.The remaining part of this paper is organized as follows.

In Section II, we briefly review some essential convex geom-etry concepts, followed by the signal model and dimension re-duction. Section III focuses on the HyperCSI algorithm devel-opment, and in Section IV, some simulation results are pre-sented for its performance comparison with some benchmarkCraig-criterion-based HU algorithms. In Section V, we furtherevaluate the effectiveness of the proposed HyperCSI algorithmwith AVIRIS [32] data experiments. Finally, we draw some con-clusions in Section VI.The following notations will be used in the ensuing presenta-

tion. is the set of real numbers ( -vectors,matrices). is the set of non-negative real

numbers ( -vectors, matrices). isthe set of positive real numbers ( -vectors, matrices).

denotes the Moore-Penrose pseudo-inverse of a matrix .and are all-one and all-zero -vectors, respectively.

denotes the unit vector of proper dimension with the th entryequal to unity. is the identity matrix. and standfor the componentwise inequality and strictly componentwiseinequality, respectively. denotes the Euclidean norm (i.e.,2-norm). The distance of a vector to a set is denoted by

[26]. denotes the cardinalityof the set . The determinant of matrix is represented by

. stands for the set of integers , for anypositive integer .

II. CONVEX GEOMETRY AND SIGNAL MODEL

In this section, a brief review on some essential convex geom-etry will be given for ease of later use. Then the signal modelfor representing the hyperspectral imaging data together withdimension reduction preprocessing will be presented.


Fig. 1. A graphical illustration in for some convex geometry concepts,where the line segment connecting and is the convex hull of ,the straight line passing and is the affine hull of , the shaded tri-angle is the convex hull of , and the plane passing the three points

is the affine hull of . As an affine hull in is calleda hyperplane if its affine dimension is 2, is a hyperplane, while

is not.

A. Convex Geometry PreliminaryThe convex hull of a given set of vectors

is defined as [26]

where (cf. Fig. 1). A convex hullis called an -dimensional simplex

with vertices if is affinelyindependent, or, equivalently, ifis linearly independent, and it is called a simplest simplex in

when [33]. For example, a triangle is a2-dimensional simplest simplex in , and a tetrahedron is a3-dimensional simplest simplex in (cf. Fig. 1).For a given set of vectors , its affine hull

is defined as [26]

where (cf. Fig. 1). This affine hull can beparameterized by a 2-tuple using thefollowing alternative representation [26]:

where (the rank of ) is the affine dimension of. Moreover, an affine hull

is called a hyperplane if its affine dimension(cf. Fig. 1).

B. Signal Model and Dimension ReductionConsider a scenario where a hyperspectral sensor measures

solar electromagnetic radiation over spectral bands fromunknown materials (endmembers) in a scene of interest. Basedon the linear mixing model (LMM) [2]–[8], [10], [28], wherethe measured solar radiations are assumed to reflect from the ex-plored scene through one single bounce, and the endmembers’spectral signature vectors are assumed to be invariant

with the pixel index , each pixel in the observeddata set can then be represented as a linear mixture of theendmembers’ spectral signatures1

(1)

where is the spectral signature ma-trix, is the abundance vector,and is the total number of pixels. The following standard as-sumptions pertaining to the model in (1), which also charac-terize the simplex structure inherent in the hyperspectral data,are used in our HU algorithm development later [2]–[8], [10],[28]:

(Non-negativity) and .(Full-additivity) .

and is full columnrank.

Moreover, like most benchmark HU algorithms (see, e.g., [23],[25], [28], [30]), the number of endmembers is assumed tobe known a priori, which can be determined beforehand byapplying model-order selection methods, such as hyperspec-tral signal subspace identification by minimum error (HySiMe)[36], and Neyman-Pearson detection theory-based virtual di-mensionality (VD) [37].We aim to blindly estimate the unknown endmembers (i.e.,

), as well as their abundances (i.e., ),from the observed spectral mixtures (i.e., ). Dueto the huge dimensionality of hyperspectral data, directly an-alyzing the data may not be very computationally efficient. In-stead, an efficient data preprocessing technique, called affine setfitting (ASF) procedure [38], can be applied to compactly rep-resent each measured pixel in a dimension-reduced(DR) space as follows:

(2)

where

(3)

(4)(5)

in which denotes the th principal eigenvector(with unit norm) of the square matrix , and

is the mean-removeddata matrix. Actually, like other dimension reduction algorithms[39], ASF also performs noise suppression in the meantime. Ithas been shown that the aboveASF best represents themeasureddata in an -dimensional space in the sense of least-squares fitting error [38], while such fitting error vanishes in the

1Note that there is a research line considering non-linear mixtures for mod-eling the effect of multiple reflections of solar radiation [34]. Moreover, theendmember spectral signatures may be spatially varying, hence leading to thefull-additivity in being violated [10]. However, studying these effects isout of the scope of this paper, and the representative LMM is sufficient for ouranalysis and algorithm development; interested readers can refer to the maga-zine papers [34] and [35], about the non-linear effect and the endmember vari-ability effect, respectively.


noiseless scenario [38]. Note that the data mean in the DR spaceis the origin (by (2) and (3)).Because of in typical HU applications, the

HyperCSI algorithm will be developed in the DR spacewherein the DR endmembers are estimated. Then,by (5), the endmember estimates in the original space canbe restored as

(6)

where ’s are the endmember estimates in the DR space.

III. HYPERPLANE-BASED CRAIG-SIMPLEX-IDENTIFICATIONALGORITHM

First of all, due to (2) and – , the true endmembers’convex hull itself is a data-enclosing sim-plex, i.e.,

(7)

According to Craig’s criterion, the true endmembers’ convexhull is estimated by minimizing the volume of the data-en-closing simplex [22], namely, by solving the following volumeminimization problem (called the CSI problem interchangeablyhereafter):

(8)

where denotes the volume of the simplex. Under some mild conditions on

data purity level [29], the optimal solution of problem (8) canperfectly yield the true endmembers .2Besides in the HU context, the NP-hard CSI problem in (8)

[31] has been studied in some earlier works in mathematicalgeology [40] and computational geometry [41]. However, theirintractable computational complexity almost disable them frompractical applications for larger problem size [41], mainly owingto calculation of the complicated nonconvex objective function[28]

in (8). Instead, the HyperCSI algorithm to be presented can judi-ciously bypass simplex volume calculations, and meanwhile theidentified simplex can be shown to be exactly the “minimum-volume” (data-enclosing) simplex in the asymptotic sense

.First of all, let us succinctly present the actual idea on which

the HyperCSI algorithm is based. As the Craig’s minimum-

2In [29], is usedto measure the data purity level of , whereand ; the geometric interpretations of canbe found in [29]. Simply speaking, one can show that , and themost heavily mixed scenario (i.e., ) will lead to thelower bound [29]. On the contrary, the pure-pixel assumption is equivalent tothe condition of (the upper bound) [29], in comparison with which a mildcondition of only is sufficient to guarantee the perfect endmemberidentifiability of problem (8) [29].

Fig. 2. An illustration of hyperplanes and DR data in for the case of, where is a purest pixel in (a purest pixel can be considered

as the pixel closest to ) but not necessarily very close to hyperplane, leading to nontrivial orientation difference between and .

However, the active pixels and identified by (17) will be very closeto (especially, for large ), and hence the orientations of and willbe almost the same. On the other hand, one can see that the pixels identified by(21) are (that are very close to each other), and thus the associatednormal vector estimate is obviously far away from the true .

volume simplex can be uniquely determined by tightly en-closed -dimensional hyperplanes, where each hyper-plane can be reconstructed from any affinely indepen-dent points of the hyperplane, we hence endeavor to search for

affinely independent pixels (referred to as active pixelsin ) that are as close to the associated hyperplane as possible.We begin with purest pixels that define disjoint proper re-gions, each centered at a different purest pixel. Then for eachhyperplane of the minimum-volume simplex, the desiredactive pixels, that are as close to the hyperplane as possible, arerespectively sifted from subsets of , each from the dataset associated with one different proper region (cf. Fig. 2). Thenthe obtained pixels are used to construct one estimatedhyperplane. Finally, the desired minimum-volume simplex canbe determined from the obtained hyperplane estimates.

A. Hyperplane Representation for Craig’s SimplexThe idea of solving the CSI problem in (8), without involving

any simplex volume computations, is based on the hyperplanerepresentation of a simplest simplex as stated in the followingproposition:Proposition 1: If is affinely inde-

pendent, i.e., is a simplestsimplex, then can be reconstructed from the associated hy-perplanes , that tightly enclose , where

.Proof: It suffices to show that can be deter-

mined by . It is known that hyperplane canbe parameterized by a normal vector and an innerproduct constant as follows:

(9)

As for all , we havefrom (9) that for all , i.e.,

(10)

where are defined as

(11)


(12)

As is a simplest simplex in must be of full rankand hence invertible [26]. Hence, we have from (10) that

(13)

implying that can be reconstructed. The proof is thereforecompleted.As it can be inferred from that the set of DR end-

members is affinely independent, one can applyProposition 1 to decouple the CSI problem (8) into sub-problems of hyperplane estimation, namely, estimation ofparameter vectors in (9). Then (13) can be utilized toobtain the desired endmember estimates. Next, let us presenthow to estimate the normal vector and the inner productconstant from the data set , respectively.

B. Normal Vector EstimationThe normal vector of hyperplane can be obtained by

projecting the vector (for any ) onto the subspacethat is orthogonal to the hyperplane [42], i.e.,

(14)

where , andis the matrix with its

th and th columns removed. Besides (14) for obtaining thenormal vector of , we also need another closed-form ex-pression of in terms of distinct points as given in thefollowing proposition.Proposition 2: Given any affinely independent set

can be alternatively obtained by(except for a positive scale factor)

(15)

where is defined in (14).The proof of Proposition 2 can be shown from the fact thatis the data mean in the DR space (by (2) and (3)), and isomitted here due to space limitation.Based on Proposition 2, we estimate the normal vector by

finding affinely independent data points

that are as close to as possible. To this end, an observationfrom (7) is needed and given in the following fact:Fact 1: Observing that (i) (i.e., all

the points lie on the same side of ; cf. (7)), andthat (ii) , the pointclosest to is exactly the one with maximum of , providedthat points outward from the true endmembers’ simplex (cf.Fig. 2 and (14)).Suppose that we are given “purest” pixels , which

basically maximize the simplex volume inscribed in , and theycan be obtained using the reliable and reproducible successiveprojection algorithm (SPA) [10], ([43], Algorithm 4). So canbe viewed as the pixel in “closest” to (cf. Fig. 2). Let

be the outward-pointing normal vector of hyperplane, i.e.,

(16)

Considering Fact 1 and the requirement that the set mustcontain distinct elements (otherwise, is not affinelyindependent), we identify the desired affinely independent set

by:

(17)

where are disjoint sets defined as

(18)

in which is theopen Euclidean norm ball with center and radius

. Note thatthe choice of the radius is to guarantee thatare non-overlapping regions, thereby guaranteeing thatcontains distinct points. Moreover, each hyperballmust contain at least one pixel (as it contains either or ;cf. (18)), i.e., , and hence problem (17) must bea feasible problem (i.e., a problem with non-empty feasible set[26]).If the points extracted by (17) are affinely independent,

then the estimated normal vector associated with can be de-termined as (cf. Proposition 2)

(19)

Fortunately, the obtained by (17) can be proved (in Theorem1 below) to be always affinely independent with one more as-sumption:3

The abundance vectors (definedbelow (1)) are independent and identically distributed(i.i.d.) following Dirichlet distribution with parameter

whose probability densityfunction (p.d.f.) is given by [44]:

(20)

where, and

denotes the gamma function.Theorem 1: Assume – hold true. Let

be a solution to (17) with defined in (18), for all

3The rationale of adopting Dirichlet distribution in is not only that it isa well known distribution that captures both the non-negativity and full-addi-tivity of [44], but because it has been used to characterize the distribution of

in the HU context [45], [46]. However, the statistical assumption isonly for analysis purpose without being involved in our geometry-oriented algo-rithm development. So even if abundance vectors are neither i.i.d. nor Dirichletdistributed, the HyperCSI algorithm can still work well (cf. Subsection IV-D).Furthermore, we would like to emphasize that, in our analysis (Theorems 1 and2), we actually only use the following two properties of Dirichlet distribution:(i) its domain is , and (ii) it is a continuousmultivariate distribution with strictly positive density on its domain [47] (cf.Appendixes A and B). Hence, any distribution with these two properties can beused as an alternative in .


and . Then, the set is affinely independent withprobability 1 (w.p.1).The proof of Theorem 1 is given in Appendix A.Note that the orientation difference between and the truemay not be small (cf. Fig. 2). Hence, itself may not be a

good estimate for either. On the contrary, it can be shown thatthe orientation difference between and tends to be smallfor large , and actually such difference vanishes as goes toinfinity (cf. Theorem 2 as well as Remark 1 in Subsection III-E).On the other hand, if the pixels with maximum inner productsin are jointly sifted from the whole data cloud , i.e.,

(21)

where and , rather thanrespectively from different regions , asgiven in (17), the identified pixels in may stay quite close,easily leading to large deviation in normal vector estimation asillustrated in Fig. 2 where are the identifiedpixels using (21). This is also a rationale of finding using(17) for better normal vector estimation.

C. Inner Product Constant EstimationFor Craig’s simplex (the minimum-volume data-enclosing

simplex), all the data in should lie on the same side of(otherwise, it is not data-enclosing), and should be as tightlyclose to the data cloud as possible (otherwise, it is not min-imum-volume); the only possibility is when the hyperplanemust be externally tangent to the data cloud. In other words,will incorporate the pixel that has maximum inner product with, and hence it can be determined as , where is

obtained by solving(22)

However, it has been reported that when the observed datapixels are noise-corrupted, the random noise may expand thedata cloud, thereby inflating the volume of the Craig’s data-en-closing simplex [21], [33]. As a result, the estimated hyper-planes are pushed away from the origin (i.e., the data mean inthe DR space) due to noise effect, and hence the estimated innerproduct constant in (22) would be larger than that of the groundtruth. To mitigate this effect, the estimated hyperplanes need tobe properly shifted closer to the origin, so instead,

, are the desired hyperplane estimates for some .Therefore, the corresponding DR endmember estimates are ob-tained by (cf. (13))

(23)

where and are given by (11) and (12) with andreplaced by and , respectively. Moreover, it

is necessary to choose such that the associated endmemberestimates in the original space are non-negative (cf. ), i.e.,

(24)

By (23) and (24), the hyperplanes should be shifted closer to theorigin with at least, where

(25)

TABLE IPSEUDO-CODE FOR HYPERCSI ALGORITHM

which can be further shown to have a closed-form solution:(26)

where is the th component of andis the th component of .Note that is just the minimum value for to yield non-nega-

tive endmember estimates. Thus, we can generally setfor some . Moreover, the value of is

empirically found to be a good choice for the scenarios wheresignal-to-noise ratio (SNR) is greater than 20 dB; typically, thevalue of SNR in hyperspectral data is much higher than 20 dB[32]. Let us emphasize that the larger the value of (or thesmaller the value of ), the farther the estimated hyperplanesfrom the origin , or the closer the estimated endmembers’simplex to the boundary of the nonnegativeorthant . On the other hand, we empirically observed thattypical endmembers in the U.S. geological survey (USGS) li-brary [48] are close to the boundary of . That is to say, areasonable choice of should be relatively large (i.e.,relatively close to 1), accounting for the reason why the presetvalue of can always yield good performance. The re-sulting endmember estimation processing of the HyperCSI al-gorithm is summarized in Steps 1 to 6 in Table I.

D. Abundance EstimationThough the abundance estimation is often done by solving

FCLS problems [15], which can be equivalently formulated inthe DR space into (cf. ([49], Lemma 3.1))

(27)

it has been reported that some geometric quantities, acquiredduring the endmember extraction stage, can be used to signif-icantly accelerate the abundance estimation procedure [50].With similar computational efficiency improvements taken intoaccount, we aim at expressing the abundance in termsof readily available quantities (e.g., normal vectors and innerproduct constants) obtained when estimating the endmembers,in this subsection. The results are summarized in the followingproposition:Proposition 3: Assume – hold true. Then

has the following closed-form expression:

(28)


Proposition 3 can be derived from some simple geometrical ob-servations (cf. items (i) and (ii) in Fact 1) and the following wellknown formula in the Algebraic Topology context

(29)

and its proof is omitted here due to space limitation; note thatthe formula (29) has been recently derived again using differentapproach in the HU context ([50], (12)).Based on (28), the abundance vector can be estimated as

(30)

where is to enforce the non-negativityof abundance fractions (cf. Step 7 in Table I). One canshow that when , the abundanceestimates obtained using (30) is exactly the solution to theFCLS problem in (27), while using (30) yields much lowercomputational cost than solving FCLS problems. Nevertheless,one should be aware of a potential limitation of using (30).Specifically, if is too far away from the endmembers’simplex (i.e., is much larger than

for some ), the zeroing operation in (30) could causenontrivial deviation in abundance estimation. This can happenif is an outlier or the SNR is very low. However, as theSNR is reasonably high (like in AVIRIS data [21], [32]), mostpixels in the hyperspectral data are expected to lie inside orvery close to the endmembers’ simplex (cf.

– )—especially when the endmembers are extractedbased on Craig’s criterion. Hence, with the endmembers esti-mated by the Craig-criterion-based HyperCSI algorithm, simplyusing (30) to enforce the abundance non-negativity is not onlycomputationally efficient, but also still capable of yielding goodabundance estimation as will be demonstrated later with thesimulation results (Table III in Subsection IV-C and Table IV inSubsection IV-D).Unlike most of the existing abundance estimation algorithms,

where all the abundance maps must be jointly estimated (e.g.,FCLS [15]), the proposed HyperCSI algorithm offers an optionof solely obtaining the abundance map of a specific material ofinterest (say the th material)

(31)

to save computational cost, or obtaining all the abundance mapsby parallel processing (cf. (30)). Moreover, when

calculating using (30), the denominator is a con-stant for all pixel indices and hence only needs to becalculated once regardless of (which is usually large).

E. Identifiability and Complexity of HyperCSIIn this subsection, let us present the identifiability and com-

plexity analyses of the proposed HyperCSI algorithm. Particu-larly, the asymptotic identifiability of the HyperCSI algorithmcan be guaranteed as stated in the following theorem with theproof given in Appendix B:Theorem 2: Under – , the noiseless assumption and

, the simplex identified by HyperCSI algorithm with

(in Step 5 in Table I) is exactly the Craig’s minimum-volumesimplex (i.e., solution of (8)) and the true endmembers’ simplex

w.p.1.Two noteworthy remarks about the philosophies and intu-

itions behind the proof of this theorem are given as follows:Remark 1: With the abundance distribution stated in ,

the pixels in can be shown to be arbitrarily close toas the pixel number , and they are affinely independentw.p.1 (cf. Theorem 1). Therefore, can be uniquely obtainedby (19), and its orientation approaches to that of w.p.1.Remark 2: Remark 1 together with (7) implies that is

upper bounded by w.p.1 (assuming without loss of generalitythat ), and this upper bound can be shown to beachievable w.p.1 as . Thus, as , we have that

w.p.1.It can be further inferred, from the above two remarks, that

is exactly the true w.p.1 (cf. (23)) as in the absenceof noise. Although the identifiability analysis in Theorem 2 isconducted for the noiseless case and , we empiricallyfound that the HyperCSI algorithm can yield good endmemberestimates for a moderate and finite SNR, to be demonstratedby simulation results and real data experiments later.Next, we analyze the computational complexity of the Hy-

perCSI algorithm. The computation time of HyperCSI is pri-marily dominated by the computations of the feasible sets

(in Step 3), the active pixels in (in Step 4), and theabundances (in Step 7), and they are respectively analyzedin the following:

Step 3: Computing the feasible sets, is equivalent to computing the

sets ; cf. (18). Since is anopen Euclidean norm ball, the computation of each set

can be done by examining inequalities. However, examining each

inequality requires (i) calculating one 2-norm (in ),which yields , and (ii) checking whether this 2-normis smaller than , which yields . Hence, Step 3 yields

.Step 4: To determine , we have to identify the pixelfrom the set , whose complexityamounts to computing inner products in(each yielding ), and performing the point-wise max-imum operation among the values of these inner products(cf. (17)), and hence the complexity of identifying iseasily verified as . More-over, gathering requires the com-plexity

; the inequality is due to that s are disjoint. Re-peating the above for , Step 4 yields

.Step 7: Estimation of the abundances requires to com-pute the fraction in (30) times. Each fraction involves2 inner products (in ), 2 scalar subtractions, and 1scalar division, and thus costs . So, this step yields

.Therefore, the overall computational complexity of HyperCSIis .


Surprisingly, the complexity order of the proposedHyperCSI algorithm is the same as (rather than much higherthan) that of some pure-pixel-based EEAs; see, e.g., [17], [21],[43], [51]. Moreover, to the best of our knowledge, the MVESalgorithm [28] for which the CSI problem in (8) is handled bysolving linear programing (LP) subproblems in an alternativefashion, and the LPs are solved using efficient primal-dual in-terior-point method [26], is the existing Craig-criterion-basedalgorithm with lowest complexity order , whereis the number of iterations [28]. Hence, the introduced hyper-plane identification approach (without simplex volume compu-tations) indeed yields much smaller complexity than most ex-isting Craig-criterion-based algorithms.Let us conclude this section with a summary of some re-

markable features of the proposed HyperCSI algorithm (givenin Table I) as follows:a) Without involving any simplex volume computations, the

Craig’s minimum-volume simplex is reconstructed fromhyperplane estimates, i.e., the estimates ,

which can be obtained in parallel (cf. Step 4 in Table I)by searching for active pixels from . Thereconstructed simplex in the DR space is actuallythe intersection of halfspaces

.b) By noting that if, and only if, , the po-

tential requirement of pixels lying on, or closeto, the associated hyperplanes is not too difficult to be metin practice because the abundance maps in hyperspectralimaging often show large sparseness. This will be furtherdiscussed in more detail in experiments with AVIRIS datain Section V.

c) All the processing steps (including SPA in Step 2 ofTable I; cf. Algorithm 4 in [43]) can be carried out eitherby simple linear algebraic formulations or by closed-formexpressions, and so its high computational efficiency canbe anticipated.

IV. COMPUTER SIMULATIONS

This section demonstrates the efficacy of the proposed Hy-perCSI algorithm by Monte Carlo simulations. In the simula-tion, endmember signatures with spectral bands ran-domly selected from USGS library [48] are used to generatenoise-free synthetic hyperspectral data according to linearmixing model in (1), where the abundance vectors are i.i.d. gen-erated following the Dirichlet distribution with (cf.(20)) as it can automatically enforce and [28], [33].Then we add i.i.d. zero-mean Gaussian noise with varianceto the noise-free synthetic data for different values of SNRdefined as , and those neg-ative entries in the generated noisy data vectors are artificiallyset to zero, so as to maintain the non-negativity nature of hyper-spectral imaging data.The root-mean-square (rms) spectral angle error between

the true endmembers and their estimatesdefined as [17], [28]

(32)

Fig. 3. The endmember identifiability of the HyperCSI algorithm with finitedata length .

is used as the performance measure of endmember esti-mation, where

is the set of all permutations of. Similarly, the performance measure of abundance

estimation is the rms angle error defined as [28]

(33)

where and are the true abundance map of th endmember(cf. (31)) and its estimate, respectively. All the HU algorithmsunder test are implemented using Mathworks Matlab R2013arunning on a desktop computer equipped with Core-i7-4790KCPU with 4.00 GHz speed and 16 GB random access memory,and all the performance results in terms of , and compu-tational time are averaged over 100 independent realizations.Next, we show some simulation results for the endmember

identifiability for moderately finite data length (cf. Theorem 2),the choice of the parameter , and the performance evaluation ofthe proposed HyperCSI algorithm, in the following subsections,respectively.

A. Endmember Identifiability of HyperCSI for Finite Data

In Theorem 2, the perfect endmember identifiability of theproposed HyperCSI algorithm (with in Step 5 in Table I)under the noise-free scenario is proved in the asymptotic sense(i.e., the data length ). In this subsection, we wouldlike to show some simulation results (cf. Fig. 3) to illustratethe asymptotic identifiability of the HyperCSI algorithm and itsgood endmember estimation accuracy even with a moderatelyfinite number of pixels .Fig. 3 shows some simulation results of versus for

. From this figure, one can observe that for agiven decreases as increases, and the HyperCSI algo-rithm indeed achieves perfect identifiability (i.e., , cf.(32)) as . On the other hand, the HyperCSI algorithmneeds to identify essential pixels for the construc-tion of the Craig’s simplex, which indicates that the HyperCSIalgorithm would need more pixels to achieve good performance


Fig. 4. The average r.m.s. spectral angle error versus different values of .

for larger . Intriguingly, the results shown in Fig. 3 are con-sistent with the above inferences, where a larger correspondsto a slightly slower convergence rate of . However, theseresults also allude to a high possibility that the HyperCSI al-gorithm can yield accurate endmember estimates with a typicaldata length (i.e., several ten thousands) for high SNR in HRSapplications.

B. Choice of the ParameterThe simulation results for versus obtained by

the proposed HyperCSI algorithm, for(dB), and are shown in

Fig. 4. From this figure, one can observe that for a fixed ,the best choice of (i.e., the one that yields the smallest )decreases as SNR decreases. The reason for this is that thelarger the noise power, the more the data cloud is expanded,and hence the more the desired hyperplanes should be shiftedtowards the data center (implying a larger or a smaller ).Moreover, one can also observe from Fig. 4 that for eachscenario of , the best choice of basically belongsto the interval [0.8, 1], a relatively large value in the interval(0, 1], as discussed in Subsection III-C. It is also interestingto note that for a given SNR, the best choice of tends toapproach the value of 0.9 as increases. For instance, for

dB, the best choices of for are, respectively.

The above observations also suggest that , the onlyparameter in the proposed HyperCSI algorithm, is a goodchoice. Next, we will evaluate the performance of the proposedHyperCSI algorithm with the parameter preset to 0.9 for allthe simulated scenarios and real data tests, though it may notbe the best choice for some scenarios.

C. Performance Evaluation of HyperCSI AlgorithmWe evaluate the performance of the proposed HyperCSI

algorithm, along with a performance comparison with fivestate-of-the-art Craig-criterion-based HU algorithms, includingMVC-NMF [23], MVSA [25], MVES [28], SISAL [30], andipMVSA [27]. As the operations of MVC-NMF, MVSA,SISAL, and ipMVSA are data-dependent, their respectiveregularization parameters have been well selected in the simu-lation, so as to yield their best performances. In particular, theregularization parameter involved in SISAL is the regression

TABLE IISIMULATION SETTINGS FOR THE ALGORITHMS UNDER TEST

weight for robustness against noise, and hence has also beentuned w.r.t. different SNRs. The implementation details andparameter settings for all the algorithms under test are listed inTable II.The purity index for each synthetic pixel [28], [29],

[33] has been defined as (due toand ); a larger index means higher pixel purity of

. Each synthetic data set in the simulationis generated with a given purity level denoted as , followingthe same data generation procedure as in [28], [29], [33], whereis a measure of mixing level of a data set. Specifically, a pool

of sufficiently large number of synthetic data is first generated,and then from the pool, pixels with the purity index notgreater than are randomly picked to form the desired data setwith a purity level of .In the above data generation, six endmembers (i.e., Jarsoite,

Pyrope, Dumortierite, Buddingtonite, Muscovite, and Goethite)with 224 spectral bands randomly selected from the USGS li-brary [48] are used to generate 10000 synthetic hyperspectraldata (i.e., ) with

and (dB). The sim-ulation results for , and computational time are dis-played in Table III, where bold-face numbers correspond to thebest performance (i.e., the smallest , and ) of all theHU algorithms under test for a specific .Some general observations from Table III are as follows. For

fixed purity level , all the algorithms under test perform betterfor larger SNR. As expected, the proposed HyperCSI algorithm


TABLE IIIPERFORMANCE COMPARISON, IN TERMS OF (DEGREES), (DEGREES), AND AVERAGE RUNNING TIME (SECONDS), OF VARIOUS HU ALGORITHMS FOR

DIFFERENT DATA PURITY LEVELS AND SNRS, WHERE ABUNDANCES ARE I.I.D. AND DIRICHLET DISTRIBUTED

rightly performs better for higher data purity level , but thisperformance behavior does not apply to the other five algo-rithms, perhaps because the non-convexity of the complicatedsimplex volume makes their performance behaviors more in-tractable w.r.t. different data purities.Among the five existing benchmark Craig-criterion-based

HU algorithms, MVC-NMF yields more accurate endmemberestimates than the other algorithms at the highest computationalcost, while ipMVSA is the most computationally efficient onebut with lower performance as a trade-off. Nevertheless, theproposed HyperCSI algorithm outperforms all the other fivealgorithms when the data are heavily mixed (i.e., )or moderately mixed (i.e., ). As for high data purity

, the HyperCSI algorithm also performs best except forthe case of . On the other hand, thecomputational efficiency of the proposed HyperCSI algorithmis about 1 to 4 orders of magnitude faster than the other five HUalgorithms under test. Note that the computational efficiency ofthe HyperCSI algorithm can be further improved by an orderof if parallel processing can be implemented in Step4 (hyperplane estimation) and Step 7 (abundance estimation)in Table I. Moreover, ipMVSA is around 4 times faster thanMVSA, but performs slightly worse than MVSA, perhapsbecause ipMVSA [27] does not adopt the hinge-type softconstraint (for noise resistance) as used in MVSA [25].

D. Performance Evaluation of HyperCSI Algorithm WithNon-i.i.d., Non-Dirichlet and Sparse Abundances

In practice, the abundance vectors may not be i.i.d. andseldom follow the Dirichlet distribution, and, moreover, theabundance maps often show large sparseness [52]. In view ofthis, as considered in [52], [53], two sets of sparse and spatiallycorrelated abundance maps displayed in Fig. 5 were used togenerate two synthetic hyperspectral images, denoted as SYN1

and SYN2 . Then allthe algorithms listed in Table II are tested again with thesetwo synthetic data sets for which the abundance vectors areobviously neither i.i.d. nor Dirichlet distributed.

The simulation results, in terms of , and com-putational time , are shown in Table IV, where bold-facenumbers correspond to the best performance among the al-gorithms under test for a particular data set and a specific

(dB). As expected, for both datasets, all the algorithms perform better for larger SNR.One can see from Table IV that for both data sets, Hy-

perCSI yields more accurate endmember estimates than theother algorithms, except for the case of (dB).As for abundance estimation, HyperCSI performs best forSYN1, while MVC-NMF performs best for SYN2. Moreover,among the five existing benchmark Craig-criterion-based HUalgorithms, ipMVSA and SISAL are the most computationallyefficient ones. However, in both data sets, the computationalefficiency of the proposed HyperCSI algorithm is at leastmore than one order of magnitude faster than the other fivealgorithms. These simulation results have demonstrated thesuperior efficacy of the proposed HyperCSI algorithm over theother algorithms under test in both estimation accuracy andcomputational efficiency.

V. EXPERIMENTS WITH AVIRIS DATA

In this section, the proposed HyperCSI algorithm along withtwo benchmark HU algorithms, i.e., the MVC-NMF algorithm[23] developed based on Craig’s criterion, and the VCA algo-rithm [17] (in conjunction with the FCLS algorithm [15] forthe abundance estimation) developed based on the pure-pixelassumption, are used to process the hyperspectral imaging datacollected by the Airborne Visible/Infrared Imaging Spectrom-eter (AVIRIS) [32] taken over the Cuprite mining site, Nevada,in 1997. We consider this mining site, not only because it hasbeen extensively used for remote sensing experiments [54],but also because the available classification ground truth in[55], [56] (though which may have coregistration issue as itwas obtained earlier than 1997, this ground truth has beenwidely accepted in the HU context) allows us to easily verifythe experimental results. The AVIRIS sensor is an imagingspectrometer with 224 channels (or spectral bands) that coverwavelengths ranging from 0.4 to 2.5 m with an approximately


TABLE IVPERFORMANCE COMPARISON, IN TERMS OF (DEGREES), (DEGREES) AND RUNNING TIME (SECONDS), OF VARIOUS HU ALGORITHMS USING

SYNTHETIC DATA SYN1 AND SYN2 FOR DIFFERENT SNRS, WHERE ABUNDANCES ARE NON-I.I.D., NON-DIRICHLET AND SPARSE (SEE FIG. 5)

Fig. 5. Two sets of sparse and spatially correlated abundance maps, where eachsubblock in subfigure (b) contains 10 10 pixels. (a) Ground truth abundancemaps of SYN1, (b) Ground truth abundance maps of SYN2.

10-nm spectral resolution. The bands with low SNR as wellas those corrupted by water-vapor absorption (including bands1–4, 107–114, 152–170, and 215–224) are removed from theoriginal 224-band imaging data cube, and hence a total of

bands is considered in our experiments. Furthermore,the selected subscene of interest includes 150 vertical lines with150 pixels per line, and its 50th band is shown in Fig. 6, wherethe 10 pixels marked with yellow color are removed from thedata set as they are outlier pixels identified by the robust affineset fitting (RASF) algorithm [57].The number of the minerals (i.e., endmembers) present in

the selected subscene is estimated using a virtual dimension-ality (VD) approach [37], i.e., the noise-whitened Harsanyi-

Fig. 6. The subimage of the AVIRIS hyperspectral imaging data cube for the50th band, where the locations of the ten outliers identified by the RASF algo-rithm are marked with yellow color.

Farrand-Chang (NWHFC) eigenvalue-thresholding-based algo-rithm with false-alarm probability . The obtainedestimate is and used in the ensuing experiments for allthe three HU algorithms under test.The estimated abundance maps are visually compared with

those reported in [17], [23], [28] as well as the ground truthreported in [55], [56], so as to determine what minerals theyare associated with. The nine abundance maps obtained by theproposed HyperCSI algorithm are shown in Fig. 7, and theyare identified as mineral maps of Muscovite, Alunite, DesertVarnish, Hematite, Montmorillonite, Kaolinite #1, Kaolinite #2,Buddingtonite, Chalcedony, respectively, as listed in Table V.The minerals identified by MVC-NMF and VCA are also listedin Table V, where MVC-NMF also identifies nine distinct min-erals, while only eight distinct minerals are retrieved by VCA,perhaps due to lack of pure pixels in the selected subscene orrandomness involved in VCA. Owing to space limitation, theirmineral maps are not shown here.The mineral spectra extracted by the three algorithms under

test, along with their counterparts in the USGS library [48], areshown in Fig. 8, where one can observe that the spectra ex-tracted by the proposed HyperCSI algorithm hold a better re-semblance to the library spectra. For instance, the spectrum ofAlunite extracted by HyperCSI shows much clearer absorptionfeature than MVC-NMF and VCA, in the bands approximately


Fig. 7. The abundance maps of minerals estimated by HyperCSI algorithm.

TABLE VTHE COMPUTATIONAL TIMES (SECONDS) AND SPECTRAL ANGLE

DISTANCE (DEGREES) BETWEEN LIBRARY SPECTRA AND ENDMEMBERSESTIMATED BY HYPERCSI, MVC-NMF, AND VCA. THE BOLD FACENUMBERS CORRESPOND TO THE SMALLEST VALUES OF OR

AMONG THE THREE ALGORITHMS UNDER TEST

from 2.3 to 2.5 m. To quantitatively compare the endmemberestimation accuracy among the three algorithms under test, thespectral angle distance between each endmember estimate andits corresponding library spectrum serves as the performancemeasure and is defined as

The values of associated with the endmember estimates forall the three algorithms under test are also shown in Table V,where the number in the parentheses is the value of associ-ated with Kaolinite #1 repeatedly classified by VCA. One cansee from Table V that the average of of the proposed Hy-perCSI algorithm is the smallest. The good performance of Hy-perCSI in endmember estimation in the experiment also implies

Fig. 8. (a) The endmember signatures taken from the USGS library, and sig-natures of the endmember estimates obtained by (b) HyperCSI, (c) MVC-NMFand (d) VCA.

that the potential requirement of sufficient number (i.e.,, in this experiment) of pixels lying close to the

hyperplanes associated with the actual endmembers’ simplex,has been met for the considered hyperspectral scene. However,we are not too surprised with this observation, since the numberof minerals present in one pixel is often small (typically,within five [10]), i.e., the abundance vector often showssparseness [52] (cf. Fig. 7), indicating that a non-trivial por-tion of pixels are more likely to lie close to the boundary ofthe endmembers’ simplex (note that if, and only if,

). Moreover, as the pure pixels may not be present inthe selected subscene, as expected the two Craig-criterion-basedHU algorithms (i.e., HyperCSI and MVC-NMF) outperformVCA in terms of endmember estimation accuracy. On the otherhand, in terms of the computation time as given in Table V, inspite of parallel processing not yet applied, the HyperCSI algo-rithm is around 2.5 times faster than VCA (note that VCA itselfonly costs 0.31 seconds (out of the 5.40 seconds), and the re-maining computation time is the cost of the FCLS) and almostfour orders of magnitude faster than MVC-NMF.

VI. CONCLUSIONBased on the hyperplane representation for a simplest sim-

plex, we have presented an effective and computationally effi-cient Craig-criterion-based HU algorithm, called HyperCSI al-gorithm, given in Table I. The proposed HyperCSI algorithmhas the following remarkable characteristics:• It never requires the presence of pure pixels in the data.• It is reproducible without involving random initialization.• It only involves simple linear algebraic computations,so suitable for parallel implementation. Its computa-tional complexity (without using parallel implementation)is , which is also the complexity of somestate-of-the-art pure-pixel-based HU algorithms.


• It estimates Craig’s minimum-volume simplex by findingonly pixels (regardless of the data length )from the data set for the construction of the associated hy-perplanes, without involving any simplex volume compu-tations, thereby accounting for its high computational effi-ciency in endmember estimation.

• The estimated endmembers are guaranteed non-negative,and the identified simplex was proven to be both Craig’ssimplex and true endmembers’ simplex w.p.1. asfor the noiseless case.

• The abundance estimation is readily fulfilled by a closed-form expression, and thus is computationally efficient.

Some simulation results were presented to demonstrate theanalytic results on the asymptotic endmember identifiability ofthe proposed HyperCSI algorithm, and its superior efficacy oversome state-of-the-art Craig-criterion-based HU algorithms inboth solution accuracy and computational efficiency. Finally,the proposed HyperCSI algorithm was tested using AVIRIS hy-perspectral data to show its applicability.

APPENDIX

A. Proof of Theorem 1For a fixed , one can see from (18) that

, implying that the pixels, identified by solving (17) must be distinct. Hence,

it suffices to show that is affinely independent w.p.1 for anythat satisfies

(34)

Then, as , we have from and(34) that there exist i.i.d Dirichlet distributed random vectors

such that (cf. (2))

(35)

For ease of the ensuing presentation, let denote the prob-ability function and define the following events:

The set is affinely dependent.The set is affinely dependent.

.Then, to prove that is affinely independent w.p.1, it sufficesto prove .Next, let us show that implies . Assume is true. Then

for some . Without loss ofgenerality, let us assume . Then,

(36)

for some , satisfying

(37)

By substituting (35) into (36), we have(38)

where . For notational simplicity,let for any given vector

. Then, from the facts of (by (37))and , (38) can be rewritten as

(39)

where . As isaffinely independent (by ), the matrix is of full columnrank [26], implying that (by (39)). Then,by the facts of and , one can readilycome up with , or, equivalently,

(by (37)), implying that is true [26].Thus we have proved that implies , and hence

(40)

As Dirichlet distribution is a continuousmultivariate distribu-tion [47] for a random vector to satisfy – withan -dimensional domain, any given affine hullwith affine dimension must satisfy [44]

(41)

Moreover, as are i.i.d. random vectors and theaffine hull must have affine dimen-sion , we have from (41) that

(42)

Then we have the following inferences:

i.e., . Therefore, the proof is completed.

B. Proof of Theorem 2It can be seen from (20) that the p.d.f. of Dirichlet distribution

satisfies

(43)

Moreover, by the facts of and

where denotes the interior of a set , the linear mapping(i.e., ) of the abundance domain full fills theinterior of the true endmembers’ simplex ,namely

(44)

Then, from (43)–(44) and , it can be inferred that

which, together with the fact that the affine mapping (cf. (2))preserves the geometric structure of [38] (notethat ), further implies

(45)


where throughout the ensuingproof. It can be inferred from (45) that there is always a pixel

that can be arbitrarily close to the extreme pointof the simplex , i.e., for all ,

(46)

Let denote the set of all minimum-volume en-closing simplexes of (i.e., Craig’s simplex containing the set). Then, one can infer from the convexity of a simplex that (cf.

[29, eq. (32)])

(47)

Moreover, by the fact that any simplex mustalso be a closed set and the fact that the closure of

is exactly , itcan be seen that if and only if

[58], and hence

(48)

Thus, it can be inferred from (45), (47) and (48) that

(49)

As itself is a simplex,, which together with

(49) yields

(50)

In other words, we have proved that the Craig’s min-imum-volume simplex is always the true endmembers’ simplex

. To complete the proof of Theorem 2, itsuffices to show that the true endmembers’ simplex is alwaysidentical to the simplex identified by the HyperCSI algorithm,i.e., for all ,

(51)

where are the estimated DR endmembers using Hy-perCSI algorithm.To this end, let us first show that, for all ,

(52)

where are the purest pixels identified by SPA (cf.Step 2 in Table I). However, directly proving (52) is difficult dueto the post-processing involved in SPA (cf. Algorithm 4 in [43]).In view of this, let be those pixels identifiedby SPA before post-processing. Because the post-processing isnothing but to obtain the purest pixel by iteratively pushingeach away from the hyperplane [43], wehave the following simplex volume inequalities

(53)

where the last inequality is due to. Hence, by (53), to prove (52), it

suffices to show that, for all ,

(54)

However, the SPA before post-processing (cf. Algorithm 4 in[43]) is exactly the same as the TRIP algorithm (cf. Algorithm2 in [51]), and it has been proven in ([51], Lemma 3) that (46)straightforwardly yields (54) for ; note that the condition“(46) with ” is equivalent to the pure-pixel assumption re-quired in ([51], Lemma 3). One can also show that (46) yields(54) for any , and the proof basically follows the sameinduction procedure as in the proof of ([51], Lemma 3) and isomitted here for conciseness. Then, recalling that (54) is a suf-ficient condition for (52) to hold, we have proven (52).By the fact that is a continuous function (cf. (14)) and by

(16) and (18), we see that

(55)

Moreover, we have from (43), (52), (55) and thatthe pixel identified by (17) can be arbitrarily close to

. Furthermore, by Theorem 1, we have that the vectorsare not only arbitrarily close to , but also

affinely independent w.p.1, which together with Proposition 2implies that the estimated (cf. (19)) is arbitrarily close tothe true w.p.1, provided that the outward-pointing normalvectors and have the same norm without loss of gen-erality. Then, from (43), (22), and the premises ofand , it can be inferred that the estimated hyperplane

is arbitrarily close to the true(cf. (9)); precisely, we have

(56)

Consequently, by comparing the formulas of (cf. (13)) and(cf. (23)), we have, from and (56), that is always

arbitrarily close to , i.e., (51) is true for all , and hencethe proof of Theorem 2 is completed.

ACKNOWLEDGMENT

The authors would like to express our sincere gratitudeto knowledgeable reviewers, providing many insightful per-spective comments that significantly improve the quality ofthis paper. The authors would also like to thank Dr. Jose M.Bioucas-Dias, Dr. Marian-Daniel Iordache, and Dr. Jie Chenfor providing sparse and spatially correlated abundance distri-bution maps on one hand, and offering valuable advice for oursimulation studies on the other hand.


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Chia-Hsiang Lin received the B.S. degree in Elec-trical Engineering from the National Tsing Hua Uni-versity (NTHU), Hsinchu, Taiwan, in 2010.He is pursuing the Ph.D. degree in Communica-

tions Engineering at NTHU. He is currently a vis-iting Doctoral Graduate Research Assistant of Vir-ginia Polytechnic Institute and State University, Ar-lington, VA, USA. His research interests are in net-work science, game theory, convex geometry and op-timization, and blind source separation.

Chong-Yung Chi (S’83–M’83–SM’89) received theB.S. degree in 1975 and M.S. degree in 1977, fromTatung Institute of Technology and National TaiwanUniversity, all in Electrical Engineering. Then hereceived the Ph.D. degree in Electrical Engineeringfrom the University of Southern California, LosAngeles, California, in 1983.From 1983 to 1988, he was with the Jet Propulsion

Laboratory, Pasadena, California. He has been a Pro-fessor with the Department of Electrical Engineeringsince 1989 and the Institute of Communications

Engineering (ICE) since 1999 (also the Chairman of ICE during 2002–2005),National Tsing Hua University, Hsinchu, Taiwan. He has published morethan 210 technical papers, including about 80 journal papers (mostly in IEEETRANSACTIONS ON SIGNAL PROCESSING), 4 book chapters and more than 130peer-reviewed conference papers, as well as a graduate-level textbook, BlindEqualization and System Identification, Springer-Verlag, 2006. His currentresearch interests include signal processing for wireless communications,convex analysis and optimization for blind source separation, biomedical andhyperspectral image analysis.Dr. Chi has been a Technical Program Committee member for many IEEE

sponsored and co-sponsored workshops, symposiums and conferences onsignal processing and wireless communications, including Co-organizer andGeneral Co-chairman of 2001 IEEE Workshop on Signal Processing Advancesin Wireless Communications (SPAWC), and Co-Chair of Signal Processing forCommunications (SPC) Symposium, ChinaCOM 2008 & Lead Co-Chair ofSPC Symposium, ChinaCOM 2009. He was an Associate Editor (AE) of IEEETRANSACTIONS ON SIGNAL PROCESSING (2001–2004, 2006, and 2012–2015),IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II (2006–2007), IEEETRANSACTIONS ON CIRCUITS AND SYSTEMS I (2008–2009), AE of IEEESIGNAL PROCESSING LETTERS (2006–2010), and a member of the EditorialBoard of Elsevier Signal Processing (2005–2008), and an editor (2003–2005)as well as a Guest Editor (2006) of the EURASIP Journal on Applied SignalProcessing. He was a member of Signal Processing Theory and MethodsTechnical Committee (SPTM-TC) (2005–2010), IEEE Signal ProcessingSociety. Currently, he is a member of Signal Processing for Communicationsand Networking Technical Committee (SPCOM-TC) and a member of SensorArray and Multichannel Technical Committee (SAM-TC), IEEE SignalProcessing Society.

Yu-Hsiang Wang received his B.S. degree from theDepartment of Communications Engineering, YuanZeUniversity, Taiwan, in 2011, andM.S. degree fromInstitute of Communications Engineering, NationalTsing Hua University, Taiwan, in 2015.He is currently a graduate student with the Institute

of Communications Engineering, National Tsing HuaUniversity, Taiwan. His research interests are convexoptimization and hyperspectral unmixing.

Tsung-Han Chan received his B.S. degree fromthe Department of Electrical Engineering, Yuan ZeUniversity, Taiwan, in 2004 and his Ph.D. degreefrom the Institute of Communications Engineering,National Tsing Hua University, Taiwan, in 2009.He is currently working as a Senior Engineer with

MediaTek Inc., Hsinchu, Taiwan. He was a co-recip-ient of a WHISPERS 2011 Best Paper Award. Hewas also recognized as an Outstanding Reviewer forCVPR 2014, and a Best Reviewer of the IEEE TGRS2014. His research interests are in image processing

and numerical optimization, with a recent emphasis on computer vision, ma-chine learning and hyperspectral remote sensing.

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