Hyperspectral Unmixing: Ground Truth Labeling, Datasets, … · 2017-10-12 · Hyperspectral...

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Hyperspectral Unmixing: Ground Truth Labeling,Datasets, Benchmark Performances and Survey

Feiyun Zhu

Abstract—Hyperspectral unmixing (HU) is a very useful andincreasingly popular preprocessing step for a wide range ofhyperspectral applications. However, the HU research has beenconstrained a lot by three factors: (a) the number of hyperspec-tral images (especially the ones with ground truths) are verylimited; (b) the ground truths of most hyperspectral images arenot shared on the web, which may cause lots of unnecessarytroubles for researchers to evaluate their algorithms; (c) the codesof most state-of-the-art methods are not shared, which may alsodelay the testing of new methods.

Accordingly, this paper deals with the above issues from thefollowing three perspectives: (1) as a profound contribution, weprovide a general labeling method for the HU. With it, we labeledup to 15 hyperspectral images, providing 18 versions of groundtruths. To the best of our knowledge, this is the first paperto summarize and share up to 15 hyperspectral images andtheir 18 versions of ground truths for the HU. Observing thatthe hyperspectral classification (HyC) has much more standarddatasets (whose ground truths are generally publicly shared)than the HU, we propose an interesting method to transformthe HyC datasets for the HU research. (2) To further facilitatethe evaluation of HU methods under different conditions, wereviewed and implemented the algorithm to generate a complexsynthetic hyperspectral image. By tuning the hyper-parameters inthe code, we may verify the HU methods from four perspectives.The code would also be shared on the web. (3) To provide astandard comparison, we reviewed up to 10 state-of-the-art HUalgorithms, then selected the 5 most benchmark HU algorithms,and compared them on the 15 real hyperspectral datasets. Theexperiment results are surely reproducible; the implementedcodes would be shared on the web.

Index Terms—Hyperspectral Unmixing (HU), Datasets, label-ing, Ground Truth, Hyperspectral Classificatin (HyC).

I. INTRODUCTION

HYPERSPECTRAL remote sensing has been widely usedin lots of applications1 since it can capture a 3D image

cube at hundreds of contiguous bands across the electromag-netic spectrum, providing substantial information of the scene[1]. However, due to the microscopic material mixing, multiplescattering and low spatial resolution of hyperspectral sensors,the pixel spectra are inevitably mixed with various substances[1, 2], resulting in lots of mixed pixels. Accordingly, hyper-spectral unmixing (HU) is an essential processing step for var-ious hyperspectral image applications, such as high-resolutionhyperspectral imaging [3–8], hyperspectral enhancement [9],sub-pixel mapping [10], hyperspectral compression and recon-struction [11], detection and identification substances in thescene [9, 12], hyperspectral visualization [13, 14], etc.

1such as mining & oil industries, agriculture, food safety, pharmaceuticalprocess monitoring and quality control, biomedical & biometric, surveillance,military, environment monitoring and forensic applications etc.

The HU aims to decompose each (mixed) pixel into a set of“pure” pixels (called endmembers such as the spectra of grass,water etc.), weighted by the corresponding proportions, calledabundances [14–16]. Formally, given a hyperspectral imagewith L channels and N pixels, each pixel y ∈ RL+ is assumedto be a composite of K endmembers {mk}Kk=1 ∈ RL+. Thelinear combinatorial model is the most commonly used one

x =

K∑k=1

mkak, s.t. ak ≥ 0 andK∑k=1

ak = 1, (1)

where ak is the composite abundance of the kth endmember.In the unsupervised setting, both endmembers {mk}Kk=1 andabundances {ak}Kk=1 are unknown, which makes the HU achallenging problem [17–26].

The HU is a very hot research topic—tens or even hundredsHU papers have been published each year. However, the HUresearch has been constrained a lot since the commonly useddatasets (especially their ground truths), are generally notshared on the web. Such case will surely hinder the devel-opment of new methods; the researchers, who are interestedin the HU, will have to make great efforts to do the preparationworks—they have to find the hyperspectral datasets and theirground truths, which generally ends up with failures. Instead,they have to label the hyperspectral images, which is verychallenging and needs lots of techniques.

In order to promote the HU research, we write this paperfrom the following five perspectrives:

1). This is the first paper to introduce a general method tolabel hyperspectral images for the HU research. We providethe method to label the endmembers & abundances (in Sec-tions IV-A, IV-B), as well as the method to evaluate the label-ing results (in Section IV-C). The hyperspectral classification(HyC) is similar to the HU task; it has much more standarddatasets with publicly available ground truths. We propose aninteresting method to transform the benchmark HyC datasetsfor the HU research. Please refer to Section IV.

2). Moreover, we are the first to summarize the informationof 15 most commonly used hypserpectral images as well astheir 18 versions of ground truths for the HU. All of them willbe shared on the web as a standard dataset for the evaluationof new HU methods (cf. Section V, Table I and Fig. 1).

3). We reviewed and implemented the method to generate acomplex synthetic hyperspectral image for the HU research inSection VI. This synthetic image is widely used in the papers[2, 12, 27–32]. The code will be also shared on the web.

4). We reviewed 10 state-of-the-art methods, summarizedtheir main ideas and algorithms as well as pointed out their

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Fig. 1. The illustration of 7 popular hyperspectral images where we select 15 subimages (i.e., ROIs) and provide 18 versions of ground truths for the HUstudy. They are [a] Samson (3 ROIs), [b] Jasper Ridge (3 ROIs), [c] Urban (the full image and 2 ROIs), [d] Cuprite (1 ROI), [e] Moffett Field (2 ROIs), [f]San Diego Airport (1 ROI), and [g] Washington DC Mall (2 ROIs). In Section V and Table I, we provide very detailed information about the 15 ROIs.

TABLE ITHE 15 REAL HYPERSPECTRAL (SUB-) IMAGES AND THEIR INFORMATION, INCLUDING THE IMAGE SIZE, THE NUMBERS OF ALL SPECTRAL BANDS AND

THE SELECTED BANDS, THE endmembers NUMBERS, THE ILLUSTRATION OF SCENES AND THEIR 18 VERSIONS OF GROUND TRUTHS ETC.

Real Datasets Image sizes Number of bands # End- Ground truth The top-left pixel in Subimages# row # column # all bands # selected bands member shown in original image shown in

1. Samson#1 95 95156 156

3 Fig. 2a (252, 332)Fig. 1a2. Samson#2 95 95 3 Fig. 2b (232, 93)

3. Samson#3 95 95 3 Fig. 2c (634, 455)4. Jasper Ridge#1 115 115

224 1985 Fig. 3a (1, 272)

Fig. 1b5. Jasper Ridge#2 100 100 4 Fig. 3b (105, 269)6. Jasper Ridge#3 122 104 4 Fig. 3c (1, 246)7, 8, 9. Urban 307 307

221 1624, 5, 6 Figs. 4a, 4b, 4c (1, 1)

Fig. 1c10. Urban#1 160 168 6 Fig. 5a (1, 1)11. Urban#2 160 168 6 Fig. 5b (40, 1)12. Cuprite 250 190 224 188 12 Fig. 6 (86, 425) Fig. 1d13. Moffett Field#1 60 60 224 196 3 Fig. 7a (54, 172) Fig. 1e14. Moffett Field#2 60 60 3 Fig. 7b (62, 146)15, 16. San Deigo Airport 160 140 224 189 4, 5 Figs. 8a, 8b (241, 261) Fig. 1f17. Washington DC Mall#1 150 150 224? 191 6 Fig. 9a (549, 160) Fig. 1g18. Washington DC Mall#2 180 160 6 Fig. 9b (945, 90)

internal relations. Besides, we provide the codes of 5 success-ful state-of-the-art HU methods. Please refer to Section III.

5). The 5 most popular methods are implemented andcompared on the 15 real hyperspectral images. We providetheir results as the benchmark HU performance in Section VII.

II. THE THREE CATEGORIES OF HU METHODS

In general, existing HU methods can be classified into threecategories: supervised methods [9, 11], weakly supervisedmethods [33, 34] and unsupervised methods [12, 15, 28, 35–37]2. The endmember is given beforehand in the supervisedsetting; only the abundance is required to estimate. Although,the HU task is simplified in this setting, it is usually intractableto obtain feasible endmembers, thus, hampering the acquisitionof good HU estimations.

2Here supervise, weakly supervise and supervise are totally different fromthe those terms in the general machine learning.

Accordingly, the weakly supervised methods [33, 34] wereproposed. A large library of material spectra had been col-lected by a field spectrometer beforehand3. Then, the HUtask becomes the problem of finding an optimal subset ofmaterial spectra in the library that can best represent allthe pixels in the hyperspectral image [33] as well as theirabundance maps. Unfortunately, the library is far from optimalbecause the spectra in it are not standardly unified. First,for different hyperspectral sensors, the spectral signatures ofthe same material can be very inconsistent. Second, for thehyperspectral images recorded by different sensors, both thenumber of spectral bands and the electromagnetic range ofrecorded spectra can be largely different as well—for example,some images (like Samson) have 156 channels covering thespectra from 401 nm to 889 nm, while other images (likeCuprite and Jasper Ridge) have 224 channels covering thespectra from 370 nm to 2, 480 nm. Finally, the recording

3e.g., the United States Geological Survey (USGS) digital spectral library.

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conditions are very different—some hyperspectral images arecaptured far from the outer space, while some hyperspectralimages are obtained from the airplane or even in the lab.Due to the atmospheric effects etc., the different recordingconditions would result in different spectral appearances. Inshort, the weakness of the spectral library brings side effectsinto this kind of methods.

More commonly, the endmembers are learned from the hy-perspectral image itself to ensure the spectral coherence [9]—the unsupervised HU methods are preferred, where both theendmember and abundances are learned from the hyperspec-tral image. Specifically, unsupervised HUs can be categorizedinto two types: geometric methods [38–43] and statisticalones [44–48]. The geometric methods usually exploit thesimplex to model the distribution of pixel spectra. The N-FINDR [49] and Vertex Component Analysis (VCA) [39] arethe most benchmark geometric methods. For the N-FINDR,the endmembers are extracted by inflating a simplex inside thehyperspectral pixel space and treating the vertices of a simplexwith the largest volume as endmembers [49]. The VCA [39]projects all the residual pixel onto a direction orthogonal tothe simplex spanned by the chosen endmembers; the newendmember is identified as the extreme of the projection.Although these methods are simple and fast, they suffer fromthe requirement of pure pixels, which is usually unreliable inpractice [12, 28, 50].

Accordingly, many statistical methods have been proposedfor or applied to the HU problem, among which the Nonneg-ative Matrix Factorization (NMF) [51] and its extensions arethe most popular. In (the following) Section III, we will reviewthe NMF and its extensions in detail.

III. REVIEW OF THE STATE-OF-THE-ART HU METHODS

Notations. For clear modeling, the boldface uppercase letter(e.g. X) and lowercase letter (x) are used to represent matricesand vectors respectively. Given a matrix X , {Xln} ∈ RL×N ,xl ∈ R1×N is the lth row vector and xn ∈ RL denotes the nth

column vector. Xln is the (l, n)-th element in the matrix. X ≥0 or X ∈ RL×N+ represent a nonnegative matrix. The `2,1-

norm of matrices is defined as ‖X‖2,1 =∑Ll

(∑Nn X

2ln

)1/2.

Formalization. In the HU modeling, the hyperspectralimage with L channles (or bands) and N pixels, is representedby a nonnegative matrix X , [x1,x2, · · · ,xN ] ∈ RL×N+ .From the perspective of the linear mixture assumption, the goalof HU is to find two nonnegative matrices to well approximateX with their product:

minM,A

loss{X, X

}+ λΨ (A) + αΦ (M) , (2)

s.t. X = MA,M ≥ 0,A ≥ 0,

where X is the approximation of the original hyperspectral im-age; M , [m1, · · · ,mK ] ∈ RL×K+ is the endmember matrixthat consists of K pure pixel spectra and K � min {L,N};A , [a1, · · · ,aN ] ∈ RK×N+ is the abundance matrix—the nth

column vector an contains all the K abundances at pixel xn;loss {·, ·} is a loss function measuring the difference between

two terms; Ψ (A) is some kind of constraints on the abundancemaps; Φ (M) is the constraint on the endmembers.

A. Nonnegative Matrix Factorization (i.e., NMF) [51, 52]

When the loss {·, ·} is the Euclidean loss, the objective (2)becomes the standard NMF problem [14, 51, 52], which iscommonly used in a wide range of applications including theHU [12, 14, 27, 28, 37, 53]. Since (2) is non-convex w.r.t. thetwo variables (i.e., M and A) together [51, 52], it is unrealisticto find global minima. Alternatively, Lee and Seung [51, 52]have proposed the multiplicative update rules as follows:

Mlk ←Mlk

(XA>

)lk

(MAA>)lk, Akn ←

Akn(M>X

)kn

(M>MA)kn, (3)

which have been proved to be non-increasing. Apart from (3),there are other optimization algorithms to solve the problem(2), such as the active-set [54], the alternation nonnegativeleast least squares [55] and the projected gradient descent [56].

Although NMF is well adapted to face analyses [57, 58]and documents clustering [59, 60], the objective function (2) isnon-convex, naturally resulting in a large solution space [52].Many extension methods have been proposed by employingsuitable priors to restrict the solution space. For the HU task,the priors are imposed either on abundances [12, 53, 61] (cf.Section III-B) or on endmembers [27, 37] (cf. Section III-C).

B. The NMF extensions with constraints on the abundances

This section reviews the NMF extensions that impose con-straints only on the abundance. That is, Ψ (A) is effectiveand Φ (M) = 0; the updating rule for the endmember is givenin (3) by default. Specifically, the sparse constraints [12, 14–16, 62, 63] and the spatial (like, manifold, graph) constraint[15, 61, 63, 64] are the most popular ones.

1) W-NMF [61] (or G-NMF [64]): The local neighborhoodweight regularized NMF (W-NMF) [61] assumes that hyper-spectral pixels are distributed on a manifold; the authors ex-ploit appropriate weights in the local neighborhood to enhancethe spectral unmixing. Specifically, W-NMF employs both thespectral and spatial information to construct the weight matrixW ∈ RN×N+ [65], where Wij is the similarity between thepixel i and j. While, G-NMF [64] is a general NMF extension,which only employs the pixel feature to form the graph.

In W-NMF (or G-NMF), the constraint on the abundanceis Ψ (A) = Tr (ALAᵀ), where L = D −W is the graphLaplacian learned from the hyperspectral image, and D is thedegree matrix whose diagonal elements are column sums ofW. The updating rule of abundances for the W-NMF is

Akn ←Akn

(M>Y + λAW

)kn

(M>MA + λAD)kn, (4)

where λ is a balancing parameter for the Graph constraint.2) Typical Sparsity constrained NMF (e.g., `1-NMF [62]

and `1/2-NMF [12]): The sparsity constrained NMFs [2, 12,14–16, 62] are the most successful methods for the HU task.Those methods assume that most hyperspectral pixels aremixed with parts of (not all) endmembers, and exploits allkinds of sparse constraints on the abundance. The `1-norm is

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one of the most benchmark sparsity constraint. Consideringit gives the `1-NMF [62], where the constraint is Ψ (A) =‖A‖1 =

∑k,n |Akn|. The updating rule for the abundance is

Akn ←Akn

(M>Y

)kn

(M>MA + λ)kn. (5)

Although the `1-NMF makes sense, the lasso constraint [66,67] could not enforce further sparse when the full additivityconstraint is used, limiting the effectiveness [12]. Accordingly,Dr. Qian proposed the state-of-the-art `1/2-norm constrainedNMF, where Ψ (A) = ‖A‖1/2 ≈

∑k,n |Akn + ξ|1/2, ξ is a

small positive value to ensure the numerical condition. Theupdating rule for the abundance is given as

Akn ←Akn

(M>Y

)lk(

M>MA + 0.5λ (A + ξ)−1/2

)lk

. (6)

where λ in Eqs. (5) and (6) are the balancing parameter forthe sparsity constraints [12, 62].

3) DgS-NMF [14] and RRLbS [16]: To reduce the solutionspace, the state-of-the-art HU methods exploit various con-straints on the abundances and on the endmembers. However,they generally employ an identical strength of constraints onall the factors, which may not meet the practical situation.Instead, Dr. Zhu [14] observed that the mixed level of eachpixel varies over image grids. Based on this prior, he proposeda novel method to learn a data-guided map (DgMap, i.e.,h ∈ RN+ ), which aims to describe the mixed level of eachpixel. Through this DgMap, the `p (0 < p < 1) constraint isapplied in an adaptive manner. For each pixel, the choice ofp is tightly related to the corresponding value in the DgMap.

In DgS-NMF, the sparsity constraint on the abundance isΨ (A) =

∑k,n (Akn+ξ)

1−Hkn , where H = 1Khᵀ ∈ RK×N+ ,h ∈ RN+ is the DgMap, and ξ is a small positive value to ensurenumerical conditions. The updating rule for the abundance is

Akn ←Akn

(M>Y

)kn(

M>MA + λ (1−H) ◦ (A + ξ)−H)kn

, (7)

where ◦ is the Hadamard product between matrices. Note thatthe `1-NMF and `1/2-NMF are special cases of the DgS-NMF.Given a constant DgMap with all elements equal to zero, theDgS-NMF turns into the `1-NMF. Whereas if each elementin the DgMap is equal to 1/2, the DgMap guided sparsityconstraint turns into the `1/2 norm based sparse regularization.

DgS-NMF [14] is an interesting method. However, a heuris-tic algorithm is proposed in [14] to learn the DgMap, whichis ineffective for the vast smooth areas in the image. It isexpected that the more accurate DgMap constraint would biasthe solution to the more satisfactory local minima. Besides,the state-of-the-art method generally ignores the badly de-graded (i.e., outlier) channels in the hyperspectral image. Toaddress the above two problems, a robust representation andlearning-based sparsity (RRLbS) method is proposed in [16]by emphasizing both robust representation and learning-basedsparsity. Specifically, the loss {·, ·} in (2) is set as the `2,1-norm based loss. The new loss is better at preventing theoutlier channels from dominating the objective. The constraint

in RRLbS is Ψ (A) =∑k,n (Akn + ξ)

1−Hkn , which is similarto the DgS-NMF. However, due to the robust loss function andthe simultaneous learning process for DgMaps, the updatingrule in RRLbS is quite different from that of DgS-NMF:

Mlk ←Mlk

(UXA>

)lk

(UMAA>)lk, (8)

Akn ←Akn

(M>UX

)kn(

M>UMA + λ (1−H) ◦ (A + ξ)−H)kn

, (9)

where U ∈ RL×L+ is a nonnegative diagonal matrix4. The mostdirect clue to estimate DgMap h ∈ RN+ is its crucial depen-dence upon abundances. Once getting the stable abundanceA, h = {hn}Nn=1 could be efficiently estimated as follows:

H = 1Kh>, hn = S (an) , ∀n ∈ {1, 2, · · · , N} (10)

where S (an) is the Gini index [16, 68] that measures thesparsity of column vectors in A. The above updating processfor {M,A} and h is iterated, which is expected to generatea sequence of ever improved estimates until convergences.

4) Structure (or Graph) and Sparsity constrained NMF (SS-NMF [15] and GL-NMF [63]): There are two methods [15,63] considering both the spatial (like graph) constraint and thesparsity constraint. In the Stucutred sparsed NMF (SS-NMF),the constraint is Ψ (A) = Tr

(ALA>

)+ α

λ ‖A‖1, where thegraph Laplacian L is learned via a novel method that considersboth the spectral and spatial information in the hyperspectralimage. The updating rule for the abundance is as follows:

Akn ←Akn

(M>X + λAW

)kn

(M>MA + λAD + α)kn. (11)

In GL-NMFE [63], the constraint is Ψ (A) =Tr(ALA>

)+ α

λ ‖A‖1/2, where the authors ignore thespatial information inherent in the hyperspectral image andonly employ the spectral information to construct the graphLaplacian L. The updating rule for the abundance is:

Akn ←Akn

(M>X + λAW

)kn(

M>MA + λAD + 1/2αA−1/2)kn

. (12)

5) Correntropy based NMF (CE-NMF) [2]: It is wellknown that the Euclidean loss is prone to outliers [69, 70].Accordingly, Dr. Wang employed the correntropy metric tomeasure the reconstruction error; the `1 norm based sparseconstraint is considered, resulting in the new robust objective

minM≥0,A≥0

L∑l=1

− exp

−∥∥∥xl−(MA)

l∥∥∥22

σ2

+2λ ‖A‖1. (13)

A half-quadratic optimization technique is proposed to convertthe complex optimization problem (13) into an iterativelyreweighted NMF problem. As a result, the optimization can

4The lth diagonal entry Ull is set as Ull=1/

(2√∥∥(MA−X)l

∥∥22+ ε

),

where ε is typically set 10−8 to avoid singular failures.

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adaptively assign small weights to noisy channels and empha-size on noise-free channels. The updating rule is

Mlk ←Mlk

(XAᵀ

)lk(

MAAᵀ)lk

, Akn ←Akn

(MᵀX

)kn(

MᵀMA + λ)kn

, (14)

where X=U1/2X, M=U1/2M and U ∈ RL×L+ is a diagonal

matrix with the lth element as Ull = exp

(−‖

xl−(MA)l‖22

σ2

).

C. The NMF extensions with constraints on the endmembers

We review the NMF extensions that impose constraints onlyon the endmember. That is, Φ (M) is effective and Ψ (A) = 0

1) Minimum Volume Constrained NMF (MVC-NMF) [27]:The MVC-NMF combines the property of both the geo-metric methods and statistical methods. It aims to find theendmembers, which compose the minimum volume simplexthat circumscribes the hyperspectral data scatters. Given theendmembers, the simplex volume is defined as

Φ (M) =1

(K − 1)!|det ([m2−m1, · · · ,mK−m1])| . (15)

The standard Euclidean is used to measure the representationerror. For the sake of easy optimization, the objective functionis finalized as minM,A f (M,A) where

f (M,A) =1

2‖X−MA‖2F+

λ

2 (K − 1)!det2

([1K , M

>]>)

where M is a low dimensional transform of M, i.e., M =Pᵀ (M− µ1ᵀ

K); P ∈ RL×(K−1) is formed by the K − 1most significant principle components of hyperspectral dataX and µ is the mean of data X. The updating rule is basedon the projected gradient algorithm, which is given as follows:

M(t+1) = max{0, M(t) − α(t)∇Mf

(M(t),A(t)

)}(16)

A(t+1) = max{0, A(t) − β(t)∇Af

(M(t+1),A(t)

)}(17)

where α(t) and β(t) are the learning rates. They can be fixedat small values or determined by the Armijo rule [27, 71].

2) Endmember Dissimilarity Constrained NMF (EDC-NMF) [37]: Inspired by the MVC-NMF [27], Dr. Wan pro-posed the Endmember Dissimilarity Constrained NMF (EDC-NMF). The core assumption is that due to the high spectralresolution of hyperspectral sensors, the endmember spectrashould be smooth itself and different as much as possible fromeach other. The EDC constraint on the endmember is

Φ (M) =

K−1∑i=1

K∑j=i+1

‖∇mi −∇mj‖22 , (18)

where ∇mi=[m2,i −m1,i, · · · ,mL,i −mL−1,i] is the gradi-ent along the spectral dimension.

The multiplicative update rule is used to optimize the objec-tive function, resulting in the algorithms for the abundancesin (3) and for the endmembers as follows

Mlk ←Mlk

(XA> − 0.5λ∇MΦ (M)

)lk

(MAA>)lk, (19)

where ∇MΦ (M) = 2HH>MT; T is a specially designedmatrix, i.e., T = KIK − 1K×K , IK is a K × K identicalmatrix and 1K×K is a matrix whose all elements are 1;H =

[−p>; IL−1

]+[−IL−1;q>

], IL−1 is a (L− 1) ×

(L− 1) identical matrix, p = [1, 0, · · · 0]> ∈ RL−1 and

q = [0, · · · , 0, 1]> ∈ RL−1.

The derivative over endmembers, i.e., ∇MΦ (M), intro-duces negative values to the updating rule (19). To make upthis problem, the negative elements in the endmember matrixare required to project to a given nonnegative value aftereach iteration. Consequently, the regularization parameter λcouldn’t be chosen freely, limiting the efficacy of EDC-NMF.

IV. HOW TO LABEL GROUND TRUTHS IN THE HU?

In the HU, the ground truth labeling consists of two parts:(1) the endmember labeling, and (2) the abundance labeling.Both parts are challenging. There is only one paper [36] brieflyintroducing the ground truth labeling on the HYDICE urbanimage. As a result, this is the first paper to provide a generalmethod to label the hyperspectral images for the HU task.

A. The Endmember labeling

The endmember is the “pure” material in the image [1].What is the “pure” material? From the Chemistry perspec-tive, the “pure” material (i.e. endmember) can be the puresubstance. However, this ideal assumption makes the HU im-possible to solve—there can be thousands of “pure” substancein the image scene. Such case leads to much more unknownvariables than the observation itself in the hyperspectral image,which is obviously a severely underconstrained problem [72].

Alternatively, it is widely accepted that the notion of “purematerial” is subjective and problem dependent [1], which issimilar with the determination of classes in the hyperspectralclassificiation task. Such setting requires lots of professionalanalyses and understandings of the image scene [73]. Takingthe Cuprite image for example, it may take Earth scientistsseveral weeks to analyze all the endmembers in the scene.The good news is that three factors make it easy to determineendmembers: (1) there are many HU papers publishing theendmember information of the common hyperspectral images,such as the HYDICE urban [12, 28, 36], Cuprite [12, 39, 63],Washington DC Mall [37], Moffett Field [45, 74, 75] etc; (2)for the images whose scenes are simple, it is easy to determinethe endmembers, e.g., the Samson [15], Japser Ridge [14, 16]and San Deigo Airport [76] etc. (3) the existing methods, likevirtual dimensionality (VD) [77], is also helpful to determinethe number of endmembers in the hyperspectral image.

After the determination of endmembers, we need theirspectral signatures, which could be obtained as below:

1). They can be manually chosen from the hyperspectralimage. In some images, the “pure” pixels have been explicitlymarked in the published papers. For example, the endmembersfor the Washington DC Mall image are explicitly marked inFig. 13 in [37]. In the HYDICE urban, the location of theendmember pixel for “asphalt” is provided in the paper [36],which is then compared with the asphalt spectrum in thestandard spectral library. In other hyperpsectral images, like

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the Jasper Ridge, Samson, etc., we may have to select theendmembers from the image based on our understandings.

2). The endmember signature can be found in the USGSmineral spectral library. The Cuprite hyperspectral image isone of the typical hyperspectral image [37] in this type.

B. The Abundance labelingThe abundance provides the composite percentage of all

endmembers at each pixel (cf. Section III). Since the abun-dance is continuous between 0 and 1, it is impossible tomanually label the abundances—Human can not tell thedifference between 30% and 40% of an endmember at a pixel;besides, it is too tedious and labor intensive to label all pixels.Instead, we propose a learning method to label the abundance.

Given the determined K endmembers, the abundance la-beling becomes a least square problem with the nonnegativeand full additivity constraints, which is indeed the supervisedunmixing problem as reviewd in Section II. Such algorithmis far more easier than the unsupervised unmixing methods.The general quadratic programming algorithm with constraints(like the quadprog in Matlab) is helpful to estimate theabundances. Moreover, we may make use of the state-of-the-art unmxing methods reviewed in Section III-A and III-B bytreating the endmembers as a fixed input rather than unknownvariables and by only updating the abundances. The state-of-the-art unmxing methods are promising to achieve very goodground truths because of the different priors on the abundance.

C. How to verify our labeling ground truth for the HU study?What is a good ground truth for the given hyperspectral

image? To verify it, we come up with two evaluation methods:1). compare our labeling result with the ground truth posted

on the published papers. It is necessary to ensure the highsimilarity between them. Note that, only the illustration ofground truths (not the ground truth itself) is posted in thepaper. Accordingly, we can only compare them visually, notquantitatively. For example, we may compare the shape of theendmember signatures and the illustration of abundances.

2). check the labeling result based on our understandingof the hyperspectral image—the labeling result should beconsistent with our understanding of the image.

3). it is reasonable to assume that similar pixels have similarabundances [15, 63]. Based on this assumption, we will clickmany pixels to evaluate their signature’s similarities, and thento check their abundance’s similarity. Those two kinds ofsimilarities should be consistent as well.

4). for the hyperspectral images, like Cuprite, the groundtruth endmembers are selected from the standard spectrallibrary [12, 14, 37, 39, 63]. We don’t have to verify it.

We may have to try the different settings on the candidateendmembers, on the supervised unmixing algorithms for theabundance labeling, and on the hyper parameters etc (as shownin Algorithm 1), to make a better ground truth.

D. How to transform the benchmark hyperspectral classifica-tion (HyC) datasets for the HU research?

Compared with the HU datasets, the datasets for the hypser-pectral classification (HyC) task have three obvious advan-

Algorithm 1 to label the ground truth for the HU studyInput a hyperspectral image X ∈ RL×N+ , with N pixels.

1: set the endmembers and the endmember number K.2: repeat3: endmember labeling via the method in Section IV-A.4: abundance labeling via the method in Section IV-B.5: verify the ground truth labeling.6: until meeting the criteria in Section IV-C

Output the ground truth M ∈ RL×K+ and A ∈ RK×N+ .

tages. (1) It is well known that the HyC task has much morebenchmark datasets than the HU task. (2) Different from theHU datasets whose ground truths are mostly unavailable onthe web, the HyC datasets and their ground truths are mostlyavailable on the web. (3) the labeling process as well as thelabel itself for the HyC image are much easier to understand.Thus, it would be promising to transform the benchmark HyCdatasets for the HU study. However, there is a big gap betweenthe HyC task and the HU task, which makes the transformationprocess challenging. In this paper, we are the first to proposea promising method to accomplish this transformation task.

Our method consists of three process steps listed as below:1). The number of classes should be consistent with the

number of endmembers. Due to the supervised informationinvolved in the training process, the HyC algorithms aregenerally able to handle more classes than the number ofendmembers in the HU. Accordingly, we have to combine thesimilar classes to reduce the number of endmembers.

2). Candidate endmember labeling. We need the endmembersignatures. Given the training labels for the HyC image, wemay select a set of high quality pixels from the pixel set ofeach class. The endmember signature can be set as the meanor median signature of all the selected pixels in that class.

3). Abundance labeling. Suppose we are given a hyperspec-tral image X ∈ RL×N , with L channels and N samples. Thelabel of all pixels, i.e., Y ∈ RK×N , could be predicted viathe state-of-the-art classification algorithms [20, 78], whereeach column yn ∈ RK ,∀n ∈ {1, · · · , N} is the 0-1 binarylabel vector. That is, if pixel n is in the kth class, we definethe label vector yn = [0, · · · , 0, 1, 0, · · · , 0]

> ∈ RK withonly the kth element equal to 1, all the others equal to 0[79]. There are two ways to treat the candidate endmemberobtained in the 2nd step. We may treat them as the finalendmembers. Accordingly, the abundance labeling becomesthe supervised unmixing problem with a special anchoringconstraint as follows:

minA

loss{X, X

}+ λΥ (A) , s.t. X = MA,M ≥ 0,A ≥ 0,

where Υ (A) = Ψ (A) + αλ ‖A−Y‖2F is the constraint on

the abundance; Ψ (A) is the state-of-the-art sparse/spatial (orboth) constraints reviewed in Section III-B; ‖A−Y‖2F is theanchoring constraint; its aim is to keeps the abundance aroundthe high accuracy classification results.

We may also treat the candidate endmember, obtained inthe 2nd step, as a good initialization of M, and update both

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Samson#3

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(c) Samson#3 and its 3 endmembers: #1 Soil, #2 Tree and #3 Water.

Fig. 2. The three subimages selected from the Samson hyperspectral image.They are Samson#1, Samson#2 and Samson#3 respectively, as shown inFigs 2a, 2b and 2c. Besides, we provide the illustration of the ground truthendmembers and abundance maps for each subimage.

the endmember M and abundance A. The general model is

minM,A

loss{X, X

}+ λΥ (A) , s.t. X = MA,M ≥ 0,A ≥ 0,

where Υ (A) is the abundance constraint reviewed right above.

V. SUMMARIZATION OF THE 15 REAL HYPERSPECTRALIMAGES AND THEIR 18 GROUND TRUTHS

A. Samson’s three subimages

The Samson5 image is illustrated in Fig. 1a, where thereare 952 × 952 pixels [14–16, 80]. Each pixel is observed at156 bands covering the electromagnetic spectra from 401nmto 889nm. As a result, the spectral resolution is highly upto 3.13nm. This data is in good condition and not degradedby the blank channels or badly noised channels. The originalimage is very large, which could be computationally expensivefor the HU study. Accordingly, three subimages are selecteddue to that they contain large transitional areas which consistof lots of mixed pixels. We name the three ROIs as Samson#1,Samson#2 and Samson#3 respectively for convenience. Pleaserefer to the Fig. 1a and Table I for their information, e.g., theirlocations, sizes, scenes, number of bands etc.

5downloaded on http://opticks.org/confluence/display/opticks/Sample+Data.

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JasperRidge #3

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(c) Jasper Ridge#3 and its four endmembers “#1 Road”, “#2 Soil”, “#3 Water”and “#4 Tree” respectively.

Fig. 3. The three subimages selected from the Jasper Ridge hyperspectralimage. They are Jasper Ridge#1, Jasper Ridge#2 and Jasper Ridge#3 respec-tively as shown in Figs 3a, 3b and 3c. Besides, we provide the illustration ofthe endmembers and the abundance maps for each subimage.

All of three ROIs have 95 × 95 pixels. The first ROI (i.e.,Samson#1) starts from the (252, 332)-th pixel in the originalimage, cf., Figs. 1a and 2a. The top-left pixel in Samson#2corresponds to the (232, 93)-th pixel in the original image,cf., Figs. 1a and 2b. Samson#3’s first pixel is the (643, 455)-th pixel in the Samson image, cf., Figs. 1a and 2c. Specifically,there are three endmembers in this image, i.e., "#1 Soil", "#2Tree" and "#3 Water" respectively, as shown in Fig. 2.

B. Jasper Ridge’s three subimages

Jasper Ridge is a popular hyperspectral image [81, 82],as shown in Fig. 1b. It was captured via the AVIRIS (Air-borne Visible/Infrared Imaging Spectrometer) sensor by theJet Propulsion Laboratory (JPL). The ground sample distance(GSD) is 20m [82]. There are 512 × 614 pixels in it. Eachpixel is recorded at 224 electromagnetic bands ranging from380nm to 2, 500nm. The spectral resolution is highly upto 9.46nm. Due to the dense water vapor and atmosphericeffects, we remove the spectral bands 1–3, 108–112, 154–166and 220–224, remaining 198 bands. This is a common pre-processing step if the HU methods do not focus on the robustlearning [2, 16, 20, 78, 83, 84].

Since the full hyperspectral image, as shown in Fig. 1b,is too complex to get the ground truth, we consider three

http://opticks.org/confluence/display/opticks/Sample+Data

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UrbanEndmembers

Abundances

4 endmembers:*. #1 Asphalt*. #2 Grass*. #3 Tree*. #4 Roof

5 endmembers:*. #1 Asphalt*. #2 Grass*. #3 Tree*. #4 Roof*. #5 Soil

6 endmembers:*. #1 Asphalt*. #2 Grass*. #3 Tree*. #4 Roof1*. #5 Roof2/Shadow*. #6 Soil

(a)

(b)

(c)

Fig. 4. The three versions of ground truths for the Urban hyperspectral image.(a) illustrates the ground truth that has 4 endmembers; (b) shows the groundtruth that has 5 endmembers; (c) the ground truth that has 6 endmembers.

subimages, named as Jasper Ridge#1, Jasper Ridge#2 andJasper Ridge#3 respectively. Please refer to the Fig. 1b andTable I for their information. The first subimage (i.e., JasperRidge#1) consists of 115 × 115 pixels, whose top-left pixelis the (1, 272)-th pixel in the original image. There are fiveendmembers in Jasper Ridge#1, i.e., “#1 Road”, “#2 Soil”, “#3Water”, “#4 Tree” and “#5 Other”, as shown in Fig. 3a.

Jasper Ridge#2 contains 100× 100 pixels. The first pixel isthe (105, 269)-th pixel in the original image. There are fourendmembers latent in this subimage: “#1 Road”, “#2 Soil”,“#3 Water” and “#4 Tree”, as shown in Fig. 3b. Since weprovided the information of this subimage in the papers [14–16] and published the data (including the ground truth) on thehomepage [85] in 2015, it has been widely used in more than10 HU papers [2, 30, 31, 82, 86–91] to verify the state-of-the-art algorithms. The third subimage (i.e., Jasper Ridge#3)consists of 122 × 104 pixels. The first (i.e., top-left) pixel isthe (1, 246)-th pixel in the original image. There are fourendmembers latent in the Jasper Ridge#3: “#1 Road”, “#2Soil”, “#3 Water” and “#4 Tree”, as shown in Fig. 3c.

C. HYDICE Urban and its two subimages

HYDICE Urban is one of the most widely used hyperspec-tral image for the HU research [2, 12, 15, 16, 28, 31, 36, 92–97]. It was recorded by the HYDICE (Hyperspectral DigitalImage Collection Experiment) sensor in October 1995, whosescene is an urban area at Copperas Cove, TX, U.S. There are307 × 307 pixels in this image; each pixel corresponds to a2 × 2m2 region in the scene. The image has 210 spectralbands ranging from 400nm to 2, 500nm, resulting in a veryhigh spectral resolution of 10nm. After removing the badlydegraded bands 1–4, 76, 87, 101–111, 136–153 and 198–210(due to dense water vapor and atmospheric effects), we remain162 bands. There are three versions of ground truths for theHYDICE Urban hyperspectral image:

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Urban#2Endmembers Abundances

(a) Urban#1 and its six endmembers “#1 Asphalt”, “#2 Grass”, “#3 Tree”, “#4Roof1”, “#5 Roof2/shado” and “6 Soil” respectively.

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Urban#1Endmembers Abundances

(b) Urban#2 and its six endmembers “#1 Asphalt”, “#2 Grass”, “#3 Tree”, “#4Roof1”, “#5 Roof2/shado” and “6 Soil” respectively.

Fig. 5. The two challenging subimages selected from the Urban hyperspectralimage, i.e., Urban#1 and Urban#2, respectively. Besides, we provide theillustration of the endmembers and their abundance maps for each subimage.

(a) Cuprite

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(b) The ground truth endmember spectrum.

Fig. 6. (a) is the Cuprite hyperspectral image, which is selected in from theimage scene in Fig. 1d, and (b) displays the ground truth endmember spectra.

1). In the early HU papers [2, 12, 15, 16, 28, 31, 36, 94],it is widely accepted that there are four endmembers in theHYDICE Urban. They are “#1 Asphalt Road”, “#2 Grass”,“#3 Tree” and “#4 Roof”, as shown in Fig. 4a.

2). Recently, the analyse on Urban image become more andmore precise. The “#1 Asphalt Road” is divided into “#1 As-phalt” and “#6 Soil” whereas the “#4 Roof” is divided into “#4Roof1” and “#5 Roof2/shadow”. In other words, the numberof endmembers rises to six [12, 97]; they are “#1 Asphalt”,“#2 Grass”, “#3 Tree”, “#4 Roof1”, “#5 Roof2/shado” and “6Soil”, as shown in shown in Fig. 4c.

3). Apart from the above two versions of ground truths, weintroduce a new ground truth that consists of 5 endmembers:“#1 Asphalt”, “#2 Grass”, “#3 Tree”, “#4 Roof” and “5 Soil”,as shown in Fig. 4b. Compared with the first version, the “#1Asphalt Road” is divided into “#1 Asphalt” and “#5 Soil”.Compared with the second version, we merge “#4 Roof1” and“#5 Roof2/shadow” into “#4 Roof”.

To make it more challenging, we select two subimages fromthe Urban image (cf. Fig. 1c). In them, the small objects, likesmall houses, vehicles and small grasslands etc., are the mainscene, causing lots of transitional areas. Such case surely leadsto lots of mixed pixels due to the low spatial resolution ofhyperspectral sensors. The first subimage (i.e., Urban#1) has160×168 pixels, whose first pixel is the (1, 1)-th pixel in theoriginal image. The second subimage (i.e., Urban#2) startsfrom the (40, 1)-th pixel in the original image. It also has160 × 168 pixels. There are six endmembers: "#1 Asphalt",

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(b) Moffett Field#2 and its three endmembers.

Fig. 7. The two subimages selected from the Moffett Field hyperspectralimage. They are Moffett Field#1 and Moffett Field#2, respectively, as shownin Figs 7a and 7b. There are 3 endmembers, i.e., “#1 Soil”, “#2 Tree” and“#3 Water” in both subimages. Besides, we provide the illustration of theendmembers and their abundance maps for each subimage.

"#2 Grass", "#3 Tree", “#4 Roof1”, "#5 Roof2/shadow", and "6Soil" in both Urban#1 (cf., Fig. 5a) and Urban#2 (cf., Fig. 5b).

D. Cuprite

Cuprite is the most benchmark and challenging hyperspec-tral image for the HU research [12, 14, 37, 39, 50, 53, 98–103],which is captured by the AVIRIS sensor. It covers a Cupritearea in Las Vegas, NV, U.S. There are 224 spectral bandsin the Cuprite image, ranging from 370nm to 2, 480nm.After removing the noisy bands (i.e., 1–2 and 221–224) andthe water absorption bands (i.e., 104–113 and 148–167), itremains 188 bands. In this paper, a subimage of 250 × 190pixels is considered, which is widely used in the state-of-the-art HU papers [12, 14, 39, 50]. Please refer to Fig. 1d for theillustration of its position, image size and scene etc.

In this subimage, there are 14 types of minerals (or endmem-bers), whose spectra can be obtained from the ENVI software.There are very minor differences between the variants of thesame type of mineral, for example Kaolinite1 and Kaolinite3are very similar in signature. The researchers have their ownthoughts, resulting in different versions of ground truths. In[39], there are 14 endmembers; while there are 10 endmembersin [12]; then Dr. Lu hold that there are 12 endmembers in theCuprite. Here we agree with Dr. Lu’s setting. Please refer toFig. 6 for the illustration of the 12 endmembers. Because thereare small differences in the setting of endmembers among thepapers [12, 37, 39, 50, 53], the results of the state-of-the-artmethods in those papers might be different from each other.

E. Moffett Field

The Moffett Field hyperspectral image is illustrated inFig. 1e. It was captured via the AVIRIS sensor by the JPLin 1997 and was used in a number of state-of-the-art papers

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San DiegoAirport#1

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4 endmembers:*. #1 Grass*. #2 Tree*. #3 Roof*. #4 Ground & Road

(a)

5 endmembers:*. #1 Grass*. #2 Tree*. #3 Roof*. #4 Ground & Road*. #5 Other

(b)

Fig. 8. Two versions of HU ground truths for the San Diego Airporthyperspectral image: (a) illustrates the ground truth that has 4 endmembers;(b) shows the ground truth that has 5 endmembers. (best viewed in color)

[104–109]. There are 512×614 pixels and 224 spectral bands,ranging the electromagnetic spectra from 400nm to 2500nm.The spectral resolution is up to 9.38nm. The full image canbe found in [110]. It consists of a large area of water (thatappears in dark pixel at the top of the image) and a coastalarea composed of trees and soil. Due to the water vapor andatmospheric effects, we remove the noisy spectral bands. Afterthis process, there remains 196 bands.

The original image is very large in size and too complex toanalyze because the spatial resolution is low and the object isvery small in the scene. In this paper, we select two subimagesfor the HU study. Please refer to Fig.1e for their locations,image sizes and scenes etc. The first subimage (named MoffettField#1) consists of 60×60 pixels, whose first (top-left) pixelis the (54, 172)-th pixel in the original image. The secondsubimage (named Moffett Field#2) also has 60×60 pixels. Itsfirst pixel is the (62, 146)-th pixel in the full image. A similarsubimage with Moffett Field#1 was used in [109, 111], wherethe authors hold that there are three endmembers. In this paper,we agree with their setting. The three endmembers for bothMoffett Field#1 and Moffett Field#2 are “#1 Soil”, “#2 Tree”and “#3 Water”, respectively, as shown in Figs. 7a and 7b.

F. San Diego Airport

San Diego Airport is a popular airborne hyperspectral imagecollected by the AVIRIS sensor [76, 112–114]. There are400 × 400 pixels and 224 spectral bands, as illustrated inFig. 1f. After removing the noisy bands with the water vaporand atmospheric effects, including the bands 1–6, 33–35, 97,107–113, 153–166, and 221–224, there remains 189 spectralbands. In this paper, we select a subimage for the HU study.It has 160×140 pixels. The top-left pixel is the (241, 261)-thpixel in the original image, as shown in Fig. 1f. There are twoendmember settings: (1) the first setting has 4 endmembers,i.e., “#1 Grass”, “#2 Tree”, “#3 Roof” and “#4 Ground &Road”, as shown in Fig. 8a; (2) the second setting has 5endmembers, i.e., “#1 Grass”, “#2 Tree”, “#3 Roof”, “#4Ground & Road” and “#5 Other”, as shown in Fig. 8b.

10

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(a) WDM#1 and its six endmembers.

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(b) WDM#2 and its six endmembers.

Fig. 9. Two challenging subimages selected from the Washington DC Mallimage: WDM#1 and WDM#2 as shown in Figs 9a and 9b. For each subimage,we give the illustration of the ground truth endmembers and abundances.

G. Washington DC Mall (WDM)

The Washington DC Mall (i.e., WDM), an airborne hyper-spectral image flightline over the Washington DC Mall (cf.Fig. 9g), is used to verify a lot of research works [28, 31,37, 102, 115, 116]. It was collected by the HYDICE sensor.The full image can be downloaded from the webpage6. Thereare 1208 × 307 pixels and 210 spectral channels, coveringthe the electromagnetic spectra from 400nm to 2400nm. Thespectral resolution is up to 9.52nm. After removing the bandswith water vapor and atmospheric effects, including 103–106,138–148, and 207–210, there remains 191 bands [102].

In this paper, we select two subimages for the HU study, i.e.,WDM#1 and WDM#2 respectively. Please refer to Fig. 9g andTable I for their information like locations and scenes etc. Thefirst subimage (i.e. WDM#1) has 150×150 pixels, whose firstpixel starts from the (549, 160)-th pixel in the original image.WDM#1 has been used in [37], where they provide the roughlocations of the six endmember; however, they did not sharethe ground truth on the Internet. The six endmembers are “#1Roof”, “#2 Grass”, “#3 Water”, “#4 Tree”, “#5 Trail” and “#6Road” respectively, as shown in Fig. 9a. The second subimage(i.e. WDM#2) has 180×160 pixels, whose left-top pixel is the(945, 90)-th pixel in the original image. The six endmembersare “#1 Road”, “#2 Grass”, “#3 Trail”, “#4 Water”, “#5 Tree”and “#6 Roof” respectively, as shown in Fig. 9b.

VI. THE SIMULATED HYPERSPECTRAL IMAGE

This section provides the method to generate a complexsynthetic image [2, 12, 27–32] for the HU research. It consistsof two steps. First, 15 true spectra are chosen from the UnitedStates Geological Survey (USGS) digital spectral library as thecandidate endmembers. Their signature are shown in Fig. 10.The first K spectra are selected as the true endmember, where2 ≤ K ≤ 15 is the number of endmembers.

Second, we use the method in [2, 12, 27, 28, 30, 32] togenerate the abundance maps, which has the following steps:

6https://engineering.purdue.edu/∼biehl/MultiSpec/hyperspectral.html

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#3 Andradite ws487#4 Diaspore HS416

#5 Erionite Merlinoite GDS144

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#11 Schwertmannite BZ93#12 Olivine GDS70

#13 Pyroxene HS119#14 Adularia GDS57 Orthoclase#15 Almandine WS477 Garnet

Fig. 10. The signature of 15 candidate endmembers used for the simulationof hyperspectral images. Each signature is observed at 480 spectral bands.

1). We divide an image of z2×z2(z ∈ Z+) pixels into z×zregions. The default setting is z = 8.

2). Each region is filled up with a type of ground cover,which means the abundance of that endmember is 1 in thewhole region. Note that, this process is randomly implemented.

3). To generate the mixed pixels, we use the (z + 1) ×(z + 1) spatial low-pass filter. Such process causes largetransitional areas, where there are lots of mixed pixels.

4). To remove pure pixels and generate sparse abundances,we replace the pixels whose largest abundance is larger than0.8 with a mixture of two endmembers with equal abundances.

Now both endmembers M and abundances A have beengenerated. Based on them, a perfect hyperspectral image iscreated as Y = MA. To simulate a more real remote sensingimage that contains complex noises, we add the zero-meanGaussian (or some other) noise to the generate the image data.The Signal-to-Noise Ratio (SNR) is defined as follows:

SNR = 10 log10

{E[y>y

]E [n>n]

}. (20)

With the above synthetic hyperspectral image, we can verifythe HU algorithmss from the following four perspectives:• Robustness analysis to the noise corruptions, e.g.,

SNR∈ {∞, 35, · · · , 10 dB} .• Generalization to the pixel number, which is set by z.• Robustness to the endmember number {K |2≤K≤15}.• Generalization to the number of bands in the image.

We will provide the full codes for this simulated dataset.

VII. EXPERIMENTS RESULTS OF 5 BENCHMARK HUMETHODS ON THE 18 HYPERSPECTRAL IMAGES

In this section, we compare 5 benchmarks HU methods onthe 15 hyperspectral images summarized in Section V. Theselected 5 methods are widely compared against the state-of-the-art HU methods. The aim is to provide benchmark HUresults, saving other’s effort to evaluate their new HU methods.

A. Five Benchmark Methods Compared in this Section

1). Vertex Component Analysis [39] (VCA) is a classicgeometric method. The code is available on http://www.lx.it.pt/bioucas/code.htm.

2). Nonnegative Matrix Factorization [51] (NMF) is abenchmark statistical method. The code is obtained fromhttp://www.cs.helsinki.fi/u/phoyer/software.html.

https://engineering.purdue.edu/~biehl/MultiSpec/hyperspectral.html

http://www.lx.it.pt/bioucas/code.htm

http://www.lx.it.pt/bioucas/code.htm

http://www.cs.helsinki.fi/u/phoyer/software.html

11

TABLE IITHE PERFORMANCES OF FIVE BENCHMARK HU ALGORITHMS ON THE SAMSON#1, SAMSON#2 AND SAMSON#3 HYPERSPECTRAL IMAGES.

Sam

son

#1 Endmembers Spectral Angle Distance (SAD) Root Mean Square Error (RMSE)VCA NMF `1-NMF `1/2-NMF G-NMF VCA NMF `1-NMF `1/2-NMF G-NMF

#1 Soil 0.1531 0.0861 0.0582 0.0902 0.0529 0.1765 0.1110 0.0930 0.0990 0.0993#2 Tree 0.0487 0.0556 0.0545 0.0410 0.0576 0.1372 0.0886 0.0798 0.0919 0.0734#3 Water 0.1296 0.1560 0.1233 0.0410 0.1609 0.1748 0.0757 0.0511 0.0344 0.0750Avg. 0.1105 0.0992 0.0787 0.0574 0.0905 0.1628 0.0918 0.0746 0.0751 0.0826

Sam

son



Sam

son



TABLE IIITHE SAD PERFORMANCES OF FIVE BENCHMARK HU ALGORITHMS ON

THE CUPRITE HYPERSPECTRAL IMAGES.

Endmembers Spectral Angle Distance (SAD)VCA NMF `1-NMF `1/2-NMF G-NMF

#1 Alunite 0.1584 0.1640 0.1618 0.1458 0.1632#2 Andradite 0.0773 0.1166 0.1253 0.0783 0.1279#3 Buddingtonite 0.0999 0.1343 0.1391 0.1456 0.1406#4 Dumortierite 0.1033 0.1202 0.1212 0.1067 0.1211#5 Kaolinite1 0.0892 0.1042 0.0951 0.1004 0.1001#6 Kaolinite2 0.0682 0.1120 0.1024 0.0763 0.1125#7 Muscovite 0.1912 0.2440 0.2490 0.2228 0.2489#8 Montmorillonite 0.0721 0.1165 0.0980 0.0769 0.1209#9 Nontronite 0.0784 0.1115 0.1036 0.0940 0.1232#10 Pyrope 0.0907 0.0966 0.0935 0.0831 0.1045#11 Sphene 0.0702 0.0952 0.0903 0.0788 0.0923#12 Chalcedony 0.0870 0.1811 0.1719 0.1542 0.1513Avg. 0.0988 0.1330 0.1293 0.1136 0.1339

3). Nonnegative sparse coding [62] (`1-NMF) is a classicsparse regularized NMF method. The code is download fromhttp://www.cs.helsinki.fi/u/phoyer/software.html.

4). `1/2 sparsity-constrained NMF [12] (`1/2-NMF) is astate-of-the-art method that could get sparser results than `1-NMF. Since the code is unavailable, we implement it.

5). Graph regularized NMF [61, 64] (G-NMF) is an inter-esting algorithm that transfers graph information inherent inthe hyperspectral image into the abundance space.

We will share the code for all the above five algorithms.

B. Experiment Settings and Experiment Results

Generally, there are one or two hyper-parameters in the 5benchmark methods. We use the coarse to fine grid searchmethod to find the best hyper-parameters [14–16]. The ini-tialization of the endmembers M and abundances A is veryimportant to the final HU results [37]. We employ the samebenchmark algorithm, i.e., VCA [39], to initialize M and Afor the five methods on all the 15 datasets. There are twoadvantages: (1) it is helpful to generate the reproducible resultsby the VCA; (2) VCA is really simple, effective and solid.

Two evaluation metrics are used to obtain the quantitativecomparison results: (1) the Spectral Angle Distance (SAD)[14–16, 53, 63] is used to verify the performance of theestimated endmembers; (2) the Root Mean Square Error(RMSE) [12, 14–16, 117] is for the evaluation of the estimated

abundances. To get a valid results, each experiment is repeated50 times and the mean result is provided in the Tables II, III,IV, V, VI, VII, VIII, IX. In short, we summarize the tablesthat display the results on all the 15 datasets:

1). Table II summarizes the experiment results of all the fivebenchmark algorithms (reviewed in Section VII-A). There arethree tables, displaying the HU results on the three subimages,i.e., Samson#1, Samson#2 and Samson#3 respectively.

2). In Table III, the HU results on the Cuprite image aresummarized. Note that the ground truth abundance is notavailable; only the endmember results are compared.

3). Table IV illustrates the HU results of 5 benchmarkalgorithms on the full HYDICE Urban image. There are 3versions of ground truths for the Urban images. Accordingly,we summarize 3 versions of HU results in the 3 sub-tables.

4). In Table V, the HU results on the two subimages,Urban#1 and Urban#2, are summarized. There are six end-members as shown in the Table V.

5). In Table VI, the experiment results on the 3 subimagesfrom the JasperRidge is displayed. The 3 subimage are Jasper-Ridge#1, JasperRidge#2 and JasperRidge#3.

6). Table VII illustrates the HU results on the 2 subimagsselected from Moffett Field. They are Moffett Field#1 andMoffett Field#2 respectively as shown in the Table.

7). Table VIII illustrates the HU results on the 2 subimagsselected from the Washington DC Mall. They are WDM#1and WDM#2 respectively.

8). Table IX displays the HU results on the two versions ofground truths on the San Diego Airport dataset.

VIII. CONCLUSION

Hyperspectral unmixing (HU) is an important preprocessingstep for a great number of hyperpsectral applications. Topromote the HU research, we accomplished this paper byemphasizing on the generation of standard HU datasets (withground truths) and the summarization of benchmark results ofthe state-of-the-art HU algorithms. Specifically, this is the firstpaper to propose a general labeling method for the HU. Viait, we summarized and labeled up to 15 hyperpsectral images.To further enrich the HU datasets, an interesting method hasbeen proposed to transform the hyperspectral classification

http://www.cs.helsinki.fi/u/phoyer/software.html

12

TABLE IVTHE PERFORMANCES OF FIVE BENCHMARK HU ALGORITHMS ON THE HYDICE URBAN IMAGE, WHICH HAS THREE VERSIONS OF GROUND TRUTHS.

V1:

4en

dmem

bers Endmembers Spectral Angle Distance (SAD) Root Mean Square Error (RMSE)

VCA NMF `1-NMF `1/2-NMF G-NMF VCA NMF `1-NMF `1/2-NMF G-NMF#1 Asphalt 0.2152 0.2114 0.1548 0.1349 0.2086 0.2884 0.2041 0.2279 0.3225 0.2056#2 Grass 0.3547 0.3654 0.2876 0.0990 0.3716 0.3830 0.2065 0.2248 0.3387 0.2078#3 Tree 0.2115 0.1928 0.0911 0.0969 0.1934 0.2856 0.1870 0.1736 0.2588 0.1824#4 Roof 0.7697 0.7370 0.7335 0.5768 0.7377 0.1638 0.1395 0.1861 0.1782 0.1389Avg. 0.3877 0.3767 0.3168 0.2269 0.3778 0.2802 0.1843 0.2031 0.2746 0.1837

V2:

5en

dmem


VCA NMF `1-NMF `1/2-NMF G-NMF VCA NMF `1-NMF `1/2-NMF G-NMF#1 Asphalt 0.3009 0.3284 0.1664 0.1837 0.3502 0.2898 0.2941 0.3080 0.3408 0.2944#2 Grass 0.3547 0.4575 0.1826 0.1143 0.5539 0.3634 0.3417 0.2048 0.3099 0.3823#3 Tree 0.4694 0.2337 0.1335 0.0658 0.2402 0.3092 0.2513 0.1245 0.2944 0.2887#4 Roof 0.4066 0.4134 0.4801 0.0834 0.4340 0.2202 0.1548 0.1101 0.1751 0.1581#5 Soil 1.0638 1.0478 1.0948 0.0290 0.9651 0.2791 0.2755 0.2797 0.3125 0.2796Avg. 0.5191 0.4962 0.4115 0.0952 0.5087 0.2923 0.2635 0.2054 0.2865 0.2806

V3:

6en

dmem

bers

Endmembers Spectral Angle Distance (SAD) Root Mean Square Error (RMSE)VCA NMF `1-NMF `1/2-NMF G-NMF VCA NMF `1-NMF `1/2-NMF G-NMF

#1 Asphalt 0.2441 0.3322 0.2669 0.3092 0.3274 0.2555 0.2826 0.2803 0.3389 0.2787#2 Grass 0.3058 0.4059 0.3294 0.0792 0.3787 0.2696 0.3536 0.2937 0.2774 0.3184#3 Tree 0.6371 0.2558 0.2070 0.0623 0.2629 0.3212 0.2633 0.1859 0.2701 0.2404#4 Roof1 0.2521 0.3701 0.4370 0.0680 0.3264 0.2500 0.1662 0.1358 0.1541 0.1725#5 Roof2/ 0.7451 0.6223 0.5330 0.1870 0.6831 0.2157 0.1603 0.2136 0.1660 0.1441Shadow#6 Soil 1.1061 0.9978 1.0371 0.0287 0.9859 0.2714 0.2505 0.2594 0.3542 0.2547Avg. 0.5484 0.4974 0.4684 0.1224 0.4941 0.2639 0.2461 0.2281 0.2601 0.2348

TABLE VTHE PERFORMANCES OF FIVE BENCHMARK HU ALGORITHMS ON THE URBAN#1 AND URBANN#2 HYPERSPECTRAL SUB-IMAGES.

Urb

an#1



Urb

an#2



datasets for the HU research. Besides, we reviewed andimplemented a widely accepted algorithm to generate a verycomplex simulated hyperpsectral image for the HU study. Suchsynthetic dataset is helpful to verify the HU algorithms underfour different conditions.

To summarize the benchmark HU results, we reviewed upto 10 state-of-the-art HU algorithms, and selected the fivemost benchmark HU algorithms for the comparison. ThoseHU algorithms are compared on the 15 hyperpsectral images.To the best of our knowledge, this is the first paper to comparethe benchmark algorithms on so many real hyperpsectralimages. The experient results on this paper may provide avalid baseline for the evaluation of new HU algorithms.

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TABLE VITHE HU RESULTS OF FIVE BENCHMARK HU ALGORITHMS ON THE JASPERRIDGE#1, JASPERRIDGE#2 AND JASPERRIDGE#3 HYPERSPECTRAL IMAGES.

Jasp

erR

idge

#1Endmembers Spectral Angle Distance (SAD) Root Mean Square Error (RMSE)

VCA NMF `1-NMF `1/2-NMF G-NMF VCA NMF `1-NMF `1/2-NMF G-NMF#1 Tree 0.3381 0.1325 0.0553 0.0458 0.0743 0.2651 0.1327 0.0928 0.1032 0.0752#2 Water 0.2573 0.2823 0.0300 0.0303 0.2923 0.2711 0.1384 0.0714 0.0911 0.1366#3 Soil 0.3062 0.2125 0.1318 0.0534 0.2005 0.2567 0.2221 0.2536 0.2631 0.1837#4 Road 0.6053 0.4843 0.6352 0.5525 0.5411 0.2361 0.1876 0.2064 0.2389 0.1908#5 Other 0.7156 0.6463 1.0503 0.8187 0.6742 0.2842 0.1601 0.1357 0.1451 0.1669Avg. 0.4445 0.3516 0.3805 0.3001 0.3565 0.2626 0.1682 0.1520 0.1683 0.1506

Jasp

erR

idge


#1 Tree 0.2565 0.2130 0.0680 0.0409 0.2781 0.3268 0.1402 0.0636 0.0707 0.1740#2 Water 0.2474 0.2001 0.3815 0.1682 0.2530 0.3151 0.1106 0.0660 0.1031 0.1375#3 Soil 0.3584 0.1569 0.0898 0.0506 0.3246 0.2936 0.2557 0.2463 0.2679 0.2840#4 Road 0.5489 0.3522 0.4118 0.3670 0.2776 0.2829 0.2450 0.2344 0.2737 0.2279Avg. 0.3528 0.2305 0.2378 0.1567 0.2833 0.3046 0.1879 0.1526 0.1789 0.2058

Jasp

erR

idge


#1 Tree 0.1168 0.0992 0.0969 0.1081 0.1117 0.1522 0.1000 0.1182 0.1017 0.1164#2 Water 0.2763 0.3124 0.0513 0.0474 0.3369 0.1467 0.1248 0.0418 0.0651 0.1298#3 Soil 0.5364 0.3748 0.3525 0.2419 0.3414 0.2475 0.2663 0.2536 0.2289 0.2247#4 Road 0.4235 0.2997 0.3906 0.4098 0.2200 0.2105 0.2163 0.1707 0.1834 0.1688Avg. 0.3383 0.2715 0.2228 0.2018 0.2525 0.1892 0.1769 0.1461 0.1448 0.1599

TABLE VIITHE PERFORMANCES OF FIVE BENCHMARK HU ALGORITHMS ON THE MOFFETT FIELD#1 AND MOFFETT FIELD#2 HYPERSPECTRAL IMAGES.

Mof

fett

Fiel

d#1



Mof

fett

Fiel

d#2



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http://arxiv.org/abs/1409.0685

14

TABLE VIIITHE HU RESULTS OF FIVE BENCHMARK ALGORITHMS ON THE WDM#1 AND WDM#2 SUBIMAGES. WDM DENOTES WASHINGTON DC MALL.

WD

M#1


#1 Roof 0.18175 0.23343 0.24818 0.22110 0.21093 0.12021 0.16432 0.16577 0.16744 0.17293#2 Grass 0.24779 0.16813 0.17883 0.17013 0.17285 0.18837 0.19286 0.19132 0.19167 0.19202#3 Water 0.18327 0.17505 0.16717 0.17077 0.17099 0.33159 0.20351 0.20051 0.20236 0.19769#4 Tree 0.13241 0.15310 0.15853 0.14407 0.15898 0.15137 0.14405 0.14417 0.14449 0.14406#5 Trail 0.17445 0.26442 0.26116 0.26343 0.27032 0.30302 0.26970 0.27523 0.27606 0.28636#6 Road 0.42313 0.35212 0.34244 0.37354 0.36481 0.36034 0.29809 0.29570 0.30967 0.29885Avg. 0.22380 0.22437 0.22605 0.22384 0.22481 0.24248 0.21209 0.21212 0.21528 0.21532

WD

M#2



TABLE IXTHE PERFORMANCES OF FIVE BENCHMARK HU ALGORITHMS ON THE SAN DIEGO AIPORT IMAGE, WHICH HAS TWO VERSIONS OF GROUND TRUTHS.

V1:

4en

dmem


VCA NMF `1-NMF `1/2-NMF G-NMF VCA NMF `1-NMF `1/2-NMF G-NMF#1 Grass 1.0476 0.7738 0.1811 0.0474 0.7763 0.3164 0.3403 0.2675 0.2692 0.3255#2 Tree 0.1891 0.1904 0.7613 0.8001 0.1709 0.2257 0.2338 0.2672 0.2851 0.2336#3 Roof 0.3844 0.5240 0.4924 0.4608 0.5224 0.4508 0.3361 0.3101 0.3491 0.3348#4 Ground 0.2187 0.2543 0.1313 0.0609 0.2487 0.3719 0.2593 0.2283 0.2964 0.2562/RoadAvg. 0.4600 0.4356 0.3915 0.3423 0.4296 0.3412 0.2924 0.2683 0.2999 0.2875

V2:

5en

dmem


VCA NMF `1-NMF `1/2-NMF G-NMF VCA NMF `1-NMF `1/2-NMF G-NMF#1 Grass 0.4806 0.3382 0.3560 0.2957 0.2544 0.3249 0.2916 0.2964 0.2754 0.2906#2 Tree 0.1592 0.1361 0.1439 0.1624 0.1430 0.2209 0.2267 0.2212 0.2176 0.2207#3 Roof 0.6239 0.6509 0.6293 0.6205 0.6409 0.3299 0.2538 0.2626 0.2715 0.2563#4 Ground 0.2509 0.2276 0.2200 0.2047 0.2501 0.3443 0.2544 0.2622 0.2566 0.2581/Road#5 Other 0.4856 0.5421 0.5586 0.5861 0.6157 0.3382 0.2665 0.2735 0.2781 0.2696Avg. 0.4000 0.3790 0.3815 0.3739 0.3808 0.3117 0.2586 0.2632 0.2599 0.2591

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