International Journal of
Geo-Information Article
An Endmember Initialization Scheme for Nonnegative Matrix Factorization and Its Application in Hyperspectral Unmixing Jingjing Cao
ID
, Li Zhuo *
ID
and Haiyan Tao
Center of Integrated Geographic Information Analysis, Guangdong Key Laboratory for Urbanization and Geo-simulation, School of Geography and Planning, Sun Yat-sen University, Guangzhou 510275, China;
[email protected] (J.C.);
[email protected] (H.T.) * Correspondence:
[email protected]; Tel.: +86-020-8411-5833 Received: 31 March 2018; Accepted: 16 May 2018; Published: 18 May 2018
Abstract: Nonnegative matrix factorization (NMF) is a blind source separation (BSS) method often used in hyperspectral unmixing. However, it tends to converge to a local optimum. To overcome this limitation, we present a simple, but effective endmember initialization scheme for NMF, which is realized by improving initial values through the application of the automatic target generation process (ATGP) algorithm. The initial spectra and abundances of target endmembers are first obtained using the ATGP algorithm and nonnegative least squares (NNLS) method, respectively. The preliminary results are then optimized through iterative application of NMF. To validate the applicability and effectiveness of the proposed method, we analyzed the improvement of NMF by the ATGP algorithm, using the synthetic hyperspectral data and real hyperspectral images. The results from the proposed method are compared with those of the vertex component analysis (VCA)-NMF algorithm, which uses the VCA algorithm to perform initialization for NMF, the minimum volume constrained NMF (MVC-NMF) algorithm, the traditional two-step VCA-fully-constrained least squares (FCLS) algorithm, which uses the VCA to extract the endmember matrix, and the FCLS algorithm to estimate the abundance matrix. The comparison results prove that proper endmember initialization can help the NMF algorithm yield better estimation results. Through the optimization of target endmembers’ initial values, the proposed ATGP-NMF algorithm can consistently produce good results at a lower computational cost, especially in the case of a real hyperspectral image for which pure pixels do not exist and there is little prior knowledge. With its high applicability and effectiveness, the ATGP-NMF algorithm has a great potential to solve hyperspectral unmixing problems. Keywords: hyperspectral remote sensing; blind unmixing; automatic target generation process (ATGP); nonnegative matrix factorization (NMF)
1. Introduction Hyperspectral imagery can provide abundant spatial and spectral information for observed ground objects. It has attracted increasingly more attention and has been widely used. However, due to the low spatial resolution of hyperspectral sensors, mixed pixels prevail in hyperspectral images. The mixed pixel problem severely influences the precision of object recognition and classification; therefore, it has become an obstacle to quantification analysis of hyperspectral images. Hence, spectral unmixing has always been an important subject in hyperspectral data analysis [1]. Traditional spectral unmixing models are usually developed based on endmember spectra and involve two steps, i.e., endmember extraction and spectral unmixing [2–5]. These methods depend heavily on the precision of endmember extraction and require certain prior knowledge [6]. Their effectiveness is greatly limited
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when there is insufficient prior information. Therefore, it is greatly important and necessary to develop data-driven blind unmixing methods for hyperspectral images. Over the past decade, the development of a new technique in the signal processing field, blind source separation (BSS) [7], has offered novel methods for spectral unmixing. Different from traditional two-stage methods, the BSS method can simultaneously obtain the endmembers and their corresponding abundances. The current blind unmixing methods mainly include independent component analysis (ICA) [8], nonnegative matrix factorization (NMF) [9], complexity-based BSS methods [10] and sparse component analysis (SCA) [11,12]. Different from other methods, NMF can obtain nonnegative results with physical significance. Therefore, NMF and its extensions have been widely used for spectral unmixing. However, these NMF algorithms often provide only “local best” solutions due to the non-convexity of objective functions [13]. Most NMF-based algorithms deal with this problem by adding different constraints, such as the smoothness constraint, the minimum volume constraint, the sparseness constraint and the regularized constraint. Representative algorithms of the smoothness constraint have been reported in [14,15]. Miao et al. developed the minimum volume constrained nonnegative matrix factorization (MVC-NMF) [16], and Li et al. introduced the sparseness constraint of abundance matrix into MVC-NMF [17]. Various sparse regression-based unmixing methods have been introduced by incorporating the sparseness constraints into spectral unmixing [18–21]. Representative research on the sparseness constraints include the coupled nonnegative matrix factorization (CNMF) [22], the substance dependence constrained sparse nonnegative matrix factorization (SDSNMF) [23], the total variation regularized reweighted sparse NMF (TV-RSNMF) [24] and the simultaneously sparse and low-rank constrained NMF (DSPLR-NMF) [25]. Recent research has also considered regularized constraints, such as the sparsity-constrained NMF with L1/2 regularization [26], the sparsity-regularized robust NMF (RNMF) [27], the Arctan-NMF with smooth and sparse regularization [28], the robust collaborative nonnegative matrix factorization (R-CoNMF) [29] and the graph regularized nonnegative matrix factorization (GNMF) [13]. These methods are mainly proposed to get rid of the local minima of NMF by adding constraints to the endmembers or the abundances. However, most statistical-based blind unmixing methods can still be trapped in local minima if the initial values are not properly provided. Researchers have also tried to solve the local optimization problem by providing reasonable initialization and have found that the proper initialization of NMF and its extensions can help them to avoid the “local best” problem and refine their solutions [30,31]. Several methods have been proposed to improve the initialization, such as singular value decomposition (SVD) [32], clustering-based initialization approaches [33] and vertex component analysis (VCA) [3,34]. Although great progress has been made in improving the local optimization problem of the NMF algorithm, these studies have mainly focused on signal processing and related areas. Further studies still need to be done to verify their applicability in solving the mixed pixel problem of a hyperspectral image. Compared to traditional supervised endmember extraction methods, such as the pixel purity index (PPI), ATGP requires less prior information to extract pure pixel vectors as the endmember spectrum from the target image [35]. Moreover, most of the ATGP-generated target pixels turn out to be endmembers [36]. In other words, ATGP can provide more accurate initial points to identify endmembers. When it was used to generate initial values for another BSS method, ICA, ICA showed a better performance [37]. Therefore, we endeavored to use the ATGP algorithm to extract better initial endmembers for the NMF algorithm. In this paper, we propose a new method that utilizes the ATGP algorithm to perform endmember initialization for NMF, with the purpose of solving the NMF’s local optimum problem. First, the ATGP algorithm is used to find initial target endmembers from a hyperspectral image, which are used to initialize the endmember matrices of NMF. The initial abundance matrices of NMF are extracted by using NNLS. Second, the endmember spectrum matrices and abundance matrices are iteratively updated by the multiplicative update algorithm until the optimization goal is achieved. Finally, the effectiveness and efficiency of the proposed method (ATGP-NMF) are verified by comparing its experimental results in different situations with the results of other methods.
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The remainder of this paper is organized as follows. Section 2 briefly describes the linear mixture model (LMM), NMF and ATGP and introduces the proposed endmember initialization scheme for NMF. Section 3 presents experimental results and evaluates the performances of the proposed method and the other three algorithms on synthetic hyperspectral data and real hyperspectral images. A summary of this study and the conclusions are presented in Section 4. 2. Methodology 2.1. Linear Mixture Model LMM assumes that the observed pixel is a linear combination of all the endmembers weighted by their corresponding abundance fractions [38,39]. In addition, LMM has been widely used for hyperspectral unmixing due to its physical effectiveness and mathematical simplicity. The LMM can be expressed as X = AS + E (1) where X = [x1 , x2 , · · · , xN ] ∈ R L× N represents the observation matrix with L bands and N pixels, and A = [a1 , a2 , · · · , aP ] ∈ R L× P and S = [s1 , s2 , · · · , s N ] ∈ RP× N represent the endmember matrix and the abundance matrix, respectively. P denotes the number of endmembers and E ∈ R L× N is the additive noise. In LMM, the abundance vectors are subjected to the abundance nonnegative constraint (ANC) and abundance sum-to-one constraint (ASC). 2.2. Nonnegative Matrix Factorization NMF is a general matrix decomposition method, which decomposes the data matrix X as a product of two nonnegative matrices A and S. Note that L denotes the number of bands, N denotes the number of pixels, P denotes the number of endmembers, and P < min ( L, N ). In essence, NMF is an optimization problem. An objective function based on Euclidean distance is presently the most widely used method; in this case the goal of the NMF model is to find the minimum reconstruction error by two nonnegative matrices A and S: minimize : Eus(X, AS) =
1 1 kX − ASk2 = ∑ (Xij −(AS)ij )2 2 2 ij
(2)
where aij ≥ 0, and sij ≥ 0, 21 k·k2F represents the Frobenius norm.
For the objective function 12 kX − ASk2F , many learning algorithms, such as the multiplicative update algorithm, alternating least squares algorithm and gradient descent algorithm, are presently often used to solve the optimization problem of NMF. Among these algorithms, the multiplicative update algorithm can guarantee the non-negativity of both the calculation process and the final results [40]. It also adopts an auto-adjusted step size in the iteration, rather than the traditional manual setting, which reduces the impacts of parameter selection. In this paper, we used the general iterative expression formula of the multiplicative update algorithm to solve the objective function. For each matrix of A and S, the objective function 12 kX − ASk2F is a convex function, but if matrices A and S are considered as a whole, the objective function is no longer convex. This makes it difficult to converge to a global optimum. Tao et al. found that the NMF algorithm can determine the optimum and physically significant solution more quickly and easily if a better starting point is provided [41]. 2.3. Automatic Target Generation Process The selection of an initial target endmember has a great impact on spectral unmixing [42]. Based on unsupervised and unconstrained orthogonal subspace projection (OSP) theory, the ATGP algorithm searches for the candidate endmembers with the maximal orthogonal projection in a sequence of orthogonal projection subspaces [43]. ATGP is an unsupervised target detection technique. It implements an orthogonal subspace projector to find a set of potential targets with the maximal
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orthogonal projections (OPs) from the image. The orthogonal subspace projector of ATGP can be defined as: −1 PU⊥ = I − U(UT U) UT (3) where the subspace U is composed of the targets previously generated by the ATGP algorithm. It should be noted that before the ATGP algorithm is applied, the number of spectral endmembers should be determined, especially in cases for which there is not sufficient prior information. The Harsanyi-Farrand-Chang (HFC) method based on the Neyman-Pearson detection theory has recently been widely used to determine the number of spectral endmembers [44]. Therefore, in this study, the HFC algorithm is used to determine the virtual dimensionality (VD), which is representative of the number of endmembers. Detailed steps of the ATGP algorithm are described as follows: 1. 2.
The observed pixel of the hyperspectral image is set to a vector x, and the VD determined by the HFC algorithm is q. Based on the convex geometry theory, we choose a target pixel with the maximum vector length, which corresponds toothe brightest pixel in the image, as the initial target pixel vector denoted by n t0 = arg max xT x . x
3.
Based on the orthogonal subspace projection theory, the ATGP algorithm begins with the initial target pixel vector t0 by applying an orthogonal subspace projector Pt⊥0 to all pixel vectors in the image and finds the first target pixel vector denotedby t1with the maximum orthogonal T
4.
⊥ ⊥ projection in the orthogonal complement space t1 = arg max ( PU x) ( PU x) . 0 0 x T ⊥ ⊥ th th Suppose that we find the i target ti generated at the i step by ti = arg max (PUi−1 x) (PUi−1 x) ,
5.
where Ui−1 = [t 0 , t1 , · · · , ti−1 ] is the target pixel vector matrix generated at the (i − 1)st step. The orthogonal subspace of Ui is given by PU⊥i = I − Ui U#i , where U#i = (UiT Ui )−1 UiT is the
x
pseudoinverse of Ui ; the notation
⊥ in PU⊥i indicates that the projector PU⊥i maps the observed Ui
⊥ , the orthogonal complement of Ui ; and I is a unit matrix. Ui The ATGP stops when i equals to the number of the endmembers q. The target pixel vector matrix [t 1 , t2 , · · · , tq which contains q target pixel vectors apart from the initial target pixel vector t0 , is then regarded as the endmember spectra.
pixel x into the range 6.
Another widely used endmember finding algorithm, i.e., the VCA algorithm, extracts endmembers by the maximal OPs from a sequence of successive orthogonal projection subspaces, which is quite similar to that of the ATGP [45,46]. Actually, the VCA algorithm can be considered as a variant of ATGP. However, the VCA algorithm does not have stable performance when random initial values are used [47]. Scholars have proven that the performance of VCA can be improved by using ATGP-generated targets as its initial conditions [48]. Compared to VCA, the ATGP algorithm is not constrained by initial conditions or dimension reduction transformations. Therefore, it serves as a more effective and accurate endmember extraction method when little or no prior information is available. 2.4. Proposed Endmember Initialization Scheme for NMF The NMF blind unmixing method improved by the ATGP-generated initial target endmembers (ATGP-NMF) can be expressed as follows: 1.
Initialization: target endmembers extracted by the ATGP algorithm are used as the initial endmember matrix A of NMF. Since the NNLS algorithm has been used for hyperspectral unmixing, especially for the initialization of NMF [33,49], the corresponding abundances of target endmembers derived by NNLS are used as the initial abundance matrix S of NMF.
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2. Objective function and optimization algorithm: The objective functions are constructed based on ISPRS Int. J. Geo-Inf. 2018, 7, 195 5 of 19 the Euclidean distance, due to its simplicity, intuition and wide application. To solve the objective functions, the multiplicative update algorithm is selected as the optimization criterion. 2. The Objective algorithm: Theabundance objective functions constructed based updatefunction formulaand of optimization endmember matrix A and matrix Sare can be expressed as on the Euclidean distance, due to its simplicity, intuition and wide application. To solve the Equations (4) and (5): objective functions, the multiplicative update algorithm is selected as the optimization criterion. (AT X)pn The update formula of endmember and abundance matrix S can be expressed(4)as Spnmatrix ←Spn A (AT AS)pn +λ Equations (4) and (5): (XST()AlpT X) pn S ← S (4) pn pn T Alp ←Alp (5) ) +λ T ) AS (ASS(A lp +λpn T
(XS )l p th nth mixed pixel, and Alp is the p(5) where Spn is the pth endmember abundance Al p ←value Al p of T the (ASS )l p +λ positive value λ is used to ensure that endmember spectrum value of the lth band. Here, a minimal th the denominator nonzero [14]. where S pn isis the p endmember abundance value of the nth mixed pixel, and Al p is the pth endmember spectrum value of the l th band. Here, a minimal positive value λ is used to ensure 3. Update: the rows of the weight coefficient matrix S are updated using Equation (4), while the that the denominator is nonzero [14]. columns of the basic matrix A are updated using Equation (5). 3. Normalization: Update: the rows of the weight coefficient are updated using the 4. according to the ASC in the matrix spectralS mixing model, and Equation following(4), thewhile methods columns of the basic matrix A are updated using Equation (5). suggested in [2,9,14], the weight coefficient matrix S is normalized after every iteration using 4. Equation Normalization: accordingthat to the spectral mixing and following the matrix methods (6) to guarantee theASC sumin ofthe each column vectormodel, in the weight coefficient S suggested in [2,9,14], the weight coefficient matrix S is normalized after every iteration using is always equal to one. Equation (6) to guarantee that the sum of each column vector in the weight coefficient matrix S is Spn always equal to one. Spn ← P S pn (6) S pn ∑ ←p Spn (6) P
5. 5.
∑ p S pn
Final endmember matrix A and endmember abundance matrix S : matrices A and S are Final endmember matrix A and endmember abundance matrix S: matrices A and S are repeatedly repeatedly updated until the maximum number of iterations is reached or until the stopping updated until the maximum number of iterations is reached or until the stopping condition is condition is satisfied, i.e., the value of the objective function is equal to zero or the prescribed satisfied, i.e., the value of the objective function is equal to zero or the prescribed error threshold ε . error threshold ε . (k) (k) Euc(k) (X, Xˆ ) =1 1 kX − Xˆ k 2 2 ≤ ε Euc (k) �X,X� (k) �= 2�X-X� (k) � ≤ε 2
(7) (7)
ˆ (k) is the reconstruction of X after the kth iteration. where (k) X � where X is the reconstruction of X after the kth iteration. The endmembers endmembers extracted extracted by by the the blind blind unmixing unmixing method method do do not about their their The not contain contain information information about categories;therefore, therefore,ititisisnecessary necessary compare endmember spectra extracted hyperspectral categories; to to compare thethe endmember spectra extracted fromfrom hyperspectral data data to the real endmember spectra and further identify their exact categories. In this study, we search to the real endmember spectra and further identify their exact categories. In this study, we search for for the match each endmember’s spectral curve thereference referencespectra spectraset setuntil until the the average average the bestbest match of of each endmember’s spectral curve ininthe correlation coefficient coefficient of correlation of the the spectra spectra set set reaches reaches the the maximum maximum value. value. The flowchart flowchart of of the The the proposed proposed method method is is shown shown in in Figure Figure 1. 1.
Figure 1. Flowchart Figure 1. Flowchart of of the the proposed proposed blind blind unmixing unmixing method method improved improved by by ATGP-generated ATGP-generated initial initial target target endmembers. endmembers.
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3. Experimental Results In this study, we carried out spectral unmixing experiments using the synthetic hyperspectral data and two real hyperspectral images to analyze the applicability and validity of the ATGP-NMF algorithm. The synthetic hyperspectral data were used to simulate a series of situations with different SNR levels, different purity levels and different numbers of endmembers. The real hyperspectral images representsituations in which pure pixels do not exist and there are strong spectral correlations between endmembers. The performance of the proposed method was compared with those of three representative algorithms, namely the VCA-NMF algorithm, the MVC-NMF algorithm and the VCA-FCLS algorithm. The VCA-NMF algorithm is included to test whether the ATGP approach is a better choice than the VCA to perform the initialization of NMF. The MVC-NMF algorithm, which adds the minimum volume constraint into NMF, is used to compare two different solutions for the NMF’s local optimum problem, i.e., to impose certain constraints and provide better initialization. In addition, it is a representative of constraint-NMF algorithms and has been widely used in comparative studies of NMF-based algorithms [23,50,51]. The VCA-FCLS algorithm, which uses VCA to identify endmembers and FCLS to calculate abundances, represents the traditional two-stage spectral unmixing method. 3.1. Performance Metrics The performances of the abovementioned algorithms are evaluated from two perspectives, the similarity to real spectra and the accuracy of the abundance estimation. The similarity of the extracted endmember’s spectra to the real spectra is evaluated via the indices of spectral angle distance (SAD) [52] and spectral information divergence (SID) [3]. The accuracy of the abundance estimation is evaluated by the root mean square error (RMSE) [53]. The equations of these indices are given as follows:
•
Spectral angle distance (SAD): SAD(A, B)= cos−1 (
AB ∑in=1 Ai Bi p ) )= cos−1 ( p n k Akk Bk ∑i=1 Ai Ai ∑in=1 Bi Bi
(8)
where A is the actual spectral vector, B is the estimated spectral vector, and n is the number of bands. The real endmember spectra can be spectra in the spectral library, observed spectra from the laboratory or field, or spectra extracted from pure pixels in remote sensing imagery.
•
Spectral information divergence (SID): SID(A, B) = D( A||B ) + D(B||A)
(9)
where A is the actual spectral vector, and B is the estimated spectral vector.
•
Root mean square error (RMSE): s RMSE =
2 ∑iN=1 ∑ Pj=1 (Sij −Sˆ ij )
n
(10)
where Sij is the actual abundance of the jth endmember abundance of the ith mixed pixel, Sˆ ij is the estimated value of Sij and n is the number of bands. The RMSE metric can partly reflect how the inversion results reconstruct the original information. 3.2. Spectral Unmixing Using Synthetic Data The hyperspectral synthetic data were generated using five spectral signatures randomly selected from the United States Geological Survey (USGS) digital spectral library [54], i.e., those for brucite, chabazite, olivine, spessartine and witherite (Figure 2). These spectral signatures are linearly independent and can be regarded as endmembers [16]. All signatures are available
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online (http://speclab.cr.usgs.gov/spectral-lib.html). The spectral data contain 420 bands covering ISPRS Int. J. Geo-Inf. 2018, 7, x FOR PEER REVIEW 7 of 19 wavelengths from 395.1–2560 nm. To create the linear mixtures, the abundance of each endmember was randomly determined using the Dirichlet distribution under under the ANC ASC. to simulate was randomly determined using the Dirichlet distribution theand ANC andMoreover, ASC. Moreover, to ISPRS Int. J. Geo-Inf. 2018, 7, x FOR PEER REVIEW 7 of 19 possible errors and sensor noise, mean noise, white Gaussian noisewhite with aGaussian 30-dB signal-to-noise simulate possible errors andzero sensor zero mean noise with ratio a (SNR), as defined in [55], was added to the mixture dataset. 30-dB ratio (SNR), defined in [55], was added tothe theANC mixture wassignal-to-noise randomly determined usingas the Dirichlet distribution under anddataset. ASC. Moreover, to simulate possible errors and sensor noise, zero mean white Gaussian noise with a 30-dB signal-to-noise ratio (SNR), as defined in [55], was added to the mixture dataset.
Figure 2. Reflectance spectra of the five materials selected from the USGS spectral library. Figure 2. Reflectance spectra of the five materials selected from the USGS spectral library. Figure 2. Reflectance spectra of the five materials selected from the USGS spectral library.
The abovementioned four algorithms, including the ATGP-NMF, VCA-NMF, MVC-NMF and The abovementioned fourfirst algorithms, the ATGP-NMF, VCA-NMF, MVC-NMF and VCA-FCLS algorithms, were applied toincluding the synthetic mixture data (SNR = 30MVC-NMF dB, purityand level The abovementioned four algorithms, including the ATGP-NMF, VCA-NMF, VCA-FCLS algorithms, were first applied to the synthetic mixture data (SNR = 30 dB, purity level ρ =VCA-FCLS 0.8) to extract the endmember spectra (Figure and calculate thedata corresponding abundances. The algorithms, were first applied to the3)synthetic mixture (SNR = 30 dB, purity level $ =similarity 0.8) to extract the spectra (Figure 3)was and calculate the corresponding abundances. ofextract the extracted spectra spectra to the real spectra evaluated by SAD and SID (Figure 4). The ρ = 0.8) to theendmember endmember (Figure 3) and calculate the corresponding abundances. The The similarity ofthe the extracted spectra to the real spectra was evaluated SAD SID4).(Figure accuracy of the abundance estimation was evaluated by RMSE (Table similarity of extracted spectra to the real spectra was evaluated by 1). SADby and SIDand (Figure The 4). The accuracy of the abundance estimation was evaluated by RMSE (Table 1). accuracy of the abundance estimation was evaluated by RMSE (Table 1).
(a) (a)
(b)
(b)
(c)(c)
(d) (d) Figure 3. Cont.
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(e) Figure 3. Comparison of the extracted endmember signatures using vertex component analysis (VCA)-
Figure 3. Comparison the extracted endmember signatures using vertex (MVC)-NMF component and analysis fully-constrained leastofsquares (FCLS), VCA-NMF, minimum volume constrained (e) from ATGP-NMF with the experiment data, spectral signals the USGSvolume digital spectral library for (a) (VCA)-fully-constrained least squares (FCLS), VCA-NMF, minimum constrained (MVC)-NMF brucite; chabazite; olivine; (d) spessartine; (e)signatures witherite. and ATGP-NMF with the(c) data, spectral signals from USGS digital analysis spectral(VCA)library for Figure 3.(b) Comparison ofexperiment the extracted endmember using the vertex component fully-constrained least(c) squares (FCLS), VCA-NMF, (e) minimum volume constrained (MVC)-NMF and (a) brucite; (b) chabazite; olivine; (d) spessartine; witherite. Table 1. Spectral distance (SAD), divergence and RMSE library values of ATGP-NMF withangle the experiment data,spectral spectralinformation signals from the USGS(SID) digital spectral forthe (a) four algorithms on synthetic data.(d) spessartine; (e) witherite. brucite; (b) chabazite; (c) olivine; Table 1. Spectral angle distance (SAD), spectral information divergence (SID) and RMSE values of the
Method SAD SID RMSE four algorithms on synthetic data. Table 1. Spectral angle distance (SAD), spectral information divergence (SID) and RMSE values of the VCA-FCLS 0.1039 0.0233 0.1002 four algorithms on synthetic data. VCA-NMF 0.0888 0.0232 0.0867 Method SAD SID RMSE MVC-NMF 0.0798 0.0113 0.0576 Method SAD SID RMSE VCA-FCLS 0.1039 0.0233 0.1002 ATGP-NMF 0.0520 0.0098 0.0549 VCA-FCLS 0.1039 0.0233 0.1002 VCA-NMF 0.0888 0.0232 0.0867 VCA-NMF 0.0888 0.0232 0.0867 MVC-NMF 0.0798 0.0113 0.0576 As shown in Figures 3 and 4, the spectra extracted 0.0113 by the proposed ATGP-NMF algorithm are ATGP-NMF 0.0520 0.0098 0.0549 MVC-NMF 0.0798 0.0576 quite similar to the real ATGP-NMF spectra, with smooth curves and little noise, followed 0.0520 0.0098 0.0549 by those extracted by the MVC-NMF, VCA-NMF and VCA-FCLS algorithms. In this experiment, although a pure pixel As shown in(purity Figures 3 and 4, the spectralevel extracted by the proposed ATGP-NMF algorithm are does notshown exist level = 0.8), of the by synthetic mixtureATGP-NMF data is higher. Therefore, As in Figures 3ρ and 4, the purity spectra extracted the proposed algorithm are quitethe similar to the real spectra, with curves and littlenoise, noise, followed by those extracted endmember spectra extracted bysmooth VCA-FCLS also resemble the real endmember spectra, with quite similar to the real spectra, with smooth curves and little followed by those extracted by by the MVC-NMF, VCA-NMF and VCA-FCLS algorithms. In this experiment, although a pure small SAD and SID values. and VCA-FCLS algorithms. In this experiment, although a pure pixelpixel the MVC-NMF, VCA-NMF does does not exist (purity level $ =ρ 0.8), ofthe thesynthetic synthetic mixture data is higher. Therefore, not exist (purity level = 0.8),the thepurity purity level level of mixture data is higher. Therefore, the endmember spectra extracted by VCA-FCLS resemble real endmemberspectra, spectra,with withsmall the endmember spectra extracted by VCA-FCLS alsoalso resemble thethe real endmember SADvalues. and SID values. SAD small and SID
(a)
(b)
Figure 4. (a) Spectral angle distance (SAD) and (b) spectral information divergence (SID) of different endmembers with the synthetic data determined using VCA-FCLS, VCA-NMF, MVC-NMF, and (a) (b) ATGP-NMF. Figure 4. (a) Spectral angle distance (SAD) and (b) spectral information divergence (SID) of different
Figure 4. (a) Spectral angle distance of (SAD) (b) spectral we information (SID) of To further evaluate robustness theseand four algorithms, compareddivergence their performances endmembers with thethe synthetic data determined using VCA-FCLS, VCA-NMF, MVC-NMF, and different endmembers withinthe synthetic VCA-NMF, MVC-NMF, under different conditions, terms of the data SNR,determined purity levelusing ρ andVCA-FCLS, number of endmembers p, by using ATGP-NMF. and ATGP-NMF. synthetic data. The comparison results are shown in Figures 5–7.
To further evaluate the robustness of these four algorithms, we compared their performances under different conditions, in terms of the ρ and number of endmembers by using To further evaluate the robustness of SNR, thesepurity four level algorithms, we compared theirp,performances data. The comparison results shown in Figures undersynthetic different conditions, in terms of theare SNR, purity level $5–7. and number of endmembers p, by using synthetic data. The comparison results are shown in Figures 5–7.
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(1) Robustness to noise interference: In this experiment, tests of the algorithms’ sensitivities to Robustness to noise interference: In level this experiment, thesynthetic algorithms’ to noise(1) were carried out by changing the SNR from 50 to 10tests dB. of The datasensitivities consist of five noise were carried out by changing the SNR level from 50 to 10 dB. The synthetic data consist of five endmembers, i.e., the spectral signatures selected from the USGS spectral library, at the purity level endmembers, i.e., thepure spectral signatures selected spectralvalues library, the purity level of of 0.8, which means pixels do not exist. The from SAD,the SIDUSGS and RMSE of at the four algorithms 0.8, which means pure pixels do not exist. The SAD, SID and RMSE values of the four algorithms are are shown in Figure 5a–c, respectively. The ATGP-NMF algorithm performed well, yielding lower SAD shown in Figure 5a–c, respectively. The ATGP-NMF algorithm performed well, yielding lower SAD and SID values than the other algorithms at different SNR levels. With the decrease in SNR, the RMSE and SID than the other algorithms at different SNR levels. thevalues decrease SNR, the RMSE values ofvalues ATGP-NMF decreased and stabilized; for MVC-NMF, theWith RMSE keptinincreasing. values of ATGP-NMF decreased and stabilized; for MVC-NMF, the RMSE values kept (2) Robustness to mixing degree: Following the method used in [56], we generated increasing. mixed spectra (2) Robustness to mixing degree: Following the method in level [56], we generated spectra at different purity levels. In this experiment, we changed theused purity ρ from 1 to 0.6mixed with an SNR at different purity levels. In this experiment, changed the purity level $from fromthe 1 tofour 0.6 with an SNRare of of 30 dB. The comparisons of the SAD, SIDwe and RMSE values derived algorithms 30 dB. The comparisons of the SAD, SID and RMSE values derived from the four algorithms are shown shown in Figure 6a–c. The SAD, SID and RMSE values of these four algorithms are inversely in Figure 6a–c. SAD, SID and values of these four inversely to proportional toThe the purity level. AtRMSE the purity level of one, i.e.,algorithms when pureare pixels exist,proportional the algorithms the purity level. At the purity level of one, i.e., when pure pixels exist, the algorithms of four methods of four methods all produced good solutions, with low SAD, SID and RMSE values. With regard to all produced good solutions, low SAD, SID and RMSE values. With regardthan to thethose purityoflevel, the purity level, the RMSEwith values of ATGP-NMF were lower overall the the RMSE values of ATGP-NMF were lower overall than those of the other algorithms. other algorithms. (3) Generalization number of endmembers: In this we generated mixed spectra Generalizationofofthe the number of endmembers: In experiment, this experiment, we generated mixed with different numbersnumbers of endmembers to test these algorithms. The endmember numbersnumbers were set spectra with different of endmembers to test these algorithms. The endmember to 3, 5set and thewhile SNR the wasSNR set towas 30 dB level waslevel set to 0.8.set When the number were to 10, 3, 5while and 10, set and to 30the dBpurity and the purity was to 0.8. When the of endmembers was set to three, the mixed spectra were generated using the spectral signatures of number of endmembers was set to three, the mixed spectra were generated using the spectral brucite, chabazite andchabazite olivine. and When the number of number endmembers was set towas 10, another spectral signatures of brucite, olivine. When the of endmembers set to 10,five another five signatures were selected from the USGS digital spectral library, in addition to the previously-selected spectral signatures were selected from the USGS digital spectral library, in addition to the previouslyfive signatures. Figure 7a–c shows the performances of VCA-NMF and ATGP-NMF are relatively selected five signatures. Figure 7a–cthat shows that the performances of VCA-NMF and ATGP-NMF are better andbetter moreand robust than those ofthose the other for thefor tested number of endmembers. relatively more robust than of thealgorithms other algorithms the tested number of endmembers.
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(c) Figure noise levels (ρ ($ = 0.8, p =p5)=in of: (a) (b) SID; and Figure 5. 5.Performance Performancecomparison comparisonatatdifferent different noise levels = 0.8, 5)terms in terms of:SAD; (a) SAD; (b) SID; (c) RMSE. and (c) RMSE.
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(c) (c) Figure Results thealgorithms algorithms withdifferent different purity levels (a)(a) SAD; (b)(b) SID; Figure 6. 6. Results ofof the levels (SNR (SNR===30 30dB, dB,pp p== =5). SAD; SID; Figure 6. Results of the algorithmswith with different purity purity levels (SNR 30 dB, 5).5). (a) SAD; (b) SID; and (c) RMSE. and (c)(c) RMSE. and RMSE.
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(c) (c) Figure 7. Performance comparison at different numbers of endmembers (SNR = 30 dB, ρ = 0.8). (a) Figure Performance 7. Performance comparison at different numbers of endmembers (SNR = 30 = 0.8). Figure at different numbers of endmembers (SNR = 30 dB,dB, $ =ρ0.8). (a) (a) SAD; SAD;7.(b) SID; and (c)comparison RMSE. SAD; (b) SID; and (c) RMSE. (b) SID; and (c) RMSE.
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3.3. Spectral Unmixing Using Berlin HyMap Image 3.3. Spectral Using Berlin Berlin HyMap HyMap Image Image 3.3. Spectral Unmixing Unmixing Using To evaluate the performance of the proposed ATGP-NMF algorithm on the real image, we To evaluate the of the proposed algorithm on the the realreal image, we To evaluate the performance performance theBerlin-Urban-Gradient proposedATGP-NMF ATGP-NMF algorithm image, conducted experiments on a subset ofofthe dataset, whichon mainly comprises a conducted experiments on a subset of the Berlin-Urban-Gradient dataset, which mainly comprises a we conducted experiments a subset of the Berlin-Urban-Gradient dataset, comprises HyMap reflectance image, aonlibrary of material spectra and the reference landwhich covermainly information [57]. reflectance image, aalibrary of material spectra and the reference land cover information [57]. aHyMap HyMap reflectance library spectra andFor thethis reference cover information [57]. More information onimage, this dataset canofbematerial found in [58–60]. work, land a subset with 80 × 80 pixels More information on this dataset can be found in [58–60]. For this work, a subset with 80 × 80 pixels More information on this dataset can be found in [58–60]. For this work, a subset with 80 × 80 pixels of the Berlin HyMap image (Figure 8a), acquired on 20 August 2009 at 9 m spatial resolution, was of the the Berlin Berlin HyMap HyMap image (Figure (Figure 8a), 8a), acquired acquired on 20 20 August August 2009 2009 at at 99 m m spatial spatial resolution, resolution, was was of selected as the study image data. The image contains 111onbands after removing noisy bands with spectral selected as the study data. The image contains 111 bands after removing noisy bands with spectral selected the studybetween data. The bands after removing bands spectral samplingasdistances 13 image and 17contains nm, and 111 spectral coverage from 440noisy to 2500 nm.with Based on the sampling distances distances between between 13 13 and and 17 17 nm, nm, and and spectral spectral coverage coverage from from 440 440 to to 2500 2500 nm. nm. Based Based on on the the sampling reference land cover information, we considered five urban land cover types in the study area, i.e., reference land landcover coverinformation, information,we weconsidered considered five urban land cover types in study the study area, i.e., reference five urban land cover types in the area, i.e., roof, roof, pavement, grass, tree and soil. The reference abundances of each land cover type (Figure 9), roof, pavement, grass, tree and soil. The reference abundances of each land cover type (Figure 9), pavement, grass, tree and soil. The reference abundances of each land cover type (Figure 9), defined as defined as the corresponding proportions at each pixel, were calculated based on the reference land defined as the corresponding proportions at each pixel, were based calculated based on theland reference land the corresponding proportions at each pixel, were calculated on the reference cover map cover map (Figure 8b). We selected the pixels with an area percentage of one and calculated their cover map (Figure 8b). Wepixels selected the pixels with an area percentage of onetheir and average calculated their (Figure We selected with an area percentage of one andtype. calculated average8b). spectral values the as the reference spectrum of the land cover Because there were spectral no pure average spectral values as the reference spectrum of the land cover type. Because there were no pure values reference the land cover type.anBecause were nogreater pure pixels the grass pixels asofthethe grass spectrum category, offour pixels with area there percentage thanof 0.7 were pixels offour thepixels grasswith category, four pixelsgreater with than an area percentage greater than 0.7 were category, an area percentage 0.7 were selected instead. selected instead. selected instead. The The comparisons comparisons of of the the VCA-FCLS, VCA-FCLS, VCA-NMF, VCA-NMF, MVC-NMF MVC-NMF and and ATGP-NMF ATGP-NMF algorithms algorithms on on the the The comparisons of the VCA-FCLS, VCA-NMF, MVC-NMF and10a,b ATGP-NMF algorithms on the Berlin HyMap image are shown in Figure 10 and Table 2. Figure shows that the proposed Berlin HyMap image are shown in Figure 10 and Table 2. Figure 10a,b shows that the proposed Berlin HyMap image are shownfive in Figure 10 andwith Table 2. Figure 10a,b shows that the proposed ATGP-NMF ATGP-NMF algorithm algorithm provided provided five endmembers endmembers with relatively relatively low low SAD SAD and and SID SID values. values. Figure Figure 11 11 ATGP-NMF algorithm provided five endmembers with relatively low SAD and SID values. Figure 11 shows the estimated abundance maps by ATGP-NMF. Comparing to the reference abundance maps shows the estimated abundance maps by ATGP-NMF. Comparing to the reference abundance maps shows the estimated abundance maps by ATGP-NMF. Comparing to the reference abundance maps (Figure (Figure 9), 9), itit is is demonstrated demonstrated that that the the ATGP-NMF ATGP-NMF algorithm algorithm can can produce produce satisfactory satisfactory abundance abundance (Figure 9), itFor is demonstrated that the algorithm can produce satisfactory abundance separation. all four four algorithms, algorithms, theATGP-NMF estimated spectra spectra of grass grass are in separation. For all the estimated of are in bad bad accordance accordance with with the the separation. For all four algorithms, the estimated spectra of grass are in bad accordance withtrees, the reference endmembers. In the study area, grasses are scattered mainly around the buildings and reference endmembers. In the study area, grasses are scattered mainly around the buildings and trees, reference endmembers. In the study area, grasses are scattered mainly the buildings and trees, which the buildings and trees andand thearound spectral similarity between the whichare aresusceptible susceptibletotothe theshadows shadowsofof the buildings and trees the spectral similarity between which are susceptible to the shadows of the buildings and trees and the spectral similarity between grasses and and trees. the grasses trees. the grasses and trees.
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Figure 8. (a) (a) A subset subset of the the Berlin HyMap HyMap image (R (R = 833 nm, nm, G == 1652 1652 nm, BB == 632 632 nm); (b) (b) the Figure Figure 8. 8. (a) A A subset of of the Berlin Berlin HyMap image image (R == 833 833 nm, G G = 1652 nm, nm, B = 632 nm); nm); (b) the the reference land cover cover map of of the study study area. reference reference land land cover map map of the the study area. area.
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(b) (b) Figure 9. Cont.
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Figure 9. Reference abundance maps of five land cover types: (a) roof; (b) pavement; (c) grass; (d)
Figure 9. abundance of five land cover types: (a)(a)roof; tree; (e)Reference soil. Figure 9. Reference abundance maps of five land cover types: roof;(b) (b)pavement; pavement;(c) (c)grass; grass;(d) (d)tree; (d)maps (e) (e) soil. tree; (e) soil. Figure 9. Reference abundance maps of five land cover types: (a) roof; (b) pavement; (c) grass; (d) tree; (e) soil.
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Figure 10. SAD (a) and SID (b) of different endmembers with the Berlin HyMap image using VCA(b) HyMap image using VCAFCLS, VCA-NMF, MVC-NMF, Figure 10. SAD (a) and SID (a) (b)and of ATGP-NMF. different endmembers with the Berlin Figure 10. SAD (a) and SID (b) of different endmembers with the Berlin HyMap image using FCLS, ATGP-NMF. FigureVCA-NMF, 10. SAD (a)MVC-NMF, and SID (b)and of different endmembers with the Berlin HyMap image using VCAVCA-FCLS, VCA-NMF, MVC-NMF, and ATGP-NMF. Table 2. SAD, SID and RMSE values of different algorithms with the Berlin HyMap image. FCLS, VCA-NMF, MVC-NMF, and ATGP-NMF. Table 2. SAD, SID and RMSE values of different algorithms with the Berlin HyMap image. SAD SID RMSE Table 2. SAD, SID and RMSE Method values of different algorithms with the Berlin HyMap image.
Table 2. SAD, SID and RMSE values of different algorithms with the Berlin HyMap image. VCA-FCLS 0.3260 Method SAD 0.2246 SID 0.3309 RMSE VCA-NMF VCA-FCLS Method Method MVC-NMF VCA-NMF VCA-FCLS VCA-FCLS ATGP-NMF MVC-NMF VCA-NMF VCA-NMF ATGP-NMF MVC-NMF MVC-NMF ATGP-NMF ATGP-NMF
(a) (a) (a)
0.2655 0.2473 0.3260 0.2246 SAD SID SAD SID 0.3508 0.2389 0.2655 0.2473 0.3260 0.2246 0.3260 0.2246 0.2407 0.0965 0.3508 0.2389 0.2655 0.2473 0.2655 0.2473 0.2407 0.0965 0.3508 0.2389 0.3508 0.2389 0.2407 0.0965 0.2407 0.0965
(b) (b) (b)
0.3249 0.3309 RMSE RMSE 0.3591 0.3249 0.3309 0.3011 0.3591 0.3309 0.3249 0.3249 0.3011 0.3591 0.3591 0.3011 0.3011
(c) (c) (c)
(d) (e) (d) (e) Figure 11. Abundance maps provided by ATGP-NMF with the Berlin (d) (e) HyMap image: (a) roof;
(b) pavement; (c) grass; (d) tree; provided (e) soil. by ATGP-NMF with the Berlin HyMap image: (a) roof; Figure 11. Abundance maps Figure 11. Abundance maps provided by ATGP-NMF with the Berlin HyMap image: (a) roof; (b) pavement; (c) grass; (d) tree; (e) soil. by ATGP-NMF with the Berlin HyMap image: (a) roof; Figure 11. Abundance maps provided (b) pavement; (c) grass; (d) tree; (e) soil.
(b) pavement; (c) grass; (d) tree; (e) soil.
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3.4. 3.4. Spectral Spectral Unmixing Unmixing Using Using aa Hyperion Hyperion Image Image We We used used aa Hyperion Hyperion image image to to further further verify verify the the applicability applicability of of the the proposed proposed ATGP-NMF ATGP-NMF algorithm in complicated situations. The Hyperion image (Figure 12a), acquired 19 January algorithm in complicated situations. The Hyperion image (Figure 12a), acquired on 19onJanuary 2011, 2011, covers parts of Guangzhou and Foshan, both of which are cities in Guangdong Province, China. covers parts of Guangzhou and Foshan, both of which are cities in Guangdong Province, China. The The image of this study a subset of 101 × 101 pixels 30-m spatial resolution. In addition, test test image of this study is a issubset of 101 × 101 pixels withwith 30-m spatial resolution. In addition, the the image contains 242 bands acquired in the spectral range of 356–2577 nm. To improve image contains 242 bands acquired in the spectral range of 356–2577 nm. To improve the the spectral spectral unmixing unmixingperformance, performance,aaseries seriesof of pretreatments pretreatmentswas wasperformed performedon onthis this image, image, including including radiometric radiometric calibration and atmospheric correction. In addition, we removed the low SNR calibration and atmospheric correction. In addition, we removed the low SNR bands bands and and the the water water vapor absorption bands. A total of 158 bands (including Bands 8–57, 79–120, 133–165 and vapor absorption bands. A total of 158 bands (including Bands 8–57, 79–120, 133–165 and 188–220) 188–220) were wereused usedin inthis thisexperiment. experiment. For the Hyperion For the Hyperion image, image, the the number number of of endmembers endmembers was was estimated estimated using using HFC HFC mentioned mentioned in in Section 2.3. Considering the actual situation of the study area, we finally determined that there are Section 2.3. Considering the actual situation of the study area, we finally determined that there are eight eight endmembers endmembers in in the the study study area. area. In In this this study, study, the the reference reference abundances abundances were were extracted extracted from from aa high-resolution remote sensing image acquired from Google Earth (Figure 12b). The classification map high-resolution remote sensing image acquired from Google Earth (Figure 12b). The classification of theofeight landland cover types of the study area (Figure 12c) was then map the eight cover types of the study area (Figure 12c) was thenacquired acquiredthrough throughsupervised supervised classification of the high-resolution remote sensing image using the k-nearest neighbor Finally, classification of the high-resolution remote sensing image using the k-nearest neighbor [61]. [61]. Finally, the area percentage of each land cover type in one 30 × 30 m pixel was calculated and considered the area percentage of each land cover type in one 30 × 30 m pixel was calculated and considered as as its endmember abundance (Figure 13). In addition, due to the variability of the endmember spectra, its endmember abundance (Figure 13). In addition, due to the variability of the endmember spectra, we weselected selectedreference referenceendmember endmemberspectra spectrafrom fromthe theHyperion Hyperionimage. image. We Weselected selected30 30samples samplesfor foreach each land cover type and used the average spectra of these 30 samples as the reference spectra for each land land cover type and used the average spectra of these 30 samples as the reference spectra for each cover type. type. land cover The values of of thethe four algorithms is shown in Figure 14, while the Thecomparison comparisonofofthe theSAD SADand andSID SID values four algorithms is shown in Figure 14, while errors of the abundance estimation areare given in in Table 3. 3. AsAsshown the errors of the abundance estimation given Table shownininFigure Figure14a,b, 14a,b,the theproposed proposed ATGP-NMF algorithm provided eight endmembers with relatively low SAD and SID values. Compared ATGP-NMF algorithm provided eight endmembers with relatively low SAD and SID values. to those of the other of three spectra extracted byextracted ATGP-NMF are closer to are those of the Compared to those the algorithms, other three the algorithms, the spectra by ATGP-NMF closer to actual land cover types. The ATGP-NMF algorithm can extract endmembers with high correlations, those of the actual land cover types. The ATGP-NMF algorithm can extract endmembers with high such as buildings white roofs blue roofs. 15 shows the maps estimated correlations, such with as buildings withand white roofs andFigure blue roofs. Figure 15abundance shows the abundance maps by ATGP-NMF. The distribution of the estimated endmembers is close to the distribution of their estimated by ATGP-NMF. The distribution of the estimated endmembers is close to the distribution corresponding land cover types. Table 3 shows that the ATGP-NMF algorithm yielded the most of their corresponding land cover types. Table 3 shows that the ATGP-NMF algorithm yielded the accurate abundance estimation results with with the lowest RMSE for the image. This is most accurate abundance estimation results the lowest RMSE for hyperspectral the hyperspectral image. This primarily because the ATGP-NMF algorithm can effectively and efficiently the mostfind representative is primarily because the ATGP-NMF algorithm can effectively andfind efficiently the most endmember spectra from hyperspectral images. Among the four different algorithms, the traditional representative endmember spectra from hyperspectral images. Among the four different algorithms, VCA-FCLS algorithm had the poorest performance, whichperformance, again proveswhich the advantages of blind the traditional VCA-FCLS algorithm had the poorest again proves the unmixing methods. advantages of blind unmixing methods.
(a)
(b)
(c)
Figure 12. 12. (a)(a)Hyperion image of the area (R = 936 G =nm, 2143Gnm, = 2345 the highFigure Hyperion image of study the study area (R nm, = 936 = B 2143 nm,nm); B =(b)2345 nm); resolution remote sensing image covering the study area; (c) the land use classification map of the (b) the high-resolution remote sensing image covering the study area; (c) the land use classification study map ofarea. the study area.
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Figure 13. 13. Reference abundance maps maps of of eight eight land land cover cover types: types: (a) (a) arable arable land; land; (b) (b) forest; forest; (c) (c) grassland; grassland; Figure Reference abundance abundance Figure 13. Reference maps of eight land cover types: (a) arable land; (b) forest; (c) grassland; (d) bare land; (e) water; (f) red brick buildings; (g) buildings with white roofs; (h) buildings with blue (d) bare land; land; (e) (e)water; water;(f)(f)red redbrick brick buildings; buildings with white roofs; (h) buildings (d) bare buildings; (g)(g) buildings with white roofs; (h) buildings with with blue roofs. blue roofs.roofs.
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(b) (b) Figure 14. SAD (a) and SID (b) of different endmembers with the Hyperion image using VCA-FCLS, Figure 14. 14. SAD SAD (a) (a) and and SID (b) of of different different endmembers endmembers with Figure SID with the the Hyperion Hyperion image image using using VCA-FCLS, VCA-FCLS, VCA-NMF, MVC-NMF and(b) ATGP-NMF. VCA-NMF, MVC-NMF and ATGP-NMF. VCA-NMF, MVC-NMF and ATGP-NMF.
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Table 3. 3. SAD, SAD, SID SID and and RMSE RMSE values values of of different different algorithms algorithms with with the the Hyperion Hyperionimage. image. Table
Method SAD SAD VCA-FCLS 0.3072 VCA-FCLS 0.3072 VCA-NMF 0.2798 VCA-NMF 0.2798 MVC-NMF 0.3368 0.3368 MVC-NMF ATGP-NMF0.1338 0.1338 ATGP-NMF Method
SID SID 0.2797 0.2797 0.3249 0.3249 0.3459 0.3459 0.1061 0.1061
RMSE 0.3101 0.2866 0.2732 0.2659
RMSE 0.3101 0.2866 0.2732 0.2659
(a)
(b)
(c)
(d)
(e)
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(g)
(h)
Figure 15. 15. Abundance ATGP-NMF with with the the Hyperion Hyperion image: image: (a) arable land; land; Figure Abundance maps maps provided provided by by ATGP-NMF (a) arable (b) forest; forest; (c) (c) grassland; grassland; (d) (d) bare bare land; land; (e) (e) water; water; (f) (f) red red brick (b) brick buildings; buildings; (g) (g) buildings buildings with with white white roofs; roofs; (h) buildings with blue roofs. (h) buildings with blue roofs.
3.5. Computational Computational Efficiency Efficiency 3.5. To evaluate evaluate the the computational computational efficiency efficiency of of the the particular particular algorithm, algorithm, the the computing computing time time was was To often used used as computational efficiency of an often as an an important importantmeasure measureininmany manyrelated relatedstudies studies[35]. [35].The The computational efficiency of algorithm can be defined as a numerical function T(n): time versus the input size n. In this study, an algorithm can be defined as a numerical function T(n): time versus the input size n. In this since the input data data are the we we used thethe average computing each study, since the input are same, the same, used average computingtime time(in (in seconds) seconds) of of each algorithm’s application application on on synthetic synthetic data, data, the the Berlin Berlin HyMap HyMap image, image, and and Hyperion Hyperion image image to to evaluate evaluate algorithm’s their computational efficiency. The four algorithms were executed in Mathworks MATLAB R2014b their computational efficiency. The four algorithms were executed in Mathworks MATLAB R2014b on a computer with Intel(R) Xeon(R) E3-1246 CPU (3.50 GHz) and 12 GB RAM. For the three NMF on a computer with Intel(R) Xeon(R) E3-1246 CPU (3.50 GHz) and 12 GB RAM. For the three NMF derivative algorithms, algorithms, i.e., i.e., VCA-NMF, VCA-NMF, MVC-NMF MVC-NMF and and ATGP-NMF, ATGP-NMF, the the endmember endmember spectral spectral and and derivative abundance estimation results were determined at the maximum iteration number of 300. abundance estimation results were determined at the maximum iteration number of 300. Table 44 shows shows that that the the computing computing time time of of the the proposed proposed ATGP-NMF ATGP-NMF algorithm algorithm is is less less than than those those Table of the traditional two-stage VCA-FCLS algorithm and the MVC-NMF algorithm. The MVC-NMF of the traditional two-stage VCA-FCLS algorithm and the MVC-NMF algorithm. The MVC-NMF algorithm required required more more processing processing time time than than the the other otherthree threealgorithms, algorithms, particularly particularly for for processing processing the the algorithm real image. image. With With the the endmember endmember initialization, initialization, the the proposed proposed ATGP-NMF ATGP-NMF and and the the existing existing VCA-NMF VCA-NMF real required less computation time. Table 4 also shows that ATGP-generated initial endmembers can required less computation time. Table 4 also shows that ATGP-generated initial endmembers can significantly reduce the computational time of the spectral unmixing process. This is mainly because significantly reduce the computational time of the spectral unmixing process. This is mainly because ATGP only needs to process previously-generated target pixels, not the whole dataset, to ATGP only needs to process previously-generated target pixels, not the whole dataset, to extract extract endmembers [42]. endmembers [42].
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Table 4. Computing time (seconds) of different methods on synthetic data, the Berlin HyMap image, and Hyperion Image. Method
VCA-FCLS
VCA-NMF
MVC-NMF
ATGP-NMF
Synthetic Data Berlin HyMap Image Hyperion Image
0.83 2.15 9.37
0.61 1.99 4.37
0.98 65.26 165.66
0.59 1.95 5.28
4. Conclusions The traditional NMF algorithm suffers from the “local best” problem, which greatly limits its applications. In this paper, via the use of ATGP-generated initial target endmembers, the ATGP-NMF algorithm is proposed to improve the performance of NMF in hyperspectral unmixing. The ATGP algorithm is introduced to provide more accurate initial points to identify endmembers, which can then be used by the NMF to obtain the global solutions for hyperspectral unmixing. Furthermore, the proposed ATGP-NMF algorithm requires little prior knowledge and can be applied in complex situations, even if there are no pure pixels. Experiments were carried out to evaluate the performances of ATGP-NMF and three other algorithms, namely, VCA-NMF, MVC-NMF, and VCA-FCLS, using synthetic hyperspectral data and real hyperspectral images. Based on analysis of the experimental results, the following conclusions can be made: (1) The traditional two-stage spectral unmixing method, VCA-FCLS, can only achieve satisfactory results when pure pixels exist. When there are little prior knowledge and no pure pixels, the studied BSS methods, namely, VCA-NMF and ATGP-NMF, and MVC-NMF without the pure pixel assumption can achieve better unmixing results than VCA-FCLS. (2) Of all the algorithms, the ATGP-NMF algorithm demonstrated the best applicability and consistently produced good results in different situations. The extracted endmember spectra were closer to the real spectra, and the abundance estimation was more accurate. (3) Proper endmember initialization can help the NMF algorithm yield better estimation results with less computation time. Author Contributions: J.C. and L.Z. conceived of and designed the experiments. J.C. performed the experiments and analyzed the results. J.C. and L.Z. wrote and revised the manuscript. H.T. provided useful suggestions for the manuscript. Acknowledgments: This paper was funded by the National Natural Science Foundation of China (Grant No. 41371499) and the Natural Science Foundation of Guangdong Province (Grant No. 2015A030313505). Conflicts of Interest: The authors declare no conflict of interest.
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