Blind Hyperspectral Unmixing with Enhanced 2DTV Regularization Term

Abstract

For the problem where the existing hyperspectral unmixing methods do not take full advantage of the correlations and differences between all these bands, resulting in affecting the final unmixing results, we design an enhanced 2DTV (E-2DTV) regularization term and suggest a blind hyperspectral unmixing method with the E-2DTV regularization term (E-gTVMBO), which adds E-2DTV regularization to the previous blind hyperspectral unmixing based on g-TV model. The E-2DTV regularization term is based on the gradient mapping of all bands of HSI, and the sparsity is calculated on the basis of the subspace, rather than applying sparsity to the gradient map itself, which can utilize the correlations and differences between all bands naturally. The experimental results prove the superiority of the E-gTVMBO method from both qualitative and quantitative perspectives. The research results can be applied to land cover classification, mineral analysis, and other fields.

1. Introduction

Due to the influence of imaging spectrometer accuracy, observation conditions, terrain factors, etc., a single pixel in HSI often contains multiple types of ground objects, which produces the so-called mixed pixel [1,2,3]. In order for HSI to be more widely used, its accuracy needs to be improved; mixed pixels need to be decomposed to extract the spectral characteristics of various basic substances that constitute the pixels and the proportion of these substances in the pixels; and this is the definition of spectral unmixing [4,5]. The signatures corresponding to various basic substances are endmembers, and the ratio of endmembers that make up each pixel is called the abundance map [6,7]. Blind hyperspectral unmixing (HU) means that the spectral information of endmembers is unknown, which requires both identification of endmembers and estimation of abundance maps [8,9]. At present, most HU methods could be divided into linear and nonlinear mixture models [3]. Since the light in the real scene will form multiple scattering nonlinear mixing effects successively with different substances, the goal of the nonlinear mixing model is to establish a reasonable mathematical expression of multiple scattering. The linear mixing model assumes that the mixed pixel spectrum is a linear combination of a set of endmembers spectra according to their corresponding abundance ratios. Because the endmember abundance represents the actual area ratio of the endmember in the pixel, it needs to conform the non-negative abundance, and the sum of abundance equals one constraint. Massively spectral unmixing methods use the linear mixing model [10] because of the simplicity and tractability of linear mixing model.

The two constraints of linear unmixing model can make the unmixing results more accurate. Therefore, when the endmembers are known, to deal with the two abundance constraints of linear unmixing model, the classical fully constrained least squares is used by iteratively updating the abundance least squares solution (FCLSU) [11]. However, the abundance properties considered by such methods such as FCLSU are very limited. To solve this issue, the fractional unmixing method (FRAC) [12] regularized by norm (q = 0.1) is used to optimize the abundance estimation problem by introducing the mixing norm. In FRAC, by using group sparsity, a new penalty is proposed to enforce both group and within-group sparsity to guarantee non-convexity costs, and the proposed penalty is compatible with abundance through one-to-one constraints. Non-negative matrix factorization (NMF) [13,14,15] is an unsupervised feature extraction method, which is comprehensively used in blind HU. In addition, there are now many sparse unmixing algorithms such as the sparse unmixing based on variable splitting and augmented Lagrangian (SUnSAL) [5], the sparse unmixing based on collaborative sparse regression (CLSUnSAL) [16], and the sparse unmixing based on joint-sparse-blocks and low-rank unmixing (JSpBLRU) [17]. Moreover, there are many HU methods based on deep learning, such as a neural architecture search based on reinforcement learning, a linear spectral mixture model based end-to-end deep neural network, etc. [18,19,20]. With the continuous development and successful application of total variation (TV) in many fields, the TV regularization term has been successfully applied to HU technology [21]. Recently, a blind HU method with g-TV regularization (gTVMBO) [22] is proposed to estimate abundance map and endmember matrix and shows good performance. However, this method does not take full advantage of the spectral and spatial information, resulting in affecting the final unmixing results. To solve this issue, based on gTVMBO, we consider the imposition of an enhanced 2DTV (E-2DTV) regularization term. Namely, we present a blind HU with enhanced 2DTV regularization (E-gTVMBO) in this paper. In E-gtvMBO, we successfully design and add E-2DTV into blind HU based on g-TV, improving the final unmixing result.

The contribution of our article is that we successfully design and add E-2DTV into blind HU based on g-TV, improving the final unmixing result. E-2DTV can naturally utilize correlations and differences between bands to better reflect the sparseness characteristics of natural HSI gradient maps. This is because adjacent bands of images of HSI are usually collected with similar sensor parameter settings, and thus with similar values, that is, the HSI has locally smoothed prior structure. This locally smoothed prior structure of HSI can be equivalently understood as the sparsity of gradient maps calculated along the spatial and spectral modes of HSI. The local smoothing prior to the HSI space and spectrum can be achieved naturally by applying E-2DTV terms to different modes of HSI. Therefore, the E-2DTV regularization term applied to the abundance matrix can achieve a local smoothness prior structure, thereby making full use of its spectral and spatial information; this improves the performance of unmixing. The results of experiments prove that the E-gTVMBO method improves unmixing results, when compared with the other traditional HU methods.

The paper structure is shown below. In Section 2, we present the E-gTVMBO method including the related work, E-2DTV, and alternating direction method of multipliers (ADMOM) optimization. In Section 3, the experiment results are tested and analyzed. Section 4 introduces the discussion of the experimental results. Conclusions are presented in Section 5.

2. Methodology

2.1. Related Work

The TV regularization term has also been successfully applied to HU technology. The TV term has an extremely wide range of applications in HU. For example, TV regularization term is applied to SUnSAL (SUnSAL-TV) [23]; SUnSAL-TV involves a 2D total variation (2DTV) regularization that could effectively improve the smoothness between pixels, thereby optimizing the unmixing accuracy. On the basis of the TV item, the 2DTV term has been successfully applied in many areas of image processing for making use of the information contained in space and spectrum, such as HU, denoising [24,25,26], medical image [27], etc. There are many regularization methods for graph signals [28]; the TV regularization terms are very widely used in graph signals. Therefore, the graph TV (g-TV) regularization term for hyperspectral imaging is also a common regular term [29], such as graph NMF (G-NMF) [30], structured sparse regularization NMF (SSNMF) [31], and superpixel-based graph regularization multilinear mixture models (G-MLM) [32]. Among them, GNMF aims to enhance the accuracy of traditional NMF methods and reduce the computational complexity, which considers not only the Euclidean distance internal structure of hyperspectral data, but also its internal Riemannian geometry. However, GNMF ignores the sparse nature of abundance. Therefore, a graph-regularized-NMF(GLNMF) method [33] is suggested for spectral unmixing, which adds abundance sparse constraints on the basis of graph regularization and obtains good results. Since the development of 2DTV and g-TV is more mature, most of the current studies applying TV items to spectral unmixing are 2DTV and g-TV [34]. However, a lot of these g-TV methods are computationally complex, specifically when calculating pairwise similarity between all pixels. To solve this problem, the Nyström method [35] is employed, and unmixing can be accomplished by using the low-rank approximation of the Laplacian matrix of the graph obtained by this method. The gTVMBO method [22] exploits the spectral information of different pixels that have similar features and preserves the abundance maps’ obvious edges, and it adopts the Nyström method to solve the computationally complexity problems.

According to the reference [22], because the $l_2$-norm operation is involved in the small dataset and because using the $l_2$-norm for the small dataset will cause over smoothing, then to mitigate over-smoothing artifacts, the g-TV regularization is proposed and is written as:

(1)${\mathrm{J}}_{\mathrm{TV}}(A)=\frac{1}{2}{{\displaystyle \sum }_{i,j=1}^s{\displaystyle \left|\right|}{\stackrel{\wedge }{a}}_i-{\stackrel{\wedge }{a}}_j{\displaystyle \left|\right|}}_1{W}_{ij}$

where $a_i$ represents the ith column of A, ${\stackrel{\wedge }{a}}_i=a_i/\sqrt{d_{ii}}$ and A$\in$ℝ${}^{p\times s}$, $W_{ij}=e^{-d{\left(x_i,x_j\right)}^2/\sigma }$, $d(i,j)$ can represent the physical distance between the vertices $x_i$ and $x_j$. In this paper, $d(i,j)$ is viewed as the cosine similarity, which is a function used to calculate the distance for hyperspectral data [36].

The margin of the abundance map of each material can be preserved in a non-local manner by minimizing the $J_{\mathrm{TV}}$. Then the addition of g-TV regularized term to the blind HU model is formulated as:

(2)$$\begin{gathered} \min \frac{1}{2}{{\displaystyle \left|\right|}Y-XA{\displaystyle \left|\right|}}_{\mathrm{F}}^2+\lambda {\mathrm{J}}_{\mathrm{TV}}(A) \\ \mathrm{s}.\mathrm{t}. x_i\ge 0,a_i\ge 0, {\displaystyle \sum }_{i=1}^p{a}_i=1 \end{gathered}$$

where Y = [$y_1,y_2,\dots ,y_s$]$\in$ℝ${}^{l\times s}$ is an HSI; $l$ represents the number of spectral bands; $s$ represents the number of pixels. Furthermore, endmembers are defined as spectral properties of pure materials and use X = [$x_1,x_2,\dots ,x_p$]$\in$ℝ${}^{l\times p}$ as the endmember matrix, which has p pure materials. In addition, λ is tuning parameter, which is positive. It is noted that the sum-to-one constraint enforces sparsity due to the $l_1$-norm. The Merriman-Bence-Osher (MBO) scheme [37] is then used to solve sub-problems in ADMOM; the sub-problems’ resolution process will be represented in Section 2.2. The g-TV regularization takes into account the similarity of spectral information between different pixels, and thus fine spatial features can be preserved in the abundance map.

2.2. E-2DTV Regularization Term

Since gTVMBO does not take full advantage of the spectral and spatial information of HSI, E-2DTV regularization term is introduced to improve the existing gTVMBO model. The $l_1$-norm is on the gradient graph $G_n(n=1,2)$ of A, where $G_1,G_2$ represent the expansion matrices of the gradient map computed by A along its spatial and spectral mode. The term is called 2DTV regularization, and it takes the form:

(3)${{\displaystyle \left|\right|}A{\displaystyle \left|\right|}}_{2\mathrm{DTV}}={{\displaystyle \sum }_{n=1}^2{\displaystyle \left|\right|}{G}_n|{|}_1}_{}{}_{}$

Extensive studies have shown that 2DTV is very helpful for various HSI processing tasks. However, 2DTV regularization has some obvious limitations, the most typical of which is to impose similar sparsity considerations across all bands of the spectrum and space. However, this always deviates from real-life situations. In different bands, there exists different sparsity of gradient maps. In addition, on account of the band-dependent properties of the original HSI, the different bands of the gradient map have non-negligible correlation. To solve this problem, we raise an enhanced 2DTV (E-2DTV) term. It is very much similar to the conventional 2DTV; the given term also applies to gradient maps along the spectrum. However, when the term is applied to the sparsity measure, which map along the subspace basis of all frequency bands, each basis is through the original gradient mapping vector linear combination of the bands: $G_n{V}_n$, where $V_n$ is a transformation matrix of size $s\times r$, promoting finding the $(r<<s)$ basis of $G_n$.

Representing the gradients with these computed few-bases provides sparse correlation insight into HSIs; their different coefficients $V_n$ represent the differences between the bands of these gradient maps. The E-2DTV item is described in detail below.

For an image A, we can obtain ${{\displaystyle \left|\right|}A{\displaystyle \left|\right|}}_{\mathrm{TV}}$ is the TV norm form. ${{\displaystyle \left|\right|}A{\displaystyle \left|\right|}}_{\mathrm{TV}}$ can be presented as ${{{\displaystyle \sum }}_{n=1}^2{\displaystyle \left|\right|}{D}_{n}A{\displaystyle \left|\right|}}_1$; $D_{\mathrm{n}}$ is the nth dimensional difference operator. Because TV norm contains the sum total of the grayscale values for the horizontal and vertical difference maps, the difference can be equated to the gradient, in the discrete case. In this way, $G_n={\nabla }_{n}A,n=1,2,$. To enhance the unmixing performance, a sparsity term is constructed for each $G_n$. In past research [38,39], we have good reason to assume that A is low-rank. Thus, $G_n$ is also low-rank property. $G_n$ can be represented as:

(4)$G_n=U_n{S}_n^{\mathrm{T}},n=1,2,$

where $U_n\in$ℝ${}^{l\times r}$,$S_n\in$ℝ${}^{s\times r}$; r represents the rank of A. Each group of basis in gradient map $G_n$ consists of columns in $U_n$. In particular, $U_n$ can be seen as the conversion result of $G_n$. Derived from the formula, we can obtain $U_n=G_n{S}_n^{}(S_n^{\mathrm{T}}{S}_n^{}{)}^{-1}$. Known from the literature [40], using a sparse regularization the result of the spectral linear transformation of the gradient map works better than imposing the regularization term directly on the gradient map itself. Hence, we directly add a sparse regularization term to the linear transformation result of $G_n,n=1,2,$ which becomes $G_n{V}_n$,$V_n$$\in$ℝ${}^{s\times r}$. The problem is as follows:

(5)$$\begin{gathered} {\mathrm{S}}_{\mathrm{E}-\mathrm{TV}}(G_n)=\underset{}{\min }{{\displaystyle \left|\right|}{G}_n{V}_n{\displaystyle \left|\right|}}_1=\underset{U_n,V_n}{\min }{{\displaystyle \left|\right|}{U}_n{\displaystyle \left|\right|}}_1 \\ \mathrm{s}.\mathrm{t}.\left|\right|G_n{V}_n|{|}_{\mathrm{F}}=|\left|G_n\right|{|}_{\mathrm{F}},G_n=U_n{V}_n{}^{\mathrm{T}},V_n{}^{\mathrm{T}}{V}_n=I. \end{gathered}$$

where $I$ is identity matrix; F represents the Frobenius Norm.

Inspired by the Equation (5), using a sparse regularization term on HSI images can achieve great results. Construct a regularization term for HSI, which can be obtained using the spectral and space. By adding the sparsity measure of the gradient map of the spatial pattern to the spectral pattern, we can model our regularization term as:

(6)${{\displaystyle \left|\right|}A{\displaystyle \left|\right|}}_{\mathrm{E}-2\mathrm{DTV}}={\displaystyle \sum }_{n=1}^2{S}_{\mathrm{E}-\mathrm{TV}}({\nabla }_{n}A)={\displaystyle \sum }_{n=1}^2{S}_{\mathrm{E}-\mathrm{T}\mathrm{V}}(G_n)$

where $S(G_n)=\left|\right|G_n|{|}_1$, is called the $l_1$-norm sparsity measure. Equation (6) can be converted to

(7)$$\begin{gathered} {{\displaystyle \left|\right|}A{\displaystyle \left|\right|}}_{\mathrm{E}-2\mathrm{DTV}}={\displaystyle \sum }_{n=1}^2\underset{U_n,V_n}{\min }{{\displaystyle \left|\right|}{U}_n{\displaystyle \left|\right|}}_1 \\ \mathrm{s}.\mathrm{t}.{\nabla }_{n}A=U_n{V}_n{}^{\mathrm{T}},V_n{}^{\mathrm{T}}{V}_n=I. \end{gathered}$$

where $U_n\in$ℝ${}^{l\times r}$, $V_n$$\in$ℝ${}^{s\times r}$,$n=1,2,$.

In this paper, E-2DTV regularization term takes full advantage of the information between all these bands with spectral and spatial information of HSI, which is introduced to improve the existing gTVMBO model. Applying an E-2DTV term to a matrix converts a matrix into the sum of gradients in its dimensions. Therefore, the Equation (2) is equivalent to

(8)$$\begin{gathered} \min \frac{1}{2}{\displaystyle \left|\right|}Y-X*{\displaystyle \left|\right|}A{{\displaystyle \left|\right|}}_{\mathrm{E}-2\mathrm{DTV}}{{\displaystyle \left|\right|}}_{\mathrm{F}}^2+\mathsf{\lambda }{\mathrm{J}}_{\mathrm{TV}}(A) \\ \mathrm{s}.\mathrm{t}.x_i\ge 0,a_i\ge 0,{\displaystyle \sum }_{i=1}^p{a}_i=1 \end{gathered}$$

where the * symbol means that two matrices are multiplied. From Equations (8) and (9), we improved the original method by applying E-2DTV terms to the abundance matrix; that is, the form A is expressed into the form ${\displaystyle {\displaystyle \sum }_{n=1}^2}{{\displaystyle \left|\right|}{U}_n{\displaystyle \left|\right|}}_1$. Therefore, the proposed E-2DTV regularization term for gTVMBO, namely, the proposed E-gTVMBO is as follows:

(9)$$\begin{gathered} \min \frac{1}{2}{\displaystyle \left|\right|}Y-X*\tau {\displaystyle {\displaystyle \sum }_{n=1}^2{\displaystyle \left|\right|}{U}_n{{\displaystyle \left|\right|}}_1}{{\displaystyle \left|\right|}}_{\mathrm{F}}^2+\lambda {\mathrm{J}}_{\mathrm{TV}}(A) \\ \mathrm{s}.\mathrm{t}.{\nabla }_{n}A=U_n{V}_n{}^{\mathrm{T}},V_n{}^{\mathrm{T}}{V}_n=I.x_i\ge 0,a_i\ge 0, {\displaystyle {\displaystyle \sum }_{i=1}^p{a}_i}=1 \end{gathered}$$

where $\tau$ is a positive parameter.

In order to solve the E-gTVMBO model, the ADMM algorithm is used; then by using indicator function ${\chi }_{set}$, ${\chi }_{set}(\mathrm{W})=\left\{\begin{array}{l}0,\mathrm{W}\in \mathrm{set}\\ \infty , \mathrm{otherwise}\end{array}\right.$, we can rewrite the Equation (9) as an unconstrained problem as:

(10)$\min \frac{1}{2}{\displaystyle \left|\right|}Y-X*\tau {\displaystyle {\displaystyle \sum }_{n=1}^2\left|\right|U_n|{|}_1}{{\displaystyle \left|\right|}}_{\mathrm{F}}^2+\mathsf{\lambda }{\mathrm{J}}_{\mathrm{TV}}(A)+\chi {\mathrm{\Omega }}_{l\times p}(X)+\chi {\mathrm{\Omega }}_{p\times s}(A)$

where ${\mathrm{\Omega }}_{m\times n}$ means that this is a non-negative matrix which has m rows and n lines. Then, we introduce two auxiliary variable B $\in$ℝ${}^{p\times s}$, C$\in$ℝ${}^{l\times p}$; the Equation (10) can be formulated as:

(11)$\begin{array}{l}L=\frac{1}{2}{\displaystyle \left|\right|}Y-C*\mathsf{\tau }{\displaystyle {\displaystyle \sum }_{n=1}^2{\displaystyle \left|\right|}{U}_n{{\displaystyle \left|\right|}}_1}{{\displaystyle \left|\right|}}_{\mathrm{F}}^2+\mathsf{\lambda }{\mathrm{J}}_{\mathrm{TV}}(B)+\chi {\mathrm{\Omega }}_{l\times p}(X)+\chi {\mathrm{\Omega }}_{p\times s}(A)+{\displaystyle {\displaystyle \sum }_{n=1}^2⟨M_n,{\nabla }_{n}A-U_n{V}_n^{\mathrm{T}}⟩}\\ +\frac{\mathsf{\mu }}{2}{\displaystyle \left|\right|}{\nabla }_{n}A-U_n{V}_n^{\mathrm{T}}{{\displaystyle \left|\right|}}_{\mathrm{F}}^2+\frac{\mathsf{\eta }}{2}{\displaystyle \left|\right|}A-B+\stackrel{\wedge }{B}{{\displaystyle \left|\right|}}_{\mathrm{F}}^2+\frac{\gamma }{2}{\displaystyle \left|\right|}X-C+\stackrel{\wedge }{C}{{\displaystyle \left|\right|}}_{\mathrm{F}}^2\end{array}$

where $\stackrel{\wedge }{B}$,$\stackrel{\wedge }{C}$ are dual variables; $M_n$(n = 1,2) is the Lagrange multiplier; γ is a positive parameter; $\mu$ is a scalar in the ADMOM algorithm, which is positive.

In the ADMOM, we should alternately optimize every variable involved in (11); while optimizing one variable we need to keep the others constant. The ADMOM algorithm in this paper needs to solve six sub-problems in each iteration; that is, when $U_n$, $V_n$, C, X, A, B are fixed, L is minimized, respectively. Appendix A demonstrates that the operational process of the six sub-problems in detail. In conclusion, each sub-problem in the ADMOM algorithm can be efficiently solved. The flow of E-gTVMBO method is shown in Algorithm 1.

Algorithm 1: E-gTVMBO

Input: The € Y; parameter $\tau ,\rho ,\mathsf{\lambda }$, $\mu ={10}^{-2}$.Output: X and A.

Initialize: Initial $X^0$, $A^0$, $U_n$, $M_n$

1: While not converge do

2: Update $U_n$, $V_n$ by Equations (12) and (14), respectively.

3: Update C, X, A by Equations (15)–(17), respectively.

4: Update B by Equation (19).

5: Check the convergence conditions${\displaystyle \left|\right|}{\mathrm{A}}^t-{\mathrm{A}}^{t+1}{\displaystyle \left|\right|}/{\displaystyle \left|\right|}{\mathrm{A}}^t{\displaystyle \left|\right|}$$\le$$\tau$${\displaystyle \left|\right|}{\mathrm{X}}^t-{\mathrm{X}}^{t+1}{\displaystyle \left|\right|}/{\displaystyle \left|\right|}{\mathrm{X}}^t{\displaystyle \left|\right|}$$\le$$\tau$

6: End while

3. Experiments

For proving the effectiveness of our proposed E-gTVMBO method, in this part, we conducted experiments on simulated data first and then on real data. We compare the E-gtvMBO method with seven other methods, such as FCLSU [11], SUnSAL-TV (short for STV) [23], GLNMF [33], fractional norm when q = 0.1 regularized unmixing method (short for FRAC) [12], NMF-QMV [41], the graph Laplacian (short for GraphL) [42], and gtvMBO [22].

For quantitatively measuring the superiority of the E-gTVMBO method, we have adopted three kinds of indicators of the error between the estimation $\stackrel{\wedge }{S}$$\in$ℝ${}^{p\times c}$ and the reference S$\in$ℝ${}^{p\times c}$, namely, the root-mean-square error (RMSE), Spectral Angel Mapper (SAM), and Normalized Mean-Square Error (nMSE) [12].

In particular, VCA [43] returns 10 p endmember candidates that are clustered into p groups. This is directly used as x for FCLSU and FRAC, while we use the mean spectrum within each group and the sum of the abundances estimated by FCLSU within each group as an initial guess of X⁰ and A⁰, respectively, for all compared methods.

3.1. USGS Library Dataset

The United States Geological Survey (USGS) has developed the USGS spectral database in conjunction with the need for remote sensing exploration of mineral resources. The spectral data include 444 mineral and typical vegetation and mixed material samples that are indicative of mineral resources. It consists of 498 spectral features and measured reflectance values for 224 spectral bands, regularly in the wavelength range of 0.4–2.5 μm.

For comparing the performance of all methods, a simulation of data, selected from the USGS library, is tested. The spectral library we chose is $\mathrm{X}\in$ℝ${}^{224\times 498}$ [44], which is the Chapter 1 of the USGS library. As shown in references [22], five spectral features are selected from the spectral library to generate hyperspectral images of 75 × 75 pixels. For this experiment, the added Gaussian noise is SNR = 20 dB.

Figure 1 takes endmember 5 as an example; it shows the results of eight unmixing methods. It is the estimated abundance maps of endmember 5 obtained when the SNR is 20 dB noisy data. From Figure 1a–i there are the Reference image, FCLSU, FRAC, SUnSAL-TV, GLNMF, NMF-QMV, the graph Laplacian, the gtvMBO, and E-gtvMBO algorithms, respectively. By observing Figure 1, we can know that the colors of SUnSAL-TV and GLNMF backgrounds are different compared to other methods. The background of the graph obtained by the NMF-QMV method has a lot of noise, and the result of GraphL loses many recognizable objects. The gtvMBO method can clearly identify the identifiable object of the endmember 5, but its background still has a lot of noise. Compared to other methods, the E-gtvMBO method implements the identifiable balance between object and background noise. Therefore, from Figure 1b–g, we can find that Figure 1g is closest to the reference image in Figure 1a. This can qualitatively illustrate the strengths of the E-gtvMBO method compared to several other methods.

Table 1 quantitatively compares the performance of all methods of SNR = 20 dB. It lists the RMSE, nMSE, and SAM values of eight unmixing methods (FCLSU, SUnSAL-TV, GLNMF, FRAC, NMF-QMV, GraphL, gtvMBO, and the proposed E-gtvMBO). The reason for the existence of Inf is that the endmember matrix obtained by FRAC and STV methods has zero samples after the operation. By observing Table 1, it shows that the RMSE, SAM, and nMSE values of abundance matrix when using E-gtvMBO method are the lowest compared to the other seven contrasting methods. The RMSE, SAM, and nMSE values being smaller means the performance of the method is better. Comprehensively considering the three evaluation indicators, E-gtvMBO is still the best method among all the methods. Therefore, this is enough to prove the advantages of the proposed E-gtvMBO method.

3.2. Urban Dataset

In this section, the real hyperspectral data we used are from the Urban dataset. The Urban dataset has four reference endmembers, which are asphalt, grass, tree, and roof. It contains 210 spectral bands, but bands 1–4, 76, 87, 101–111, 136–153, and 198–210 were removed due to dense water vapor and atmospheric effects, leaving 162 spectral bands in the end. The hyperspectral images contain 307 × 307 pixels; the spectral resolution is 10 nm.

In this experiment, our unmixing results are presented in Figure 2 and Figure 3 and Table 2. In detail, Figure 2 is endmember maps of the Urban dataset; Figure 3 is abundance maps of the Urban dataset; Table 2 quantitatively compares the performance of all methods. In Table 2, we list the RMSE, nMSE, and SAM values of eight unmixing methods (FCLSU, SUnSAL-TV, GLNMF, FRAC, NMF-QMV, the graph Laplacian, the gtvMBO, and the E-gtvMBO). From Figure 2, we can know that the endmember map of gtvMBO is the closest to the reference image. Figure 3 shows the estimated abundance maps of eight unmixing methods for the Urban dataset. In Figure 3, from above and below are FCLSU, SUnSAL-TV, GLNMF, FRAC, NMF-QMV, GraphL, gtvMBO, and the proposed E-gtvMBO methods, respectively. Most of these methods, such as FCLSU, FRAC, SUnSAL-TV, GLNMF, and NMF-QMV, especially when the endmembers are asphalt and roofs, produced abundance maps with low image contrast. The E-gtvMBO, gtvMBO, and GraphL methods can enhance the contrast of abundance maps. It is worth noting that since NMF-QMV does not enforce the abundance matrix to be non-negative; the resulting spectrum of pitch is below zero in NMF-QMV.

About the abundance maps of roofs, only GraphL, gtvMBO, and E-gtvMBO can capture those fragmentary roof self-approximation maps. Although it is observed from Figure 3 that the effect of the gtvMBO method is similar to that of the E-gtvMBO method, combining the four types of ground objects, we can observe that the results of E-gtvMBO method unmixing are more similar to the reference image.

By observing Table 2, we can know that E-gtvMBO method has the lowest values of RMSE and nMSE when compared to the other seven methods. Although the SAM value of E-gtvMBO is not the highest, it is the closest to the SAM value of the NMF-QMV method which is the best value. Overall, the E-gtvMBO method can recover abundance maps and endmember matrices more accurately.

3.3. Samson Dataset

In this part, we use the Samson data with 95 × 95 pixels and 156 spectral bands after preprocessing, whose reference has three endmembers. Its wavelength range is 401 nm to 889 nm, and the spectral resolution of Samson dataset is 10 nm. The unmixing results are given in Figure 4 and Figure 5 and Table 3 for endmembers, abundance maps, and quantitative metrics, respectively.

From Figure 4, all endmember plots can capture the rough shape and discontinuities in the ground truth but with different heights. The endmember map of gtvMBO is the closest to the reference image. This also illustrates the superiority of the proposed method. Figure 5 shows the estimated abundance maps of eight unmixing methods for the Samson dataset. In Figure 5, the STV result looks fuzzy, and the GLNMF result is noisy in the homogeneous region because its Laplacian plot is based on the entire dataset, which contains a certain amount of noise. The result graphs obtained by GraphL, gtvMBO, and E-gtvMBO are similar, but E-gtvMBO produces a sharper edge. By observing Table 3, although SAM(X), nMSE(X), and RMSE(X) in FCLSU are better than those in the proposed E-gtvMBO for the quantitative index of endmember matrix X, RMSE(A) and nMSE(A) for the quantitative indices of the abundance matrix A obtained by the proposed E-gtvMBO are the smallest. Therefore, from the quantitative point of view, the proposed E-gtvMBO still shows good performance.

4. Discussion

4.1. Performance of E-gtvMBO for Parameter $\tau$

In the proposed E-gTVMBO model in Equation (9), $\tau$ is a positive parameter. In the process of the above experiments, we find that different parameter settings in the program will bring different unmixing effects. Therefore, we discuss the performance of E-gtvMBO for different values of $\tau$. In this section, we use the urban dataset experiment as an example; the results of unmixing are obtained by setting the range from 0 to 0.1 of parameter $\tau$. From Figure 6, we can find that the value of $\tau$ is in the range of 0.08 to 0.1; the RMSE and nMSE are becoming more stable and optimal. However, the SRE is smaller as the $\tau$ value increases. Therefore, as a trade-off, the setting range of parameter $\tau$ is preferably between 0.05 and 0.08.

4.2. Performance of E-gtvMBO for Different SNR Value

For the simulated dataset in experiment 1, we only give the unmixing results of SNR = 20 dB. To identify the performance of E-gtvMBO in different SNR values, we add the experiments for other SNR value, namely, SNR = 10 dB and SNR = 30 dB here. As shown in Table 4 and Table 5, they list the RMSE, nMSE, and SAM values of eight different unmixing methods (FCLSU, STUnSAL-TV, GLNMF, FRAC, NMF-QMV, GraphL, gtvMBO, and the proposed E-gtvMBO) in different SNR values. By observing Table 3 and Table 4, the E-gtvMBO method has the lowest RMSE values of abundance map compared to the other seven contrasting methods. Although its nMSE and SAM values are not the greatest, it is close to the method of best value. By comparing Table 1, Table 4 and Table 5, we can find that the E-gtvMBO method works has better performance when SNR = 20 dB.

4.3. Performance of E-gtvMBO for Running Time

Finally, the running time of methods is also a non-negligible index. Table 6 records the running times of the eight unmixing methods on the dataset. By observing Table 6, compared with the other unmixing methods, the E-gtvMBO method is slower than FRAC, GraphL, and gtvMBO but much quicker than the other four methods. The most important reason is that the E-gtvMBO method needs more processing steps compared to FRAC, GraphL, and gtvMBO methods, while it is much more advanced than other methods.

5. Conclusions

In this paper, we suggest a new HU method with E-2DTV (E-gtvMBO). E-2DTV can naturally utilize correlations and differences between bands, which can better reflect the sparseness characteristics of natural HSI gradient maps. Based on the gtvMBO method, we impose an E-2DTV on the abundance matrix. For using the E-2DTV regularization term to solve the problem of unmixing, we derived an effective method based on ADMM. By using ADMM algorithm, all the sub-problems in it are solved efficiently. The simulated dataset and real dataset results show that when compared with the other seven HU methods, the E-gtvMBO method could improve the unmixing results. Under the situation of unknown spectral information of endmembers, the proposed method obtains the better unmixing results than the other spectral unmixing methods. Because the simulated experimental data and real data that we tested belong to urban data, the proposed method always is applied in an advantageous way of the type of urban data.

Author Contributions

Conceptualization, P.W.; Methodology, P.W.; Software, X.S.; Validation, Y.K., X.Z. and L.W.; Formal analysis, P.W.; Investigation, Y.K.; Resources, X.Z.; Data curation, P.W.; Writing—original draft preparation, X.S.; Writing—review and editing, P.W. and Y.K.; Visualization, P.W.; Supervision, Y.K.; Project administration, P.W.; Funding acquisition, X.Z. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The dataset used can be obtained from the corresponding authors according to reasonable requirements.

Acknowledgments

The authors sincerely thank the academic editors and reviewers for their useful comments and constructive suggestions.

Conflicts of Interest

The authors declare no conflict of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. Estimated abundance maps for endmember 5 with an SNR of 20 dB where the data are noisy. From left to right are eight methods (Reference image, FCLSU, FRAC, STV, GLNMF, NMF-QMV, GraphL, gtvMBO, and E-gtvMBO).

Figure 1. Estimated abundance maps for endmember 5 with an SNR of 20 dB where the data are noisy. From left to right are eight methods (Reference image, FCLSU, FRAC, STV, GLNMF, NMF-QMV, GraphL, gtvMBO, and E-gtvMBO).

Figure 2. Endmember profiles (x) of the Urban dataset.

Figure 3. Abundance maps of Urban dataset from up to down are reference image and eight unmixing methods (FCLSU, FRAC, STV, GLNMF, NMF-QMV, GraphL, gtvMBO, and E-gtvMBO), respectively.

Figure 3. Abundance maps of Urban dataset from up to down are reference image and eight unmixing methods (FCLSU, FRAC, STV, GLNMF, NMF-QMV, GraphL, gtvMBO, and E-gtvMBO), respectively.

Figure 4. Endmember profiles (x) of the Samson dataset.

Figure 5. Abundance maps of Samson dataset from up to down are reference image and eight unmixing methods (FCLSU, FRAC, STV, GLNMF, NMF-QMV, GraphL, gtvMBO, and E-gtvMBO), respectively.

Figure 5. Abundance maps of Samson dataset from up to down are reference image and eight unmixing methods (FCLSU, FRAC, STV, GLNMF, NMF-QMV, GraphL, gtvMBO, and E-gtvMBO), respectively.

Figure 6. (a) RMSE (b) nMSE (c) SRE evaluating at different [Forumla omitted. See PDF.] for the urban data.

Table 1

Quantitative evaluation for unmixing results in USGS dataset with SNR = 20 dB.

	Methods	FCLSU	FRAC	STV	GLNMF	NMF-QMV	GraphL	gtvMBO	E-gtvMBO
Indicators
RMSE(X)		0.057	inf	inf	0.073	0.049	0.082	0.071	0.052
nMSE(X)		0.089	inf	inf	0.091	0.072	0.092	0.084	0.079
SAM(X)		2.75	inf	inf	3.27	2.91	2.82	3.94	2.47
RMSE(A)		0.2591	0.2568	0.2672	0.4425	0.2579	0.2501	0.2439	0.2215
nMSE(A)		0.8682	0.8981	0.8519	1.3728	0.8966	0.8182	0.7938	0.7492

Table 2

Quantitative evaluation for unmixing results in urban dataset.

Indicators	FCLSU	FRAC	STV	GLNMF	NMF-QMV	GraphL	gtvMBO	E-gtvMBO
RMSE(X)	0.109	inf	inf	0.188	0.211	0.099	0.099	0.082
nMSE(X)	0.635	inf	inf	1350	1200	0.636	0.639	0.617
SAM(X)	2.39	inf	inf	3.58	2.09	3.89	3.93	2.56
RMSE(A)	0.145	0.153	0.288	0.175	0.245	0.184	0.180	0.142
nMSE(A)	0.437	0.450	0.788	0.554	0.655	0.520	0.512	0.395

Table 3

Quantitative evaluation for unmixing results in Samson dataset.

	Methods	FCLSU	FRAC	STV	GLNMF	NMF-QMV	GraphL	gtvMBO	E-gtvMBO
Indicators
RMSE(X)		0.044	inf	inf	0.036	0.073	0.055	0.058	0.052
nMSE(X)		0.169	inf	inf	0.153	0.302	0.221	0.249	0.189
SAM(X)		3.64	inf	inf	4.49	12.82	8.43	8.54	9.36
RMSE(A)		0.180	0.165	0.164	0.187	0.148	0.139	0.128	0.119
nMSE(A)		0.455	0.429	0.375	0.502	0.428	0.302	0.299	0.276

Table 4

Quantitative evaluation for unmixing results in the USGS dataset with SNR = 10 dB.

	Methods	FCLSU	FRAC	STV	GLNMF	NMF-QMV	GraphL	gtvMBO	E-gtvMBO
Indicators
RMSE(X)		0.153	inf	inf	0.233	0.196	0.183	0.172	0.164
nMSE(X)		0.227	inf	inf	0.358	0.728	0.248	0.245	0.239
SAM(X)		11.91	inf	inf	16.24	24.73	10.65	8.82	9.64
RMSE(A)		0.435	0.432	0.389	0.433	0.254	0.314	0.388	0.232
nMSE(A)		1757	1751	1642	1558	1393	1015	1371	1101

Table 5

Quantitative evaluation for unmixing results in the USGS dataset with SNR = 30 dB.

	Methods	FCLSU	FRAC	STV	GLNMF	NMF-QMV	GraphL	gtvMBO	E-gtvMBO
Indicators
RMSE(X)		0.244	inf	inf	0.271	0.225	0.205	0.261	0.211
nMSE(X)		0.591	inf	inf	0.729	0.686	0.523	0.682	0.9137
SAM(X)		4.92	inf	inf	4.81	6.09	4.53	4.86	4.73
RMSE(A)		0.369	0.352	0.367	0.386	0.244	0.227	0.307	0.214
nMSE(A)		0.954	0.928	0.938	1278	0.956	0.978	0.905	0.914

Table 6

The processing time of different unmixing methods on datasets.

Time(s)	FCLSU	FRAC	STV	GLNMF	NMF-QMV	GraphL	gtvMBO	E-gtvMBO
Simulated data	1221	0.069	2318	1241	1519	0.083	0.412	0.787
Urban data	34,804	7539	81,717	107,56	126,79	18,392	26,893	31,247
Cuprite data	550,032	154,130	1051.7	579,835	534,221	478,03	488,062	512,372

Appendix A

To solve $U_n$ sub-problem efficiently, we can utilize the known soft threshold operator S [45]. By utilizing the soft threshold operator S, we can obtain the solution of $U_n$ as:(A1)$$U_n=(C^{\mathrm{T}}Y-\mu ({\nabla }_{n}A+\frac{M_n}{\mu })V_n){(\tau C^{\mathrm{T}}C-\mu )}^{-1}$$

Then, extracting all terms containing orthogonal $V_n$ from the Equation (11), we attain that the $V_n$ sub-problem reads as:(A2)$${\displaystyle {\displaystyle \sum }_{n=1}^2⟨M_n,{\nabla }_{n}A-U_n{V}_n^{\mathrm{T}}⟩}+\frac{\mu }{2}{\displaystyle \left|\right|}{\nabla }_{n}A-U_n{V}_n^{\mathrm{T}}{{\displaystyle \left|\right|}}_{\mathrm{F}}^2$$

The realization of the global solution of $V_n$ sub-problem can be based on the theorem [46]; the theorem is defined as follows:

Theorem: For $P\in {\mathrm{W}}_{m,n}$, the global solution of $\underset{{\mathrm{VV}}^{\mathrm{T}}=\mathrm{I}}{\min }⟨P,V⟩$ is $V^{*}=B{C}^{\mathrm{T}},\mathrm{where} P=BD{C}^{\mathrm{T}}$ denotes the condensed singular value decomposition (SVD) of P. Then, we can obtain the updated equation of $V_n$ as:(A3)$$\left\{\begin{array}{l}[Q,D,F]=\mathrm{svd}({({\nabla }_{n}A+M_n/\mu )}^{\mathrm{T}}{U}_n)\\ V_{\mathrm{n}}=Q{F}^{\mathrm{T}}\end{array}\right.$$

The solution of C is as follows:(A4)$$C=(Y*(U_n{)}^{\mathrm{T}}+\mathsf{\gamma }(X+\stackrel{\wedge }{C}))/(U_n*{(U_n)}^{\mathrm{T}}+\mathsf{\gamma }I)$$

The X sub-problem seeks the projection of $C-\stackrel{\wedge }{C}$ onto the set of non-negative matrices; we solve this problem by using a hard thresholding method. Then, the solution of X reads as:(A5)$$\max (C-\stackrel{\wedge }{C},0)$$

As for A sub-problem, the closed-form solution for A can be derived as:(A6)$$\begin{gathered} \mathrm{H}=\frac{\mu }{\lambda }{{\displaystyle {{\displaystyle \sum }}_{n=1}^{2}F({\mathrm{D}}_n)}}^{*}\odot F(\mathrm{fold}(U_n{V}_n^{\mathrm{T}})-M_n) \\ {\mathrm{T}}_x=|F({\mathrm{D}}_1){|}^2+|F({\mathrm{D}}_2){|}^2 \\ A=F^{-1}(\frac{F(fold(\eta \stackrel{}{B}-\eta \tilde{B})\mathrm{H})}{\mu +\mu {\mathrm{T}}_x}) \end{gathered}$$ where ${\mathrm{D}}_n$ is the correlation difference filter.

About the B sub-problem, we approximate the non-differentiable gTV with the Graph GL function. We approximate any real number in [0.1] with the most suitable M-bit binary representation, which can lift the binary restriction of MBO. MBO schemes can be implemented in parallel across channels. In the end, combining all the channels, we can obtain an approximate solution to the element [0.1] in the B sub-problem. We use a group of M binary matrices $B_m\in$ℝ${}^{p\times s}$ with m = 1, …, M (M is the total number of bits) to represent the matrix B; then $B_{ij}\approx {\displaystyle {\displaystyle \sum }_{m=1}^M{2}^{-m}{(B_m)}_{ij}}$, $B_m$ is the mth bit channel of the matrix B.

Therefore, the minimization problem of each $B_m$ is as follows:(A7)$$\underset{B_m}{\min }\epsilon \mathrm{tr}(B_m{L}_s{B}_m^{\mathrm{T}})+\frac{1}{\epsilon }\mathrm{W}(B_m)+\frac{\eta }{2\mathsf{\lambda }}{\displaystyle \left|\right|}{B}_m-A_m-\widetilde{B_m}{{\displaystyle \left|\right|}}_{\mathrm{F}}^2$$

To solve (18), we use the MBO scheme, at the the i + 1 iteration, we update $B_m$ as follows:(A8)$$\begin{array}{l}{\mathrm{Z}}_m^{i+1}={\mathrm{Z}}_m^i(I-di\mathrm{\Lambda })-di*D_m^i\\ B_m^{i+1/2}=V{\mathrm{Z}}_m^{i+1}\\ D_m^{i+1}=\frac{\eta }{\mathsf{\lambda }}(B_m^{i+1/2}-(A_m+\widetilde{B_m}))V\\ B_m^{i+1}=H_{1/2}(B_m^{i+1/2})\end{array}$$ where ${\mathrm{Z}}_m=B_{m}V$ and $D_m=\frac{\eta }{\mathsf{\lambda }}(B_{m}V-(A_m+\widetilde{B_m})V)$.

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