*3.2. DGC-NMF Algorithm*

Using the sparsity information learnt in Section 3.1, the proposed NMF with data-guided constraints decides whether the *L*1/2 constraint or the *L*2 constraint should be assigned to a pixel. In the DGC-NMF algorithm, both the *L*1/2 regularizer and the *L*2 regularizer are adopted to achieve better control of sparsity in each pixel's abundances. Pixels are split into two categories according to sparsity levels. For pixels with high sparsity levels, the *L*1/2 regularizer is adopted to constrain their abundance. For pixels with low sparsity levels, the *L*2 regularizer is applied. The model of DGC-NMF is as follows:

$$f(\mathcal{W}, H) = \frac{1}{2} \|X - \mathcal{W}H\|\_F^2 + \lambda \|\mathbb{C}.\*H\|\_{1/2} + \mu \|\|D.\*H\|\_2 \tag{13}$$

where *λ* and *μ* are the regularization parameter, and C =**1***P***s***<sup>T</sup>* and *D* = **<sup>1</sup>***P*(**<sup>1</sup>***N* − **s**)<sup>T</sup> are the indictor matrices which decide whether an *L*1/2 constraint constraint or an *L*2 constraint is imposed or not for each pixel. **s** ∈ R *N* + is obtained by evaluating the sparse levels of abundances of all pixels

$$\mathbf{s}(n) = \begin{cases} 1 & \text{sparseness}(n) > \delta \\ 0 & \text{sparseness}(n) \le \delta \end{cases} \tag{14}$$

where *δ* is a threshold that controls which kind of constraint should be imposed. The threshold *δ* is decided by applying Otsu's method to maximize the separability of pixels with high sparsity level and pixels with low sparsity level [38]. Figure 4 shows the histogram of estimated sparseness values for pixels in a synthetic image and the selected value of threshold *δ* for this synthetic image. The sparseness histogram is obtained by counting the sparseness of estimated abundance of pixels. When the sparseness of a pixel's abundance is higher than *δ*, the pixel's abundance will be constrained by an *L*1/2 sparsity regularization. Otherwise, the pixel's abundance will enjoy an *L*2 constraint to promote evenness.

**Figure 4.** The histogram of estimated sparseness values for pixels in a synthetic image and the selected threshold value.

Based on the cost function in Equation (13), the update rules are derived as follows

$$\mathcal{W} = \mathcal{W}.\*(XH^T)./\mathcal{W}HH^T\tag{15}$$

$$H = H.\*(\mathcal{W}^T X)./\left(\mathcal{W}^T W H + \frac{\lambda}{2} \mathbb{C}.\* H^{-\frac{1}{2}} + 2\mu D.\*H\right) \tag{16}$$

The procedure of the proposed DGC-NMF is described in Algorithm 1.

#### **Algorithm 1** DGC-NMF algorithm

**Input:** Hyperspectral data *X* ∈ *RL*×*<sup>N</sup>* ; the number of endmembers *P*. **Initialization:** Initialize endmember matrix *W*1 and abundance matrix *H*1 by SGA-FCLS. 1: **repeat**


The update rule for *W* in Equation (15) is just the same as that in [18]. The authors of [18] have proved objective (2) is nonincreasing under the update rule in Equation (3). Therefore, we only need to focus on proving objective (13) is nonincreasing under the update rule for *H* in Equation (16).

#### **Theorem 1.** *The objective (13) is nonincreasing under the update rule in (16).*

Since the objective function in Equation (13) is separable by columns, for each column of *H*, we could consider each column of *H* individually. For convenience, let *h* denote a column of *H*, *x* denotes the corresponding columns in *X*, and *c*, *d* denote the corresponding column in *C*, *D*, respectively. *c* and *d* are vectors with all ones or zeros. The objective function by column is expressed as follows:

$$F(h) = \frac{1}{2} \left\| \mathbf{x} - \mathcal{W}h \right\|\_2^2 + \lambda \left\| \mathbf{c}.\*h \right\|\_{\frac{1}{2}} + \mu \left\| \mathbf{d}.\*h \right\|\_2 \tag{17}$$

An auxiliary function similar to that used in the expectation-maximization algorithm is defined to prove Theorem 1 [39,40].

**Definition 1.** *<sup>G</sup>*(*h*, *h* ) *is an auxiliary function of F*(*h*) *with*

$$\mathcal{G}(h, h') \ge F(h), \quad \mathcal{G}(h, h) = F(h) \tag{18}$$

*satisfied*

**Lemma 1.** *If <sup>G</sup>*(*h*, *h* ) *is an auxiliary function of <sup>F</sup>*(*h*)*, F*(*h*) *is nonincreasing under the update*

$$h^{t+1} = \arg\min\_{h} G(h, h^t) \tag{19}$$

**Proof.**

$$F(h^{t+1}) \le G(h^{t+1}, h^t) \le G(h^t, h^t) = F(h^t)$$

Following [29], we define the function *G* as:

$$\mathcal{G}(h, h^t) = \mathcal{F}(h, h^t) + (h - h^t)^T \nabla F(h^t) + \frac{1}{2} (h - h^t)^T \mathcal{K}(h^t) (h - h^t) \tag{20}$$

where *K*(*h<sup>t</sup>*) is a diagonal matrix with diagonal *k*

$$k = (\mathcal{W}^T \mathcal{W} h^T + \frac{\lambda}{2} \mathfrak{c}.\*(h^t)^{-\frac{1}{2}} + 2\mu d.\*h^t)./h^t \tag{21}$$

Obviously, the second property of *G* defined in Definition 1 is satisfied. Writing out the Taylor expansion of *F*(*h*)

$$\begin{aligned} F(h) &= F(h^t) + (h - h^t)^T \nabla F(h^t) \\ &+ \frac{1}{2} (h - h^t)^T [\mathcal{W}^T \mathcal{W} - \frac{\lambda}{4} \text{diag} (\mathbf{c}.\*(h^t)^{-\frac{3}{2}}) + 2\mu \text{diag}(d)](h - h^t) \\ &+ R(\nabla^{(n \ge 3)} F(h^t)) \end{aligned}$$

where the function *R* denotes the Lagrange remainder term, which can be omitted.

Comparing *F*(*h*) with *<sup>G</sup>*(*h*, *h<sup>t</sup>*) in Equation (20), we find the first property *<sup>G</sup>*(*h*, *h<sup>t</sup>*) ≥ *F*(*h*) is satisfied when

$$0 \le (h - h^t)^T [\mathcal{K}(h^t) - \mathcal{W}^T \mathcal{W} + \frac{\lambda}{4} \text{diag} \{ c.\* (h^t)^{-\frac{3}{2}} \} - 2\mu \text{diag}(d) [(h - h^t) \tag{22}$$

Equivalent to

$$0 \le (h - h^t)^T [K'(h^t) + \frac{3\lambda}{4} \text{diag}(\mathbf{c}.^\ast (h^t)^{-\frac{3}{2}})](h - h^t) \tag{23}$$

where *K* is is a diagonal matrix with diagonal *k* 

$$k' = \left(\mathsf{W}^T \mathsf{W} h^T. / h^t\right) - \mathsf{W}^T \mathsf{W} \tag{24}$$

The positive semidefiniteness of *K* has been proved in [18]. Another term in Equation (23) is nonnegative since *c* and *h* both are nonnegative. Thus, Equation (22) holds due to the sum of two positive semidefinite matrices is also positive semidefinite.

It remains to select the minimum of *G* by taking the gradient and equating to zero

$$\nabla\_s \mathbf{G}(h, h^t) = \mathsf{W}^T(\mathsf{W}h^t - \mathbf{x}) + \frac{\lambda}{2} \mathsf{c.} \ast (h^t)^{-\frac{1}{2}} + 2\mu \mathsf{d.} \ast h^t + \mathsf{K}(h^t)(h - h^t) = 0 \tag{25}$$

Solving *h* gets *h<sup>t</sup>*+<sup>1</sup>

$$\begin{split} h^{t+1} &= h^t - K(h^t)^{-1} (\mathcal{W}^T (\mathcal{W}h^t - \mathbf{x}) + \frac{\lambda}{2} \mathbf{c} .\* (h^t)^{-\frac{1}{2}} + 2\mu \mathbf{d} .\* h^t) \\ &= h^t - h^t ./ (\mathcal{W}^T \mathcal{W}h^T + \frac{\lambda}{2} \mathbf{c} .\* (h^t)^{-\frac{1}{2}} + 2\mu \mathbf{d} .\* h^t) \\ &\quad .\* (\mathcal{W}^T (\mathcal{W}h^t - \mathbf{x}) + \frac{\lambda}{2} \mathbf{c} .\* (h^t)^{-\frac{1}{2}} + 2\mu \mathbf{d} .\* h^t) \\ &= h^t .\* (\mathcal{W}^T \mathbf{x}) ./ (\mathcal{W}^T \mathcal{W}h^T + \frac{\lambda}{2} \mathbf{c} .\* (h^t)^{-\frac{1}{2}} + 2\mu \mathbf{d} .\* h^t) \end{split} \tag{26}$$

which is the desired columnwise form of update rule in Equation (16). The proof of Theorem 1 is completed.

#### **4. Experimental Results and Analysis**

#### *4.1. Experiments on Synthetic Data*

In this section, the proposed DGC-NMF algorithm is tested on synthetic data to evaluate its performance. Three related methods, including NMF [20], *L*1/2-NMF [35] and *L*2-NMF [41] are used for comparison with the proposed method. The synthetic data used to test is generated following [29]. The spectral signatures are randomly selected from the United States Geological Survey (USGS) digital spectral library to simulate synthetic images [42]. The abundances are generated as follows. Firstly, a *z*2 × *z*2 size image is divided into *z* × *z* regions. Each region is initialized with the same kind of ground material. Secondly, a (*z* + 1) × (*z* + 1) low-pass filter is applied to generated mixed pixels and make the abundance variation smooth. Finally, a threshold *θ* (0 < *θ* ≤ 1) is used to reject pixels with high purity. The pixels with abundance larger than *θ* will be replaced by mixtures of all endmembers with equal abundance. *θ* can be used as the parameter to generate synthetic data with various sparseness levels. In addition, zero-mean white Gaussian noise is added into the synthetic data to simulate possible noise. In the experiments on synthetic data and real data, DGC-NMF and the compared NMF based algorithms are all initialized using SGA-FCLS. SGA-FCLS provides a more accurate initialization than random initialization. We also compared the unmixing performance of our proposed DGC-NMF with that of SGA-FCLS in Section 4.2.

Two criteria, spectral angle distance (SAD) and root-mean-square error (RMSE), are adopted to evaluate the unmixing performance of algorithms. They are defined as follows:

$$\text{SAD}\_p = \cos^{-1}(\frac{\mathbf{w}\_i^T \hat{\mathbf{w}}\_i}{||\mathbf{w}\_i|| \ ||\hat{\mathbf{w}}\_i||})\tag{27}$$

$$\text{RMSE}\_{p} = \sqrt{\frac{1}{N} \sum\_{n=1}^{N} \left( h\_{pn} - \hat{h}\_{pn} \right)^{2}} \tag{28}$$

where **w***i* and **wˆ** *i* are the reference endmember signatures and their estimates. Respectively, *hpn* and ˆ *hpn* are the reference and estimated abundances. Before calculating evaluation criteria, the estimated endmembers should firstly be reordered to match the reference endmembers. The estimated abundances should also be reordered respectively.

To present the effects of algorithms on the sparseness of unmixing results intuitively, we compare the sparseness histograms of different algorithms in Figure 5. The histogram are obtained by counting the sparseness levels of pixels' abundance estimated by different algorithms. From the histogram in Figure 5b, it can be seen that the abundance result achieved by *L*2-NMF generally tends to be smoother. The histogram of *L*2-NMF owns more pixels with low sparseness levels compared to other algorithms. In Figure 5c, the whole histogram of *L*1/2-NMF has the tendency of a right shift, which demonstrates that *L*1/2-NMF can effectively promote sparsity in the unmixing process. The sparseness of pixels' abundances will be raised when applying *L*1/2-NMF. The pixels with various sparseness are not able to receive constraints accommodated to their sparsity levels in *L*2-NMF and *L*1/2-NMF. For the histogram of DGC-NMF in Figure 5d, the right part of the histogram has the tendency towards a right shift and the left part has a tendency towards a left shift. This validates that the DGC-NMF algorithm can impose adaptive constraints on pixels according to their sparseness of abundances. Figure 6 shows the abundance maps of NMF, *L*1/2-NMF, *L*2-NMF, and DGC-NMF, respectively, when applied on synthetic data. It can be seen that DGC-NMF achieves sparser abundance results than NMF and *L*2-NMF in areas possessing high sparsity levels. Meanwhile, DGC-NMF obtains more accurate abundance results than *L*1/2-NMF in evenly mixed areas.

**Figure 5.** Comparison of sparseness histograms for true abundances different algorithms' estimated abundance. (**a**) Ground truth; (**b**) *L*2-NMF; (**c**) *<sup>L</sup>*1/2-NMF; (**d**) DGC-NMF. NMF: nonnegative matrix factorization; DGC-NMF: NMF with data-guided constraints.

**Figure 6.** Abundance maps of synthetic data estimated by NMF, *<sup>L</sup>*1/2-NMF, *L*2-NMF, and DGC-NMF, respectively. Each row shows the corresponding abundance maps of a same endmember by different algorithms.

Due to the first unmixing process for learning sparseness maps from data, the proposed DGC-NMF is more computationally expensive than *L*2-NMF and *L*1/2-NMF. However, it is still in the same order of magnitude as *L*2-NMF and *L*1/2-NMF. Table 1 shows the running time of different algorithms on a 100 × 100 size synthetic image. For each algorithm, 20 independent runs are carried out and the results are averaged. All experiments are performed using a laptop PC with an Intel Core I7 CPU and 8 GB of RAM. The iteration number of the two steps in DGC-NMF is set as 200. The iteration number of comparative algorithms is also set as 200.

**Table 1.** Comparison of the time cost of different algorithms.


To further analyze the performance of algorithms, five experiments are conducted with respect to the following: (1) sparseness; (2) size of image; (3) number of endmembers; and (4) the signal-to-noise ratio (SNR). For each experiment, 20 independent runs are carried out and the results are averaged. Considering DGC-NMF has the same parameters *λ* and *μ* as *L*2-NMF and *L*1/2-NMF, we set *λ* and *μ* in DGC-NMF to the same values as those of *L*1/2-NMF and *L*2-NMF to make fair comparisons. The values of parameters *λ* and *μ* for *L*1/2-NMF and *L*2-NMF, respectively, are carefully determined to achieve best results as in [33]. DGC-NMF adopts the same values of parameters to validate the effectiveness.

*Experiment 1:* In this experiment, we investigate the performance of algorithms under various sparsity levels. Since the real abundance maps of synthetic data are available, we also make comparison between DGC-NMF with a real sparseness map and DGC-NMF with an estimated sparseness map. The algorithms are tested on synthetic data with different average sparseness levels of abundances. The size of data used here and in the following experiments is 100 × 100, except in Experiment 2. The endmember number K = 6 and SNR = 20 dB. Figure 7 shows that the proposed DGC-NMF performs the best at various sparseness levels. The DGC-NMF with estimated sparseness map performs quite closely with the DGC-NMF with the real sparseness map, which proves the effectiveness of the proposed method for estimating the sparsity levels of pixels' abundances. For SAD, DGC-NMF performs the best, while *L*2-NMF has the poorest performance. With the sparseness level rises to 0.6, *L*1/2-NMF achieves better performance than *L*2-NMF and NMF, while still being inferior to DGC-NMF. Considering RMSE, DGC-NMF also achieves the best performance under different sparseness levels. *L*2-NMF achieves more accurate results than *L*1/2-NMF when applied to data with relatively low sparsity levels.

*Experiment 2:* The algorithms are also tested on synthetic data with different sizes to validate the performance. The image size is set as 36 × 36, 49 × 49, ..., 144 × 144, respectively, with *K* = 6, and SNR = 20 dB. In this experiment and following experiments, the threshold *θ* is set as 0.91. Figure 8 shows that the proposed DGC-NMF achieves best results for either SAD or RMSE when applied to different sizes of images. For larger images, *L*1/2-NMF and *L*2-NMF may not obtain a satisfactory result since the images consist of areas with various sparsity levels and a simple constraint is not applicable. The proposed method provides a reliable way for images possessing areas with various sparsity levels and requiring adaptive constraints.

**Figure 7.** Performance comparison of the algorithms when sparseness level of abundance varies. (**a**) spectral angle distance (SAD); (**b**) root-mean-square error (RMSE).

**Figure 8.** Performance comparison of the algorithms with respect to the different sizes of images. (**a**) SAD; (**b**) RMSE.

*Experiment 3:* The algorithms' performance when the number of endmembers changes is presented in Figure 9a,b. The number of endmembers is set from 4 to 8 and the SNR is also set as 20 dB. Generally, DGC-NMF performs the best while *L*2-NMF performs the worst when the number of endmembers varies. For SAD, DGC-NMF still gains the best results, while *L*2-NMF performs the worst and *L*1/2-NMF and NMF have similar performance. From Figure 9b, we can see that DGC-NMF also achieves the lowest RMSE values when applied to data with different number of endmembers.

*Experiment 4:* To test the robustness of the proposed method, synthetic data with different noise levels are used to examine the performance of algorithms. We change the SNR of synthetic data from 10 dB to 30 dB at the steps of 5 dB. With the increase of noise level, the performance of algorithms degrades as expected. The DGC-NMF shows the best performance as the SNR varies. For SAD, *L*1/2-NMF yields better results than *L*2-NMF and NMF when SNR = 20. For RMSE, NMF is better than *<sup>L</sup>*1/2-NMF and *L*2-NMF. It can be seen from the Figure 10 that the proposed DGC-NMF is not sensitive to noise compared to other three algorithms.

**Figure 9.** Performance comparison of the algorithms when the number of endmembers varies. (**a**) SAD; (**b**) RMSE.

**Figure 10.** Performance comparison of the algorithms under various noise levels. (**a**) SAD; (**b**) RMSE.

#### *4.2. Experiments on Real Data*

In this section, we present the experimental results of the proposed method on real hyperspectral data. Two hyperspectral datasets which include regions with different sparsity levels in an urban scene and a regional mineral scene are used in the experiments. To verify the performance of the proposed method, the results of DGC-NMF are compared with NMF [20], *L*1/2-NMF [35], and *L*2-NMF [41]. VCA-FCLS and SGA-FCLS are also adopted to compare with the proposed method. The dimensionality reduction (DR) method adopted for SGA here is principal component analysis (PCA) [43]. The initial condition for SGA in this paper is set as starting with two endmembers with maximal segmen<sup>t</sup> produced by the one-dimensional two-vertex simplex with maximal distance. The experiment for VCA-FCLS is repeated 10 times. The results are averaged values and the standard deviations are

taken. Since the result of SGA is consistent, there is no standard deviation reported for SGA-FCLS and NMF-based methods. The standard deviation of VCA-FCLS comes from the randomness of VCA.

The first hyperspectral scene to be used is the urban dataset collected by a Hyperspectral Digital Imagery Collection Experiment (HYDICE) sensor over an area located at Copperas Cove near Fort Hood, TX, U.S., in October 1995. The spectral and spatial resolutions are 10 nm and 2 m, respectively. After the bands with low SNR are removed from the original dataset, only 162 bands remain in the experiment (i.e., L = 162). The image is 307 × 307 pixels in size and consists of a suburban residential area as shown in Figure 11a. There are four targets of interest existing in this area: asphalt, grass, roofs, and trees. Since the ground truth of Urban dataset is not available. We use the reference abundance maps obtained from [44] to evaluate the algorithms' performance. Two criteria, spectral angle distance (SAD) and root-mean-square error (RMSE), are adopted to evaluate the accuracy of estimated endmembers and abundances, respectively.

**Figure 11.** The two real hyperspectral data used in the experiments. (**a**) The Hyperspectral Digital Imagery Collection Experiment (HYDICE) urban dataset; (**b**) The airborne visible/infrared imaging spectrometer (AVIRIS) Cuprite dataset.

Table 2 represents the mean values and standard deviations of SAD of different methods on urban data. The rows respectively show the results of four targets of interest, i.e., 'asphalt', 'grass', 'trees' and 'roofs', along with the mean values. From the Table 2, it can be seen that the SAD results achieved by DGC-NMF are better than those of other methods in general. For target 'roofs' and the mean value, the proposed DGC-NMF achieves the best results. For 'asphalt' and 'trees', DGC-NMF achieves the second best result. The RMSE results of algorithms are illustrated in Table 3. We can also find that the DGC-NMF's results are generally better than those yielded by the other algorithms. For 'asphalt', 'grass' and the mean value, DGC-NMF achieves the best results. For 'trees' and 'roofs', DGC-NMF achieves the second best results.

In Figure 12, the endmember signatures obtained by different methods are displayed with reference to the ground truth for visual comparison. It is shown that the endmember signatures obtained by DGC-NMF are in good accordance with the ground truth. Figure 13 shows the sparseness maps of abundance results obtained by *L*2-NMF, *L*1/2-NMF, and DGC-NMF, respectively. It can be seen that the sparseness values in the map of *L*2-NMF are low as a whole, while those of *L*1/2-NMF show relatively high levels. For the proposed DGC-NMF, the sparseness values are in better accordance with the distribution of ground covers in hyperspectral data. In high sparsity level areas such as the areas composed of asphalt, DGC-NMF acts in a similar manner to *<sup>L</sup>*1/2-NMF. In these areas, DGC-NMF promotes *L*1/2 constraint and obtains sparser abundance results. In areas with low sparsity levels that are evenly mixed with signatures such as the areas with both trees and grass, DGC-NMF

promotes *L*2 constraint adaptively and obtains smoother abundance results of pixels. Therefore, the sparseness values are lower than those of *L*1/2-NMF, similar to *L*2-NMF. Figure 14 shows the separated abundance maps of each endmember by VCA-FCLS, SGA-FCLS, NMF, *L*2-NMF, *L*1/2-NMF, and DGC-NMF, respectively. As shown in the figure, all algorithms separate out the four targets successfully. Through visual comparison, we can see that *L*2-NMF and *L*1/2-NMF obtain smoother and sparser results than NMF, respectively. *<sup>L</sup>*1/2-NMF achieves grea<sup>t</sup> results in high sparsity level areas, but fails to capture mixed information in evenly mixed areas. The proposed DGC-NMF achieves sparser abundance maps than *L*2-NMF, and has better abundance estimation than *L*1/2-NMF in transition regions. Generally, Figures 13 and 14 demonstrate that the proposed DGC-NMF could promote adaptive constraints on areas in hyperspectral images with various sparsity levels and achieve better unmixing results of abundance.


**Table 2.** The spectral angle distance and their standard deviations of algorithms on the urban dataset. Numbers in bold and red color represent the best results, numbers in bold and blue color represent the second-best results. FCLS: full constrained least squares.

**Table 3.** RMSEs and their standard derivations of algorithms on the urban dataset.


**Figure 12.** Comparison of endmember signatures estimated by different methods over urban data. (**a**) asphalt; (**b**) grass; (**c**) trees; (**c**) roofs. VCA: vertex component analysis; SGA: simplex growing algorithm.

**Figure 13.** The sparseness maps of abundance results obtained by different algorithms. (**a**) *L*2-NMF; (**b**) *<sup>L</sup>*1/2-NMF; (**c**) DGC-NMF.

*Remote Sens.* **2017**, *9*, 1074

**Figure 14.** Abundance maps of urban data estimated by VCA-FCLS, SGA-FCLS, NMF, *L*2-NMF, *<sup>L</sup>*1/2-NMF, and DGC-NMF, respectively, from right column to left column. Each row shows the corresponding abundance maps of a same endmember.

To validate the performance of our proposed method on hyperspectral data with various sparsity levels, we also conduct an experiment on the Cuprite data. The well known Cuprite dataset is collected by an airborne visible/infrared imaging spectrometer (AVIRIS) sensor over Cuprite mining site, Nevada. The raw images have 224 spectral bands covering the wavelength ranging from 0.4 μm to 2.5 μm. The spatial resolution is 20 m and the spectral resolution is 10 nm. Approximate distributions of the minerals have been illustrated in many pieces of research [10,22,26]. The image used in our experiment is a 250 × 190 pixel subset of the Cuprite scene, as shown in Figure 11b. Due to the water absorption and low SNR, several bands are removed, including bands 1–2, 104–113, 148–167, and 221–224. Hence, 188 bands are used in the experiment. According to [10], there are 14 kinds of minerals existing in the scene. However, the variants of the same mineral have minor differences between each other and could be considered as the same endmember. Therefore, we set the number of endmembers in the scene to 12 [25,33]. Figure 15 presents the extracted endmembers and their corresponding abundance maps by DGC-NMF. In the figure, the extracted signatures are compared with USGS library spectra and show good accordance with them. Table 4 presents the SAD results of the proposed DGC-NMF, along with those of other methods. It shows that DGC-NMF achieves the greatest number of cases of best SAD results, outperforming NMF, *L*1/2-NMF, and *L*2-NMF. *L*1/2-NMF obtains the most second-best SAD results of endmembers. In the terms of mean value, DGC-NMF performs the best.

**Figure 15.** The extracted endmembers by DGC-NMF and their corresponding United States Geological Survey (USGS) library signatures, along with the estimated abundance maps. (**a**) alunite; (**b**) andradite; (**c**) buddingtonite; (**d**) dumortierite; (**e**) kaolinite #1; (**f**) kaolinite #2; (**g**) muscovite; (**h**) montmorillonite; (**i**) nontronite; (**j**) pyrope; (**k**) sphene; and (**l**) chalcedony.


**Table 4.** The spectral angle distance and their standard derivations of algorithms on the Cuprite data. Numbers in bold and red color represent the best results; numbers in bold and blue color represent the second-best results.
