# Hyperspectral Unmixing via Low-Rank Representation with Space Consistency Constraint and Spectral Library Pruning

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

#### 2.1. Linear Spectral Unmixing

**y**is an L × 1 column vector (the mixed pixel), L is the number of bands,

**A**is an L × m matrix containing m spectral signatures (endmembers),

**x**is a m × 1 vector containing the fractional abundances of the endmembers, and

**n**is an L × 1 vector denoting the noise and model error.

#### 2.2. Abundance Estimation via LRR

**Theorem**

**1.**

**X**) = rank(

**Y**) = k

**A**is extracted from the hyperspectral image itself. Matrix

**A**usually satisfies the full column rank property as the spectra of the extracted pure endmembers are distinct from each other and the number of bands L is much larger than the number of endmembers m. Normally, the columns of

**Y**are highly correlated and it means that the matrix

**Y**is a low-rank matrix. Thus, we can infer that the corresponding representation matrix

**X**is also low rank.

#### 2.3. Low-Rank Representation of Coefficient Constraints

**B**is $n\times n$, and it can be calculated by the method that, if ${y}_{j}$ is similar to ${y}_{i}$, then the matrix

**B**equals to ${\omega}_{ij}$, or equals to zero, where the ${\omega}_{ij}$ is the similarity of the observation vector ${y}_{i}$ and ${y}_{j}$, according to the different situations having different construction methods, generally in the range between 0 and 1, more commonly using a heat kernel function and Gauss kernel function.

## 3. Unmixing via LRR Based on Space Consistency Constraint with Spectral Library Pruning

#### 3.1. Space Consistency Constraint

- (1)
- The pixels of the near neighbourhood in the image have similar endmembers and their corresponding abundance.
- (2)
- The pixels of the near spectral distance have similar endmembers and their corresponding abundance.
- (3)
- At the same time, combined with the above two points, selecting p pixels with the nearest distance to the current pixels in the n neighbourhood as its real neighbourhood, constrain its abundances to be similar.

**B**is $n\times n$, and it can be calculated by the method that if ${y}_{j}$ is similar to ${y}_{i}$, then the matrix

**B**equals to ${\omega}_{ij}$, or equals to zero, where ${\omega}_{ij}$ is the similarity of the observation vector ${y}_{i}$ and ${y}_{j}$. We set ${\omega}_{ij}$ to be 1.

Algorithm 1: Solve the SCC-LRR problem by ALM |

Input: library A, data matrix Y, near neighbourhood consistency constraint matrix H, and regularization parameter λ, β.Initialize: X = J = L = 0, E = 0, Y1 = 0, Y2 = 0, Y3 = 0, µ = 10–6, maxu = 1010, ρ = 1.1, ε = 10–8.while not converged do1: Fix the others and update J by $J=\mathrm{arg}\mathrm{min}\frac{1}{\mu}{\Vert J\Vert}_{\ast}+\frac{1}{2}{\Vert J-(X+{Y}_{2}/\mu )\Vert}_{F}^{2}$, $J=\mathrm{max}(J,0)$.2: Fix the others and update X by $X={(2I+{A}^{t}A)}^{-1}({A}^{t}Y-{A}^{t}E+J+L+({A}^{t}{Y}_{1}-{Y}_{2}-{Y}_{3})/\mu )$.3: Fix the others and update E by $E=\mathrm{arg}\mathrm{min}\frac{\lambda}{\mu}{\Vert E\Vert}_{2,1}+\frac{1}{2}{\Vert E-(Y-AX+{Y}_{1}/\mu )\Vert}_{F}^{2}$.4: Fix the others and update L by $L=(\mu X+{Y}_{3}){(2\beta H{H}^{T}+\mu I)}^{-1}$.5: Update the multipliers ${Y}_{1}={Y}_{1}+\mu (Y-AX-E){Y}_{2}={Y}_{2}+\mu (X-J){Y}_{3}={Y}_{3}+\mu (X-L)$. 6: Update the parameter μ by $\mu =\mathrm{min}(\rho \mu ,ma{x}_{\mu})$.7: check the convergence condition ${\Vert Y-AX-E\Vert}_{\infty}<\epsilon \text{}{\Vert X-J\Vert}_{\infty}\epsilon \text{}{\Vert X-L\Vert}_{\infty}\epsilon $. endOutput: representation coefficient X. |

#### 3.2. Spectral Library Pruning

**X**corresponding to the spectral library

**A**. Compared with the size of the spectral library, the number of endmembers in a given scene is usually much smaller. Thus, sparsity exists among the lines of the obtained abundances matrix

**X**and each line of

**X**denotes the abundance map corresponds to one endmember. To utilize the sparsity, we propose a dictionary pruning strategy to prune the abundances matrix

**X**and its corresponding spectral signatures in

**A**.

- (1)
- Compute the number of the pixels whose abundance value corresponding to one endmember is smaller than a preset threshold denoted by $\epsilon $.
- (2)
- If the number is equal to the total number of pixels in the scene, we will get rid of the spectral signature from the endmember matrix.

**A**. In the experiments on simulated data, we set $\epsilon $ to be 0.02 and then if one endmember is distributed under 2% over the scene, it will be removed from the library. The value of $\epsilon $ should be set to a small value to avoid removing the true endmembers from the library. Thus, we set $\epsilon $ to 0.01 in the experiments of real data because real data are more complex than simulated data. In each iteration, the size of spectral library will decrease and the abundances of the retained endmembers will increase. Thus, in each iteration, the value of $\epsilon $ will be updated by $\epsilon \times iterations$.

**A**as the endmember matrix in which the number of spectral signatures is smaller than the original one. By using this pruned spectral library

**A**, we solve the optimization (Equation (9)) again and obtain a new fractional abundances matrix

**Y**corresponding to the pruned spectral library

**A**. According to the dictionary pruning strategy, we can prune the spectral library

**A**repeatedly until the number of spectral signatures retained meets the stopping condition.

## 4. Experiments with Simulated Data and Real Data

#### 4.1. Simulated Datasets

- (1)
- Simulated Data Cube 1 (DC1): This simulated data cube is generated following the methodology of [26], using five randomly selected spectral signatures from A1. DC1 has 75 × 75 pixels and each simulated pixel was generated using a LMM, with the five endmembers and imposing the ASC in it. In the resulting simulated image, shown in Figure 3a, there are pure regions as well as mixed regions constructed using mixtures ranging between two and five endmembers, distributed spatially in the form of distinct square regions. Figure 3b–f, respectively, shows the true fractional abundances for each of the five endmembers. The background pixels are made up of mixtures of the same five endmembers, but their respective fractional abundances values were randomly fixed as 0.1149, 0.0741, 0.2003, 0.2055 and 0.4051, respectively. The obtained data cube was then contaminated with white noise, having different levels of the signal-to-noise ratio (SNR): 20, 30 and 40 dB.
- (2)
- Simulated Data Cube 2 (DC2): Using the library A1, we generated a data cube of 48 × 48 pixels and it contains six endmembers. The endmembers were randomly selected from library A1. In each simulated pixel, the fractional abundances of the endmembers follow a Dirichlet distribution [14]. As DC1, the scene was again contaminated with white noise using the same SNR value adopted for DC1.
- (3)
- Simulated Data Cube 3 (DC3): Using the library
**A1**, we generate various data cubes of 75 × 75 pixels, each containing five endmembers. The simulated method is similar to the first simulated data. There are pure regions as well as mixed regions constructed using mixtures ranging between two and five endmembers, distributed spatially in the form of distinct square regions. The background pixels are made up of mixtures of the same five endmembers, but their respective fractional abundances values were randomly fixed as 0.1149, 0.0741, 0.2003, 0.2055 and 0.4051, respectively. The obtained data cube was then contaminated with white noise

**X**is the true fractional abundance vector and $\text{}$ is the estimated fractional abundance vector. $E(\cdot )$ stands for mean value. The higher the SRE is, the better the quality of the unmixing is. As it can give more information regarding the power of the signal in relation with the power of the error, we use this measure instead of the classical root mean squared error (RMSE) [34]. We also compute the abundance angle distance (AAD) as another performance discriminator adopted in this paper. AAD can express as follows:

**X**

_{i}is the true fractional abundance vector of the i-th endmember and ${\widehat{x}}_{i}$ is the estimated fractional abundance vector of the i-th endmember. m stands for the number of the endmembers.

**A1**library. The datasets contain six true endmembers and the positions of the true endmembers in the original library are 5, 6, 26, 34, 67 and 177, as shown in Figure 5a. In Figure 5, it can be seen graphically that the lines (denoting the abundance of a certain endmember in all pixels of the image) estimated by CLSUnSAL are more sparse than the ones estimated by SUnSAL. After applying SUnSAL, as shown in Figure 5b, there are many low abundance values estimated for endmembers which are not actually present in the image. Due to the high mutual coherence of the library signatures, SUnSAL does not perform well. Unlike SUnSAL, which employs pixelwise independent regression, CLSUnSAL enforces joint sparsity among all the pixels. Thus, it improves significantly the accuracy of the unmixing solutions over those provided by SUnSAL. Although the abundances obtain by CLSUnSAL are similar to those in the ground-truth, there are also several lines that are not true abundances in Figure 5c. Figure 5d shows the estimated abundances obtained by the proposed algorithm. After applying this method, the remained subset of the spectral library contains six endmembers. However, almost every line of the image is much brighter than the one in the true abundance map. This is because that the image in DC2 contains very few homogeneous regions, which leads to the failure of spatial consistency constraints. However, it improves the accuracy in the whole, which is shown in Table 3.

**A1**as the endmembers matrix, CLSUnSAL can obtain the highest SRE (dB) values. This is because CLSUnSAL constrains the pixels to share the same set of endmembers. AAD is defined as the angle between estimated abundances and the true abundances. The AAD values achieved by LRR are lower than those of the other methods when the observed data have a lower SNR. This is because LRR can capture the global structures of the abundance matrix. Some other methods exploit pruning methods in unmixing with given spectral library. Zortea and Plaza [34] use the array processing methodologies to prune the spectral library effectively while the greedy algorithms proposed in [43,44] can also be taken as dictionary pruning methods to boost sparse unmixing algorithms. Thus, we compare the results obtained by our proposed dictionary pruning method to those obtained with subspace matching pursuit (SMP) algorithm [43] and regularized simultaneous forward-backward greedy algorithm (RSFoBa) [44]. The numbers of potential endmembers retained from the original library by the SMP method are 148 (SNR = 20 dB), 167 (SNR = 30 dB) and 162 (SNR = 40 dB), respectively. The numbers of endmembers retained by the RSFoBa method are 100 (SNR = 20 dB), 64 (SNR = 30 dB) and 42 (SNR = 40 dB), respectively. In Table 3, we can conclude that both SMP and RSFoBa can improve the unmixing algorithm performance compared with those algorithms using the original large spectral library for unmixing. Since the size of sub-library obtained by the RSFoBa is smaller than that by SMP and both the sub-libraries contain true endmembers, RSFoBa makes the unmixing algorithms more accurate than SMP, as shown in Table 3. In our proposed algorithm, threshold T acting as the stop condition controls the size of retained subset of the spectral library. We set it to be 1 to 10 in the experiments so that the retained endmembers will not be 10 more than the estimated number of endmembers in a scene. The parameters of the DC2 are shown in Table 4. Especially, when the observed data have high SNR levels, our proposed dictionary pruning method can select the true endmembers accurately from the original large spectral library. Thus, we can also find the potential of our proposed dictionary pruning method can select the true endmembers accurately from the original large spectral library. Besides, we can also find the potential of our proposed dictionary pruning method which can boost the performances of the other unmixing algorithms.

#### 4.2. Real Datasets

**A**, is the USGS library containing 498 pure endmember signatures. Before unmixing the real hyperspectral data, essential calibration was undertaken to mitigate the mismatches between the hyperspectral image and the signatures in the library [17].

^{−3}for SUnSAL, λ = 10 for LRR and λ = 5 × 10

^{−3}for CLSUnSAL. For the proposed algorithm, we set threshold ε to be 0.01, λ = 10 and T = 25. In addition, the ASC constraint is not applied in our models in this section.

**A**. Because each endmember has several variations, the spectral library

**A**which contains 498 members, includes about 229 kinds of materials. Thus, the mutual coherence of the library is very close to one. In Figure 9, it can be observed that the fractional abundances estimated by CLSUnSAL are generally higher in the regions assigned to the respective materials than SUnSAL. Due to the high correlation of the spectral library, the traditional unmixing algorithm SUnSAL may not obtain good performance. CLSUnSAL constrains the pixels to share the same set of endmembers. Thus, it obtains better results than SUnSAL. It is also worth noting that the abundance maps inferred by LRR do not exhibit good spatial consistency of minerals of interest compared with our proposed method. This is because the proposed method can better capture the global structure of the abundance matrix by using LRR model based on space consistency constraint as well as identify a subset of the spectral library by the spectral dictionary pruning, which only contains 42 pure endmembers. Compared with the size of the original spectral library, the number of endmembers of the subset is much smaller. Thus, using the subset as the endmember matrix improves the accuracy of the unmixing solutions significantly. Overall, the qualitative results reported in this section indicate that our proposed method which combines LRR model based on space consistency constraint and the dictionary pruning strategy, can obtain more accurate results than other methods. Further experiments with additional hyperspectral scenes and quantitative comparisons should be conducted in future work to fully objectify our findings.

#### 4.3. Discussion of the Parameters Setting and Time Complexity

**J**,

**X**, and

**E**can be achieved, and the unmixing results are very close to the true abundances.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 3.**True fractional abundances of endmembers in the simulated data cube 1 (DC1): (

**a**) simulated image; (

**b**) the true abundance of endmember 1; (

**c**) the true abundance of endmember 2; (

**d**) the true abundance of endmember 3; (

**e**) the true abundance of endmember 4; and (

**f**) the true abundance of endmember 5.

**Figure 4.**Abundance maps obtained by different unmixing methods for endmember #5 in DC1 and, from top to bottom, SNR is 20 dB, 30 dB, and 40 dB.

**Figure 5.**Ground-truth and estimated abundances obtained by different unmixing methods in the scene DC2, with SNR = 40 dB: (

**a**) ground-truth abundance; (

**b**) estimated abundances obtained by SUnSAL; (

**c**) estimated abundances obtained by CLSUnSAL; and (

**d**) estimated abundances obtained by SLP-SCC-LRR.

**Figure 6.**True fractional abundances of endmembers in the simulated data cube 3 (DC3): (

**a**) simulated image; (

**b**) the true abundance of endmember 1; (

**c**) the true abundance of endmember 2; (

**d**) the true abundance of endmember 3; (

**e**) the true abundance of endmember 4; and (

**f**) the true abundance of endmember 5.

**Figure 7.**Abundance maps obtained by the proposed unmixing method in DC3 and the SNR is 40 dB: (

**a**) the true abundance of endmember 1; (

**b**) the true abundance of endmember 2; (

**c**) the true abundance of endmember 3; (

**d**) the true abundance of endmember 4; and (

**e**) the true abundance of endmember 5.

**Figure 8.**USGS map showing the location of different minerals in the Cuprite mining district in Nevada. The map is available online at http://speclab.cr.usgs.gov/cuprite95.tgif.2.2um_map.gif.

**Figure 9.**Abundance maps estimated for the minerals alunite, buddingtonite, and chalcedony by applying the SUnSAL, CLSUnSAL, SCC-LRR, and SLP-SCC-LRR algorithms to the AVIRIS Cuprite scene using the library

**A**.

**Table 1.**SRE (dB) and AAD values achieved after applying different unmixing methods to Simulated DATA 1.

Methods | DC1 | SNR = 20 dB | SNR = 30 dB | SNR = 40 dB |
---|---|---|---|---|

NCLS | SRE | 1.2648 | 2.5000 | 7.5332 |

AAD | 0.8512 | 0.5657 | 0.2211 | |

SUnSAL | SRE | 1.5753 | 3.2432 | 8.2820 |

AAD | 0.8276 | 0.6414 | 0.1853 | |

SUnSAL-TV | SRE | 5.5956 | 15.0211 | 23.6639 |

AAD | 0.4530 | 0.0486 | 0.0206 | |

LRR | SRE | 1.6232 | 3.5426 | 6.7140 |

AAD | 0.4777 | 0.2779 | 0.1401 | |

SCC-LRR | SRE | 2.8598 | 5.1026 | 4.8676 |

AAD | 0.1838 | 0.0753 | 0.1823 | |

SLP-SCC-LRR | SRE | 21.8418 | 32.7520 | 44.5256 |

AAD | 0.0318 | 0.0221 | 0.0074 |

Methods | DC1 | SNR = 20 dB | SNR = 30 dB | SNR = 40 dB |
---|---|---|---|---|

NCLS | Times | 41.8314 | 28.7967 | 23.2396 |

SUnSAL | Parameters | λ = 8 × 10^{−2} | λ = 8 × 10^{−2} | λ = 1 × 10^{−2} |

Times | 21.2719 | 22.4660 | 24.2318 | |

SUnSAL-TV | Parameters | λ = 5 × 10^{−2} λ _{TV} = 5 × 10^{−2} | λ = 5 × 10^{−3} λ _{TV} = 1 × 10^{−2} | λ = 3 × 10^{−3} λ _{TV} = 5 × 10^{−3} |

Times | 523.0716 | 527.0992 | 611.4388 | |

LRR | Parameters | λ = 0.5 | λ = 4 | λ = 70 |

Times | 861.1286 | 936.9994 | 784.9693 | |

SCC-LRR | Parameters | λ = 6 β = 100 | λ =15 β = 110 | λ = 6 β = 100 |

Times | 232.0626 | 246.2892 | 264.2972 | |

SLP-SCC-LRR | Parameters | λ = 6 β = 100 T = 3 | λ = 15 β = 110 T = 2 | λ = 6 β = 100 T = 1 |

Times | 400.3627 | 374.6602 | 412.6240 |

**Table 3.**SRE (dB) and AAD values achieved after applying different unmixing methods to Simulated DATA 2.

Methods (DC2) | SNR = 20 dB | SNR = 30 dB | SNR = 40 dB | |||
---|---|---|---|---|---|---|

SRE | AAD | SRE | AAD | SRE | AAD | |

SUnSAL | 1.7227 | 0.9619 | 3.1024 | 0.7060 | 6.0602 | 0.3963 |

CLSUnSAL | 2.1666 | 0.7251 | 4.0155 | 0.4013 | 9.4707 | 0.0958 |

LRR | 1.7299 | 0.5822 | 3.0060 | 0.3143 | 4.0263 | 0.1781 |

SCC-LRR | 0.6004 | 0.4881 | 1.3476 | 0.2931 | 1.7065 | 0.2102 |

SMP + SUnSAL | 2.1126 | 0.8923 | 3.4765 | 0.6654 | 6.9170 | 0.3579 |

SMP + CLSUnSAL | 2.9256 | 0.8083 | 4.3575 | 0.2092 | 7.6023 | 0.2367 |

SMP + LRR | 1.8926 | 0.5424 | 3.0206 | 0.3123 | 4.7494 | 0.1866 |

SMP + SCC-LRR | 0.6582 | 0.4881 | 1.4118 | 0.2931 | 1.7759 | 0.2102 |

RSFoBa + SUnSAL | 2.3032 | 0.8520 | 5.4326 | 0.4982 | 11.5092 | 0.2134 |

RSFoBa + CLSUnSAL | 2.8469 | 0.6162 | 9.6058 | 0.1628 | 20.3783 | 0.0521 |

RSFoBa + LRR | 2.5989 | 0.4672 | 5.4969 | 0.3503 | 9.8677 | 0.1443 |

RSFoBa + SCC-LRR | 0.7044 | 0.4881 | 1.6437 | 0.2931 | 1.9177 | 0.2102 |

SLP + SUnSAL | 4.9159 | 0.5632 | 18.9343 | 0.1100 | 28.5571 | 0.0363 |

SLP + CLSUnSAL | 3.6273 | 0.3010 | 19.0567 | 0.1090 | 28.4343 | 0.0367 |

SLP-LRR | 5.3920 | 0.3386 | 19.0560 | 0.1090 | 28.5731 | 0.0363 |

SLP-SCC-LRR | 4.4974 | 0.4054 | 19.0002 | 0.1097 | 28.4715 | 0.0366 |

Methods (DC2) | SNR = 20 dB | SNR = 30 dB | SNR = 40 dB | |||
---|---|---|---|---|---|---|

Parameters | Times | Parameters | Times | Parameters | Times | |

SUnSAL | λ = 0.05 | 16.9724 | λ = 0.01 | 11.6911 | λ = 1 × 10^{−3} | 8.7732 |

CLSUnSAL | λ = 0.9 | 40.8787 | λ = 0.2 | 41.5278 | λ = 0.05 | 41.3956 |

LRR | λ = 2 | 206.4049 | λ = 20 | 182.7591 | λ = 50 | 164.0270 |

SCC-LRR | λ = 2 β = 0.01 | 122.5415 | λ = 36 β = 0.1 | 109.7295 | λ = 6 β = 0.01 | 117.2215 |

SMP + SUnSAL | λ = 0.08 | 6.4694 | λ = 0.01 | 6.1948 | λ = 9 × 10^{−4} | 4.6662 |

SMP + CLSUnSAL | λ = 2 | 11.8081 | λ = 0.2 | 12.5174 | λ = 8 × 10^{−3} | 12.5302 |

SMP + LRR | λ = 2 | 64.0299 | λ = 10 | 73.1744 | λ = 80 | 79.6736 |

SMP + SCC-LRR | λ =2 β = 0.01 | 114.3858 | λ =36 β = 0.1 | 111.7261 | λ = 6 β = 0.01 | 123.7080 |

RSFoBa + SUnSAL | λ = 0.03 | 3.1553 | λ = 4 × 10^{−3} | 1.9873 | λ = 6 × 10^{−4} | 0.8679 |

RSFoBa + CLSUnSAL | λ = 0.2 | 19.9172 | λ = 0.2 | 9.6373 | λ = 0.1 | 6.0128 |

RSFoBa + LRR | λ = 2 | 54.3540 | λ = 35 | 35.4342 | λ = 80 | 21.1428 |

RSFoBa + SCC-LRR | λ = 2 β = 0.01 | 118.9212 | λ = 36 β = 0.1 | 118.7259 | λ = 6 β = 0.01 | 128.7818 |

SLP + SUnSAL | λ = 0.05 T = 10 | 76.3745 | λ = 6 × 10^{−3} T = 1 | 45.0329 | λ = 1 × 10^{−3} T = 1 | 19.0163 |

SLP + CLSUnSAL | λ = 2 T = 10 | 25.0145 | λ = 0.1 T = 1 | 33.0251 | λ = 0.04 T = 1 | 23.7061 |

SLP-LRR | λ = 2 T = 10 | 689.9156 | λ = 8 T = 1 | 471.5313 | λ = 80 T = 1 | 324.2536 |

SLP-SCC-LRR | λ = 2 β = 0.01 T = 10 | 290.9236 | λ = 36 β = 0.1 T = 1 | 489.6472 | λ = 6 β = 0.01 T = 1 | 323.6368 |

**Table 5.**SRE (dB) and AAD values to Simulated Dataset 3 with the parameter settings and running times.

Methods | DC3 | SNR = 40 dB | Parameters | Times |
---|---|---|---|---|

LRR | SRE | 0.2997 | λ = 1 × 10^{−1} | 486.9841 |

AAD | 1.1210 | |||

SUnSAL | SRE | 0.2419 | λ = 1 × 10^{−2} | 11.4843 |

AAD | 1.2584 | |||

SUnSAL-TV | SRE | 0.0971 | λ = 1 λ _{TV} = 5 × 10^{−2} | 228.4922 |

AAD | 0.9397 | |||

SCC-LRR | SRE | 0.4061 | λ = 10 β = 50 | 232.9945 |

AAD | 0.8596 | |||

SLP-SCC-LRR | SRE | 22.3349 | λ = 1 β = 111 T = 1 | 840.1768 |

AAD | 0.0903 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhang, X.; Li, C.; Zhang, J.; Chen, Q.; Feng, J.; Jiao, L.; Zhou, H.
Hyperspectral Unmixing via Low-Rank Representation with Space Consistency Constraint and Spectral Library Pruning. *Remote Sens.* **2018**, *10*, 339.
https://doi.org/10.3390/rs10020339

**AMA Style**

Zhang X, Li C, Zhang J, Chen Q, Feng J, Jiao L, Zhou H.
Hyperspectral Unmixing via Low-Rank Representation with Space Consistency Constraint and Spectral Library Pruning. *Remote Sensing*. 2018; 10(2):339.
https://doi.org/10.3390/rs10020339

**Chicago/Turabian Style**

Zhang, Xiangrong, Chen Li, Jingyan Zhang, Qimeng Chen, Jie Feng, Licheng Jiao, and Huiyu Zhou.
2018. "Hyperspectral Unmixing via Low-Rank Representation with Space Consistency Constraint and Spectral Library Pruning" *Remote Sensing* 10, no. 2: 339.
https://doi.org/10.3390/rs10020339