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使用简单线性迭代聚类[17](simple linear iterative clustering, SLIC)算法来分割高光谱图像。SLIC计算量小且只有要划分的超像素个数这一个参量需要确定。假设高光谱图像中共有n个像元,要划分的超像素的个数为M,则每个超像素包含的像元个数大约为$\frac{n}{\mathit{\boldsymbol{M}}} $聚类中心之间的初始距离为${s_0} = \sqrt {\frac{n}{\mathit{\boldsymbol{M}}}} $,每个聚类中心的搜索范围为2s0×2s0。值得注意的是需要对SLIC算法进行扩展以适应高光谱问题。用像元之间的光谱距离代替原有的RGB距离。对于像元i和j,其距离计算公式为:
$ {s_1}{\rm{ = }}\parallel {\mathit{\boldsymbol{y}}_i}-{\mathit{\boldsymbol{y}}_j}{\parallel _2} $
(1) $ {s_2} = \sqrt {{{({a_i} - {a_j})}^2} + {{({b_i} - {b_j})}^2}} $
(2) $ s = \sqrt {\alpha {s_1}^2 + {s_2}^2} $
(3) 式中,yi和yj代表像元i和j的光谱向量,s1表示光谱距离;(ai, bi)和(aj, bj)分别代表像元i和j的空间位置,s2表示空间距离;α为空间距离与光谱距离之间的权重,s表示像元i和j的综合距离。一般而言,α的值为常数,这里设定为0.5。首先选定M个初始聚类中心,在每个聚类中心的搜索区域内计算各个像元到聚类中心的距离。把像元归为距离聚类中心最近的那一类,每一轮类别划分结束之后计算各个类别的均值作为新的聚类中心,若新聚类中心与原来的相同则聚类结束,否则重新聚类。关于SLIC算法更详细的介绍可以参见参考文献[17]。
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对高光谱图像进行超像素分割之后,对每个超像素施加协同约束,并添加低秩性约束来利用空间信息, 其解混模型如下:
$ \begin{array}{c} \mathop {\min }\limits_X \left[ {\frac{1}{2}{\rm{ }}\parallel {\rm{ }}\mathit{\boldsymbol{Y}}{\rm{ }} - {\rm{ }}\mathit{\boldsymbol{AX}}{\rm{ }}\parallel _{_F}^{^2} + } \right.\\ \lambda \sum\limits_{j = 1}^M {{\rm{ }}\parallel {\rm{ }}{\mathit{\boldsymbol{X}}_j}} \left. {{\parallel _{2,1}} + \tau {\rm{rank}}\left( {{\rm{ }}\mathit{\boldsymbol{X}}{\rm{ }}} \right)} \right]{\rm{ }},\\ \left( {{\rm{ }}\mathit{\boldsymbol{X}}{\rm{ }} \ge 0} \right) \end{array} $
(4) 式中,$ \parallel .{\parallel _{}}_F$代表F范数,Y∈RL×n为观测所得的混合像元矩阵,L为波段数目,n为像元数目,RL×n为L×n矩阵实数集;A∈RL×m为预先获得的包含m个原子的光谱库;X∈Rm×n为对应的丰度矩阵;$\parallel {\mathit{\boldsymbol{X}}_j}{\parallel _{2, 1}} = \sum\limits_{k = 1}^m {} \parallel {\mathit{\boldsymbol{X}}_{j, k}}{\parallel _2} $,其中Xj, k为第j个超像素的第k行;rank(X)为丰度矩阵X的秩,λ和τ为正则项参量。参考文献[18]~参考文献[20]中,在求解(4)式之前用核范数$ \parallel \mathit{\boldsymbol{X}}{\parallel _*} = \sum\limits_{i = 1}^r {{\mathit{\sigma }_i}\left( \mathit{\boldsymbol{X}} \right)} $代替低秩正则项rank(X)。其中σi(X)为X的第i个奇异值,r=rank(X)。另外把非负性约束添加到目标函数中,则(4)式等价为:
$ \begin{array}{c} \mathop {{\rm{min}}}\limits_X \left[ {} \right.\frac{1}{2}\parallel \mathit{\boldsymbol{Y}}{\rm{ }} - {\rm{ }}\mathit{\boldsymbol{AX}}{\rm{ }}\parallel _F^{^2} + \lambda \sum\limits_{j = 1}^M {\parallel {\mathit{\boldsymbol{X}}_j}{\parallel _{2,1}} + } \\ \tau \parallel \mathit{\boldsymbol{X}}{\parallel _*} + {l_{{\bf{R}}{\rm{ }} + }}\left( {{\rm{ }}\mathit{\boldsymbol{X}}{\rm{ }}} \right)\left. {} \right] \end{array} $
(5) 式中,lR+(X)为正向量空间R+的指示函数,当X∈R+时, lR+(X)=0, 否则lR+(X)=+∞。直接求解(5)式是非常困难的,为此采用分离变量增广拉格朗日算法进行求解。令V1=AX,V2=X,V3=X,V4=X,则(5)式可等效为:
$ \begin{array}{c} {\rm{ }}\mathop {{\rm{min}}}\limits_{X,{\rm{ }}{V_1},{\rm{ }}{V_2},{\rm{ }}{V_3},{\rm{ }}{V_4}} \left[ {\frac{1}{2}\parallel {\mathit{\boldsymbol{V}}_1} - \mathit{\boldsymbol{Y}}\parallel _F^{^2} + } \right.\\ \lambda \sum\limits_{{\rm{ }}j = 1}^M {} \parallel {\mathit{\boldsymbol{V}}_{2,j}}\left. {{\parallel _{2,1}} + \mathit{\tau }\parallel {\mathit{\boldsymbol{V}}_3}{\parallel _*} + {l_{{\bf{R}}{\rm{ }} + }}\left( {{\mathit{\boldsymbol{V}}_4}} \right)} \right]{\rm{ }},\\ ({\mathit{\boldsymbol{V}}_1} = \mathit{\boldsymbol{AX}},{\mathit{\boldsymbol{V}}_2} = \mathit{\boldsymbol{X}},{\rm{ }}{\mathit{\boldsymbol{V}}_3} = \mathit{\boldsymbol{X}},{\rm{ }}{\mathit{\boldsymbol{V}}_4} = \mathit{\boldsymbol{X}}) \end{array} $
(6) 令V≡(V1, V2, V3, V4),B=diag(-I),G=[A, I, I, I]T,其中I为n×n的单位矩阵, 则可构建如下拉格朗日函数:
$ \begin{array}{l} L\left( {\mathit{\boldsymbol{X}},\mathit{\boldsymbol{V}},\mathit{\boldsymbol{D}}{\rm{ }}} \right) = {\rm{ }}\frac{1}{2}\parallel {\rm{ }}\mathit{\boldsymbol{V}}{_1} - {\rm{ }}\mathit{\boldsymbol{Y}}{\rm{ }}\parallel _{_F}^2 + \lambda \sum\limits_{j = 1}^M {\parallel {\rm{ }}{\mathit{\boldsymbol{V}}_{2,j}}{\parallel _{2,1}} + } \\ \;\;\;\;\;\tau \parallel {\rm{ }}\mathit{\boldsymbol{V}}{_3}{\parallel _*} + {l_{{\bf{R}}{\rm{ }} + }}({\rm{ }}\mathit{\boldsymbol{V}}{_4}) + \frac{\mu }{2}{\rm{ }}\parallel {\rm{ }}\mathit{\boldsymbol{GX}}{\rm{ }} + {\rm{ }}\mathit{\boldsymbol{BV}}{\rm{ }} - {\rm{ }}\mathit{\boldsymbol{D}}{\rm{ }}\parallel _{_F}^{^2} \end{array} $
(7) 式中,μ>0是一个正常数,$\frac{\mathit{\boldsymbol{D}}}{\mu }$表示关于约束条件GX+BV=0的拉格朗日乘子。然后采用交替方向乘子法[21]进行优化求解。在求解V2时,首先将V2按照高光谱图像进行超像素分割,然后逐个求解每个子问题。在求解V3时,先定义奇异值分解(singular value decomposition, SVD)。假设有矩阵Q,则Q的奇异值分解为Q=φdiag(σ1, …, σr)ψ,其中r=rank(Q),φ和ψ为矩阵Q奇异值分解对应的酉矩阵,σi表示矩阵Q的第i个奇异值。定义奇异值阈值(singular value threshold, SVT)算子为:
$ S\left( {\mathit{\boldsymbol{Q}}, \varepsilon } \right) = \mathit{\boldsymbol{\varphi }}{\rm{soft}}({\rm{diag}}\left( {{\sigma _1}, \ldots , {\sigma _r}} \right), \varepsilon )\mathit{\boldsymbol{\psi }} $
(8) 式中,soft表示软阈值函数$ {\rm{soft}}(y,\tau ) = {\rm{sign}}(y)\cdot max\{ \left| y \right|{\rm{ }}\tau ,0\} $则求解(7)式的算法流程及各个变量的迭代更新公式见表 1。
Table 1. Algorithm flow of alternating direction multiplier method to solve equation (7)
1. Initialize V(0), D(0). Select parameters λ, τ, μ and set k=0. 2. Repeat: $ {\mathit{\boldsymbol{V}}_{2, j, i(k + 1)}} \leftarrow {\rm{ vect}} - {\rm{soft}}\left( {{\mathit{\boldsymbol{X}}_{j, i(k + 1)}} - {\mathit{\boldsymbol{D}}_{2, j, i(k)}}, \frac{\lambda }{\mu }} \right)$ $ {\mathit{\boldsymbol{V}}_{3(k + 1)}} \leftarrow {\rm{ }}S\left( {{\mathit{\boldsymbol{X}}_{(k + 1)}} - {\mathit{\boldsymbol{D}}_{3(k)}}, \frac{\tau }{\mu }} \right)$ $ {\mathit{\boldsymbol{V}}_{4(k + 1)}} \leftarrow {\rm{max}}({\mathit{\boldsymbol{X}}_{(k + 1)}} - {\mathit{\boldsymbol{D}}_{4(k)}}, 0)$ $ {\mathit{\boldsymbol{D}}_{1(k + 1)}} \leftarrow {\rm{ }}{\mathit{\boldsymbol{D}}_{1(k)}} - \mathit{\boldsymbol{A}}{\mathit{\boldsymbol{X}}_{(k + 1)}} + {\mathit{\boldsymbol{V}}_{1(k + 1)}}$ $ {\mathit{\boldsymbol{D}}_{j(k + 1)}} \leftarrow {\rm{ }}{\mathit{\boldsymbol{D}}_{j(k)}} - {\mathit{\boldsymbol{X}}_{(k + 1)}} + {\mathit{\boldsymbol{V}}_{j(k + 1)}}, {\rm{ }}(j = 2, 3, 4)$ 3. Until the stopping criterion is met. 表中,vect-soft为著名的向量软阈值函数[8],V2, j, i为V2的第j个超像素的第i行。令$ {\mathit{\boldsymbol{\zeta }}_{j, i}} = {\mathit{\boldsymbol{X}}_{j, i(k + 1)}} - {\mathit{\boldsymbol{D}}_{2, j, i(k)}}\beta = \frac{\lambda }{\mu }$则V2, j, i的迭代更新公式为:
$ {\mathit{\boldsymbol{V}}_{2, j, i(k + 1)}} \leftarrow \frac{{{\rm{max}}(\parallel {\mathit{\boldsymbol{\zeta }}_{j, i}}{\parallel _2} - \beta , 0)}}{{{\rm{max}}\left( {\parallel {\mathit{\boldsymbol{\zeta }}_{j, i}}{\parallel _2} - \beta , 0} \right) + \beta }} \times {\zeta _{j, i}} $
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为验证所提算法的有效性和可靠性,采用一组模拟数据和一组真实数据进行实验, 并与CLSUnSAL算法、SUnSAL-TV算法和S2WSU算法进行对比。在这两组实验中均使用USGS光谱库,共包含224个波段、498种物质,光谱范围为0.4μm~2.5μm。在模拟数据实验中使用信号重构误差(signal-to-reconstruction error, SRE)定量评价解混性能。在真实数据实验中通过比较解混所得到的丰度图定性评价解混性能。令xi表示真实的丰度向量,$ {\mathit{\boldsymbol{\hat x}}_i}$为xi的估计值, E(·)表示期望函数, 则信号重构误差可定义为:
$ {R_\rm {SRE}}({\rm{dB}}) = 10{\rm{lg}}[\frac{{E(\parallel {\mathit{\boldsymbol{x}}_i}\parallel _2^2)}}{{E(\parallel {\mathit{\boldsymbol{x}}_{^i}} - {\mathit{\boldsymbol{\hat x}}_{^i}}\parallel _2^2)}}] $
(10) RSRE的值越大解混精度越高。所有算法均在配备有Intel core 5处理器、2.3GHz主频率、12GB内存的笔记本电脑上通过MATLAB R2018a运行。
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所构建的模拟数据为包含75×75个像元的数据立方体。为了简化计算对光谱库进行裁剪,使光谱库中任意两物质的夹角不小于4.44°。最终得到包含240个原子的光谱库。从光谱库中随机选取5个原子作为本次实验的端元。丰度满足非负性约束以及和为1约束,图 1为各个端元的真实丰度图。图中方块区域可能是纯净的或者是由几个端元混合而成。背景区域由5个端元混合而成,所占的比例分别为0.1149,0.0741,0.2003,0.2055和0.4051。为了更加符合真实地物的分布情况以及检测各个算法的抗噪声能力,在模拟数据中分别添加信噪比(signal-to-noise ratio, SNR)为20dB,30dB和40dB的高斯白噪声。各个算法的参量均调整到合适大小以得到每个算法的最佳性能。表 2为各个算法所得到的RSRE值、所需要的时间以及对应的参量值。其中λTV为SUnSAL-TV算法对应的全变差正则项参量。图 2中为信噪比30dB时各算法的解混丰度图的对比。
Table 2. RSRE, time and parameter values of each algorithm for simulated data
SNR/
dBcriteria CLSUnSAL SUnSAL-TV S2WSU SLRCSU 20 RSRE /dB 8.5435 9.4239 7.6971 13.4488 time/s 4.2877 122.6244 30.0975 58.7048 parameter λ=1.6 λ=0.05,
λTV=0.05λ=0.1 M=225,
λ=0.05,
τ=0.00530 RSRE /dB 11.1263 14.4408 15.4868 19.4420 time/s 5.7490 118.8624 29.4267 57.1883 parameter λ=0.5 λ=0.007,
λTV=0.01λ=0.005 M=225,
λ=0.01,
τ=0.00140 RSRE /dB 13.6605 22.0968 28.2348 30.2934 time/s 7.5112 121.7894 30.1787 58.5224 parameter λ=0.1 λ=0.001,
λTV=0.003λ=0.001 M=225,
λ=0.007,
τ=0.001从表 2中能够看出,在所有信噪比下SLRCSU都能够获得最好的解混性能。相比经典的SUnSAL-TV算法,其解混精度提高了40%左右,并且SLRCSU所用的时间仅为SUnSAL-TV的一半。从图 2中能够看出,SUnSAL-TV和SLRCSU都能够获得清晰纯净的背景,但在端元2的解混丰度图中SLRCSU所得方块数量明显多于SUnSAL-TV。因此,模拟数据实验能够证明所提算法的有效性。
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真实数据使用机载可见/红外成像光谱仪于1997年收集的Cuprite数据集。它被广泛应用于解混算法对比。为了简化计算,使用大小为250×191的数据子集。该子集共包含224个波段,去除低信噪比以及水吸收较高的波段(1~2, 105~115, 150~170, 223~224)后仅保留188个波段。图 3为Cuprite数据集各物质的分布图。它被制作于1995年,因此只能用来定性评价解混性能。各个算法的参量均调整到合适大小。表 3为各个算法的参量值及所用时间。图 4为从数据集中选取的2种典型代表物质(buddingtonite,chalcedony)的解混丰度图。
Table 3. Parameters and times corresponding to each algorithm for real data
CLSUnSAL SUnSAL-TV S2WSU SLRCSU time/s 195.7 1219.7 485.6 624.8 parameter λ=0.01 λ=0.001,
λTV=0.001λ=7×10-5 M=1910,
λ=0.01, τ=0.005从图 4中能够看出,SUnSAL-TV算法所得到的解混丰度图存在明显的边缘模糊现象,并且其背景存在较多杂质。其它算法的背景均较为清晰纯净。从矿物Chalcedony的解混丰度图中能够看出,SLRCSU算法的解混结果在相对应的区域丰度更高,更符合真实地物的分布情况。因此,真实数据实验同样能证明所提算法的有效性。
基于超像素和低秩的协同稀疏高光谱解混
Superpixels and low rank for collaborative sparse hyperspectral unmixing
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摘要: 为了克服经典协同稀疏解混算法的不足以及全变差正则项引起的边缘模糊问题,同时考虑到稀疏性和空间信息对解混精度提高的重要性,采用结合超像素和低秩的协同稀疏高光谱解混算法,进行了理论分析和实验验证。该算法对高光谱图像进行超像素分割,并对每个超像素施加协同稀疏性约束。此外使用低秩正则项代替传统的全变差正则项来利用空间信息,选取一组模拟数据和一组真实数据进行了实验。结果表明,模拟实验中信噪比为30dB时得到的信号重构误差为19.4,比经典的变量分裂增广拉格朗日全变差算法提高了35%左右;真实数据实验直观地反映出了该算法能有效地克服边缘模糊问题,具有更好的解混性能。该研究为如何综合利用稀疏性和空间信息提供了参考。Abstract: To overcome the shortcomings of the classic collaborative sparse unmixing algorithm and the edge blur problem caused by the total variation regular term, considering the importance of sparsity and spatial information to improve the accuracy of unmixing, a novel algorithm called superpixel and low rank for collaborative sparse unmixing was proposed. The unmixing algorithm was theoretically analyzed and experimentally verified. The superpixel segmentation was performed on the hyperspectral images, and then collaborative sparsity constraints were imposed on each superpixel. In addition, a low-rank regular term was used instead of the traditional total variation regular term to utilize spatial information. A set of simulated data and a set of real data were selected for experiments. These results show that the signal reconstruction error obtained in the simulated experiment is 19.4 when the signal-to-noise ratio is 30dB, which is about 35% higher than that of the classic sparse unmixing via variable splitting augmented Lagrangian and total variation algorithm. Real data experiment intuitively reflects that the algorithm can effectively overcome the problem of edge blur. The proposed algorithm has better unmixing performance. This research provides a reference for how to use sparsity and spatial information comprehensively.
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Key words:
- spectroscopy /
- hyperspectral image /
- sparse unmixing /
- superpixel /
- low rank
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Table 1. Algorithm flow of alternating direction multiplier method to solve equation (7)
1. Initialize V(0), D(0). Select parameters λ, τ, μ and set k=0. 2. Repeat: $ {\mathit{\boldsymbol{V}}_{2, j, i(k + 1)}} \leftarrow {\rm{ vect}} - {\rm{soft}}\left( {{\mathit{\boldsymbol{X}}_{j, i(k + 1)}} - {\mathit{\boldsymbol{D}}_{2, j, i(k)}}, \frac{\lambda }{\mu }} \right)$ $ {\mathit{\boldsymbol{V}}_{3(k + 1)}} \leftarrow {\rm{ }}S\left( {{\mathit{\boldsymbol{X}}_{(k + 1)}} - {\mathit{\boldsymbol{D}}_{3(k)}}, \frac{\tau }{\mu }} \right)$ $ {\mathit{\boldsymbol{V}}_{4(k + 1)}} \leftarrow {\rm{max}}({\mathit{\boldsymbol{X}}_{(k + 1)}} - {\mathit{\boldsymbol{D}}_{4(k)}}, 0)$ $ {\mathit{\boldsymbol{D}}_{1(k + 1)}} \leftarrow {\rm{ }}{\mathit{\boldsymbol{D}}_{1(k)}} - \mathit{\boldsymbol{A}}{\mathit{\boldsymbol{X}}_{(k + 1)}} + {\mathit{\boldsymbol{V}}_{1(k + 1)}}$ $ {\mathit{\boldsymbol{D}}_{j(k + 1)}} \leftarrow {\rm{ }}{\mathit{\boldsymbol{D}}_{j(k)}} - {\mathit{\boldsymbol{X}}_{(k + 1)}} + {\mathit{\boldsymbol{V}}_{j(k + 1)}}, {\rm{ }}(j = 2, 3, 4)$ 3. Until the stopping criterion is met. Table 2. RSRE, time and parameter values of each algorithm for simulated data
SNR/
dBcriteria CLSUnSAL SUnSAL-TV S2WSU SLRCSU 20 RSRE /dB 8.5435 9.4239 7.6971 13.4488 time/s 4.2877 122.6244 30.0975 58.7048 parameter λ=1.6 λ=0.05,
λTV=0.05λ=0.1 M=225,
λ=0.05,
τ=0.00530 RSRE /dB 11.1263 14.4408 15.4868 19.4420 time/s 5.7490 118.8624 29.4267 57.1883 parameter λ=0.5 λ=0.007,
λTV=0.01λ=0.005 M=225,
λ=0.01,
τ=0.00140 RSRE /dB 13.6605 22.0968 28.2348 30.2934 time/s 7.5112 121.7894 30.1787 58.5224 parameter λ=0.1 λ=0.001,
λTV=0.003λ=0.001 M=225,
λ=0.007,
τ=0.001Table 3. Parameters and times corresponding to each algorithm for real data
CLSUnSAL SUnSAL-TV S2WSU SLRCSU time/s 195.7 1219.7 485.6 624.8 parameter λ=0.01 λ=0.001,
λTV=0.001λ=7×10-5 M=1910,
λ=0.01, τ=0.005 -
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