﻿ 基于混合像元空间与谱间相关性模型的NMF线性盲解混
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NMF linear blind unmixing method based on mixed pixel's spatial and spectral correlation model
YUAN Bo
School of Computer and Information Engineering, Nanyang Institute of Technology, Nanyang 473000, China
Abstract: The present hyperspectral unmixing methods based on correlation analysis, either lack of comprehensive analysis and utilization of hyperspectral image's spatial & spectral correlation characteristics, or have a high dependence degree on prior knowledge. This paper proposes a NMF linear blind unmixing method based on mixed pixel's spatial and spectral correlation model. The method sets up spatial correlation model of adjacent pixels by improving Markov Random Filed(MRF) model, sets up spectral correlation model of adjacent bands by adopting complexity mapping technology, and introduces the two models respectively into NMF objective function externally and internally, as the constraints of the blind unmixing method. Experimental result indicates that, the proposed method can significantly reduced the degree of dependence on prior knowledge, comparing with other representative NMF reference methods including area-correlated spectral unmixing method based on Bayesian nonnegative matrix factorization(ACBNMF), minimum spectral correlation constraint NMF(MSCCNMF) and minimum volume constrained nonnegative matrix factorization(MVCNMF), the unmixing accuracy is also improved.
Key words: nonnegative matrix factorization    spatial correlation    spectral correlation    Markov random field    complexity mapping

1 标准NMF线性盲分解模型及存在的主要问题 1.1 NMF线性盲分解模型

1.1.1 线性光谱混合模型(linear spectral mixture model，LSMM)

LSMM的数学形式如式(1)所示

(1)

(2)

1.1.2 NMF解混模型

(3)

(4)

(5)
(6)
1.2 NMF线性盲分解存在的主要问题

NMF存在局部极小问题，如果不采取对应措施，将损害NMF线性盲分解的精度和稳定性。

NMF的求解过程等价于通过迭代求目标函数最小值的过程，理想情况是目标函数为凸函数。函数凸性是数学分析中的一个重要概念，凸函数的重要性质是：任何局部极小同时也是全局最小。只要NMF的目标函数取得局部极小收敛，就说明获取了全局最小值，也即全局最优解。文献[19]给出了标准NMF目标函数(5)的凸性判定过程与收敛性证明，结论如下：NMF目标函数分别对于端元光谱矩阵和丰度矩阵都是凸函数，但同时对于二者是非凸函数。也就是说，式(5)取局部极小值时，端元光谱矩阵和丰度矩阵的解并不是二者的全局最优解。“非凸性”会使收敛结果沦为局部极小，增大解的不确定性并降低算法整体精度。

2 基于MRF改进模型的空间相关性模型

MRF包含“Markov性质”和“随机场”两个要素，可简单解释为具有Markov性质的随机场。MRF的具体定义与主要性质可参考文献[20]。二维数字图像可看作随机场，空间相关性质也可类比于Markov性质(仅相邻时刻间的状态相关)，则空间相关特征显著的高光谱遥感图像可用MRF近似表示。图像的空间能量大小与地物的空间变化频率和幅度成正比。对于空间相关特征显著的高光谱图像，解混结果中空间能量越小，越可能接近地面空间能量分布的真实情况。本文以文献[21]关于MRF的图像分割模型为基础，建立反映图像空间相关特征的能量函数模型如下

(7)

(8)

(9)

U为分离矩阵，即丰度矩阵S的逆，Y为端元矩阵M的估计，由式(3)可得

(10)
(11)

(12)

(13)
3 基于复杂度映射的谱间相关性模型

(14)
(15)

(1) Mlp存在一定的变化范围；

(2) Mlp的值在时间上变化“缓慢”。

(16)

(17)
(18)

(19)
4 基于空间与谱间相关性模型的NMF线性盲分解

(1) 利用基于最小误差的高光谱信号识别法(hyperspectral signal identification by minimum error，HySime)估算端元数量P

(2) 初始化端元矩阵M和丰度矩阵S

(3) 含有谱间相关约束的NMF迭代：根据更新规则式(17)和式(18)分别计算端元矩阵M和丰度矩阵S的迭代结果，并利用MS计算分离矩阵U的迭代结果；

(4) 空间相关约束：归一化分离矩阵U的每一列，同时估计像元特征向量的均值矩阵W，然后计算U

(5) 重复步骤(3)和(4)，继续迭代，直到各自的停止准则同时满足，得到一个估计的成分，将其转换为矩阵，即获得一个端元分布。继续迭代，直至满足阈值条件(预先设置的一个很小的正数如10-4等，作为迭代停止条件)，得到端元分布的估计结果。

5 试验与分析 5.1 试验1

 图 1 美国华盛顿特区HYDICE数据 Fig. 1 HYDICE data of Washington D.C., USA

(20)

 图 2 降分辨率后的美国华盛顿特区HYDICE数据 Fig. 2 Resolution descended HYDICE data of Washington D.C., USA

 图 3 降分辨率后的美国华盛顿特区HYDICE数据丰度参考值 Fig. 3 Abundance reference of resolution descended HYDICE data of Washington D.C., USA

 图 4 降分辨率后美国华盛顿特区HYDICE数据S2CNMF丰度估计 Fig. 4 S2CNMF's abundance estimation of resolution descended HYDICE data of Washington D.C., USA

 图 5 降分辨率后美国华盛顿特区HYDICE数据ACBNMF丰度估计 Fig. 5 ACBNMF's abundance estimation of resolution descended HYDICE data of Washington D.C., USA

 图 6 降分辨率后美国华盛顿特区HYDICE数据MSCCNMF丰度估计 Fig. 6 MSCCNMF's abundance estimation of resolution descended HYDICE data of Washington D.C., USA

 图 7 降分辨率后美国华盛顿特区HYDICE数据MVCNMF丰度估计 Fig. 7 MVCNMF's abundance estimation of resolution descended HYDICE data of Washington D.C., USA

(21)

εi, jk是第k个端元在图像位置(i, j)处像元的丰度余差，即丰度估计值与丰度真实值(参考值)的差，则第k个端元在整幅图像所有像元中均方根误差如式(22)所示，代表第k个端元的丰度分解精度

(22)

 解混方法 植被 水体 裸土 平均 S2CNMF 0.152 3 0.126 1 0.148 2 0.142 2 ACBNMF 0.176 8 0.146 7 0.165 5 0.163 0 MSCCNMF 0.172 9 0.156 4 0.170 6 0.166 6 MVCNMF 0.182 5 0.143 7 0.175 8 0.167 3

 解混方法 植被 水体 裸土 平均 S2CNMF 0.125 8 0.121 5 0.140 1 0.129 1 ACBNMF 0.155 6 0.138 9 0.142 6 0.145 7 MSCCNMF 0.150 8 0.144 5 0.151 9 0.149 1 MVCNMF 0.155 3 0.143 4 0.153 7 0.150 8

 解混方法 S2CNMF ACBNMF MSCCNMF MVCNMF 运行时间/s 85.67 99.72 186.64 232.73

5.2 试验2

 图 8 Cuprite采矿区AVIRIS高光谱数据 Fig. 8 AVIRIS hyperspectral data of Cuprite mining field

 图 9 Cuprite采矿区AVIRIS数据S2CNMF丰度估计 Fig. 9 S2CNMF's abundance results of AVIRIS hyperspectral data of Cuprite mining field

 解混方法 S2CNMF ACBNMF MSCCNMF MVCNMF Alunite 0.212 2 0.225 7 0.237 5 0.240 1 Buddingtointe 0.198 7 0.213 8 0.220 8 0.226 7 Calcite 0.193 1 0.229 5 0.219 4 0.210 3 Kaolinite 0.198 5 0.205 6 0.223 6 0.223 8 Muscovite 0.212 4 0.237 9 0.235 5 0.217 5 平均 0.202 8 0.222 5 0.227 4 0.223 7

 解混方法 S2CNMF ACBNMF MSCCNMF MVCNMF Alunite 0.212 2 0.219 3 0.223 6 0.241 2 Buddingtointe 0.199 8 0.222 5 0.219 4 0.217 9 Calcite 0.220 1 0.235 7 0.230 8 0.233 7 Kaolinite 0.215 2 0.231 9 0.227 7 0.235 8 Muscovite 0.206 7 0.215 3 0.230 1 0.217 5 平均 0.210 8 0.224 9 0.226 3 0.229 2

6 结论与展望

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http://dx.doi.org/10.11947/j.AGCS.2019.20180054

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#### 文章信息

YUAN Bo

NMF linear blind unmixing method based on mixed pixel's spatial and spectral correlation model

Acta Geodaetica et Cartographica Sinica, 2019, 48(9): 1151-1160
http://dx.doi.org/10.11947/j.AGCS.2019.20180054