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 应用科技  2020, Vol. 47 Issue (3): 46-50  DOI: 10.11991/yykj.201908011 0

### 引用本文

FANG Sen, JIAO Shuhong. Hyperspectral image unmixing method combining hierarchy and alternating direction method of multipliers (ADMM)[J]. Applied Science and Technology, 2020, 47(3): 46-50. DOI: 10.11991/yykj.201908011.

### 文章历史

Hyperspectral image unmixing method combining hierarchy and alternating direction method of multipliers (ADMM)
FANG Sen, JIAO Shuhong
College of Information and Communication Engineering, Harbin Engineering University, Harbin 150001, China
Abstract: Hyperspectral image has a higher spectral resolution, but the area covered by a single pixel is relatively large, resulting in more than one material exist in a single pixel, called a mixed pixel. The presence of mixed pixels severely affects the subsequent use of hyperspectral data. The purpose of hyperspectral image unmixing technique is to determine the materials (endmembers) present in the mixed pixels and their corresponding proportions (abundance). Because the coverage of hyperspactral data is relatively large, the phenomenon of endmember variation exists inevitably. In order to take endmember variation into consideration, an extended linear mixed model is used to describe the hyperspectral data. Based on the hierarchical unmixing technique, the alternating direction method of multipliers is used to optimize the result of unmixing. The experimental results show that the unmixing effect has been greatly improved.
Keywords: mixed pixel    endmember    abundance    linear model    alternating direction method of multipliers    unmixing    spectral variation    multiple endmembers

1 扩展的线性混合模型

ELMM主要用于克服由光照条件以及地形变化引起的SV，而且同时保持了LMM的特点，并证明了该模型的有效性。传统的线性混合模型表达式为

 ${{{x}}_k} = \sum\limits_{p = 1}^P {{a_{pk}}{{{s}}_p}} + {{{e}}_k}$ (1)

 ${{X}} = {{SA}} + {{E}}$

 ${{{x}}_k} = \sum\limits_{p = 1}^P {a_{pk}{f_{pk}}\left( {{{{s}}_p}} \right)} + {{{e}}_k}$ (2)

 $\begin{array}{l} {{{x}}_k} = \displaystyle\sum\limits_{p = 1}^P {{a_{pk}}{\psi _k}{{{s}}_p} + {{{e}}_k}} = {\psi _k}\displaystyle\sum\limits_{p = 1}^P {{a_{pk}}{{{s}}_p} + {{{e}}_k}} = \\ \;\;\;\;\;\;{\psi _k}{{S}}{a_k} + {{{e}}_k} = {{S}}{\psi _k}{a_k} + {{{e}}_k} \\ \end{array}$ (3)

 ${{{x}}_k} = \sum\limits_{p = 1}^P {{a_{pk}}{\psi _{pk}}{{{s}}_p} + {{{e}}_k}} = {{S}}{{{\varphi}} _k}{\partial _k} + {{{e}}_k}$ (4)

 ${{X}} = {{S}}\left( {{{\varPsi}} \odot {{A}}} \right) + {{E}}$

2.1 基于分层的高光谱解混算法

1)在第一层利用端元集E对像元 $y_{{\rm{pixel}}}^k$ 进行解混，选出每个地物种类所对应的最大非零系数，假设确定的地物种类以及类内光谱为 $\left\{ {{M_{1,1}},{M_{2,1}},} \right.$ $\left. {{M_{3,1}}} \right\}$ ，然后分别计算 ${M_{1,1}}$ ${M_{2,1}}$ ${M_{3,1}}$ $y_{{\rm{pixel}}}^k$ 之间的光谱角距离，假设光谱角距离最小的端元为 ${M_{1,1}}$

2)在第一层确定的端元集合的基础上分别与 ${M_{1,1}}$ 进行两端元组合，分别记为 $\left\{ {{M_{1,1}},{M_{2,1}}} \right\}$ $\left\{ {{M_{1,1}},} \right.$ $\left. {{M_{3,1}}} \right\}$ ，然后分别计算与 ${{y}}_{{\rm{pixel}}}^k$ 之间的光谱角距离，选择出光谱角距离的最小值所对应的端元组合，在此处假设为 $\left\{ {{M_{1,1}},{M_{2,1}}} \right\}$

3)在 $\left\{ {{M_{1,1}},{M_{2,1}}} \right\}$ 的基础上与不包含以上2种地物的其他端元进行组合，构造三端元组合，记为 $\left\{ {{M_{1,1}},{M_{2,1}},{M_{3,1}}} \right\}$ ，然后计算其与 ${{y}}_{{\rm{pixel}}}^k$ 之间的光谱角距离；

4)对步骤1)~3)得到的光谱角距离进行排序，将光谱角距离最小所对应的端元组合作为该像元中实际的地物混合，以此端元组合对 ${{y}}_{{\rm{pixel}}}^k$ 进行解混计算得到最终的丰度。

 $\mathop {\min }\limits_{x \in {{\bf{R}}^n}} {f_1}\left( x \right) + {f_2}\left( {Gx} \right)$ (5)

1) 令 $k = 0$ ，取 $\mu > 0$ ${u_0}$ ${d_0}$

2) 执行步骤3)~6)，直到满足迭代终止条件；

3) ${x_{k + 1}} \in {\rm{arg }}\mathop {\min }\limits_x {f_1}\left( x \right) + \dfrac{\mu }{2}\left\| {Gx - {u_k} - {d_k}} \right\|_2^2$

4) ${u_{k + 1}} \in {\rm{arg }}\mathop {\min }\limits_u {f_2}\left( u \right) + \dfrac{\mu }{2}\left\| {G{x_{k + 1}} - u - {d_k}} \right\|_2^2$

5) ${d_{k + 1}} \leftarrow {d_k} - \left( {G{x_{k + 1}} - {u_{k + 1}}} \right)$

6) $k \leftarrow k + 1$

3 合成数据实验

4 真实数据实验

5 结论