﻿ 基于大规模多目标优化的高光谱稀疏解混算法
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 哈尔滨工程大学学报  2019, Vol. 40 Issue (7): 1354-1360  DOI: 10.11990/jheu.201807075 0

### 引用本文

BI Xiaojun, ZHOU Zeyu. Sparse unmixing of hyperspectral images based on large-scale many-objective optimization algorithm[J]. Journal of Harbin Engineering University, 2019, 40(7), 1354-1360. DOI: 10.11990/jheu.201807075.

### 文章历史

1. 中央民族大学 信息工程学院, 北京 100081;
2. 哈尔滨工程大学 信息与通信工程学院, 黑龙江 哈尔滨 150001

Sparse unmixing of hyperspectral images based on large-scale many-objective optimization algorithm
BI Xiaojun 1, ZHOU Zeyu 2
1. School of Information Engineering, Minzu University of China, Beijing 100081, China;
2. Department of Information and Communication Engineering, Harbin Engineering University, Harbin 150001, China
Abstract: The existing multi-objective sparse unmixing algorithm has the defect of random grouping strategy and simplicity of the knee point selection, which leads to low accuracy of the hyperspectral data unmixing. Considering this problem, this paper proposes a hyperspectral sparse unmixing algorithm based on the large-scale many-objective evolutionary optimization (LMEA) algorithm. Based on the decision variable grouping strategy of the LMEA, a constrained knee point area selection strategy is used to obtain the abundance optimal solution to improve the unmixing accuracy. Experiments on simulated and real hyperspectral data show that the proposed algorithm greatly improved the unmixing accuracy. Compared with other algorithms, the abundance map edge details obtained by this algorithm were better processed and the anti-noise performance was stronger; this verifies the effectiveness and advancement of the proposed algorithm.
Keywords: hyperspectral image    linear spectral unmixing model    sparse unmixing    multi-objective optimization    large-scale many-objective evolutionary optimization (LMEA) algorithm    knee point area

1 理论基础知识 1.1 高光谱图像的稀疏解混模型

 ${y_i} = \sum\limits_{j = 1}^q {{m_{ij}}} {\alpha _j} + {n_i}$ (1)

 $\mathit{\boldsymbol{y}} = \mathit{\boldsymbol{M\alpha }} + \mathit{\boldsymbol{n}}$ (2)

 $\left\{ {\begin{array}{*{20}{l}} {{\rm{ASC}}:\sum\limits_{j = 1}^q {{\alpha _j}} = 1}\\ {{\rm{ANC}}:{\alpha _j} \ge 0} \end{array}} \right.j = 1, 2, \cdots , q$ (3)

 $\mathit{\boldsymbol{y}}{\rm{ = }}\mathit{\boldsymbol{A x}}{\rm{ + }}\mathit{\boldsymbol{n}}$ (4)

 $\left\{ {\begin{array}{*{20}{l}} {\mathop {\min }\limits_\mathit{x} {{\left\| \mathit{\boldsymbol{x}} \right\|}_0}}\\ {{\rm{ s}}{\rm{.t}}{\rm{. }}{{\left\| {\mathit{\boldsymbol{y - Ax}}} \right\|}_2} \le \delta , \mathit{\boldsymbol{x}} \ge 0, {\bf{1}}_{m \times 1}^{\rm{T}}\mathit{\boldsymbol{x}} = 1} \end{array}} \right.$ (5)

 $\left\{ {\begin{array}{*{20}{l}} {\mathop {\min }\limits_x {{\left\| \mathit{\boldsymbol{x}} \right\|}_1}}\\ {{\rm{ s}}{\rm{.t}}{\rm{. }}{{\left\| {\mathit{\boldsymbol{y}} - \mathit{\boldsymbol{Ax}}} \right\|}_2} \le \delta , \mathit{\boldsymbol{x}} \ge 0, {\bf{1}}_{m \times 1}^{\rm{T}}\mathit{\boldsymbol{x}} = 1} \end{array}} \right.$ (6)

 $\mathop {\min }\limits_x \frac{1}{2}\left\| {\mathit{\boldsymbol{y}} - \mathit{\boldsymbol{Ax}}} \right\|_2^2 + \lambda {\left\| \mathit{\boldsymbol{x}} \right\|_1} + {\mathit{\boldsymbol{\iota }}_{\{ 1\} }}\left( {{\bf{1}}_{m \times 1}^{\rm{T}}\mathit{\boldsymbol{x}}} \right) + {\mathit{\boldsymbol{\iota }}_{R_ + ^m}}(\mathit{\boldsymbol{x}})$ (7)

1.2 多目标优化算法

 $\left\{ {\begin{array}{*{20}{l}} {\min F(\mathit{\boldsymbol{x}}) = {{\left( {{f_1}(\mathit{\boldsymbol{x}}), {f_2}(\mathit{\boldsymbol{x}}), \cdots , {f_m}(\mathit{\boldsymbol{x}})} \right)}^{\rm{T}}}}\\ {{\rm{ s}}{\rm{.t}}{\rm{. }}\mathit{\boldsymbol{x}} \in \mathit{\Omega }} \end{array}} \right.$ (8)

 ${P^*} = \left\{ {{\mathit{\boldsymbol{x}}^*} \in \mathit{\Omega }|\neg \;\exists \mathit{\boldsymbol{x}} \in \mathit{\Omega }, \mathit{\boldsymbol{x}} \succ {\mathit{\boldsymbol{x}}^*}} \right\}$ (9)

Pareto最优解集在目标函数空间上对应的图像称为Pareto最优前沿面(pareto front, PF)，其数学表达式为：

 $\begin{array}{l} {\rm{P}}{{\rm{F}}^*} \buildrel \Delta \over = \left\{ {F\left( {{x^*}} \right) = \left( {{f_1}\left( {{x^*}} \right), {f_2}\left( {{x^*}} \right), \cdots } \right.} \right., \\ {f_m}\left( {{x^*}} \right))|{x^*} \in {P^*}\} \end{array}$ (10)
2 基于LMEA多目标优化稀疏解混算法 2.1 多目标稀疏解混框架

2.2 基于LMEA多目标优化的稀疏解混算法

2.2.1 决策变量的分组策略

 $\left\{ {\begin{array}{*{20}{l}} {{f_k}{{\left. {(\mathit{\boldsymbol{x}})} \right|}_{{\mathit{\boldsymbol{x}}_i} = {\mathit{\boldsymbol{a}}_2}, {\mathit{\boldsymbol{x}}_j} = {\mathit{\boldsymbol{b}}_1}}} < {f_k}{{\left. {(\mathit{\boldsymbol{x}})} \right|}_{{\mathit{\boldsymbol{x}}_i} = {\mathit{\boldsymbol{a}}_1}, {\mathit{\boldsymbol{x}}_j} = {\mathit{\boldsymbol{b}}_1}}}}\\ {{f_k}{{\left. {(\mathit{\boldsymbol{x}})} \right|}_{{\mathit{\boldsymbol{x}}_i} = {\mathit{\boldsymbol{a}}_2}, {\mathit{\boldsymbol{x}}_j} = {\mathit{\boldsymbol{b}}_2}}} > {f_k}{{\left. {(\mathit{\boldsymbol{x}})} \right|}_{{\mathit{\boldsymbol{x}}_i} = {\mathit{\boldsymbol{a}}_1}{\rm{, }}{\mathit{\boldsymbol{x}}_j} = {\mathit{\boldsymbol{b}}_2}}}} \end{array}} \right.$ (11)

2.2.2 收敛性优化策略和多样性优化策略

 $\left\{ {\begin{array}{*{20}{c}} {{\rm{ANC}}:{x_i} = \left\{ {\begin{array}{*{20}{l}} {0\;\;\;\;{\mathit{x}_\mathit{i}}{\rm{ < 0}}}\\ {{x_i}\;\;\;其他} \end{array}} \right.}\\ {{\rm{ASC}}:{x_i} = \frac{{{x_i}}}{{\sum\limits_{i = 1}^n {{x_i}} }}\;\;\;\;\;\;\;\;\;} \end{array}} \right.$ (12)

2.2.3 有约束拐点区域选择策略

 Download: 图 3 有约束拐点区域选择示意 Fig. 3 Diagram of constrained knee point region selection

 ${\rm{Error}} = \left\| {\mathit{\boldsymbol{A\tilde x}} - \mathit{\boldsymbol{y}}} \right\|_2^2$ (13)

3 实验仿真与分析

 ${\rm{SRE}} = 10\lg \left( {\frac{{{\rm{E}}\left[ {\left\| \mathit{\boldsymbol{x}} \right\|_2^2} \right]}}{{{\rm{E}}\left[ {\left\| {\mathit{\boldsymbol{x}} - \mathit{\boldsymbol{\widehat x}}} \right\|_2^2} \right.}}} \right)$ (14)
 ${\mathop{\rm RMSE}\nolimits} = \frac{1}{q}\sum\limits_{i = 1}^q {\sqrt {\frac{1}{L}\sum\limits_{j = 1}^L {\left( {{\mathit{\boldsymbol{x}}_{ij}} - {{\mathit{\boldsymbol{\widehat x}}}_{ij}}} \right)} } }$ (15)

3.1 模拟数据仿真实验

 Download: 图 4 模拟数据集端元1、2、5的真实丰度图像及4种算法的解混结果 Fig. 4 The real-abundance images of the endmembers 1, 2 and 5 of the simulated dataset and the results of the unmixing of the four algorithms
3.2 真实数据仿真实验

 Download: 图 5 AVIRIS Cuprite数据矿物质分布 Fig. 5 Distribution of different minerals for AVIRIS Cuprite data obtained by Tricorder software

 Download: 图 6 4种算法获得的AVIRIS Cuprite数据集中明矾石和水铵长石估计丰度 Fig. 6 The abundance maps of alumite and buddingtonite for the four algorithms of the AVIRIS Cuprite dataset

4 结论

1) 本文提出将LMEA算法引入高光谱稀疏解混模型，将决策变量分为收敛性相关变量和多样性相关变量，并分别进行优化，充分考虑了决策变量之间的相关性。

2) 本文提出有约束拐点区域选择策略在得到的PF上进行选解，提升了算法的解混精度。

3) 本文通过模拟和真实的高光谱数据集进行实验，本文算法和其他3种对比算法相比，得到了最优的高光谱数据解混精度，并且随着信噪比的增加，本文算法的解混精度得到了明显的提升，这对解决实际复杂的高光谱图像的解混问题具有重要的意义。

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