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 哈尔滨工程大学学报  2018, Vol. 39 Issue (9): 1582-1588  DOI: 10.11990/jheu.201707008 0

### 引用本文

ZHAO Chunhui, MENG Meiling, YAN Yiming. Sub-pixel target detection on hyperspectral image based on dictionary reconstruction[J]. Journal of Harbin Engineering University, 2018, 39(9), 1582-1588. DOI: 10.11990/jheu.201707008.

### 文章历史

Sub-pixel target detection on hyperspectral image based on dictionary reconstruction
ZHAO Chunhui, MENG Meiling, YAN Yiming
College of Information and Communication Engineering, Harbin Engineering University, Harbin 150001, China
Abstract: The introduction of sparse representation provides a new way for the target detection of hyperspectral remote sensing images. However, in the detection process, because the structure of an over-complete dictionary is obtained directly from the hyperspectral image, there are uncertainties and it is hard to realize the accurate detection for the sub-pixels. In order to solve the above-mentioned problems, this paper proposed a sub-pixel target detection algorithm on hyperspectral image based on dictionary reconstruction. In the algorithm, an unsupervised method is used to complete the construction of an over-complete dictionary, so as to ensure that the dictionary contains the spectral information of partial target pixels; meanwhile, binary alternative hypothesis model is introduced to detect the sub-pixel target in hyperspectral image. Simulation experiments were carried out respectively for the simulative and real hyperspectral remote-sensing image data. By carrying out comparison and analysis for the three-dimensional drawing, ROC curve and AUC value in the test results, it is known that the algorithm proposed in the paper not only increased the detection precision, but also properly constrained the background noise.
Keywords: hyperspectral image    sub-pixel target    target detection    sparse representation    dictionary reconstruction

1 稀疏表示目标检测原理

 $\begin{array}{l} \mathit{\boldsymbol{x}} \approx {\mathit{\boldsymbol{\alpha }}_1}\mathit{\boldsymbol{a}}_1^b + {\mathit{\boldsymbol{\alpha }}_2}\mathit{\boldsymbol{a}}_2^b + \cdots + {\mathit{\boldsymbol{\alpha }}_{{N_b}}}\mathit{\boldsymbol{a}}_{{N_b}}^b = \\ \;\;\;\;\;[\mathit{\boldsymbol{a}}_1^b\;\;\mathit{\boldsymbol{a}}_2^b \cdots \mathit{\boldsymbol{a}}_{{N_b}}^b]{[{\mathit{\boldsymbol{a}}_1}{\mathit{\boldsymbol{a}}_2} \cdots {\mathit{\boldsymbol{a}}_{{N_b}}}]^{\rm{T}}} = {\mathit{\boldsymbol{A}}_b}\mathit{\boldsymbol{\alpha }} \end{array}$ (1)

 $\begin{array}{l} \mathit{\boldsymbol{x}} \approx {\mathit{\boldsymbol{\beta }}_1}\mathit{\boldsymbol{a}}_1^t + {\mathit{\boldsymbol{\beta }}_2}\mathit{\boldsymbol{a}}_2^t + \cdots + {\mathit{\boldsymbol{\beta }}_{{N_t}}}\mathit{\boldsymbol{a}}_{{N_t}}^t = \\ \;\;\;\;\;\;[\mathit{\boldsymbol{a}}_1^t\mathit{\boldsymbol{a}}_2^t \cdots \mathit{\boldsymbol{a}}_{{N_t}}^t]{[{\mathit{\boldsymbol{\beta }}_1}{\mathit{\boldsymbol{\beta }}_2} \cdots {\beta _{{N_i}}}]^{\rm{T}}} = {\mathit{\boldsymbol{A}}_t}\mathit{\boldsymbol{\beta }} \end{array}$ (2)

 $\mathit{\boldsymbol{x}} = {\mathit{\boldsymbol{A}}_b}\mathit{\boldsymbol{\alpha }} + {\mathit{\boldsymbol{A}}_t}\mathit{\boldsymbol{\beta }} = \underbrace {[{\mathit{\boldsymbol{A}}_b}\;\;{\mathit{\boldsymbol{A}}_t}]}_\mathit{\boldsymbol{A}}\underbrace {\left[ \begin{array}{l} \mathit{\boldsymbol{\alpha }}\\ \mathit{\boldsymbol{\beta }} \end{array} \right]}_\gamma = \mathit{\boldsymbol{A\gamma }}$ (3)

 $\mathit{\boldsymbol{\hat \gamma }} = \arg \min {\left\| \mathit{\boldsymbol{\gamma }} \right\|_0}\;\;\;{\rm{subject}}\;{\rm{to}}\;\;\;\mathit{\boldsymbol{x}} = \mathit{\boldsymbol{A\gamma }}$ (4)

 ${\left\| \mathit{\boldsymbol{s}} \right\|_0} = \mathop {\lim }\limits_{p \to 0} \left\| \mathit{\boldsymbol{s}} \right\|_p^p = \mathop {\lim }\limits_{p \to 0} \sum\limits_{k = 1}^m {|{\mathit{\boldsymbol{s}}_k}{|^p}}$ (5)

 $\mathit{\boldsymbol{\hat \gamma }} = \arg \min {\left\| \mathit{\boldsymbol{\gamma }} \right\|_0}\;\;\;{\rm{subject}}\;{\rm{to}}\;\;\;{\left\| {\mathit{\boldsymbol{A\gamma }} - \mathit{\boldsymbol{x}}} \right\|_2} \le \sigma$ (6)

 $\mathit{\boldsymbol{\hat \gamma }} = \arg \min {\left\| {\mathit{\boldsymbol{A\gamma }} - \mathit{\boldsymbol{x}}} \right\|_2}\;\;\;{\rm{subject}}\;{\rm{to}}\;\;\;{\left\| \mathit{\boldsymbol{\gamma }} \right\|_0} \le K$ (7)

 ${r_b}\left( \mathit{\boldsymbol{x}} \right) = {\left\| {\mathit{\boldsymbol{x}} - {\mathit{\boldsymbol{A}}_b}\mathit{\boldsymbol{\hat \alpha }}} \right\|_2}$ (8)
 ${r_t}\left( \mathit{\boldsymbol{x}} \right) = {\left\| {\mathit{\boldsymbol{x}} - {\mathit{\boldsymbol{A}}_t}\mathit{\boldsymbol{\hat \beta }}} \right\|_2}$ (9)
 $R\left( \mathit{\boldsymbol{x}} \right) = {r_b}\left( \mathit{\boldsymbol{x}} \right) - {r_t}\left( \mathit{\boldsymbol{x}} \right)$ (10)

2 字典重构亚像元检测算法 2.1 字典重构

 $\mathit{\boldsymbol{R}} = \frac{1}{N}\sum\limits_{i = 1}^N {{\mathit{\boldsymbol{r}}_i}\mathit{\boldsymbol{r}}_i^{\rm{T}}}$ (11)

 $\mathit{\boldsymbol{P}} = \mathit{\boldsymbol{I}} - \mathit{\boldsymbol{U}}{({\mathit{\boldsymbol{U}}^{\rm{T}}}\mathit{\boldsymbol{U}})^{ - 1}}{\mathit{\boldsymbol{U}}^{\rm{T}}} = \mathit{\boldsymbol{I}} - \mathit{\boldsymbol{U}}{\mathit{\boldsymbol{U}}^\# }$ (12)

 $\mathit{\boldsymbol{D}} = \mathit{\boldsymbol{Ipx}}$ (13)

 $\mathit{\boldsymbol{x}} = {\mathit{\boldsymbol{D}}_b}\mathit{\boldsymbol{\alpha }} + {\mathit{\boldsymbol{D}}_t}\mathit{\boldsymbol{\beta }} = \underbrace {[{\mathit{\boldsymbol{D}}_b}\;\;{\mathit{\boldsymbol{D}}_t}]}_\mathit{\boldsymbol{D}}\underbrace {\left[ \begin{array}{l} \mathit{\boldsymbol{\alpha }}\\ \mathit{\boldsymbol{\beta }} \end{array} \right]}_\mathit{\boldsymbol{\gamma }} = \mathit{\boldsymbol{D\gamma }}$ (14)
2.2 假设检验模型

 $\begin{array}{l} \mathit{\boldsymbol{x}} \approx {\mathit{\boldsymbol{\alpha }}_1}\mathit{\boldsymbol{d}}_1^b + {\mathit{\boldsymbol{\alpha }}_2}\mathit{\boldsymbol{d}}_2^b + \cdots + {\mathit{\boldsymbol{\alpha }}_{{N_b}}}\mathit{\boldsymbol{d}}_{{N_b}}^b = \\ \;\;\;\;\;[\mathit{\boldsymbol{d}}_1^b\;\;\mathit{\boldsymbol{d}}_2^b \cdots \mathit{\boldsymbol{d}}_{{N_b}}^b]{[{\mathit{\boldsymbol{\alpha }}_1}{\mathit{\boldsymbol{\alpha }}_2} \cdots {\mathit{\boldsymbol{\alpha }}_{{N_b}}}]^{\rm{T}}} = {\mathit{\boldsymbol{D}}_b}\mathit{\boldsymbol{\alpha }} \end{array}$ (15)

 $\begin{array}{l} \mathit{\boldsymbol{x}} \approx ({\mathit{\boldsymbol{\alpha }}_1}\mathit{\boldsymbol{d}}_1^b + {\mathit{\boldsymbol{\alpha }}_2}\mathit{\boldsymbol{d}}_2^b + \cdots + {\mathit{\boldsymbol{\alpha }}_{{N_b}}}\mathit{\boldsymbol{d}}_{{N_b}}^b) + \\ \;\;\;\;\;\;({\mathit{\boldsymbol{\beta }}_1}\;\mathit{\boldsymbol{d}}_1^t + {\mathit{\boldsymbol{\beta }}_2}\mathit{\boldsymbol{d}}_2^t + \cdots + {\mathit{\boldsymbol{\beta }}_{{N_t}}}\mathit{\boldsymbol{d}}_{{N_t}}^t) = \\ \;\;\;\;\;\;[\mathit{\boldsymbol{d}}_1^b\;\mathit{\boldsymbol{d}}_2^b \cdots \mathit{\boldsymbol{d}}_{{N_b}}^b]{[{\mathit{\boldsymbol{\alpha }}_1}\;{\mathit{\boldsymbol{\alpha }}_2} \cdots {\mathit{\boldsymbol{\alpha }}_{{N_b}}}]^{\rm{T}}} + \\ \;\;\;\;\;\;[\mathit{\boldsymbol{d}}_1^t\;\mathit{\boldsymbol{d}}_2^t \cdots \mathit{\boldsymbol{d}}_{{N_t}}^t]{[{\mathit{\boldsymbol{\beta }}_1}\;{\mathit{\boldsymbol{\beta }}_2} \cdots {\mathit{\boldsymbol{\beta }}_{{N_t}}}]^{\rm{T}}} = \\ \;\;\;\;\;{\mathit{\boldsymbol{D}}_b}\mathit{\boldsymbol{\alpha }} + {\mathit{\boldsymbol{D}}_t}\mathit{\boldsymbol{\beta }} = \mathit{\boldsymbol{D\gamma }} \end{array}$ (16)

 $\mathit{\boldsymbol{x}} = \left\{ \begin{array}{l} {\mathit{\boldsymbol{D}}_b}\mathit{\boldsymbol{\alpha }}, \;\;{H_0}\\ {\mathit{\boldsymbol{D}}_b}\mathit{\boldsymbol{\alpha }} + {\mathit{\boldsymbol{D}}_t}\mathit{\boldsymbol{\beta }} = \mathit{\boldsymbol{D\gamma }}, \;\;{H_1} \end{array} \right.$ (17)

2.3 算法流程图

3 仿真实验结果及分析 3.1 高光谱图像实验数据

3.2 仿真结果与分析

 Download: 图 5 模拟数据实验结果三维图 Fig. 5 Experimental results for synthetic dataset in 3-D version

 Download: 图 6 真实数据实验结果三维图 Fig. 6 Experimental results for real dataset in 3-D version