﻿ 采用PPI算法改进的一种数学形态学端元提取方法
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1. 西安航空学院电子工程学院, 陕西 西安 710077;
2. 西安石油大学计算机学院, 陕西 西安 710065

An improved endmember extraction method of mathematical morphology based on PPI algorithm
XU Jun1, WANG Cailing2, WANG Li1
1. School of Electronic Engineering, Xi'an Aeronautical University, Xi'an 710077, China;
2. School of Computer Science, Xi'an Shiyou University, Xi'an 710065, China
Abstract: Automated morphological endmember extraction(AMEE) algorithm defines the spectral angular distance between the purest pixel and the most mixed pixel in the structural element as the morphological eccentricity index(MEI) to quantitatively denote the purity of the pixel. However, the most mixed pixels as the reference standard are not the same in different structural elements, especially when the pure pixels account for the majority of the structural elements, the mean spectrum of all the pixels will be closer to the pure pixels. At this time, the higher the MEI, the lower the purity of the pixel. To solve this problem, a novel endmember extraction algorithm is proposed in this paper which combines the pixel purity index (PPI) algorithm with AMEE algorithm and is named PPI-AMEE. In the structural element, the PPI is used to replace the MEI index in the AMEE algorithm to find the purest pixel. When the structural element is transformed, only the purest pixel can always be projected to the two ends of the randomly generated line, therefore the PPI value of the purest pixel will increase continuously, while the PPI value of the other pixels will not increase continuously. The PPI value of each pixel is accumulated and recorded until the iterative termination condition is satisfied, and a PPI image is finally obtained. The endmembers are selected from the pixels with higher PPI value. The PPI-AMEE algorithm runs the PPI algorithm in relatively small structural elements, and then processes the whole image with the expansion operation of mathematical morphology, which takes into account both the spectral and spatial information of the image. In the experiment, AVIRIS hyperspectral data from Cuprite area, Nevada, USA are used to validate the proposed PPI-AMEE algorithm. The experimental results show that the endmember extraction accuracy of PPI-AMEE algorithm is better than that of AMEE algorithm and PPI algorithm on the whole.
Key words: hyperspectral image    endmember extraction    pure pixel index    mathematical morphology

1 现有的AMEE算法与PPI算法 1.1 AMEE算法

AMEE算法将传统数学形态学理论进行了拓展性定义，给出了高光谱图像的形态学膨胀和腐蚀操作方法。AMEE算法中定义了两种腐蚀和膨胀操作[5]

(1) 将结构元素内某像元到其他像元的距离累计之和作为纯度指标。结构元素K内某个像元f(x, y)到其他像元的距离累计之和用式(1)表示

(1)

(2)
(3)

(2) 将结构元素内像元到结构元素中心的距离作为纯度指标。假设结构元素K像元数量为M，那么K的中心可定义为

(4)

(5)

(6)
(7)

AMEE算法将d(x, y)与e(x, y)之间的光谱夹角距离定义为MEI，MEI用来定量表示像元的相对纯度，并赋值给结构元素中最纯的像元，计算公式如式(8)所示

(8)

AMEE算法可以描述如下：从最小结构元素Kmin开始，在整幅影像中移动，每移动一次，都会在结构元素内找到一个最纯净像元和一个混合度最大的像元，计算MEI值并将其赋给最纯净像元。然后逐渐增大结构元素的尺寸，并重复执行上述操作，达到预设的最大迭代次数Imax后终止，最终获得一幅MEI图像，参考MEI的值由高到低选取端元。

1.2 PPI算法

PPI算法的思想是基于凸面几何学理论。凸面几何学理论认为高光谱影像的所有像元在高维光谱特征空间中对应的样本点呈散点图分布，所有样本点包含在一个凸面单形体内部，那些纯净像元(端元)位于凸面单形体的顶点处[12-17]

 图 1 PPI算法中像元投影原理 Fig. 1 Schematic diagram of pixel projection principle in PPI algorithm

PPI算法虽然原理比较简单直观，但有PPI算法没有利用图像的空间信息，需要利用可视化工具手动选择端元，自动化程度不高，而且投影直线都是随机生成的，导致每次算法运行所提取的端元并不一定相同。

2 PPI-AMEE算法

 图 2 不同结构元素覆盖像元的空间分布 Fig. 2 Spatial distribution of pixels covered by different structural elements

(9)

(10)
(11)

PPI-AMEE算法的具体步骤如下：

(1) 先对整幅图像进行最小噪声分离变换(minimum noise fraction，MNF)变换进行降维和去噪处理，估算端元数目m

(2) 设定结构元素的最小尺寸Kmin和最大尺寸Kmax，得出最大迭代次数Imax

(3) 令i=1，所有像元的PPI初值P(f(x, y), K)=0，从最小结构元素Kmin开始执行；

(4) 按照PPI算法拓展定义的形态学结构算子进行膨胀操作，得到结构元素内各像元的纯度指数；

(5) i=i+1，如果i=Imax，顺序执行步骤(6)，否则增大结构元素K，跳回步骤(4)执行；

(6) 输出PPI图像，参考PPI值较大的m个像元确定为端元。

 图 3 PPI-AMEE算法流程 Fig. 3 Flow chart of PPI-AMEE algorithm

3 试验结果与分析

 图 4 构建模拟数据的4种光谱 Fig. 4 Four spectra of simulated data

 SNB/db PPI AMEE PPI-AMEE SAD/rad SID/bit SAD/rad SID/bit 时间/s SAD/rad SID/bit 时间/s 10 0.158 0.052 0.149 0.050 65.26 0.142 0.048 72.53 20 0.113 0.041 0.132 0.047 64.83 0.126 0.043 66.78 30 0.107 0.037 0.127 0.042 61.62 0.109 0.038 55.96 40 0.102 0.035 0.102 0.033 60.59 0.093 0.030 52.67 50 0.092 0.028 0.096 0.031 59.93 0.086 0.026 51.98

 图 5 Cuprite地区的AVIRIS数据假彩色合成图 Fig. 5 False-color composition image of AVIRIS data in Cuprite region

 矿物种类 PA1 PA2 PA3 PA4 PA5 PA6 PA7 PA8 PA9 alunite 0.173 0.185 0.049 0.205 0.156 0.253 0.262 0.166 0.172 buddingtonite 0.196 0.233 0.272 0.183 0.123 0.178 0.212 0.231 0.196 calcite 0.229 0.234 0.237 0.062 0.167 0.196 0.182 0.239 0.217 kaolinite 0.156 0.243 0.236 0.193 0.184 0.198 0.217 0.249 0.053

 图 6 PPI、AMEE、PPI-AMEE提取的端元与USGS标准光谱的对比 Fig. 6 Comparison of enmembers extracted by PPI, AMEE, PPI-AMEE with USGS standard spectra

 矿物种类 PPI AMEE PPI-AMEE SAD/rad SID/bit SAD/rad SID/bit SAD/rad SID/bit alunite 0.074 0.022 0.052 0.020 0.049 0.019 buddingtonite 0.165 0.053 0.136 0.047 0.123 0.043 calcite 0.085 0.029 0.070 0.025 0.062 0.023 kaolinite 0.095 0.031 0.061 0.022 0.053 0.020

4 结论

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http://dx.doi.org/10.11947/j.AGCS.2019.20180475

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#### 文章信息

XU Jun, WANG Cailing, WANG Li

An improved endmember extraction method of mathematical morphology based on PPI algorithm

Acta Geodaetica et Cartographica Sinica, 2019, 48(8): 996-1003
http://dx.doi.org/10.11947/j.AGCS.2019.20180475