1. 辽宁工程技术大学矿业技术学院, 辽宁葫芦岛 125105;
2. 辽宁工程技术大学测绘与地理科学学院, 辽宁阜新 123000
 收稿日期: 2015-06-10; 修订日期: 2015-09-09; 优先数字出版日期: 2015-09-16 基金项目: 教育部高等学校博士学科点专项科研基金资助(编号:20122121110007);国家自然科学基金面上项目(编号:41271435) 中图分类号: P237 文献标识码: A 文章编号: 1007-4619(2016)01-0103-11

# 2 算法描述

## 2.1 区间二型模糊集合

$\begin{array}{*{20}{c}}{\tilde A = }\\{\left\{ {\left({\left({z，u} \right)，{u_{\tilde A\left({z，u} \right)}}|\forall z \in Z，\forall u \in {J_z} \subseteq \left[ {0，1} \right]} \right)} \right\}}\end{array}$ (1)

## 2.2 模糊图像模型

(1)一型模糊模型

$F = {\left[ {{F_{ji}}} \right]_{n \times m}}$ (2)

${F_{ji}} = {\beta _i} \times \frac{1}{{\sqrt {2\pi } {\sigma _i}}}\exp \left\{ { - \frac{{{{\left({{x_j} - {\mu _i}} \right)}^2}}}{{2\sigma _i^2}}} \right\}$ (3)

(2)具有不确定均值和标准差的区间二型模糊模型。

$F{'_{ji}} = {\beta _i} \times \frac{1}{{\sqrt {2\pi } {\sigma _i}}}\exp \left\{ { - \frac{{{{\left({{x_j} - {\mu _{i1}}} \right)}^2}}}{{2\sigma _i^2}}} \right\}$ (4)

$\begin{array}{*{20}{c}}{\mu _i^ - = {\mu _i} - {\alpha _i} \times {\sigma _i}}\\{\mu _i^ + = {\mu _i} + {\alpha _i} \times {\sigma _i}\;\;\;{\alpha _i} \in \left[ {0，3} \right]}\end{array}$ (5)

$F_{ji}^ + = \left\{ {\begin{array}{*{20}{c}} {F_{ji}^{, - }}&{{x_j}AAA\mu _i^ - }\\ 1&{\mu _i^ - \le {x_j} \le \mu _i^ + }\\ {F_{ji}^{, + }}&{{x_j} > \mu _i^ + } \end{array}} \right.$ (6)

$F_{ji}^ - = \left\{ {\begin{array}{*{20}{c}} {F_{ji}^{, - }}&{{x_j} \le \frac{{\mu _i^ - + \mu _i^ + }}{2}}\\ {F_{ji}^{, + }}&{{x_j}AAA\frac{{\mu _i^ - + \mu _i^ + }}{2}} \end{array}} \right.$ (7)

$F{'_{ji}} = {\beta _i} \times \frac{1}{{\sqrt {2\pi } {\sigma _{i1}}}}\exp \left\{ { - \frac{{{{\left({{x_j} - {\mu _i}} \right)}^2}}}{{2\sigma _{i1}^2}}} \right\}$ (8)

$\sigma _i^ - = \frac{{{\sigma _i}}}{{{c_i}}}\;\;\;\sigma _i^ + = {\sigma _i} \times {c_i}\;\;\;{c_i} \in \left[ {0.3，1} \right]$ (9)

$F_{ji}^ + = {\beta _i} \times \frac{1}{{\sqrt {2\pi } \sigma _{ji}^ - }}\exp \left\{ { - \frac{{{{\left({{x_j} - {\mu _i}} \right)}^2}}}{{2\sigma _i^{ - 2}}}} \right\}$ (10)

$F_{ji}^ - = {\beta _i} \times \frac{1}{{\sqrt {2\pi } \sigma _{ji}^ + }}\exp \left\{ { - \frac{{{{\left({{x_j} - {\mu _i}} \right)}^2}}}{{2\sigma _i^{ + 2}}}} \right\}$ (11)

## 2.3 模糊决策模型

$F'{'_{ji}} = \frac{1}{{{N_j}}}\sum\limits_{k \in {N_j}} {\left({W_{ki}^ + F_{ki}^ + + {W_{ki}}{F_{ki}} + W_{ki}^ - F_{ki}^ - } \right)}$ (12)

$\begin{array}{*{20}{c}}{{A_{ki}} = \frac{1}{{{{\left({{y_{ki}} - {F_{ki}}} \right)}^2}}};}\\{A_{ki}^ + = \frac{1}{{{{\left({{y_{ki}} - F_{ki}^ + } \right)}^2}}};A_{ki}^ - = \frac{1}{{{{\left({{y_{ki}} - F_{ki}^ - } \right)}^2}}}}\end{array}$ (13)

${W_{ki}} = \left\{ {\begin{array}{*{20}{c}}{\frac{{{A_{ki}}}}{{A_{ki}^ - + {A_{ki}} + A_{ki}^ + }}}&{{y_{ji}} \ne {F_{ji}}}\\1&{{y_{ji}} = {F_{ji}}}\end{array}} \right.$ (14)

$W_{ji}^ - = \left\{ {\begin{array}{*{20}{c}}{\frac{{A_{ki}^ - }}{{A_{ki}^ - + {A_{ki}} + A_{ki}^ + }}}&{{y_{ji}} \ne F_{ji}^ - }\\1&{{y_{ji}} = F_{ji}^ - }\end{array}} \right.$ (15)

$W_{ji}^ + = \left\{ {\begin{array}{*{20}{c}}{\frac{{A_{ki}^ + }}{{A_{ki}^ - + {A_{ki}} + A_{ki}^ + }}}&{{y_{ji}} \ne F_{ji}^ + }\\1&{{y_{ji}} = F_{ji}^ + }\end{array}} \right.$ (16)

${F^*} = {\left[ {F'{'_{ji}}} \right]_{n \times m}}$ (17)

${Z_j} = {\arg _i}\left\{ {\max \left\{ {F'{'_{ji}}} \right\}} \right\}，\;\;\;i = 1，\cdots，m;j = 1，\cdots，n$ (18)

# 3 实验结果与讨论

## 3.2 合成影像

Table 1 Comparison of precision and Kappa value
 方法 精度指标 同质区域 调节因子 Ⅰ Ⅱ Ⅲ Ⅳ 最大似然 用户精度 0.857 0.986 0.721 0.603 产品精度 0.947 0.990 0.256 0.927 总精度=0.780；Kappa=0.707 区间二型 (标准差) 用户精度 0.924 0.992 0.776 0.719 c1=c2=c3=c4=0.4 产品精度 0.925 0.993 0.586 0.893 总精度=0.849；Kappa=0.800 邻域区间二型 (标准差) 用户精度 0.980 0.992 0.988 0.971 c1=c2=c3=c4=0.4 产品精度 0.995 0.999 0.953 0.985 总精度=0.983；Kappa=0.977 区间二型 (均值) 用户精度 0.924 0.992 0.763 0.728 α1=α2=α3=α4=3 产品精度 0.925 0.993 0.606 0.876 总精度=0.850；Kappa = 0.800 邻域区间二型 (均值) 用户精度 0.980 0.992 0.987 0.971 α1=α2=α3=α4=3 产品精度 0.995 0.998 0.953 0.984 总精度=0.983；Kappa=0.977

(1)图 3中，居民地和林区两种地物同质区域内灰度差异较大，几乎覆盖整个灰度区间，并且通过观察可以看出两类地物的灰度特征接近。图 4中第三类(居民地)和第四类(林区)两种地物的拟合模型存在相当大的重叠区域，这部分重叠区域存在的像素类属的不确定性显著，分割困难。

(2)图 5图 6中大部分训练数据直方图被包围在区间二型模糊模型FOU内，相对于一型模糊模型区间二型模糊模型提供的信息更加丰富。图 7图 8为没有融入空间关系最优模糊分割模型，该模型使用区间二型模糊模型的上、下隶属函数对拟合模型(一型模糊图像模型)进行全局约束，并以不规则形状拟合训练样本直方图。观察图 9(b)(c)(e)可以看出，基于区间二型模糊模型的分割方法(c)和(e)对Ⅲ类居民地的分割结果得到很大程度上的改善。表 1中的最大似然分割方法居民地的用户精度为0.721，而产品精度仅为0.256，对于此类地物基于区间二型模糊模型的分割其用户精度为0.763和0.776，产品精度为0.606和0.586，总体精度提高7%。而对Ⅰ类中，农田的不同块交接处误分情况(被误分为居民地)无法处理，并且两种方法都不能对Ⅳ类林区实现很好的划分。

(3)图 9(d)(f)为使用本文方法得到的分割结果，相对于以上两种方法，类别Ⅰ中，由于在二型模糊模型的基础上考虑了邻域像素的作用，Ⅰ类农田中只存在零星误分像素，Ⅱ类水域中的噪声被消除，Ⅲ类居民地和Ⅳ类林区的改善十分显著，对于居民地中存在的极少的其他像素，是因为居民区内会存在少量植被以及小块种植区，故有少量林区及农田像素存在，而房屋间的遮挡阴影灰度特征与水域显示，因此会出现被误分的零星水域像素。

(4)表 1中，使用最大似然分割其最低分割精度为0.256，总精度为0.780。使用基于区间二型模糊模型的分割，其最低分割精度为0.606和0.586，总精度为0.850，相对于最大似然精度有所提高。而本文方法由于融入邻域关系，最低分割精度为0.953，总精度达到0.983，分割精度提高显著，分割结果可靠。

# 参考文献

Segmentation of high-resolution remote sensing images with type-2 fuzzy model based on spatial relationship
WANG Chunyan1 , XU Aigong2 , LI Yu2 , SUI Xin2
1. School of Mining Industry and Technology, Liaoning Technical University, Huludao 125105, China;
2. School of Geomatics, Liaoning Technical University, Fuxin, 123000, China

# Abstract

Image segmentation is a significant step in image processing. High-resolution remote sensing images can clearly characterize landscape information and eliminate the membership uncertainty caused by mixed pixels. It has considerable advantages and potential in precise segmentation. Nonetheless, the spatial complexity caused by spectral measurement and the enhanced differences among pixels in the same object may reduce segmentation result accuracy. Thus, this study establishes an interval type-2 fuzzy model for detected images. Interval type-2 fuzzy theory considers the main and the secondary membership functions; the latter is represented by the label "membership 1" to express the uncertainty of pixel membership and of a segmentation decision. The aforementioned problem is solved effectively with the proposed method. The drop type problem in type-2 fuzzy theory is another focus of research in this field; a fuzzy decision model is established by reducing the type-2 fuzzy model to a type-1 model. The fuzzy decision model directly influences segmentation accuracy, and recent studies are all based on the upper and lower membership functions. These models can improve decision quality to some extent; however, they do not consider the main membership function. This neglect may significantly influence decision results, particularly when the influence of neighborhood pixels cannot be incorporated into the supervised image segmentation algorithm. To overcome these shortcomings, we proposed high-resolution remote sensing image segmentation by introducing a spatial relationship into the interval type-2 fuzzy model. The proposed algorithm considers the influences of the upper, lower, and the main membership functions in establishing the fuzzy decision model.First, a type-1 fuzzy model is built with the Gauss function to characterize the uncertainty of pixel membership. Then, we extend the mean and variance of this model to construct the type-2 fuzzy model, which improves the expression of the membership function in the type-1 fuzzy model and serves as the knowledge basis to enhance segmentation decision accuracy. A segmentation decision model is then established based on the information derived from the upper, lower, and main membership functions of the trained data. Finally, the membership of a pixel is decided by the membership functions of both the pixel itself and its neighbors to optimize the segmentation of high-resolution remote sensing images.We compare this method with maximum likelihood segmentation and an interval type-2 fuzzy model segmentation without a spatial relationship via high-resolution real images. Qualitative and quantitative analysis findings indicate that the method applied in this study generates high segmentation accuracy.This study proposes a supervised image segmentation method based on an interval type-2 fuzzy model with a spatial relationship. This method improves the uncertainty expression of pixel membership, solves the problems caused by complicated spatial relevance, and enhances the accuracy of the segmentation strategy. Furthermore, the experiments show that this method is effective and feasible. In the future, the Gauss mixture model will be used as a type-1 fuzzy model to potentially improve the accurate characterization of landscape features.

# Key words

interval type-2 fuzzy model; trajectory of uncertainty; high resolution; remote sensing image segmentation; secondary membership function