﻿ 一种融合邻域信息的模糊C-均值图像分割算法
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 智能系统学报  2019, Vol. 14 Issue (2): 273-280  DOI: 10.11992/tis.201712035 0

### 引用本文

DI Lan, LIU Haitao, HE Ruibo. Fuzzy C-means image segmentation algorithm incorporating neighborhood information[J]. CAAI Transactions on Intelligent Systems, 2019, 14(2): 273-280. DOI: 10.11992/tis.201712035.

### 文章历史

Fuzzy C-means image segmentation algorithm incorporating neighborhood information
DI Lan , LIU Haitao , HE Ruibo
College of Digital Media, Jiangnan University, Wuxi 214122, China
Abstract: The fuzzy C-means algorithm (FCM) is sensitive to image noise; in addition, it only considers the image numerical information and ignores the neighborhood spatial information, resulting in inaccurate final image segmentation result. To overcome this drawback, an FCM image segmentation algorithm is proposed in which the local information and non-local information of the image are integrated into a multidimensional model, which extends the original single dimension of clustering. In addition, a prior probability is introduced into the membership matrix, so that the neighborhood information of the pixel in the membership matrix is fully considered before each iteration, and then a neighborhood membership penalty is added to correct the clustering result. Finally, a penalty of neighborhood membership degree is used to modify the clustering results. Experimental results demonstrate that the algorithm is robust against noise and achieves an ideal image segmentation effect.
Key words: fuzzy C-means    image segmentation    spatial information    local information    non-local information    multidimensional model    neighborhood membership degree    penalty term

1 融入邻域信息的多测度图像模型 1.1 多测度模型

 Download: 图 1 多测度模型 Fig. 1 Multidimensional model
1.2 图像像素点相似度 ${S_{ij}}$

 ${S_{ij}} = \left\{ {\begin{array}{*{20}{l}} {\exp (\displaystyle\frac{{ - {{\left\| {{x_i} - {x_j}} \right\|}^2}}}{{{\sigma ^2}}})}, \,\,\,\,{i \ne j,{\sigma ^2} \ne 0}\\ 0,\,\,\,\, {i = j}\\ 1,\,\,\,\, {{\sigma ^2} = 0} \end{array}} \right.$ (1)
 $\sigma = \sqrt {\frac{{\sum\limits_{{x_j} \in {N_i}} {{{\left\| {{x_i} - {x_j}} \right\|}^2}} }}{{{N_R}}}}$ (2)

2 模糊C-均值聚类算法及应用 2.1 传统模糊C-均值聚类算法

FCM(fuzzy C-means)是一种经典的模糊聚类算法，该算法根据带聚类数据的隶属度将数据划分成几类，即将相似度较高的划分到一类中。

 ${u_{ij}} \in [0,1], \forall j = 1,2, \cdots ,n,\,\, \forall i = 1, 2, \cdots ,c$
 $\sum\limits_{i = 1}^c {{u_{ij}}} = 1, \;\; 0 < \sum\limits_{j = 1}^n {{u_{ij}}} < n$

FCM算法通过最小化目标函数来得到最优隶属度矩阵及聚类中心，一般表现形式为

 $J({{U}},{{V}}) = \sum\limits_{i = 1}^c {\sum\limits_{j = 1}^n {u_{ij}^m} {{\left\| {{x_j} - {v_i}} \right\|}^2}}$ (3)
 ${d_{ij}} = \left\| {{x_j} - {v_i}} \right\|$ (4)

 ${u_{ij}} = {\left[ {\sum\limits_{k = 1}^c {{{\left( {\frac{{{d_{ij}}}}{{{d_{kj}}}}} \right)}^{\frac{2}{{m - 1}}}}} } \right]^{ - 1}}$ (5)
 ${v_i} = \displaystyle\frac{{\sum\limits_{j = 1}^n {{u_{ij}}^m{x_j}} }}{{\sum\limits_{j = 1}^n {{u_{ij}}^m} }}$ (6)

2.2 FCM_S1与FCM_S2算法

 Download: 图 2 FCM算法处理结果 Fig. 2 results of FCM algorithm

 $J = \sum\limits_{i = 1}^c {\sum\limits_{j = 1}^n {u_{ij}^m} {{\left\| {{x_j} - {v_i}} \right\|}^2}} + \alpha \sum\limits_{i = 1}^c {\sum\limits_{j = 1}^n {u_{ij}^m} {{\left\| {{{\overline x }_j} - {v_i}} \right\|}^2}}$ (7)

2.3 FCM算法改进

2.3.1 引入先验概率的隶属度矩阵

 ${p_{ij}} = \frac{{N_R^i}}{{{N_R}}}$ (8)
 $u_{ij}^ * = \frac{{{u_{ij}}{p_{ij}}}}{{\sum\limits_{k = 1}^c {{u_{kj}}{p_{kj}}} }}$ (9)

2.3.2 邻域隶属度差异惩罚项及算法模型

 $G = \sum\limits_{j = 1}^n {\sum\limits_{i = 1}^c {\sum\limits_{k = 1}^n {{u^*}_{ij}^m{{\left( {1 - {u^*}_{ik}} \right)}^m}{Q_{jk}}} } }$ (10)

 ${{{Q}}_{jk}} = \left\{ {\begin{array}{*{20}{c}} {1,}\\ 0, \end{array}\begin{array}{l} { j{\text{与}}k{\text{相邻}},\;\;{\text{且}}j \ne k}\\ {j{\text{与}}k{\text{不相邻}}} \end{array}} \right.$ (11)

 $J = \sum\limits_{i = 1}^c {\sum\limits_{j = 1}^n {{u^{*m}_{ij}}} d_{ij}^2} + \lambda \sum\limits_{j = 1}^n {\sum\limits_{i = 1}^c {\sum\limits_{k = 1}^n {{u^{*m}_{ij}}{{\left( {1 - {u^{*}}_{ik}} \right)}^m}{{{Q}}_{jk}}} } }$ (12)

 ${u^*}_{ij} = {\left[ {\sum\limits_{r = 1}^c {{{\left( {\frac{{{d_{ij}}^2 + \lambda \sum\limits_{k = 1}^n {{{(1 - {u_{ik}})}^m}{Q_{jk}}} }}{{{d_{rj}}^2 + \lambda \sum\limits_{k = 1}^n {{{(1 - {u_{rk}})}^m}{Q_{jk}}} }}} \right)}^{\frac{1}{{m - 1}}}}} } \right]^{ - 1}}{p_{ij}}$ (13)
 ${v_i} = \frac{{\sum\limits_{j = 1}^n {{u^{*m}_{ij}}{x_j}} }}{{\sum\limits_{j = 1}^n {{u^{*m}_{ij}}} }}$ (14)
2.3.3 算法流程

1) 算法思想

2) 算法步骤

1)获取待分割图像 ${{I}}$ 的灰度值向量X= ${[{x_1}\,\,\,\,{x_2}\,\,\,\, \cdots\,\,\,\,{x_n}]^{\rm{T}}}$ 作为第一测度向量，采用均值滤波得到第二测度向量 ${{Y}} = {[{y_1}\,\,\,\,{y_2}\,\,\,\,\cdots \,\,\,\,{y_n}]^{\rm{T}}}$ ，通过考虑邻域相似度处理得到第三测度向量 ${{Z}} = {[{z_1}\,\,\,\,{z_2}\,\,\,\, \cdots \,\,\,\,{z_n}]^{\rm{T}}}$

2)取 ${\bf{Data}} = [{{X}};{{Y}};{{Z}}]$ 组成多测度模型；

3)根据人工视觉预先确定聚类个数，设定聚类中心个数c,隶属度模糊参数m=2，迭代停止条件 $\varepsilon = 0.01$ ，取 ${\textit{λ}} = 400$ (从文献[17]中获取)；

4)初始化隶属度矩阵；

5)根据初始化得到的隶属度矩阵计算多测度模型先验概率矩阵 ${{{p}}_{ij}}$

6)更新隶属度矩阵；

7)更新聚类中心；

8)若 $\left| {{v^k} - {v^{k + 1}}} \right| < \varepsilon$ ，即两次迭代差值小于 $\varepsilon$ ，算法终止；

9)根据最终聚类中心 ${{V}}$ ，以及隶属度 ${{U}}$ 第一维信息，完成最终的图像分割。

3 实验结果与分析

3.1 合成图像去噪声算法对比 3.1.1 合成图像添加椒盐噪声

 Download: 图 3 图像1加椒盐噪声效果对比 Fig. 3 Image 1 with pepper salt noise
 Download: 图 4 图像2加椒盐噪声效果对比 Fig. 4 Image 2 with pepper salt noise

 Download: 图 5 图像3加椒盐噪声效果对比 Fig. 5 Image 3 with pepper salt noise
 Download: 图 6 图像4加高斯噪声效果对比 Fig. 6 Image 4 with gaussian noise
 Download: 图 7 图像1加高斯噪声效果对比 Fig. 7 Image 1 with gaussian noise
3.1.2 合成图像添加高斯噪声

 ${\rm{SA}} = \frac{{{\text{分割正确像素点个数}}}}{{{\text{总像素点个数}}}} \times 100{\text{% }}$ (15)
3.1.3 合成图像去噪算法评价

3.2 自然图像去噪算法对比 3.2.1 自然图像添加椒盐噪声

 Download: 图 8 图像11加椒盐噪声效果对比 Fig. 8 Image 11 with pepper salt noise
 Download: 图 9 图像12加椒盐噪声效果对比 Fig. 9 Image 12 with pepper salt noise
 Download: 图 10 图像13加椒盐噪声效果对比 Fig. 10 Image 13 with pepper salt noise
 Download: 图 11 图像14加椒盐噪声效果对比 Fig. 11 Image 14 with pepper salt noise
3.2.2 自然图像添加高斯噪声

 Download: 图 12 图像11加高斯噪声效果对比 Fig. 12 Image 11 with gaussian noise
 Download: 图 13 图像12加高斯噪声效果对比 Fig. 13 Image 12 with gaussian noise
 Download: 图 14 图像13加高斯噪声效果对比 Fig. 14 Image 13 with gaussian noise
 Download: 图 15 图像14加高斯噪声效果对比 Fig. 15 Image 14 with gaussian noise
3.2.3 自然图像去噪算法比对评价

4 结束语

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