发布时间: 2018-12-16 摘要点击次数: 全文下载次数: DOI: 10.11834/jig.180332 2018 | Volume 23 | Number 12 图像分析和识别

 收稿日期: 2018-05-23; 修回日期: 2018-07-19 基金项目: 国家自然科学基金项目（61671377，51709228）；陕西省自然科学基金项目（2017JM6107）；研究生创新基金项目（CXL2016-14） 第一作者简介: 吴其平, 1992年生, 女, 西安邮电大学电子工程学院电子与通信工程专业硕士研究生, 主要研究方向为图像处理。E-mail:1739565890@qq.com. 中图法分类号: TP391.41 文献标识码: A 文章编号: 1006-8961(2018)12-1838-14

# 关键词

Fast robust kernel-based fuzzy C-means clustering segmentation
Wu Qiping, Wu Chengmao
School of Electronic Engineering, Xi'an University of Posts and Telecommunications, Xi'an 710121, China
Supported by: National Natural Science Foundation of China (61671377, 51709228);Natural Science Foundation of Shaanxi Province, China (2017JM6107)

# Abstract

Objective In existing image segmentation theories and methods, the fuzzy clustering segmentation method has been widely studied and applied in image segmentation and object detection in images with intrinsic fuzziness and uncertainty. Fuzzy C-means clustering and its related improvement is a typical unsupervised clustering learning method, which has become an important tool in solving problems in image processing, machine vision, and remote sensing image interpretation. Many relevant scholars, local and abroad, have paid high attention to this research field for a long time. The classical fuzzy C-means clustering applied in image segmentation has only considered the clustering problem of pixels and cannot overcome the influence of noise on image segmentation results and thus cannot meet the requirements for target extraction, recognition, and interpretation of industrial, medical, and high-resolution remote sensing images. Hence, a robust fuzzy C-means clustering segmentation algorithm that is embedded in pixel spatial neighborhood or local information is a popular topic in image segmentation theory in recent years. The existing robust kernel-based fuzzy C-means clustering segmentation algorithms cannot meet the requirement of fast and robust segmentation of large images with strong interferences, such as Gaussian, salt-and-pepper, and mixed noise due to the large time cost and weak ability to suppress the aforementioned noise. Hence, a fast robust kernel-based fuzzy C-means clustering segmentation algorithm, which is based on a two-dimensional histogram, is proposed in view of pixel clustering accumulation. Method The linear weighted filtering image is first obtained by combining the gray and spatial location information of neighboring pixels with other related information in the segmenting gray image. The weighted filtering image is segmented by the existing robust kernel-based fuzzy C-means clustering to obtain positive segmentation quality and strong robustness against Gaussian, salt-and-pepper, and mixed noise. Second, to improve the operational efficiency of the robust fuzzy clustering segmentation algorithm in further segmenting large images, the two-dimensional histogram between the current clustered pixel and mean value of its neighborhood pixels is introduced into the existing robust kernel-based fuzzy C-means clustering method with spatial information. The two-dimensional histogram-based fuzzy clustering optimization mathematical model of the robust kernel-based fuzzy C-means clustering method is constructed in view of pixel clustering accumulation. In summary, the iterative expression of the gray level robust fuzzy clustering segmentation algorithm to segment image is rapidly deduced by the Lagrange multiplier method to solute the new optimization model of the robust kernel-based fuzzy clustering segmentation method embedded in the two-dimensional histogram of the weighted filtered image. Meanwhile, a fast robust kernel-based fuzzy C-means clustering segmentation in the two-dimensional histogram is designed to implement image segmentation. Result Segmentation results of the noiseless standard medical and large remote sensing images that are interrupted by Gaussian, salt-and-pepper, and mixed noise, show that the proposed fast algorithm in the two-dimensional histogram can obtain better segmentation performance than those of the existing robust kernel-based fuzzy C-means and other kernel-based fuzzy clustering segmentation algorithms. In detail, the proposed fast robust kernel-based fuzzy clustering segmentation algorithm has increased the peak signal-to-noise ratio by nearly 1.5 dB, reduced at least 5% of the misclassification rate, and increased approximately 10% of the partition coefficient compared with the existing robust kernel-based fuzzy clustering segmentation algorithms with spatial neighborhood information. In the meantime, the method can reduce the computational complexity to a considerable extent, and the running speed of the proposed robust kernel-based fuzzy clustering segmentation algorithm is similar to that of the classical kernel-based fuzzy C-means clustering segmentation method based on a one-dimensional histogram. Furthermore, the result of the proposed fast robust kernel-based fuzzy C-means clustering segmentation algorithm has better segmentation effects than those of the existing kernel-based fuzzy C-means clustering segmentation algorithms from the perspective of human visual perception. Conclusion Compared with the existing robust kernel-based fuzzy clustering segmentation algorithm, which is based on spatial constraints of neighborhood information, the kernel-based fuzzy C-means and classical fuzzy clustering segmentation algorithms and the proposed algorithm with spatial neighborhood formation not only have stronger ability against noise but has also improved segmentation performance and real time. Meanwhile, it has a positive effect on rapid interpretation of images, such as large high resolution remote sensing and medical images. The proposed algorithm can meet the requirements of large remote sensing and medical image segmentation with high real-time demands.

# Key words

image segmentation; kernel function; fuzzy clustering; linear weighted image; two-dimension histogram; robustness; clustering validity

# 1 核空间模糊聚类方法

 $\begin{array}{l} \min J\left( {U, V} \right) = \sum\limits_{i = 1}^n {\sum\limits_{k = 1}^c {u_{i, k}^md_\mathit{\Phi }^2\left( {{x_i}, {v_k}} \right)} } \\ {\rm{s}}.\;{\rm{t}}{\rm{.}}\\ \left\{ \begin{array}{l} 0 \le {u_{i, k}} \le 1, 1 \le i \le n, 1 \le k \le c\\ \sum\limits_{k = 1}^c {{u_{i, k}}} = 1, 1 \le i \le n\\ 0 < \sum\limits_{k = 1}^c {{u_{i, k}}} < n, 1 \le k \le c \end{array} \right. \end{array}$ (1)

 $\begin{array}{*{20}{c}} {d_\mathit{\Phi }^2\left( {{x_i}, {v_k}} \right) = K\left( {{x_i}, {x_i}} \right) - }\\ {2K\left( {{x_i}, {v_k}} \right) + K\left( {{v_k}, {v_k}} \right)} \end{array}$ (2)

 $u_{i,k}^{\left( t \right)} = \frac{{{{\left( {1 - {K_\sigma }\left( {{x_i},v_k^{\left( t \right)}} \right)} \right)}^{ - \frac{1}{{\left( {m - 1} \right)}}}}}}{{\sum\limits_{j = 1}^c {{{\left( {1 - {K_\sigma }\left( {{x_i},v_j^{\left( t \right)}} \right)} \right)}^{ - \frac{1}{{\left( {m - 1} \right)}}}}} }}$ (3)

 ${u_{i, k}} = \frac{{{{\left( {d_\mathit{\Phi }^2\left( {{x_i}, {v_k}} \right) + \alpha d_\mathit{\Phi }^2\left( {{{\bar x}_i}, {v_k}} \right)} \right)}^{ - \frac{1}{{\left( {m - 1} \right)}}}}}}{{\sum\limits_{j = 1}^c {{{\left( {d_\mathit{\Phi }^2\left( {{x_i}, {v_j}} \right) + \alpha d_\mathit{\Phi }^2\left( {{{\bar x}_i}, {v_j}} \right)} \right)}^{ - \frac{1}{{\left( {m - 1} \right)}}}}} }}$ (6)

 $\mathit{\Phi }\left( {{v_k}} \right) = \frac{{\sum\limits_{i = 1}^n {u_{i, k}^m\left( {\mathit{\Phi }\left( {{x_i}} \right) + \alpha \mathit{\Phi }\left( {{{\bar x}_i}} \right)} \right)} }}{{\sum\limits_{i = 1}^n {u_{i, k}^m\left( {1 + \alpha } \right)} }}$ (7)

 $K\left( {{x_i}, {v_k}} \right) = \frac{{\sum\limits_{l = 1}^n {u_{l, k}^m\left( {\mathit{K}\left( {{x_l}, {x_i}} \right) + \alpha \mathit{K}\left( {{{\bar x}_l}, {x_i}} \right)} \right)} }}{{\sum\limits_{l = 1}^n {u_{l, k}^m\left( {1 + \alpha } \right)} }}$ (8)

 $K\left( {{{\bar x}_i}, {v_k}} \right) = \frac{{\sum\limits_{l = 1}^n {u_{l, k}^m\left( {\mathit{K}\left( {{x_l}, {{\bar x}_i}} \right) + \alpha \mathit{K}\left( {{{\bar x}_l}, {{\bar x}_i}} \right)} \right)} }}{{\sum\limits_{l = 1}^n {u_{l, k}^m\left( {1 + \alpha } \right)} }}$ (9)

 $\begin{array}{*{20}{c}} {K\left( {{v_k}, {v_k}} \right) = \frac{{\sum\limits_{i = 1}^n {\sum\limits_{l = 1}^n {u_{l, k}^mu_{i, k}^m\left( {K\left( {{x_i}, {x_l}} \right) + \alpha \left( {K\left( {{x_i}, {{\bar x}_l}} \right)} \right)} \right.} } }}{{{{\left( {\sum\limits_{i = 1}^n {u_{i, k}^m\left( {1 + \alpha } \right)} } \right)}^2}}} + }\\ {\frac{{\sum\limits_{i = 1}^n {\sum\limits_{l = 1}^n {u_{l, k}^mu_{i, k}^m\left( {\alpha K\left( {{{\bar x}_i}, {x_l}} \right) + {\alpha ^2}K\left( {{{\bar x}_i}, {{\bar x}_l}} \right)} \right)} } }}{{{{\left( {\sum\limits_{i = 1}^n {u_{i, k}^m\left( {1 + \alpha } \right)} } \right)}^2}}}} \end{array}$ (10)

 $\begin{array}{*{20}{c}} {u_{i,k}^{\left( t \right)} = }\\ {\frac{{{{\left( {1 - {K_{{\sigma _1}}}\left( {{x_i},v_k^{\left( t \right)}} \right) + \alpha \left( {1 - {K_{{\sigma _2}}}\left( {{{\bar x}_i},v_k^{\left( t \right)}} \right)} \right)} \right)}^{ - \frac{1}{{\left( {m - 1} \right)}}}}}}{{\sum\limits_{j = 1}^c {{{\left( {1 - {K_{{\sigma _1}}}\left( {{x_i},v_j^{\left( t \right)}} \right) + \alpha \left( {1 - {K_{{\sigma _2}}}\left( {{{\bar x}_i},v_j^{\left( t \right)}} \right)} \right)} \right)}^{ - \frac{1}{{\left( {m - 1} \right)}}}}} }}} \end{array}$ (11)

 $\begin{array}{*{20}{c}} {v_k^{\left( {t + 1} \right)} = }\\ {\frac{{\sum\limits_{i = 1}^n {{{\left( {u_{i, k}^{\left( t \right)}} \right)}^m}\left( {{K_{{\sigma _1}}}\left( {{x_i}, v_k^{\left( t \right)}} \right){x_i} + \alpha {K_{{\sigma _2}}}\left( {{{\bar x}_i}, v_k^{\left( t \right)}} \right){{\bar x}_i}} \right)} }}{{\sum\limits_{i = 1}^n {{{\left( {u_{i, k}^{\left( t \right)}} \right)}^m}\left( {{K_{{\sigma _1}}}\left( {{x_i}, v_k^{\left( t \right)}} \right) + \alpha {K_{{\sigma _2}}}\left( {{{\bar x}_i}, v_k^{\left( t \right)}} \right)} \right)} }}} \end{array}$ (12)

 ${u_{i, k}} = \frac{{{{\left( {d_\mathit{\Phi }^2\left( {i, {v_k}} \right) + \alpha d_\mathit{\Phi }^2\left( {j, {v_k}} \right)} \right)}^{ - \frac{1}{{\left( {m - 1} \right)}}}}}}{{\sum\limits_{r = 1}^c {{{\left( {d_\mathit{\Phi }^2\left( {i, {v_r}} \right) + \alpha d_\mathit{\Phi }^2\left( {j, {v_r}} \right)} \right)}^{ - \frac{1}{{\left( {m - 1} \right)}}}}} }}$ (18)

# 5.1 高斯噪声干扰图像测试与分析

Table 1 Signal-to-noise ratio and misclassification rate of different algorithms to suppress Gaussian noise

 加噪图 指标 算法1 算法2 算法3 本文算法 合成规则图 信噪比/dB 10.443 3 15.932 4 15.538 2 20.382 1 误分率/% 32.38 30.73 32.32 27.17 合成不规则图 信噪比/dB 10.874 5 16.890 1 17.450 2 18.435 2 误分率/% 46.51 37.99 37.23 35.99 河流遥感图 信噪比/dB 3.826 5 11.464 8 4.729 3 13.206 6 误分率/% 51.51 22.04 44.75 19.77 机场遥感图 信噪比/dB 4.882 6 6.284 2 8.764 3 10.142 0 误分率/% 45.78 38.09 29.43 26.42 注：加粗字体为每行最优值。

Table 2 partition coefficient of different algorithms to suppress gaussion noise

 图像 指标 算法1 算法2 算法3 本文算法 规则图 ${V_{{\rm{PC}}}}$ 0.637 3 0.700 1 0.692 9 0.779 3 不规则图 ${V_{{\rm{PC}}}}$ 0.583 9 0.708 6 0.736 6 0.810 6 河流图 ${V_{{\rm{PC}}}}$ 0.515 1 0.736 9 0.610 5 0.895 6 机场图 ${V_{{\rm{PC}}}}$ 0.651 3 0.776 9 0.666 8 0.818 6 注：加粗字体为每行最优值。

# 5.2 椒盐噪声干扰图像测试与分析

Table 3 Signal-to-noise ratio and misclassification rate of different algorithms to suppress salt and pepper noise

 加噪图 指标 算法1 算法2 算法3 本文算法 合成规则图 信噪比/dB 8.166 6 14.467 7 12.969 16.945 4 误分率/% 48.53 43.21 450.11 39.14 合成不规则图 信噪比/dB 8.149 9 15.956 2 16.026 6 18.100 2 误类率/% 50.3 42.96 42.31 39.05 河流遥感图 信噪比/dB 2.820 9 5.943 2 4.335 2 8.524 2 误分率/% 61.08 36.45 47.52 27.36 机场遥感图 信噪比/dB 4.539 2 7.982 9 8.328 2 11.256 6 误分率/% 47.71 31.46 30.53 24.59 注：加粗字体为每行最优值。

Table 4 Partition coefficient of different algorithms to suppress salt and pepper noise

 图像 指标 算法1 算法2 算法3 本文算法 规则图 ${V_{{\rm{PC}}}}$ 0.753 2 0.860 3 0.800 6 0.920 5 不规则图 ${V_{{\rm{PC}}}}$ 0.561 3 0.744 9 0.712 3 0.901 1 河流图 ${V_{{\rm{PC}}}}$ 0.586 9 0.468 7 0.663 9 0.801 6 机场图 ${V_{{\rm{PC}}}}$ 0.512 5 0.736 5 0.661 0 0.860 8 注：加粗字体为每行最优值。

# 5.3 混合噪声干扰图像测试与分析

Table 5 Signal-to-noise ratio and misclassification rate of different algorithms to suppress mixed noise

 加噪图 指标 算法1 算法2 算法3 本文算法 合成规则图 信噪比/dB 8.955 5 15.145 7 14.465 2 17.753 7 误分率/% 45.67 42.59 47.54 36.31 合成不规则图 信噪比/dB 9.834 1 16.601 1 16.568 0 18.209 9 误分率/% 52.68 41.53 41.98 39.67 河流遥感图 信噪比/dB 3.802 9 5.442 9 4.937 6 8.499 3 误分率/% 51.94 40.43 43.42 27.67 机场遥感图 信噪比/dB 4.430 7 5.471 7 7.520 1 9.092 9 误分率/% 48.79 42.18 33.11 28.55 注：加粗字体为每行最优值。

Table 6 Partition coefficient of different algorithms to suppress mixed noise

 图像 指标 算法1 算法2 算法3 本文算法 规则图 ${V_{{\rm{PC}}}}$ 0.617 3 0.333 0 0.518 2 0.751 2 不规则图 ${V_{{\rm{PC}}}}$ 0.851 3 0.660 3 0.700 6 0.920 5 河流图 ${V_{{\rm{PC}}}}$ 0.536 9 0.736 9 0.610 5 0.811 9 机场图 ${V_{{\rm{PC}}}}$ 0.557 1 0.770 2 0.713 1 0.853 2 注：加粗字体为每行最优值。

# 5.4 无噪声干扰图像测试与分析

Table 7 Misclassification rate of different algorithms

 /% 加噪图 算法1 算法2 算法3 本文算法 CT图 1 37.91 37.83 37.32 36.54 CT图 2 36.25 24.57 34.57 19.38 遥感图 1 44.63 43.24 42.39 20.81 遥感图 2 22.16 22.47 20.55 17.91 注：加粗字体为每行最优值。

Table 8 Partition coefficient of different segmentation algorithms

 图像 指标 算法1 算法2 算法3 本文算法 CT图 1 ${V_{{\rm{PC}}}}$ 0.617 3 0.333 0 0.518 2 0.751 2 CT图 2 ${V_{{\rm{PC}}}}$ 0.851 3 0.660 3 0.700 6 0.920 5 遥感图 1 ${V_{{\rm{PC}}}}$ 0.536 9 0.736 9 0.610 5 0.811 9 遥感图 2 ${V_{{\rm{PC}}}}$ 0.557 1 0.7702 1 0.713 1 0.853 2 注：加粗字体为每行最优值。

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