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发布时间: 2018-12-16
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DOI: 10.11834/jig.180332
2018 | Volume 23 | Number 12




    图像分析和识别    




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快速鲁棒核空间模糊聚类分割
expand article info 吴其平, 吴成茂
西安邮电大学电子工程学院, 西安 710121

摘要

目的 传统模糊C-均值聚类应用于图像分割仅考虑像素本身的聚类问题,无法克服噪声干扰对图像分割结果的影响,不利于受到噪声干扰的工业图像、医学影像和高分遥感影像等进行目标提取、识别和解译。嵌入像素空间邻域信息或局部信息的鲁棒模糊C-均值聚类分割算法是近年来图像分割理论研究中的热点课题。为此,针对现有的鲁棒核空间模糊聚类算法非常耗时且抑制噪声能力弱、不适合强噪声干扰下大幅面图像快速分割等问题,提出一种快速鲁棒核空间模糊聚类分割算法。方法 利用待分割图像中像素邻域的灰度信息和空间位置等信息构建线性加权滤波图像,对其进行鲁棒核空间模糊聚类。为了进一步提高算法实时性,引入当前聚类像素与其邻域像素均值所对应的2维直方图信息,构造一种基于2维直方图的鲁棒核空间模糊聚类快速分割最优化数学模型,采用拉格朗日乘子法获得图像分割的像素聚类迭代表达式。结果 对大幅面图像添加一定强度的高斯、椒盐以及混合噪声,以及未加噪标准图像的分割测试结果表明,本文算法比基于邻域空间约束的核模糊C-均值聚类等算法的峰值信噪比至少提高1.5 dB,误分率降低约5%,聚类性能评价的划分系数提高约10%,运行速度比核模糊C-均值聚类和基于邻域空间约束的鲁棒核模糊C-均值聚类算法至少提高30%,与1维直方图核空间模糊C-均值聚类算法具有相当的时间开销,所得分割结果具有较好的主观视觉效果。结论 通过理论分析和实验验证,本文算法相比现有空间邻域信息约束的鲁棒核空间模糊聚类等算法具有更强的抗噪鲁棒性、更优的分割性能和实时性,对大幅面遥感、医学等影像快速解译具有积极的促进作用,能更好地满足实时性要求较高场合的图像分割需要。

关键词

图像分割; 核函数; 模糊聚类; 线性加权和图像; 2维直方图; 鲁棒性; 聚类有效性

Fast robust kernel-based fuzzy C-means clustering segmentation
expand article info 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

0 引言

图像分割[1]是将图像分割成不同区域,即解决具有相似特性的像素分类问题,它是图像处理和图像分析的一个关键步骤。由于图像从3维目标投影为平面对象时会有信息损失,人眼对相邻灰度级的区分存在不确定性,从而使得模糊理论[2]在图像分割领域得到了广泛应用。目前,模糊C-均值(FCM)[3]聚类是图像分割中最常用的聚类方法之一,它采用欧氏距离构造聚类目标函数,将样本空间中相近的样本聚在一起,易于陷入局部极小值点且对初始化值较为敏感,主要适合于类样本数相差不悬殊的团状凸数据集,而对于那些非凸数据的聚类,其聚类性能显著下降[4]。而核模糊C-均值(KFCM)聚类算法[5],能有效解决非凸数据聚类问题,它将样本数据通过非线性映射至高维特征空间并改善样本的可分性,从而达到改善聚类性能的目的。但是,KFCM聚类应用于像素聚类分割时,未考虑到相邻像素之间的依赖关系,导致对噪声干扰图像的分割缺乏抗噪鲁棒性。为此,邻域信息KFCM(KFCM_S)[6-8]算法将图像的邻域信息引入到聚类目标函数中,从而克服噪声对图像分割效果的影响。但是,核函数的引入导致算法时间复杂度的增加, 不利于实时性要求较高的大幅面遥感等影像解译场合的应用需要。为此,Cai等人[9-10]提出了空间信息约束的快速FCM分割算法(FGFCM),该算法利用原始图像的每个像素邻域窗内的灰度和空间位置信息构造一个新的线性加权和图像,然后在该图像的灰度直方图上进行图像聚类,减少运行时间,并且提高了抑制噪声能力,但无法改善该类分割算法的分割性能,不利于诸如医学和遥感等复杂场合的图像分割需要。

为了提高鲁棒核空间模糊聚类分割法的分割性能和噪声抑制能力;同时,降低大幅面遥感或医学等影像分割的时间开销,本文将鲁棒核空间模糊聚类KFCM_S分割法和FGFCM分割算法相结合,提出一种改进的鲁棒核空间模糊聚类分割算法,并将像素与其邻域像素紧密关联的2维直方图引入新的鲁棒分割算法中,获得一种快速鲁棒核空间模糊聚类分割算法。测试结果表明,本文提出的算法能有效地提高大幅面图像分割速度;同时,相比现有的鲁棒核空间模糊聚类分割法有更加良好的分割性能。

1 核空间模糊聚类方法

传统FCM算法是基于平方欧氏距离的非监督聚类方法,主要适合团状数据聚类需要。为了增强FCM算法的适应性,将样本通过非线性变换$\mathit{\Phi }\left(x \right)$映射到高维希尔伯特再生核空间,可获得适合非凸数据聚类的KFCM算法,该算法所对应最优化模型为

$ \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)

式中,$n$表示样本数,$c$表示聚类数,${u_{i, k}}$表示第$i$个样本$x_i$属于第$k$类的模糊隶属度,$v_k$代表第$k$类的聚类中心,$m$是模糊指数且常选取1.5、2.0和2.5,$d_\mathit{\Phi }^2\left({{x_i}, {v_k}} \right) = {\left\| {\mathit{\Phi }\left({{x_i}} \right) - \mathit{\Phi }\left({{v_k}} \right)} \right\|^2}$表示第$i$个样本$x_i$所对应高维非线性映射像${\mathit{\Phi }\left({{x_i}} \right)}$与第$k$类的聚类中心$v_k$所对应高维非线性映射像${\mathit{\Phi }\left({{v_k}} \right)}$之间的平方欧氏距离。

利用非线性映射$\mathit{\Phi }\left(x \right)$和核函数$K\left({x, y} \right)$之间的内积关系$K\left({x, y} \right) = < \mathit{\Phi }\left(x \right), \mathit{\Phi }\left(y \right) > $,获得核空间样本与聚类之间的距离

$ \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)

式中,核函数$K\left({x, y} \right)$常选取全局多项式核函数、局部高斯核函数,以及多种核函数的加权组合等。但是,若选取一般核函数时,则最优化问题式(1)所对应迭代求解算法[11]的计算复杂度为${\rm{O}}\left({{n^2}} \right)$,并不适合大规模数据快速聚类。特别地,若选取高斯核函数${K_\sigma }\left({x, y} \right) = \exp \left({ - {\sigma ^{ - 2}}{{\left\| {x - y} \right\|}^2}} \right)$,则能获得更加简单的迭代表达式,即

$ 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)

式中,${K_\sigma }\left({{x_i}, v_k^{\left(t \right)}} \right) = \exp \left({ - {\sigma ^{ - 2}}{{\left\| {{x_i} - v_k^{\left(t \right)}} \right\|}^2}} \right)$$t$为迭代次数,$v_k$的计算可利用以下迭代法获取, 即

$ v_k^{\left( {t + 1} \right)} = \frac{{\sum\limits_{i = 1}^n {{{\left( {u_{i, k}^{\left( t \right)}} \right)}^m}\exp \left( { - {\sigma ^{ - 2}}{{\left\| {{x_i} - v_k^{\left( t \right)}} \right\|}^2}} \right){x_i}} }}{{\sum\limits_{i = 1}^n {{{\left( {u_{i, k}^{\left( t \right)}} \right)}^m}\exp \left( { - {\sigma ^{ - 2}}{{\left\| {{x_i} - v_k^{\left( t \right)}} \right\|}^2}} \right)} }} $ (4)

可将上述模型式(1)推广至1维直方图上获得一种快速核模糊$\text{C}$-均值(FGKFCM)聚类分割算法,时间复杂度由${\rm{O}}\left({n \times c \times r} \right)$变成${\rm{O}}\left({L \times c \times r} \right)$。因图像大小$n$远大于图像灰度级总数$L$,从而使得1维直方图上的核模糊聚类算法远快于核空间模糊聚类算法。

虽然核聚类通过非线性函数把输入空间样本映射到高维特征空间,改变不同类样本之间的可分性,增强模糊聚类性能。但是,将其用于图像像素聚类分割时,直接聚类像素而未考虑像素之间的关联性,导致核空间模糊聚类算法对噪声干扰图像分割的鲁棒性差,难以满足诸如医学、遥感等影像的分割需要。

2 鲁棒核空间模糊聚类分割法

为了增强核空间模糊聚类分割对噪声干扰图像的鲁棒性,文献[7]提出基于邻域空间信息约束的核模糊$\text{C}$-均值聚类算法(KFCM_S),其最优化模型描述目标函数为

$ \begin{array}{*{20}{c}} {\min J\left( {U, V} \right) = }\\ {\sum\limits_{i = 1}^n {\sum\limits_{k = 1}^c {u_{i, k}^m\left[ {d_\mathit{\Phi }^2\left( {{x_i}, {v_k}} \right) + \alpha d_\mathit{\Phi }^2\left( {{{\bar x}_i}, {v_k}} \right)} \right]} } } \end{array} $ (5)

式中,${\overline x _i}$表示当前像素$x_i$所对应的$\text{K}$近邻或邻域窗内样本的均值、中值或非局部均值等信息,$\alpha $为当前像素近邻或邻域窗内像素影响参数,常选取$\alpha $的值为大于0.5且小于1.0。特别地,当$\alpha $趋近于0时,鲁棒核空间模糊聚类分割算法退化为核空间模糊聚类分割算法。

针对聚类分割最优化模型式(5),利用拉格朗日乘子法将其转化为无约束优化问题,对无约束优化目标函数关于${u_{i, k}}$${\mathit{\Phi }\left({{v_k}} \right)}$求偏导数并令其等于零,可获得隶属度${u_{i, k}}$和非线性映射聚类中心${\mathit{\Phi }\left({{v_k}} \right)}$迭代求解的表达式为

$ {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)

针对非线性映射聚类中心式(7),由于非线性函数$\mathit{\Phi }\left(x \right)$是隐函数而不便于计算,利用式(2)和$K\left({x, y} \right) = < \mathit{\Phi }\left(x \right), \mathit{\Phi }\left(y \right) > $,可获得式(6)有效计算的相关核函数为

$ 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)

为了降低式(10)的计算时间开销,大量文献选取具有局部逼近能力的高斯核函数${K_\sigma }\left({x, y} \right) = \exp \left({ - {\sigma ^{ - 2}}{{\left\| {x - y} \right\|}^2}} \right)$($\sigma $是核函数参数),可获得迭代求解式(6)隶属度${u_{i, k}}$和聚类中心$v_k$表达式为

$ \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)

式中,核函数参数${\sigma _1}$${\sigma _2}$可利用被分割图像及其平滑图像的均方差来估计。

利用式(11)(12)可构造鲁棒核空间模糊聚类分割算法,该算法的时间复杂度为${\rm{O}}\left({n{w^2} + ncr} \right)$($r$为算法迭代次数,${{w^2}}$是当前样本$x_i$所对应的 $ \text{K}$近邻或邻域窗大小)。虽然该鲁棒聚类分割算法对噪声干扰具有一定的抑制能力;但是,难以满足强噪声干扰图像分割的需要。

3 改进鲁棒核空间模糊聚类分割法

为了提高模糊$\text{C}$-均值聚类进行图像分割的鲁棒性,文献[9]提出了图像邻域像素平滑滤波信息的鲁棒模糊聚类分割最优化模型

$ \min J\left( {U, V} \right) = \sum\limits_{i = 1}^n {\sum\limits_{k = 1}^c {u_{i, k}^m{d^2}\left( {{\xi _i}, {v_j}} \right)} } $ (13)

式中,${\xi _i}$是邻域信息线性加权和图像像素值,它是根据当前像素$x_i$所对应的邻域窗内所有像素估计的,即

$ {\xi _i} = \frac{{\sum\limits_{j \in {N_i}} {{S_{ij}}{x_j}} }}{{\sum\limits_{j \in {N_i}} {{S_{ij}}} }} $ (14)

式中,$S_{ij}$表示当前像素$x_i$所对应的邻域窗口样本$x_j$与当前样本$x_i$之间的灰度信息与空间信息所对应的相似性乘积,详细描述过程见文献[9]。

虽然最优化问题式(13)对应的鲁棒模糊聚类分割算法对噪声具有一定的抑制能力,但难以满足医学、遥感等复杂场合影像分割的需要。为此,将最优化模型式(5)(13)相结合,构造了一种更强噪声抑制能力的鲁棒核空间模糊聚类最优化模型

$ \begin{array}{*{20}{c}} {\min J\left( {U, V} \right) = }\\ {\sum\limits_{i = 1}^n {\sum\limits_{k = 1}^c {u_{i, k}^m\left[ {d_\mathit{\Phi }^2\left( {{\xi _i}, {v_k}} \right) + \alpha d_\mathit{\Phi }^2\left( {{{\bar \xi }_i}, {v_k}} \right)} \right]} } } \end{array} $ (15)

式中,${\xi _i}$是利用当前像素$x_i$所对应的邻域窗内像素估计的,${\xi _i}$是当前像素$x_i$所对应的邻域窗内所有估计${\xi _r}\left({r \in {N_i}} \right)$经均值、中值或非局部均值等估计所得。

针对最优化模型式(15),若核函数选取局部逼近能力的高斯核函数${K_\sigma }\left({x, y} \right) = \exp \left({ - {\sigma ^{ - 2}}{{\left\| {x - y} \right\|}^2}} \right)$,则可获得与式(11)(12)相类似的模糊隶属度${u_{i, k}}$和聚类中心$v_k$表达式。这种具有强噪声抑制能力的鲁棒模糊聚类分割算法需要对图像中每个像素进行多次重复遍历,导致算法时间开销仍较大,不利于幅面远大于256×256像素的图像快速分割需要。

4 快速鲁棒核空间模糊聚类分割法

为了提高像素邻域信息加权图像分割的快速性,文献[9]将直方图模糊聚类分割法[12]引入线性加权图像模糊聚类分割中并得到一种快速的鲁棒模糊聚类分割算法。为此,探索一种基于2维直方图的鲁棒核空间模糊聚类分割快速方法。

假设大小为$n$的灰度图像$\mathit{\boldsymbol{G}} = {\left({{g_l}} \right)_n}$,其灰度级数为$L$,采用3×3像素或5×5像素大小的邻域模板平滑所得图像为$\mathit{\boldsymbol{G'}} = {\left({{{g'}_l}} \right)_n}$,其相应灰度级数为$L$且大小为$n$。图像$\mathit{\boldsymbol{G}}$与平滑滤波图像$\mathit{\boldsymbol{G'}}$相同位置不同像素对所对应的2维直方图描述为

$ H\left( {i, j} \right) = \sum\limits_{l = 1}^n {\delta \left( {{g_l} - i} \right)\delta \left( {{{g'}_l} - j} \right)} $ (16)

它描述了图像$\mathit{\boldsymbol{G}}$与其平滑滤波图像$\mathit{\boldsymbol{G'}}$相同位置不同灰度级对的分布情况, 具体如图 1所示。

图 1 Lena图及其2维直方图
Fig. 1 Lena image and its two-dimensional histogram

图 1可见,该2维直方图描述了图像任意像素与其邻域像素滤波信息的空间分布关系,利用它可增强鲁棒模糊聚类等[13-16]分割算法对噪声干扰的抑制能力。针对鲁棒核空间模糊聚类分割最优化模型式(15),将2维直方图引入并得到等价模型为

$ \begin{array}{l} \begin{array}{*{20}{c}} {\min J\left( {U, V} \right) = }\\ {\sum\limits_{i = 0}^{L - 1} {\sum\limits_{j = 0}^{L - 1} {\sum\limits_{k = 1}^c {H\left( {i, j} \right)u_{i, k}^m\left( {d_\mathit{\Phi }^2\left( {i, {v_k}} \right) + \alpha d_\mathit{\Phi }^2\left( {j, {v_k}} \right)} \right)} } } } \end{array}\\ {\rm{s}}.\;{\rm{t}}{\rm{.}}\\ \left\{ \begin{array}{l} 0 \le {u_{i, k}} \le 1, 0 \le i \le L - 1, 1 \le k \le c\\ \sum\limits_{k = 1}^c {{u_{i, k}}} = 1, 0 \le i \le L - 1\\ 0 < \sum\limits_{i = 0}^{L - 1} {{u_{i, k}}} < L, 1 \le k \le c \end{array} \right. \end{array} $ (17)

式中,${H_{\left({i, j} \right)}} = \sum\limits_{l = 1}^n {\delta \left({{\xi _l} - i} \right)\delta \left({{{\overline \xi }_l} - j} \right)} $$n$表示图像总像素数。

针对最优化模型式(17),选取整体逼近或局部逼近的任意核函数或组合加权核函数,可获得与式(6)—式(10)类似的利用2维直方图信息的快速迭代的表达

$ {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)

式中,$d_\mathit{\Phi }^2\left({i, {v_k}} \right)$$d_\mathit{\Phi }^2\left({j, {v_k}} \right)$根据式(2)由相关的$K\left({i, {v_k}} \right)$$K\left({j, {v_k}} \right)$$K\left({{v_k}, {v_k}} \right)$计算得到

$ K\left( {i, {v_k}} \right) = \frac{{\sum\limits_{l = 0}^{L - 1} {\sum\limits_{j = 0}^{L - 1} {H\left( {l, j} \right)u_{l, k}^m\left( {K\left( {l, i} \right) + \alpha K\left( {j, i} \right)} \right)} } }}{{\sum\limits_{l = 0}^{L - 1} {\sum\limits_{j = 0}^{L - 1} {H\left( {l, j} \right)u_{l, k}^m\left( {1 + \alpha } \right)} } }} $

$ K\left( {j, {v_k}} \right) = \frac{{\sum\limits_{l = 0}^{L - 1} {\sum\limits_{r = 0}^{L - 1} {H\left( {l, r} \right)u_{l, k}^m\left( {K\left( {l, i} \right) + \alpha K\left( {r, j} \right)} \right)} } }}{{\sum\limits_{l = 0}^{L - 1} {\sum\limits_{r = 0}^{L - 1} {H\left( {l, r} \right)u_{l, k}^m\left( {1 + \alpha } \right)} } }} $

$ \begin{array}{*{20}{c}} {K\left( {{v_k}, {v_k}} \right) = }\\ {\frac{{\sum\limits_{{l_1} = 0}^{L - 1} {\sum\limits_{{r_1} = 0}^{L - 1} {\sum\limits_{l = 0}^{L - 1} {\sum\limits_{r = 0}^{L - 1} {H\left( {{l_1}, {r_1}} \right)H\left( {l, r} \right)u_{{l_1}, k}^mu_{l, k}^m\left( {K\left( {{l_1}, l} \right)} \right.} } } } }}{{{{\left( {\sum\limits_{l = 0}^{L - 1} {\sum\limits_{r = 0}^{L - 1} {H\left( {l, r} \right)u_{l, k}^m\left( {1 + \alpha } \right)} } } \right)}^2}}} + }\\ {\alpha \frac{{\sum\limits_{{l_1} = 0}^{L - 1} {\sum\limits_{{r_1} = 0}^{L - 1} {\sum\limits_{l = 0}^{L - 1} {\sum\limits_{r = 0}^{L - 1} {H\left( {{l_1}, {r_1}} \right)H\left( {l, r} \right)u_{{l_1}, k}^mu_{l, k}^m\left( {K\left( {{l_1}, r} \right)} \right.} } } } }}{{{{\left( {\sum\limits_{l = 0}^{L - 1} {\sum\limits_{r = 0}^{L - 1} {H\left( {l, r} \right)u_{l, k}^m\left( {1 + \alpha } \right)} } } \right)}^2}}} + }\\ {\alpha \frac{{\sum\limits_{{l_1} = 0}^{L - 1} {\sum\limits_{{r_1} = 0}^{L - 1} {\sum\limits_{l = 0}^{L - 1} {\sum\limits_{r = 0}^{L - 1} {H\left( {{l_1}, {r_1}} \right)H\left( {l, r} \right)u_{{l_1}, k}^mu_{l, k}^m\left( {K\left( {{r_1}, l} \right)} \right.} } } } }}{{{{\left( {\sum\limits_{l = 0}^{L - 1} {\sum\limits_{r = 0}^{L - 1} {H\left( {l, r} \right)u_{l, k}^m\left( {1 + \alpha } \right)} } } \right)}^2}}} + }\\ {{\alpha ^2}\frac{{\sum\limits_{{l_1} = 0}^{L - 1} {\sum\limits_{{r_1} = 0}^{L - 1} {\sum\limits_{l = 0}^{L - 1} {\sum\limits_{r = 0}^{L - 1} {H\left( {{l_1}, {r_1}} \right)H\left( {l, r} \right)u_{{l_1}, k}^mu_{l, k}^m\left( {K\left( {{r_1}, r} \right)} \right.} } } } }}{{{{\left( {\sum\limits_{l = 0}^{L - 1} {\sum\limits_{r = 0}^{L - 1} {H\left( {l, r} \right)u_{l, k}^m\left( {1 + \alpha } \right)} } } \right)}^2}}}} \end{array} $

式中,${u_{i, k}}$的计算需要利用$K\left({{v_k}, {v_k}} \right)$,而$K\left({{v_k}, {v_k}} \right)$的时间复杂度为${\rm{O}}\left({{L^4}} \right)$,若${L^4} \ll n$时,则2维直方图上鲁棒核空间模糊聚类分割法时间开销就会显著降低。但是,若满足${L^2} \ll n$,则可选择核函数为高斯核函数,得到复杂度为${\rm{O}}\left({{L^2}} \right)$的2维直方图上鲁棒核空间模糊聚类分割最优化模型,即

$ \begin{array}{*{20}{c}} {\min J\left( {U, V} \right)}\\ {\sum\limits_{i = 0}^{L - 1} {\sum\limits_{j = 0}^{L - 1} {\sum\limits_{k = 1}^c {H\left( {i, j} \right)u_{i, k}^m\left( {1 - {K_{{\sigma _1}}}\left( {i, {v_k}} \right)} \right)} } } + }\\ {\alpha \sum\limits_{i = 0}^{L - 1} {\sum\limits_{j = 0}^{L - 1} {\sum\limits_{k = 1}^c {H\left( {i, j} \right)u_{i, k}^m\left( {1 - {K_{{\sigma _2}}}\left( {j, {v_k}} \right)} \right)} } } } \end{array} $ (19)

式中,${\sigma _1} = {\left({\frac{1}{{n - 1}}\sum\limits_{i = 1}^n {{{\left({{d_i} - \hat d} \right)}^2}} } \right)^{\frac{1}{2}}}$$d_i$表示线性加权和图像$\mathit{\boldsymbol{G}} = {\left({{g_l}} \right)_n}$中像素值${\xi _i}$与其灰度均值$\hat \xi = \frac{1}{{n - 1}}\sum\limits_{i = 0}^{L - 1} {\sum\limits_{j = 0}^{L - 1} {\left({H\left({i, j} \right)i} \right)} } $之间的距离,${d_i} = \left\| {{\xi _i} - \hat \xi } \right\|$${\hat d}$$d_i$的均值,$\hat d = \frac{1}{n}\sum\limits_{i = 1}^n {{d_i}} $${\sigma _2} = {\left({\frac{1}{{n - 1}}\sum\limits_{i = 1}^n {{{\left({{{\overline d }_i} - \tilde d} \right)}^2}} } \right)^{\frac{1}{2}}}$${{{\overline d }_i}}$表示图像$\mathit{\boldsymbol{G}} = {\left({{g_l}} \right)_n}$所对应的邻域均值平滑图像$\mathit{\boldsymbol{\overline G}} = {\left({{{\overline \xi }_l}} \right)_n}$中像素值${{{\overline \xi }_i}}$和其灰度均值$\tilde \xi = \frac{1}{{n - 1}}\sum\limits_{i = 0}^{L - 1} {\sum\limits_{j = 0}^{L - 1} {H\left({i, j} \right)j} } $之间的距离${\overline d _i} = \left\| {{{\overline \xi }_i} - \tilde \xi } \right\|$${\tilde d}$${\overline d _i}$的均值,$\tilde d = \frac{1}{n}\sum\limits_{i = 1}^n {{{\overline d }_i}} $$n = {M_1} \times {N_1} = \sum\limits_{i = 0}^{L - 1} {\sum\limits_{j = 0}^{L - 1} {H\left({i, j} \right)} } $

利用约束优化问题求解的拉格朗日乘子法,可获得迭代优化求解所对应的隶属度和聚类中心表达式分别为

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

$ \begin{array}{*{20}{c}} {u_k^{\left( {t + 1} \right)} = }\\ {\frac{{\sum\limits_{i = 0}^{L - 1} {\sum\limits_{r = 0}^{L - 1} {H\left( {i, r} \right){{\left( {u_{i, k}^{\left( t \right)}} \right)}^m}\left( {{K_{{\sigma _1}}}\left( {i, v_k^{\left( t \right)}} \right)i + \alpha {K_{{\sigma _2}}}\left( {r, v_k^{\left( t \right)}} \right)r} \right)} } }}{{\sum\limits_{i = 0}^{L - 1} {\sum\limits_{r = 0}^{L - 1} {H\left( {i, r} \right){{\left( {u_{i, k}^{\left( t \right)}} \right)}^m}\left( {{K_{{\sigma _1}}}\left( {i, v_k^{\left( t \right)}} \right) + \alpha {K_{{\sigma _2}}}\left( {r, v_k^{\left( t \right)}} \right)} \right)} } }}} \end{array} $ (21)

利用式(20)(21)可构造鲁棒核空间模糊聚类分割快速算法,其详细步骤描述如下:

1) 利用文献[9]方法获取被分割原始图像$\mathit{\boldsymbol{G}} = {\left({{g_l}} \right)_n}$所对应的线性加权和图像$\mathit{\boldsymbol{\xi = }}{\left({{\xi _l}} \right)_n}$,以及对加权和图像$\mathit{\boldsymbol{\xi = }}{\left({{\xi _l}} \right)_n}$获取局部均值滤波图像$\mathit{\boldsymbol{\overline \xi }} \mathit{\boldsymbol{ = }}{\left({{{\overline \xi }_l}} \right)_n}$

2) 利用线性加权和图像$\mathit{\boldsymbol{\xi = }}{\left({{\xi _l}} \right)_n}$和局部均值滤波图像$\mathit{\boldsymbol{\overline \xi }} \mathit{\boldsymbol{ = }}{\left({{{\overline \xi }_l}} \right)_n}$获取不同位置像素对所对应的2维直方图$H\left({i, j} \right)\left({i, j = 0, 1, \cdots, L - 1} \right)$

3) 利用2维直方图$H\left({i, j} \right)$分别求出加权图像的高斯核函数参数${\sigma _1}$和局部均值滤波图像的高斯核函数参数${\sigma _2}$

4) 初始化聚类中心$v_k^{\left(0 \right)}$、选取聚类数$c$、模糊指数$m$和迭代终止误差$\varepsilon $,迭代次数$t$=0,聚类迭代次数最大次数$T$=1 000。

5) 利用式(20)计算不同灰度级的隶属度$u_{i, k}^{\left(t \right)}$

6) 采用式(21)更新聚类中心$v_k^{\left({t + 1} \right)}$

7) 若$\mathop {\max }\limits_{1 \le k \le c} \left\{ {\left| {v_k^{\left({t + 1} \right)} - v_k^{\left(t \right)}} \right|} \right\} > \varepsilon $$t + 1 < T$,执行迭代次数递增$t=t+1$,转步骤5);否则,聚类分割算法终止。

5 测试结果与分析

为了客观评价不同分割算法的分割性能,本文将误分率MR作为分割算法性能好坏的评定指标,该值越小表示分割结果与理想分割结果越相近,分割算法性能越好;反之,分割算法性能越差。为了定量描述分割算法的抗噪性能,本文将改进的峰值信噪比PSNR[17]作为分割算法噪声抑制能力的评价指标,该改进PSNR值越大表明分割算法抑制噪声能力越强。为了定量评价不同模糊聚类算法性能,本文将划分系数${V_{{\rm{PC}}}}$作为模糊聚类算法的评价指标[18],该值越大表明模糊聚类性能越好;反之,模糊聚类性能越差。为此,本文将聚类分割算法的误分率MR、改进的PSNR和模糊划分系数${V_{{\rm{PC}}}}$相结合,对不同模糊聚类分割算法的性能进行客观评价[19],以便评判本文提出的鲁棒快速模糊聚类分割算法具有潜在优势。

为了比较KFCM(算法1)、FGKFCM(算法2)、KFCM_S1(算法3)及本文算法的效果,选取大小为625×625像素的规则图、626×626像素的不规则图、514×514像素的河流遥感图和729×603像素的机场遥感图(如图 2所示),对其添加高斯噪声、椒盐噪声, 以及椒盐与高斯的混合噪声进行分割测试与分析。测试环境选取计算机OPTIPLEX 360, 处理器Intel(R) Core(TM)2 Duo CPU,工作频率2.66 GHz,内存2.0 GB, Matlab2014a。

图 2 灰度图像
Fig. 2 Grayscale images((a) synthetic regular image; (b) synthetic irregular image; (c) river remote sensing image; (d) airport remote sensing image)

为了后续叙述的方便,将KFCM、FGKFCM、KFCM_S1算法及本文算法分别简记为算法1、算法2、算法3及本文算法。

5.1 高斯噪声干扰图像测试与分析

针对图 2中4幅灰度图像,将其分别添加均值为0且均方差为81、67、57和81(归一化方差分别为0.1、0.07、0.05、0.1)的高斯噪声,采用算法1、算法2、算法3及本文算法对其分割测试,测试所得结果如图 3表 1表 2图 4所示。

图 3 高斯噪声干扰图及分割结果
Fig. 3 Images interfered by Gaussion noise and their segmentation results ((a) noise images; (b) algorithm 1; (c) algorithm 2; (d) algorithm 3; (e) the proposed algorithm)

表 1 不同算法抑制高斯噪声结果的信噪比及误分率
Table 1 Signal-to-noise ratio and misclassification rate of different algorithms to suppress Gaussian noise

下载CSV
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。

表 2 不同算法抑制高斯噪声结果的划分系数
Table 2 partition coefficient of different algorithms to suppress gaussion noise

下载CSV
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。
图 4 不同算法抑制高斯噪声过程的时间开销
Fig. 4 Time cost of different algorithms to suppress Gaussian noise

图 3所示的4种分割算法所得结果来看,前3种算法均不能有效抑制噪声,分割结果存在明显噪声,而本文算法克服了噪声对分割结果的影响,分割结果存在的噪声颗粒明显减少,且获得目标轮廓清晰,不模糊。从表 1所示的评价结果也可看出,本文算法相比算法1、算法2、算法3的峰值信噪比值(误分率)更高(更低),抗噪性能得到了明显增强。

表 2所示不同聚类算法抑制高斯噪声所对应聚类结果的划分系数值可看出,本文算法相比其他3种算法的划分系数值更大,表明本文算法的聚类性能优于其他3种算法。

图 4可见,本文算法与算法2的时间开销基本相当,相比算法1、算法3而言,本文算法有了明显减少,从而表明本文算法具有较高的执行效率。

5.2 椒盐噪声干扰图像测试与分析

针对图 2所示灰度图像,对其分别添加30%、25%、30%、30%的椒盐噪声,采用算法1、算法2、算法3和本文算法进行图像分割,测试所得结果如图 5表 3表 4图 6所示。

图 5 椒盐噪声干扰图像及分割结果
Fig. 5 Images interfered by salt and pepper noise and their segmentation results ((a) noise images; (b) algorithm 1; (c) algorithm 2; (d) algorithm 3; (e) the proposed algorithm)

表 3 不同算法抑制椒盐噪声结果的信噪比及误分率
Table 3 Signal-to-noise ratio and misclassification rate of different algorithms to suppress salt and pepper noise

下载CSV
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。

表 4 不同算法抑制椒盐噪声结果的划分系数
Table 4 Partition coefficient of different algorithms to suppress salt and pepper noise

下载CSV
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。
图 6 不同算法抑制椒盐噪声过程的时间开销
Fig. 6 Time cost of different algorithms to suppress salt and pepper noise

表 3的峰值信噪比值和误分率值可以看出,相比其他3种算法,本文算法具有更强的抗椒盐噪声能力。另外,根据图 5所示的4种分割算法所得效果图上看,算法1、算法2、算法3所得结果存在明显的噪声颗粒,部分细节信息丢失,而本文算法不仅有效抑制分割图像中的噪声,且分割结果能保留原图像中丰富的细节信息,获得的目标轮廓更为完整。

表 4中所示不同聚类算法抑制椒盐噪声所对应聚类结果的划分系数值可看出,本文算法相比其他3种算法的划分系数值更大,表明本文算法的聚类性能优于其他3种算法。

图 6所示的不同算法抑制椒盐噪声过程的时间开销可见,本文算法与算法2的时间开销基本相当,相比算法1、算法3而言,本文算法有了明显减少,从而表明本文算法具有较高的执行效率。

5.3 混合噪声干扰图像测试与分析

针对图 2所示的灰度图像,将其分别添加均值为0,且均方差分别为81、57、44以及81(归一化方差分别为0.1、0.05、0.03以及0.1)的高斯噪声和10%、10%、10%、10%的椒盐噪声,对其采用算法1、算法2、算法3和本文算法进行图像分割测试,测试所得结果如图 7表 5表 6图 8所示。

图 7 混合噪声干扰图及分割结果
Fig. 7 Images interfered by mixed noise and their segmentation results ((a) noise images; (b) algorithm 1; (c) algorithm 2; (d) algorithm 3; (e) the proposed algorithm)

表 5 不同算法抑制混合噪声结果的信噪比及误分率
Table 5 Signal-to-noise ratio and misclassification rate of different algorithms to suppress mixed noise

下载CSV
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。

表 6 不同算法抑制混合噪声结果的划分系数
Table 6 Partition coefficient of different algorithms to suppress mixed noise

下载CSV
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。
图 8 不同算法抑制混合噪声过程的时间开销
Fig. 8 Time cost of different algorithms to suppress mixed noise

图 7的分割结果可见,算法1、算法2、算法3所得结果存在的噪声污染仍比本文算法严重,获得的目标边缘轮廓模糊,而本文算法所获得的分割图像更为清晰且噪声颗粒明显减少。另外,从表 5的定量评价结果来看,本文算法较其他3种算法的峰值信噪比值(误分率)更高(低)。测试结果表明本文算法更适合高斯和椒盐混合噪声干扰图像分割的需要,说明本文算法具有更强的鲁棒抗噪性能。

表 6中所示不同聚类算法抑制混合噪声所对应聚类结果的划分系数值可看出,本文算法相比其他3种算法的划分系数值更大,表明本文算法的聚类性能优于其他3种算法。

图 8所示的不同算法抑制混合噪声过程的时间开销可见,本文算法与算法2的时间开销基本相当,相比算法1、算法3而言,本文算法明显减少,从而表明本文算法具有较高的执行效率。

5.4 无噪声干扰图像测试与分析

为进一步验证本文算法对大幅面医学和遥感等影像具有良好的分割性能、实时性和适应性,本文选取大小为507×621像素的CT切片图 1,512×512像素大小CT切片图 2,768×768像素大小遥感图 1和680×680像素大小遥感图 2,采用算法1、算法2、算法3及本文算法进行分割,测试所得分割结果如图 9表 7表 8图 10所示。

图 9 不同灰度图像及分割结果
Fig. 9 Segmentation results of different gray images ((a) original images; (b) algorithm 1; (c) algorithm 2; (d) algorithm 3; (e) the proposed algorithm)

表 7 不同分割算法分割结果的误分率
Table 7 Misclassification rate of different algorithms

下载CSV
/%
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。

表 8 不同聚类分割算法结果的划分系数
Table 8 Partition coefficient of different segmentation algorithms

下载CSV
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。
图 10 不同聚类分割算法的时间开销
Fig. 10 Time cost of different clustering algorithms

图 9所示的分割效果可以看出,本文算法相比算法1、算法2、算法3分割得到的图像轮廓更为清晰,且有效提取目标并保留原图像中丰富的细节信息,尤其是遥感图 1和遥感图 2,本文算法能够去除更多的杂乱背景,从而得到更为理想的分割效果。

表 7所示的4幅图像分割结果的误分率可见,本文算法相比其他3种聚类分割算法的误分率更低,表明本文算法相比而言具有良好的分割性能。另外,由表 8可知,本文算法分割测试结果的划分系数值明显大于其他3种算法,表明本文算法的模糊聚类性能优于其他3种算法,进一步证实本文算法具有良好的分割性能。

图 10所示不同聚类算法的时间开销可见,本文算法的时间开销和算法2相当,远比算法1、算法3的时间开销少很多。表明本文算法具有较好的实时性。

6 结论

为了增强鲁棒核空间模糊聚类分割算法的分割性能,噪声抑制能力和实时性,将原图像像素与其邻域像素信息相结合并得到线性加权和图像,对其进行改进的鲁棒核空间模糊聚类分割;同时,将2维直方图引入改进鲁棒核空间模糊聚类分割算法,获得一种快速鲁棒核空间模糊聚类分割算法。一定强度的高斯、椒盐和混合噪声干扰图像的分割测试结果表明,该算法相比于KFCM、FGKFCM以及KFCM_S1算法具有更强的抗噪鲁棒性,良好的分割性能和实时性,尽管运行速度和FGKFCM算法相近,但本文算法的分割效果远比FGKFCM算法更好,从而具有一定的应用前景。

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