CN106408587B - Regard SAR image segmentation method and device more - Google Patents

Regard SAR image segmentation method and device more Download PDF

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CN106408587B
CN106408587B CN201610833053.0A CN201610833053A CN106408587B CN 106408587 B CN106408587 B CN 106408587B CN 201610833053 A CN201610833053 A CN 201610833053A CN 106408587 B CN106408587 B CN 106408587B
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赵泉华
李晓丽
李玉
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Liaoning Technical University
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Abstract

本发明提供了一种多视SAR图像分割方法及装置,其中,该方法包括:读取待分割多视SAR图像;初始化双权重w;重复执行下述步骤:计算Gamma分布尺度参数β;计算上述图像的或然率p(z|w);计算上述双权重w的分布函数p(w);计算品质函数L;根据梯度法更新上述双权重w;将更新后的双权重w代入品质函数L;直到|L(t+1)‑L(t)|小于预设的阈值ε时,停止执行上述步骤,根据当前的双权重w确定上述图像中各个像素所属的类别;按照各个像素所属的类别输出分割结果。本发明分割图像的抗噪性能较好,分割结果误分现象较少,分割边界拟合精确。

The present invention provides a multi-view SAR image segmentation method and device, wherein the method includes: reading the multi-view SAR image to be segmented; initializing the double weight w; repeatedly performing the following steps: calculating the Gamma distribution scale parameter β; calculating the above The probability of the image p(z|w); calculate the distribution function p(w) of the above double weight w; calculate the quality function L; update the above double weight w according to the gradient method; substitute the updated double weight w into the quality function L; until When |L (t+1) ‑L (t) | is less than the preset threshold ε, stop performing the above steps, and determine the category of each pixel in the above image according to the current double weight w; output segmentation according to the category of each pixel result. The anti-noise performance of the segmented image of the invention is better, the phenomenon of misclassification of the segmented result is less, and the fitting of the segmented boundary is accurate.

Description

多视SAR图像分割方法及装置Multi-view SAR image segmentation method and device

技术领域technical field

本发明涉及图像分割技术领域,具体而言,涉及一种多视SAR图像分割方法及装置。The present invention relates to the technical field of image segmentation, in particular to a multi-view SAR image segmentation method and device.

背景技术Background technique

合成孔径雷达(Synthetic Aperture Radar,SAR)是一种高分辨率成像雷达,通过接收目标散射的电磁波并转换成图像以记录地物形态,其特有的成像机制所导致的固有斑点噪声给图像分割带来了巨大的困难。尽管多视技术能够减少部分噪声,但是,在实际应用中多视SAR图像仍然存在大量斑点噪声,因此,多视SAR图像分割方法的抗噪性及准确性一直是研究的热点问题。Synthetic Aperture Radar (SAR) is a high-resolution imaging radar that records the shape of ground objects by receiving the electromagnetic waves scattered by the target and converting them into images. The inherent speckle noise caused by its unique imaging mechanism gives the image segmentation a Great difficulty came. Although multi-look technology can reduce part of the noise, there are still a lot of speckle noise in multi-look SAR images in practical applications. Therefore, the anti-noise and accuracy of multi-look SAR image segmentation methods have always been a hot research issue.

目前,多视SAR图像分割的方法主要有:阈值法、边界法、聚类法及统计模型法等。其中,应用最广泛的为统计法,其通常采用混合模型刻画图像的复杂分布情况。混合模型中最常用的为高斯混合模型(Gaussian Mixture Model,GMM),其假设图像中像素的灰度值服从高斯分布,但是,传统GMM中的权重系数为用向量表示的仅与聚类相关的单权重,且SAR图像服从Gamma分布,上述问题均会导致传统GMM对SAR图像建模不够准确。At present, the methods of multi-view SAR image segmentation mainly include: threshold method, boundary method, clustering method and statistical model method. Among them, the most widely used is the statistical method, which usually uses a mixture model to describe the complex distribution of images. The most commonly used mixture model is the Gaussian Mixture Model (Gaussian Mixture Model, GMM), which assumes that the gray value of the pixel in the image obeys the Gaussian distribution. However, the weight coefficient in the traditional GMM is represented by a vector and only related to clustering. Single weight, and the SAR image obeys the Gamma distribution, the above problems will cause the traditional GMM to model the SAR image inaccurately.

针对上述多视SAR图像分割方法抗噪性能差及分割结果不理想的问题,目前尚未提出有效的解决方案。Aiming at the problems of poor anti-noise performance and unsatisfactory segmentation results of the multi-view SAR image segmentation method mentioned above, no effective solution has been proposed yet.

发明内容Contents of the invention

有鉴于此,本发明的目的在于提供一种多视SAR图像分割方法及装置,能够提高图像分割的抗噪性能且增强分割精度。In view of this, the object of the present invention is to provide a multi-view SAR image segmentation method and device, which can improve the anti-noise performance of image segmentation and enhance the segmentation accuracy.

为了实现上述目的,本发明实施例采用的技术方案如下:In order to achieve the above object, the technical solution adopted in the embodiment of the present invention is as follows:

第一方面,本发明实施例提供了一种多视SAR图像分割方法,包括:In a first aspect, an embodiment of the present invention provides a multi-view SAR image segmentation method, including:

读取待分割多视SAR图像;上述待分割多视SAR图像使用特征场z表示:Read the multi-view SAR image to be segmented; the above multi-view SAR image to be segmented is represented by the characteristic field z:

z={zi(xi,yi):i=1,…,n}z={z i (x i ,y i ):i=1,…,n}

其中,i为像素索引,n为总像素数,zi为像素i的强度,(xi,yi)∈D为像素i的格点位置,D为图像域;where i is the pixel index, n is the total number of pixels, z i is the intensity of pixel i, ( xi , y i ) ∈ D is the grid position of pixel i, and D is the image domain;

初始化双权重w:Initialize the dual weight w:

wi=(wil:l=1,…,k)w i =(w il :l=1,...,k)

其中,wil为类属l包含像素i的类属权重,满足k为类数;Among them, w il is the category weight of category l containing pixel i, which satisfies k is the number of classes;

重复执行下述步骤:计算Gamma分布尺度参数β:Repeat the following steps: Calculate the Gamma distribution scale parameter β:

其中,α为Gamma分布形态参数;Among them, α is the Gamma distribution shape parameter;

计算上述图像的或然率p(z|w);Calculate the probability p(z|w) of the above image;

其中p(zi|wil)为以双权重Gamma混合模型定义的zi的概率密度,如下:Where p(z i |w il ) is the probability density of z i defined by the double-weight Gamma mixture model, as follows:

计算上述双权重w的分布函数p(w):Calculate the distribution function p(w) of the above double weight w:

其中,A为归一化系数,η为邻域作用系数,Ni为以(xi,yi)为中心的8邻域像素集合,且满足i’∈Ni,i’≠i;Among them, A is the normalization coefficient, η is the neighborhood effect coefficient, N i is a set of 8 neighborhood pixels centered on ( xi , y i ), and satisfies i'∈N i , i'≠i;

计算品质函数L,上述品质函数L为或然率p(z|w)与分布函数p(w)联合概率分布函数的对数函数;Calculating the quality function L, the above-mentioned quality function L is the logarithmic function of the joint probability distribution function of the probability p(z|w) and the distribution function p(w);

根据梯度法更新上述双权重w:Update the above double weight w according to the gradient method:

w(t+1)=w(t)+ξΔw(t) w (t+1) = w (t) +ξΔw (t)

其中,t为迭代次数,ξ为步长,Δw(t)为梯度,如下:Among them, t is the number of iterations, ξ is the step size, and Δw (t) is the gradient, as follows:

将更新后的双权重w代入品质函数L;Substitute the updated double weight w into the quality function L;

直到|L(t+1)-L(t)|小于预设的阈值ε时,停止执行上述步骤,根据当前的双权重w确定上述图像中各个像素所属的类别;Until |L (t+1) -L (t) | is less than the preset threshold ε, stop performing the above steps, and determine the category to which each pixel in the above image belongs according to the current double weight w;

按照各个像素所属的类别输出分割结果。Output the segmentation result according to the class to which each pixel belongs.

结合第一方面,本发明实施例提供了第一方面的第一种可能的实施方式,其中,计算品质函数L包括:With reference to the first aspect, the embodiment of the present invention provides a first possible implementation manner of the first aspect, wherein calculating the quality function L includes:

计算或然率p(z|w)与分布函数p(w)联合概率分布函数p(z,w):Calculate the probability p(z|w) and the distribution function p(w) joint probability distribution function p(z,w):

p(z,w)=p(z|w)p(w);p(z,w)=p(z|w)p(w);

对该联合概率分布函数取对数:Take the logarithm of the joint probability distribution function:

L(w)=log p(z,w)=log p(z|w)+log p(w)。L(w)=log p(z,w)=log p(z|w)+log p(w).

结合第一方面,本发明实施例提供了第一方面的第二种可能的实施方式,其中,确定上述像素所属的类别包括:With reference to the first aspect, the embodiment of the present invention provides a second possible implementation manner of the first aspect, wherein determining the category to which the foregoing pixel belongs includes:

计算双权重的最大值:Compute the maximum of double weights:

ci=arg max{wil,l=1,...,k};c i =arg max{w il ,l=1,...,k};

其中,ci为第i个像素所属类别的标号;Among them, c i is the label of the category to which the i-th pixel belongs;

将上述最大值作为上述像素所属的类别。The above-mentioned maximum value is taken as the category to which the above-mentioned pixel belongs.

结合第一方面,本发明实施例提供了第一方面的第三种可能的实施方式,其中,按照各个像素所属的类别输出分割结果包括:用图像中同一类别中的所有像素的强度的均值更新该同一类别中的所有像素的强度,得到均值图像;通过显示装置显示该均值图像。In combination with the first aspect, the embodiment of the present invention provides a third possible implementation manner of the first aspect, wherein outputting the segmentation result according to the category to which each pixel belongs includes: updating with the mean value of the intensities of all pixels in the same category in the image The intensities of all pixels in the same category are obtained as an average image; the average image is displayed by a display device.

结合第一方面,本发明实施例提供了第一方面的第四种可能的实施方式,还包括:设置常数,上述常数包括:类数k、Gamma分布形态参数α、归一化常数A和邻域作用系数η。In combination with the first aspect, the embodiment of the present invention provides a fourth possible implementation of the first aspect, which also includes: setting constants, the above-mentioned constants include: the number of classes k, the shape parameter α of Gamma distribution, the normalization constant A and the neighbor Domain effect coefficient η.

第二方面,本发明实施例还提供了一种多视SAR图像分割装置,包括:读取模块,用于读取待分割多视SAR图像;上述待分割多视SAR图像使用特征场z表示:In the second aspect, the embodiment of the present invention also provides a multi-view SAR image segmentation device, including: a reading module for reading the multi-view SAR image to be segmented; the above-mentioned multi-view SAR image to be segmented is represented by a characteristic field z:

z={zi(xi,yi):i=1,…,n}z={z i (x i ,y i ):i=1,…,n}

其中,i为像素索引,n为总像素数,zi为像素i的强度,(xi,yi)∈D为像素i的格点位置,D为图像域;where i is the pixel index, n is the total number of pixels, z i is the intensity of pixel i, ( xi , y i ) ∈ D is the grid position of pixel i, and D is the image domain;

双权重初始化模块,用于初始化双权重w:The double weight initialization module is used to initialize the double weight w:

wi=(wil:l=1,…,k)w i =(w il :l=1,...,k)

其中,wil为类属l包含像素i的类属权重,满足k为类数;Among them, w il is the category weight of category l containing pixel i, which satisfies k is the number of classes;

权重参数迭代更新模块,用于重复执行下述计算:计算Gamma分布尺度参数β:The weight parameter iterative update module is used to repeatedly perform the following calculations: Calculate the Gamma distribution scale parameter β:

其中,α为Gamma分布形态参数;Among them, α is the Gamma distribution shape parameter;

计算图像的或然率p(z|w);Calculate the probability p(z|w) of the image;

其中p(zi|wil)为以双权重Gamma混合模型定义的zi的概率密度,如下:Where p(z i |w il ) is the probability density of z i defined by the double-weight Gamma mixture model, as follows:

计算上述双权重w的分布函数p(w):Calculate the distribution function p(w) of the above double weight w:

其中,A为归一化系数,η为邻域作用系数,Ni为以(xi,yi)为中心的8邻域像素集合,且满足i’∈Ni,i’≠i;Among them, A is the normalization coefficient, η is the neighborhood effect coefficient, N i is a set of 8 neighborhood pixels centered on ( xi , y i ), and satisfies i'∈N i , i'≠i;

计算品质函数L,上述品质函数L为或然率p(z|w)与分布函数p(w)联合概率分布函数的对数函数;Calculating the quality function L, the above-mentioned quality function L is the logarithmic function of the joint probability distribution function of the probability p(z|w) and the distribution function p(w);

根据梯度法更新上述双权重w:Update the above double weight w according to the gradient method:

w(t+1)=w(t)+ξΔw(t) w (t+1) = w (t) +ξΔw (t)

其中,t为迭代次数,ξ为步长,Δw(t)为梯度,如下:Among them, t is the number of iterations, ξ is the step size, and Δw (t) is the gradient, as follows:

将更新后的双权重w代入品质函数L;Substitute the updated double weight w into the quality function L;

直到|L(t+1)-L(t)|小于预设的阈值ε时,停止执行上述计算;Until |L (t+1) -L (t) | is less than the preset threshold ε, stop performing the above calculation;

类别确定模块,用于根据当前的双权重w确定图像中各个像素所属的类别;A category determination module, configured to determine the category to which each pixel in the image belongs according to the current double weight w;

输出模块,用于按照各个像素所属的类别对图像进行分割,输出分割结果。The output module is configured to segment the image according to the category to which each pixel belongs, and output the segmentation result.

结合第二方面,本发明实施例提供了第二方面的第一种可能的实施方式,其中,上述权重参数迭代更新模块包括:第一计算单元,用于计算或然率p(z|w)与分布函数p(w)的联合概率分布函数p(z,w):In combination with the second aspect, the embodiment of the present invention provides a first possible implementation manner of the second aspect, wherein the above-mentioned weight parameter iterative update module includes: a first calculation unit for calculating the probability p(z|w) and the distribution The joint probability distribution function p(z,w) of the function p(w):

p(z,w)=p(z|w)p(w);p(z,w)=p(z|w)p(w);

第二计算单元,用于对上述联合概率分布函数取对数:The second calculation unit is used to take the logarithm of the above joint probability distribution function:

L(w)=log p(z,w)=log p(z|w)+log p(w)。L(w)=log p(z,w)=log p(z|w)+log p(w).

结合第二方面,本发明实施例提供了第二方面的第二种可能的实施方式,其中,类别确定模块包括:最大值计算单元,用于计算双权重的最大值:In combination with the second aspect, an embodiment of the present invention provides a second possible implementation manner of the second aspect, wherein the category determination module includes: a maximum value calculation unit, configured to calculate the maximum value of the double weight:

ci=arg max{wil,l=1,...,k};c i =arg max{w il ,l=1,...,k};

其中,ci为第i个像素所属类别的标号;Among them, c i is the label of the category to which the i-th pixel belongs;

类别确定单元,用于将上述最大值作为像素所属的类别。A class determining unit, configured to use the above maximum value as the class to which the pixel belongs.

结合第二方面,本发明实施例提供了第二方面的第三种可能的实施方式,其中,上述输出模块包括:均值单元,用于图像中同一类别中的所有像素的强度的均值更新该同一类别中的所有像素的强度,得到均值图像;显示单元,用于通过显示装置显示该均值图像。In combination with the second aspect, the embodiment of the present invention provides a third possible implementation of the second aspect, wherein the above-mentioned output module includes: an average value unit, which is used to update the average value of the intensities of all pixels in the same category in the image The intensities of all pixels in the category are used to obtain an average image; the display unit is configured to display the average image through a display device.

结合第二方面,本发明实施例提供了第二方面的第四种可能的实施方式,还包括:常数设置模块,用于设置常数,上述常数包括:类数k、Gamma分布形态参数α、归一化常数A和邻域作用系数η。In combination with the second aspect, the embodiment of the present invention provides a fourth possible implementation manner of the second aspect, which also includes: a constant setting module for setting constants, the above constants include: class number k, Gamma distribution morphological parameter α, regression The normalization constant A and the neighborhood interaction coefficient η.

本发明实施例提供的多视SAR图像分割方法及装置,采用基于类属场表示的双权重Gamma混合模型对图像特征场进行建模;为了引入邻域关系,基于马尔科夫随机场模型(Markov Random Filed,MRF),结合误差平方和理论将中心像素类属权重视为均值,构造其与邻域像素类属权重差的平方和函数以描述邻域窗口内像素的差异性,用误差平方和表征差异能充分利用类属信息,增强了图像的分割精度。同时在估计Gamma混合模型中的双权重时,本发明实施例采用梯度法对双权重进行求解,迭代速度快、不易出现局部最优问题。The multi-view SAR image segmentation method and device provided by the embodiments of the present invention use a double-weight Gamma mixture model based on generic field representation to model the image feature field; in order to introduce neighborhood relations, a Markov random field model (Markov Random Filed, MRF), combined with the error sum of squares theory, regards the central pixel’s category weight as the mean value, and constructs the sum of squares function of the difference between it and the neighborhood pixel’s category weight to describe the difference of pixels in the neighborhood window, and uses the error sum of squares Representing the difference can make full use of the category information and enhance the segmentation accuracy of the image. At the same time, when estimating the dual weights in the Gamma mixture model, the embodiment of the present invention uses the gradient method to solve the dual weights, which has a fast iteration speed and is less prone to local optimal problems.

附图说明Description of drawings

为了更清楚地说明本发明实施例的技术方案,下面将对实施例中所需要使用的附图作简单地介绍,应当理解,以下附图仅示出了本发明的某些实施例,因此不应被看作是对范围的限定,对于本领域普通技术人员来讲,在不付出创造性劳动的前提下,还可以根据这些附图获得其他相关的附图。In order to illustrate the technical solutions of the embodiments of the present invention more clearly, the accompanying drawings used in the embodiments will be briefly introduced below. It should be understood that the following drawings only show some embodiments of the present invention, and thus It should be regarded as a limitation on the scope, and those skilled in the art can also obtain other related drawings based on these drawings without creative work.

图1示出了本发明实施例所提供的一种多视SAR图像分割装置的流程图;Fig. 1 shows a flow chart of a multi-view SAR image segmentation device provided by an embodiment of the present invention;

图2示出了本发明实施例所提供的一种多视SAR图像分割装置的结构示意图。Fig. 2 shows a schematic structural diagram of a multi-view SAR image segmentation device provided by an embodiment of the present invention.

具体实施方式Detailed ways

为使本发明实施例的目的、技术方案和优点更加清楚,下面将结合本发明实施例中附图,对本发明实施例中的技术方案进行清楚、完整地描述,显然,所描述的实施例仅仅是本发明一部分实施例,而不是全部的实施例。通常在此处附图中描述和示出的本发明实施例的组件可以以各种不同的配置来布置和设计。因此,以下对附图中提供的本发明的实施例的详细描述并非旨在限制要求保护的本发明的范围,而是仅仅表示本发明的选定实施例。基于本发明的实施例,本领域技术人员在没有做出创造性劳动的前提下所获得的所有其他实施例,都属于本发明保护的范围。In order to make the purpose, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only It is a part of embodiments of the present invention, but not all embodiments. The components of the embodiments of the invention generally described and illustrated in the figures herein may be arranged and designed in a variety of different configurations. Accordingly, the following detailed description of the embodiments of the invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely represents selected embodiments of the invention. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without making creative efforts belong to the protection scope of the present invention.

考虑到现有技术中的多视SAR图像分割方法存在的抗噪性能及准确性差的问题,本发明实施例提供了一种多视SAR图像分割方法及装置,该技术可以采用相应的软件和硬件实现。下面通过实施例进行描述。Considering the problem of poor anti-noise performance and poor accuracy in the multi-view SAR image segmentation method in the prior art, the embodiment of the present invention provides a multi-view SAR image segmentation method and device, which can use corresponding software and hardware accomplish. The following is described by way of examples.

实施例1Example 1

图1示出了本发明实施例提供的多视SAR图像分割方法的流程示意图。下面将对图1所示方法的具体流程进行详细阐述。Fig. 1 shows a schematic flowchart of a multi-view SAR image segmentation method provided by an embodiment of the present invention. The specific flow of the method shown in FIG. 1 will be described in detail below.

步骤S110,读取待分割多视SAR图像。Step S110, read the multi-view SAR image to be segmented.

该待分割多视SAR图像使用特征场z表示:The multi-view SAR image to be segmented is represented by the feature field z:

z={zi(xi,yi):i=1,…,n}z={z i (x i ,y i ):i=1,…,n}

其中,i为像素索引,n为总像素数,zi为像素i的强度,(xi,yi)∈D为像素i的格点位置,D为图像域。where i is the pixel index, n is the total number of pixels, z i is the intensity of pixel i, ( xi , y i )∈D is the grid position of pixel i, and D is the image domain.

步骤S120,初始化双权重w。Step S120, initialize the double weight w.

在本实施例中,定义双权重w表示像素与类属之间关系,具体如下:In this embodiment, a double weight w is defined to represent the relationship between a pixel and a category, as follows:

wi=(wil:l=1,…,k)w i =(w il :l=1,...,k)

其中,wil为类属l包含像素i的类属权重,满足k为类数。其中,初始双权重随机生成,行表示像素,列表示类别,矩阵中元素取值范围为0~1,并且满足每一行和为1。Among them, w il is the category weight of category l containing pixel i, which satisfies k is the number of classes. Among them, the initial double weight is randomly generated, the row represents the pixel, and the column represents the category. The value range of the elements in the matrix is 0-1, and the sum of each row is 1.

步骤S130,计算Gamma分布尺度参数β。Step S130, calculating the Gamma distribution scale parameter β.

在本实施例中,定义Gamma分布尺度参数βl为关于类属权重wil的函数,具体如下:In this embodiment, the Gamma distribution scale parameter β l is defined as a function of the category weight w il , specifically as follows:

其中,α为Gamma分布形态参数,对于多视SAR图像,α等于其视数。基于双权重,结合Gamma混合模型定义特征场概率分布,其中,混合模型中分布尺度参数与分布形态参数的乘积为均值。Among them, α is the shape parameter of Gamma distribution, and for multi-view SAR images, α is equal to the number of views. Based on the double weights, the probability distribution of the characteristic field is defined in combination with the Gamma mixture model, where the product of the distribution scale parameter and the distribution shape parameter in the mixture model is the mean value.

步骤S140,计算上述图像的或然率p(z|w)。Step S140, calculating the probability p(z|w) of the above image.

假设图像中各像素特征值相互独立,定义上述或然率为:Assuming that the eigenvalues of each pixel in the image are independent of each other, the above probability rate is defined as:

其中p(zi|wil)为以双权重Gamma混合模型定义的zi的概率密度,如下:Where p(z i |w il ) is the probability density of z i defined by the double-weight Gamma mixture model, as follows:

步骤S150,计算上述双权重w的分布函数p(w)。Step S150, calculating the distribution function p(w) of the above-mentioned double weight w.

具体地,以误差平方和理论为基础,结合MRF定义中心像素与其邻域像素差异性,进而得到双权重w的分布函数p(w)为:Specifically, based on the error sum of squares theory, combined with MRF to define the difference between the central pixel and its neighboring pixels, the distribution function p(w) of the double weight w is obtained as:

其中,A为归一化系数,控制聚类尺度;η为邻域作用系数,表征邻域影响强度;Ni为以(xi,yi)为中心的8邻域像素集合,且满足i’∈Ni,i’≠i。Among them, A is the normalization coefficient, which controls the clustering scale; η is the neighborhood effect coefficient, which characterizes the strength of neighborhood influence; N i is a set of 8 neighborhood pixels centered on ( xi , y i ), and satisfies i '∈N i , i'≠i.

步骤S160,计算品质函数L,该品质函数L为或然率p(z|w)与分布函数p(w)联合概率分布函数的对数函数。Step S160, calculating a quality function L, which is a logarithmic function of the joint probability distribution function of the probability p(z|w) and the distribution function p(w).

其中,计算品质函数L具体包括:Among them, the calculation of the quality function L specifically includes:

计算或然率p(z|w)与分布函数p(w)的联合概率分布函数p(z,w):Calculate the joint probability distribution function p(z,w) of the probability p(z|w) and the distribution function p(w):

p(z,w)=p(z|w)p(w);p(z,w)=p(z|w)p(w);

品质函数L定义为p(z,w)的对数函数,即对上式取对数:The quality function L is defined as the logarithmic function of p(z,w), that is, take the logarithm of the above formula:

L(w)=log p(z,w)=log p(z|w)+log p(w)。L(w)=log p(z,w)=log p(z|w)+log p(w).

步骤S170,根据梯度法更新上述双权重w。Step S170, updating the above-mentioned double weight w according to the gradient method.

w(t+1)=w(t)+ξΔw(t) w (t+1) = w (t) +ξΔw (t)

其中,t为迭代次数,ξ为步长,Δw(t)为梯度,该梯度定义为上述品质函数L的导数如下:Among them, t is the number of iterations, ξ is the step size, and Δw (t) is the gradient, which is defined as the derivative of the above quality function L as follows:

步骤S180,将更新后的双权重w代入品质函数L。Step S180, substituting the updated double weight w into the quality function L.

步骤S190,重复执行上述步骤S130-S180,直到|L(t+1)-L(t)|小于预设的阈值ε时,停止执行上述步骤,根据当前的双权重w确定图像中各个像素所属的类别。Step S190, repeat the above steps S130-S180, until |L (t+1) -L (t) | is less than the preset threshold ε, stop performing the above steps, and determine each pixel in the image according to the current double weight w category.

确定上述各个像素所属的类别具体包括以下步骤:Determining the category to which each of the above pixels belongs specifically includes the following steps:

(1)计算上述双权重的最大值:(1) Calculate the maximum value of the above double weights:

ci=arg max{wil,l=1,...,k};c i =arg max{w il ,l=1,...,k};

其中,ci为第i个像素所属类别的标号;Among them, c i is the label of the category to which the i-th pixel belongs;

(2)将该最大值作为该像素所属的类别。(2) Use the maximum value as the class to which the pixel belongs.

由上述步骤S120中双权重的定义可知,其是像素与类属的关系矩阵,每一行中最大值对应的列即为像素所属类别。From the definition of the double weight in step S120 above, it can be seen that it is a relationship matrix between pixels and categories, and the column corresponding to the maximum value in each row is the category to which the pixel belongs.

步骤S200,按照各个像素所属的类别输出分割结果。Step S200, outputting the segmentation result according to the class to which each pixel belongs.

具体地,通过上述步骤确定待分割图像的每个像素属于哪个类别后,将同一类中所有像素的强度(即像素的灰度值)求均值,将该均值作为此类的强度值即得到分割结果。用上述图像中同一类别中的所有像素的强度的均值更新该同一类别中的所有像素的强度,得到均值图像;通过显示装置显示该均值图像。Specifically, after determining which category each pixel of the image to be segmented belongs to through the above steps, the intensity of all pixels in the same category (that is, the gray value of the pixel) is averaged, and the average is used as the intensity value of this category to obtain the segmentation result. updating the intensities of all pixels in the same category with the mean value of the intensities of all pixels in the same category in the above image to obtain a mean value image; displaying the mean value image through a display device.

在本实施例的方法实际实施时,在输入待分割的图像后执行上述步骤前,还包括设置常数的步骤,该常数具体包括:类数k、Gamma分布形态参数α、归一化常数A和邻域作用系数η。When the method of this embodiment is actually implemented, before performing the above steps after inputting the image to be segmented, it also includes the step of setting a constant, which specifically includes: the number of classes k, the Gamma distribution morphological parameter α, the normalization constant A and Neighborhood effect coefficient η.

本发明实施例提供的多视SAR图像分割方法,通过采用基于类属场表示的双权重Gamma混合模型对图像特征场进行建模;为了引入邻域关系,基于马尔科夫随机场模型(Markov Random Filed,MRF),结合误差平方和理论将中心像素类属权重视为均值,构造其与邻域像素类属权重差的平方和函数以描述邻域窗口内像素的差异性,用误差平方和表征差异能充分利用类属信息,增强了图像的分割精度;同时在估计Gamma混合模型中的双权重时,本发明实施例采用梯度法对双权重进行求解。具体如下:In the multi-view SAR image segmentation method provided by the embodiment of the present invention, the image feature field is modeled by using a double-weight Gamma mixture model based on the generic field representation; Filed, MRF), combined with the error sum of squares theory, treat the central pixel category weight as the mean value, and construct the sum of squares function of the difference between it and the neighborhood pixel category weights to describe the difference of pixels in the neighborhood window, and use the error sum of squares to represent The difference can make full use of the category information and enhance the segmentation accuracy of the image; at the same time, when estimating the double weights in the Gamma mixture model, the embodiment of the present invention uses the gradient method to solve the double weights. details as follows:

(1)利用Gamma混合模型刻画SAR图像特征场的概率分布,以类属场表征像素与类属间关系,该关系用矩阵表示,并将该类属矩阵作为Gamma混合模型的双权重,从像素与聚类两个角度共同作用于Gamma分布,较传统混合模型中以向量表示的仅与聚类相关的单权重能够更好地刻画特征场;(1) Use the Gamma mixture model to describe the probability distribution of the SAR image feature field, and use the category field to represent the relationship between pixels and categories. The relationship is represented by a matrix, and the category matrix is used as the double weight of the Gamma mixture model. The two angles of clustering and clustering work together on the Gamma distribution, which can better describe the characteristic field than the single weight that is only related to clustering in the traditional mixed model;

(2)基于误差平方和及马尔科夫随机场(Markov Random Field,MRF)理论,以中心像素与其8邻域像素类属权重差的平方和刻画邻域窗口内特征差异程度,进而定义双权重的分布函数,当平方和越大时,说明窗口内像素特征差异越大,即其属于不同类别的概率越大;(2) Based on the error sum of squares and Markov Random Field (MRF) theory, the sum of the squares of the difference between the central pixel and its 8 neighboring pixels’ category weights is used to describe the degree of feature difference in the neighborhood window, and then the double weight is defined The distribution function of , when the sum of squares is larger, it means that the pixel feature difference in the window is greater, that is, the probability of belonging to different categories is greater;

(3)利用贝叶斯定理定义特征场和双权重的联合概率分布函数,以其对数函数作为品质函数,并利用梯度法求解双权重,以获得最小化品质函数对应的最佳估计值。(3) Use Bayesian theorem to define the joint probability distribution function of eigenfield and double weight, use its logarithmic function as the quality function, and use the gradient method to solve the double weight to obtain the best estimated value corresponding to the minimized quality function.

本发明实施例的效果可通过如下仿真实验进一步说明:The effects of the embodiments of the present invention can be further illustrated by the following simulation experiments:

(1)仿真实验条件(1) Simulation experiment conditions

本实施例在CPU为Core(TM)i5-3470 3.20GHz的Windows 7旗舰版系统上使用MATLAB 2011a软件编程实现仿真。In this embodiment, the simulation is realized by using MATLAB 2011a software programming on a Windows 7 ultimate system with Core(TM) i5-3470 3.20GHz CPU.

仿真数据1为模拟SAR图像,包含3个同质区域,其由标准模板图像添加形态参数为4,尺度参数分别为2、10、20的Gamma分布随机数得到。仿真数据2为真实SAR影像,包含2个同质区域,由于真实遥感图像无标准模板,因此将手绘模板视为标准模板。以上仿真数据图像大小均为128×128,图像总像素数n=16384。The simulation data 1 is a simulated SAR image, which contains 3 homogeneous regions, which are obtained by adding random numbers from the Gamma distribution with a shape parameter of 4 and a scale parameter of 2, 10, and 20, respectively, to the standard template image. The simulation data 2 is the real SAR image, which contains two homogeneous regions. Since the real remote sensing image has no standard template, the hand-painted template is regarded as the standard template. The image size of the above simulation data is 128×128, and the total number of pixels of the image is n=16384.

(2)仿真实验结果(2) Simulation results

为了证明本实施例算法的有效性,分别以模板图像为标准,对本实施例算法及对比算法分割结果生成混淆矩阵,并计算其用户精度、产品精度及Kappa值,以对本实施例方法进行定量分析(如表1所示),其中“-”表示图像中无此区域。由表1可知,本实施例的总精度及Kappa值均高于对比算法,从数字角度精确地验证了本实施例方法的有效性。本实施例的方法抗噪性能较好,分割结果误分现象较少,分割边界拟合精确;而对比算法不能有效克服复杂噪声,分割结果中误分像素较多,导致视觉效果极差。In order to prove the effectiveness of the algorithm of this embodiment, the template image is used as a standard to generate a confusion matrix for the segmentation results of the algorithm of this embodiment and the comparison algorithm, and calculate its user accuracy, product accuracy and Kappa value, so as to quantitatively analyze the method of this embodiment (as shown in Table 1), where "-" indicates that there is no such area in the image. It can be seen from Table 1 that the total precision and Kappa value of this embodiment are higher than those of the comparison algorithm, which accurately verifies the validity of the method of this embodiment from a numerical point of view. The method of this embodiment has better anti-noise performance, less misclassification of segmentation results, and accurate segmentation boundary fitting; while the comparison algorithm cannot effectively overcome complex noise, and there are many misclassified pixels in the segmentation results, resulting in extremely poor visual effects.

表1Table 1

实施例2Example 2

结合前述实施例,本实施例提供了一种多视SAR图像分割装置,参见图2所示的多视SAR图像分割装置的结构示意图,该装置包括:读取模块301、双权重初始化模块302、权重参数迭代更新模块303、类别确定模块304和显示模块305。In combination with the foregoing embodiments, this embodiment provides a multi-view SAR image segmentation device, referring to the schematic structural diagram of the multi-view SAR image segmentation device shown in FIG. 2 , the device includes: a reading module 301, a dual weight initialization module 302, Weight parameter iterative update module 303 , category determination module 304 and display module 305 .

具体介绍如下:The details are as follows:

读取模块301,用于读取待分割多视SAR图像。The reading module 301 is used for reading the multi-view SAR image to be segmented.

该待分割多视SAR图像使用特征场z表示:The multi-view SAR image to be segmented is represented by the feature field z:

z={zi(xi,yi):i=1,…,n}z={z i (x i ,y i ):i=1,…,n}

其中,i为像素索引,n为总像素数,zi为像素i的强度,(xi,yi)∈D为像素i的格点位置,D为图像域。where i is the pixel index, n is the total number of pixels, z i is the intensity of pixel i, ( xi , y i )∈D is the grid position of pixel i, and D is the image domain.

双权重初始化模块302,用于初始化双权重w。The dual weight initialization module 302 is configured to initialize the dual weight w.

在本实施例中,定义双权重w表示像素与类属之间关系,具体如下:In this embodiment, a double weight w is defined to represent the relationship between a pixel and a category, as follows:

wi=(wil:l=1,…,k)w i =(w il :l=1,...,k)

其中,wil为类属l包含像素i的类属权重,满足k为类数。其中,初始双权重随机生成,行表示像素,列表示类别,矩阵中元素取值范围为0~1,并且满足每一行和为1。Among them, w il is the category weight of category l containing pixel i, which satisfies k is the number of classes. Among them, the initial double weight is randomly generated, the row represents the pixel, and the column represents the category. The value range of the elements in the matrix is 0-1, and the sum of each row is 1.

权重参数迭代更新模块303,用于重复执行下述计算:The weight parameter iterative update module 303 is used to repeatedly perform the following calculations:

计算Gamma分布尺度参数β。Computes the gamma distribution scale parameter β.

定义Gamma分布尺度参数βl为关于类属权重wil的函数,具体如下:Define the Gamma distribution scale parameter β l as a function of the category weight w il , as follows:

其中,α为Gamma分布形态参数,对于多视SAR图像,α等于其视数。基于双权重,结合Gamma混合模型定义特征场概率分布,其中,混合模型中分布尺度参数与分布形态参数的乘积为均值。Among them, α is the shape parameter of Gamma distribution, and for multi-view SAR images, α is equal to the number of views. Based on the double weights, the probability distribution of the characteristic field is defined in combination with the Gamma mixture model, where the product of the distribution scale parameter and the distribution shape parameter in the mixture model is the mean value.

计算图像的或然率p(z|w)。Calculate the probability p(z|w) of the image.

假设图像中各像素特征值相互独立,定义上述或然率为:Assuming that the eigenvalues of each pixel in the image are independent of each other, the above probability rate is defined as:

其中p(zi|wil)为以双权重Gamma混合模型定义的zi的概率密度,如下:Where p(z i |w il ) is the probability density of z i defined by the double-weight Gamma mixture model, as follows:

计算上述双权重w的分布函数p(w)。Calculate the distribution function p(w) of the above double weight w.

具体地,以误差平方和理论为基础,结合MRF定义中心像素与其邻域像素差异性,进而得到双权重w的分布函数p(w)为:Specifically, based on the error sum of squares theory, combined with MRF to define the difference between the central pixel and its neighboring pixels, the distribution function p(w) of the double weight w is obtained as:

其中,A为归一化系数,控制聚类尺度;η为邻域作用系数,表征邻域影响强度;Ni为以(xi,yi)为中心的8邻域像素集合,且满足i’∈Ni,i’≠i。Among them, A is the normalization coefficient, which controls the clustering scale; η is the neighborhood effect coefficient, which characterizes the strength of neighborhood influence; N i is a set of 8 neighborhood pixels centered on ( xi , y i ), and satisfies i '∈N i , i'≠i.

计算品质函数L,上述品质函数L为或然率p(z|w)与分布函数p(w)联合概率分布函数的对数函数。Calculate the quality function L, which is the logarithmic function of the joint probability distribution function of the probability p(z|w) and the distribution function p(w).

上述权重参数迭代更新模块303包括:第一计算单元,用于计算或然率p(z|w)与分布函数p(w)的联合概率分布函数p(z,w):The above weight parameter iterative update module 303 includes: a first calculation unit for calculating the joint probability distribution function p(z,w) of the probability p(z|w) and the distribution function p(w):

p(z,w)=p(z|w)p(w);p(z,w)=p(z|w)p(w);

第二计算单元,用于对上式取对数:The second calculation unit is used to take the logarithm of the above formula:

L(w)=log p(z,w)=log p(z|w)+log p(w)。L(w)=log p(z,w)=log p(z|w)+log p(w).

根据梯度法更新上述双权重w。The above dual weight w is updated according to the gradient method.

w(t+1)=w(t)+ξΔw(t) w (t+1) = w (t) +ξΔw (t)

其中,t为迭代次数,ξ为步长,Δw(t)为梯度,该梯度定义为上述品质函数L的导数如下:Among them, t is the number of iterations, ξ is the step size, and Δw (t) is the gradient, which is defined as the derivative of the above quality function L as follows:

将更新后的双权重w代入品质函数L;Substitute the updated double weight w into the quality function L;

直到|L(t+1)-L(t)|小于预设的阈值ε时,停止执行上述步骤。Until |L (t+1) -L (t) | is less than the preset threshold ε, stop performing the above steps.

类别确定模块304,用于根据当前的双权重w确定图像中各个像素所属的类别。A category determining module 304, configured to determine the category to which each pixel in the image belongs according to the current dual weight w.

上述类别确定模块304包括:最大值计算单元,用于计算双权重的最大值:The above-mentioned category determination module 304 includes: a maximum value calculation unit, which is used to calculate the maximum value of the double weight:

ci=arg max{wil,l=1,...,k}c i =arg max{w il ,l=1,...,k}

其中,ci为第i个像素所属类别的标号;Among them, c i is the label of the category to which the i-th pixel belongs;

类别确定单元,用于将上述最大值作为像素所属的类别。A class determining unit, configured to use the above maximum value as the class to which the pixel belongs.

由上述双权重的定义可知,其是像素与类属的关系矩阵,每一行中最大值对应的列即为像素所属类别。From the definition of the above double weight, it can be seen that it is a relationship matrix between pixels and categories, and the column corresponding to the maximum value in each row is the category to which the pixel belongs.

输出模块305,用于按照各个像素所属的类别输出分割结果。The output module 305 is configured to output the segmentation result according to the category to which each pixel belongs.

具体地,上述输出模块305包括:Specifically, the above-mentioned output module 305 includes:

均值单元,用于图像中同一类别中的所有像素的强度的均值更新该同一类别中的所有像素的强度,得到均值图像;Mean value unit, for the mean value of the intensity of all pixels in the same category in the image to update the intensity of all pixels in the same category, to obtain the mean value image;

显示单元,用于通过显示装置显示该均值图像。The display unit is used for displaying the mean value image through a display device.

在本实施例的装置实际实施时,还包括:常数设置模块,用于设置常数,上述常数包括:类数k、Gamma分布形态参数α、归一化常数A和邻域作用系数η。When the device of this embodiment is actually implemented, it also includes: a constant setting module, which is used to set constants, and the above constants include: the number of classes k, the Gamma distribution shape parameter α, the normalization constant A and the neighborhood interaction coefficient η.

本实施例所提供的多视SAR图像分割装置的实现原理及产生的技术效果和前述实施例相同,为简要描述,本实施例部分未提及之处,可参考前述实施例中相应内容。The implementation principle and technical effects of the multi-view SAR image segmentation device provided in this embodiment are the same as those of the foregoing embodiments. For brief description, for parts not mentioned in this embodiment, reference may be made to the corresponding content in the foregoing embodiments.

上述功能如果以软件功能单元的形式实现并作为独立的产品销售或使用时,可以存储在一个计算机可读取存储介质中。基于这样的理解,本发明的技术方案本质上或者说对现有技术做出贡献的部分或者该技术方案的部分可以以软件产品的形式体现出来,该计算机软件产品存储在一个存储介质中,包括若干指令用以使得一台计算机设备(可以是个人计算机,服务器,或者网络设备等)执行本发明各个实施例方法的全部或部分步骤。而前述的存储介质包括:U盘、移动硬盘、只读存储器(ROM,Read-Only Memory)、随机存取存储器(RAM,Random Access Memory)、磁碟或者光盘等各种可以存储程序代码的介质。If the above functions are realized in the form of software functional units and sold or used as independent products, they can be stored in a computer-readable storage medium. Based on this understanding, the essence of the technical solution of the present invention or the part that contributes to the prior art or the part of the technical solution can be embodied in the form of a software product, and the computer software product is stored in a storage medium, including Several instructions are used to make a computer device (which may be a personal computer, a server, or a network device, etc.) execute all or part of the steps of the methods in various embodiments of the present invention. The aforementioned storage medium includes: U disk, mobile hard disk, read-only memory (ROM, Read-Only Memory), random access memory (RAM, Random Access Memory), magnetic disk or optical disk and other media that can store program codes. .

以上所述,仅为本发明的具体实施方式,但本发明的保护范围并不局限于此,任何熟悉本技术领域的技术人员在本发明揭露的技术范围内,可轻易想到变化或替换,都应涵盖在本发明的保护范围之内。因此,本发明的保护范围应以权利要求的保护范围为准。The above is only a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Anyone skilled in the art can easily think of changes or substitutions within the technical scope disclosed in the present invention. Should be covered within the protection scope of the present invention. Therefore, the protection scope of the present invention should be based on the protection scope of the claims.

Claims (10)

1. a kind of regarding SAR image segmentation method more, which is characterized in that including:
It reads to be split mostly regarding SAR image;It is described to be split mostly to be indicated using Characteristic Field z depending on SAR image:
Z={ zi(xi,yi):I=1 ..., n }
Wherein, i is pixel index, and n is total pixel number, ziFor the intensity of pixel i, (xi, yi) ∈ D be pixel i lattice site, D For image area;
The double weight w of initialization:
W=(wil:L=1 ..., k)
Wherein, wilInclude the generic weight of pixel i for generic l, meetsK is class number;
Repeat following step:Calculate Gamma distribution scale parameters βl
Wherein, α is Gamma distributional pattern parameters;
Calculate the probability p (z | w) of described image;
Wherein p (zi|wil) it is the z defined with double weight Gamma mixed modelsiProbability density, it is as follows:
Wherein Ga (zil) it is with ziFor independent variable, βlFor the Gamma distribution probability density functions of scale parameter, Γ (α) is with α For the Gamma functions of parameter;
Calculate the distribution function p (w) of double weight w:
Wherein, A is normalization coefficient, and η is neighborhood function coefficient, NiFor with (xi,yi) centered on 8 neighborhood territory pixel set, and it is full Sufficient i ' ∈ Ni,i'≠i;
Calculate merit functions L, the merit functions L is the probability p (z | w) and distribution function p (w) joint probability point The logarithmic function of cloth function;
Double weight w are updated according to gradient method:
w(t+1)=w(t)+ξΔw(t)
Wherein, t is iterations, and ξ is step-length, Δ w(t)It is as follows for gradient:
Updated double weight w are substituted into the merit functions L;
Until | L(t+1)-L(t)| when being less than preset threshold epsilon, stop executing above-mentioned steps, it is true according to current double weight w Determine the classification belonging to each pixel in described image;
Segmentation result is exported according to the classification belonging to each pixel.
2. according to the method described in claim 1, it is characterized in that, calculating merit functions L includes:
Calculate joint probability distribution function ps (z, w) of the probability p (z | w) with distribution function p (w):
P (z, w)=p (z | w) p (w);
Logarithm is taken to the joint probability distribution function:
L (w)=logp (z, w)=logp (z | w)+logp (w).
3. according to the method described in claim 1, it is characterized in that, determining that the classification belonging to the pixel includes:
Calculate the maximum value of double weights:
ci=arg max { wil, l=1 ..., k };
Wherein, ciFor the label of ith pixel generic;
Using the maximum value as the classification belonging to the pixel.
4. according to the method described in claim 1, it is characterized in that, according to the classification output segmentation knot belonging to each pixel Fruit includes:All pictures in the same category are updated with the mean value of the intensity of all pixels in same category in described image The intensity of element, obtains mean value image;The mean value image is shown by display device.
5. according to the method described in claim 1, it is characterized in that, further including:
Constant is set, and the constant includes:Class number k, Gamma distributional pattern parameter alpha, normaliztion constant A and neighborhood function coefficient η。
6. a kind of regarding SAR image segmenting device more, which is characterized in that including:
Read module, it is to be split mostly regarding SAR image for reading;It is described to be split mostly to be indicated using Characteristic Field z depending on SAR image:
Z={ zi(xi,yi):I=1 ..., n }
Wherein, i is pixel index, and n is total pixel number, ziFor the intensity of pixel i, (xi, yi) ∈ D be pixel i lattice site, D For image area;
Double weights initialisation modules, for initializing double weight w:
W=(wil:L=1 ..., k)
Wherein, wilInclude the generic weight of pixel i for generic l, meetsK is class number;
Weight parameter iteration update module, for repeating following step:Calculate Gamma distribution scale parameters βl
Wherein, α is Gamma distributional pattern parameters;
Calculate the probability p (z | w) of described image;
Wherein p (zi|wil) it is the z defined with double weight Gamma mixed modelsiProbability density, it is as follows:
Wherein Ga (zil) it is with ziFor independent variable, βlFor the Gamma distribution probability density functions of scale parameter, Γ (α) is with α For the Gamma functions of parameter;
Calculate the distribution function p (w) of double weight w:
Wherein, A is normalization coefficient, and η is neighborhood function coefficient, NiFor with (xi,yi) centered on 8 neighborhood territory pixel set, and it is full Sufficient i ' ∈ Ni,i'≠i;
Calculate merit functions L, the merit functions L is the probability p (z | w) and distribution function p (w) joint probability point The logarithmic function of cloth function;
Double weight w are updated according to gradient method:
w(t+1)=w(t)+ξΔw(t)
Wherein, t is iterations, and ξ is step-length, Δ w(t)It is as follows for gradient:
Updated double weight w are substituted into the merit functions L;
Until | L(t+1)-L(t)| when being less than preset threshold epsilon, stop executing above-mentioned steps;
Category determination module, for determining the classification in described image belonging to each pixel according to current double weight w;
Output module, for exporting segmentation result according to the classification belonging to each pixel.
7. device according to claim 6, which is characterized in that the weight parameter iteration update module includes:
First computing unit, the joint probability distribution function p (z, w) for calculating probability p (z | w) and distribution function p (w):
P (z, w)=p (z | w) p (w);
Second computing unit, for taking logarithm to the joint probability distribution function:
L (w)=logp (z, w)=logp (z | w)+logp (w).
8. device according to claim 6, which is characterized in that the category determination module includes:
Maximum value calculation unit, the maximum value for calculating double weights:
ci=arg max { wil, l=1 ..., k };
Wherein, ciFor the label of ith pixel generic;
Classification determination unit, for using the maximum value as the classification belonging to the pixel.
9. device according to claim 6, which is characterized in that the output module includes:
The mean value of equal value cell, the intensity for all pixels in same category in described image updates in the same category All pixels intensity, obtain mean value image;
Display unit shows the mean value image for passing through display device.
10. device according to claim 6, which is characterized in that further include:
Constant setup module, for constant to be arranged, the constant includes:Class number k, Gamma distributional pattern parameter alpha, normalization are normal Number A and neighborhood function coefficient η.
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