CN113779502A - A Correlation Vector Machine-Based Evidence Function Estimation Method for Image Processing - Google Patents

A Correlation Vector Machine-Based Evidence Function Estimation Method for Image Processing Download PDF

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CN113779502A
CN113779502A CN202110963746.2A CN202110963746A CN113779502A CN 113779502 A CN113779502 A CN 113779502A CN 202110963746 A CN202110963746 A CN 202110963746A CN 113779502 A CN113779502 A CN 113779502A
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邹大伟
马春华
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Abstract

本发明公开了一种基于相关向量机的图像处理证据函数估计方法。步骤1:将图像内的数据根据修正后的权参数先验形式,利用正态分布中均值与协方差,证明权参数的后验分布是正态分布;步骤2:将图像内的数据根据多变量泰勒公式、似然函数与权重的先验分布的乘积对权参数做积分得到证据函数的具体表达式,即边缘似然函数;步骤3:基于步骤2图像内的数据的边缘似然函数,利用矩阵微积分、矩阵代数以及最优化的方法最大化含有超参数的证据函数,从而得到图像的各个超参数的优化迭代算法。本发明用以解决图像处理中似然函数与权重的先验分布的乘积对权参数做积分求得证据函数的过程中,不得不面对复杂的积分,比较难计算的问题。

Figure 202110963746

The invention discloses an image processing evidence function estimation method based on a correlation vector machine. Step 1: Prove that the posterior distribution of the weight parameters is a normal distribution by using the mean and covariance in the normal distribution according to the modified prior form of the weight parameters; Step 2: Analyze the data in the image according to multiple The product of the variable Taylor formula, the likelihood function and the prior distribution of the weight is integrated with the weight parameter to obtain the specific expression of the evidence function, that is, the edge likelihood function; Step 3: Based on the edge likelihood function of the data in the image in step 2, Using matrix calculus, matrix algebra and optimization methods to maximize the evidence function containing hyperparameters, the optimal iterative algorithm for each hyperparameter of the image is obtained. The invention is used to solve the problem of complex integration and difficult calculation in the process of integrating the product of the likelihood function and the prior distribution of the weight to obtain the evidence function in image processing.

Figure 202110963746

Description

一种基于相关向量机的图像处理证据函数估计方法A Correlation Vector Machine Based Evidence Function Estimation Method for Image Processing

技术领域technical field

本发明属于图像处理领域,具体涉及一种基于相关向量机的图像处理证据函数估计方法。The invention belongs to the field of image processing, in particular to an image processing evidence function estimation method based on a correlation vector machine.

背景技术Background technique

在图像处理领域与相关向量机相关的证据函数的估计过程中,需要证明后验分布是正态分布,并且求正态分布的均值与协方差,目前传统方法不得不寻找完全平方项,这样的方法不但难操作,而且缺少逻辑。同时目前在图像处理领域的相关向量机相关的证据函数原理中,权参数的先验分布是一些均值为零的正态分布的乘积,这缺少一般性。In the process of estimating the evidence function related to the correlation vector machine in the field of image processing, it is necessary to prove that the posterior distribution is a normal distribution, and to find the mean and covariance of the normal distribution. At present, the traditional method has to find the perfect square term, such as The method is not only difficult to operate, but also lacks logic. At the same time, in the current evidence function principle related to correlation vector machines in the field of image processing, the prior distribution of weight parameters is the product of some normal distributions with zero mean, which lacks generality.

在似然函数与权重的先验分布的乘积对权参数做积分求得证据函数的过程中,不得不面对复杂的积分,比较难计算。如何找到一种图像处理的新方法与逻辑框架来更简单更有逻辑的求图像处理的相关积分,更有效地最大化证据函数,目前这方面研究还比较少。In the process of obtaining the evidence function by integrating the weight parameter with the product of the likelihood function and the prior distribution of the weight, it has to face the complex integration, which is difficult to calculate. How to find a new method and logical framework for image processing to obtain the correlation integral of image processing more simply and logically, and to maximize the evidence function more effectively, there are still relatively few researches in this area.

发明内容SUMMARY OF THE INVENTION

本发明提供一种基于相关向量机的图像处理证据函数估计方法,用以解决图像处理中似然函数与权重的先验分布的乘积对权参数做积分求得证据函数的过程中,不得不面对复杂的积分,比较难计算的问题。The present invention provides an image processing evidence function estimation method based on a correlation vector machine, which is used to solve the problem in the process of integrating the product of the likelihood function and the prior distribution of the weight in image processing to obtain the evidence function by integrating the weight parameters. For complex integrals, it is more difficult to calculate.

本发明通过以下技术方案实现:The present invention is achieved through the following technical solutions:

一种基于相关向量机的图像处理证据函数估计方法,所述证据函数估计方法包括以下步骤:A correlation vector machine-based image processing evidence function estimation method, the evidence function estimation method comprising the following steps:

步骤1:将图像内的数据根据修正后的权参数先验形式,利用正态分布中均值与协方差,证明权参数的后验分布是正态分布;Step 1: Prove that the posterior distribution of the weight parameters is a normal distribution by using the mean and covariance in the normal distribution according to the modified prior form of the weight parameters in the data in the image;

步骤2:将图像内的数据根据多变量泰勒公式、似然函数与权重的先验分布的乘积对权参数做积分得到证据函数的具体表达式,即边缘似然函数;Step 2: Integrate the data in the image with the weight parameter according to the multivariate Taylor formula, the product of the likelihood function and the prior distribution of the weight to obtain the specific expression of the evidence function, that is, the edge likelihood function;

步骤3:基于步骤2图像内的数据的边缘似然函数,利用矩阵微积分、矩阵代数以及最优化的方法最大化含有超参数的证据函数,从而得到图像的各个超参数的优化迭代算法。Step 3: Based on the edge likelihood function of the data in the image in Step 2, the method of matrix calculus, matrix algebra and optimization is used to maximize the evidence function containing hyperparameters, so as to obtain an optimization iterative algorithm for each hyperparameter of the image.

进一步的,所述步骤1具体为,当

Figure BDA0003223100000000011
其中x为标量,A为n×n可逆对称矩阵,Tr(·)为矩阵的迹、
Figure BDA0003223100000000021
Tr(xyT)=xTy,其中Tr(·)为矩阵的迹,x为向量,y为向量;算子
Figure BDA0003223100000000022
定义为
Figure BDA0003223100000000023
Figure BDA0003223100000000024
其中k∈N;Further, the step 1 is specifically, when
Figure BDA0003223100000000011
where x is a scalar, A is an n×n invertible symmetric matrix, Tr( ) is the trace of the matrix,
Figure BDA0003223100000000021
Tr(xy T )=x T y, where Tr( ) is the trace of the matrix, x is the vector, and y is the vector; the operator
Figure BDA0003223100000000022
defined as
Figure BDA0003223100000000023
and
Figure BDA0003223100000000024
where k∈N;

令h=(h1,h2)T

Figure BDA0003223100000000025
其中h1为向量h的第一个分量,h2为向量h的第二个分量,x1为向量x的第一个分量,x2为向量x的第二个分量,Let h=(h 1 , h 2 ) T and
Figure BDA0003223100000000025
where h 1 is the first component of the vector h, h 2 is the second component of the vector h, x 1 is the first component of the vector x, and x 2 is the second component of the vector x,

可得

Figure BDA0003223100000000026
Figure BDA0003223100000000027
Available
Figure BDA0003223100000000026
and
Figure BDA0003223100000000027

当算子

Figure BDA0003223100000000028
作用在x,得到when operator
Figure BDA0003223100000000028
acting on x, we get

Figure BDA0003223100000000029
Figure BDA0003223100000000029

Figure BDA00032231000000000210
Figure BDA00032231000000000210

则依据算子

Figure BDA00032231000000000211
对x作用的公式,可得then according to the operator
Figure BDA00032231000000000211
The formula acting on x can be obtained

Figure BDA00032231000000000212
Figure BDA00032231000000000212

Figure BDA00032231000000000213
Figure BDA00032231000000000213

其中h=(h1,h2)T

Figure BDA00032231000000000214
where h=(h 1 , h 2 ) T and
Figure BDA00032231000000000214

利用

Figure BDA00032231000000000215
可得use
Figure BDA00032231000000000215
Available

Figure BDA00032231000000000216
Figure BDA00032231000000000216

Figure BDA00032231000000000217
Figure BDA00032231000000000217

Figure BDA0003223100000000031
Figure BDA0003223100000000031

若f:B(w,r)→R被定义为f(w)=wTAw+wTb+c;其中A是n×n可逆对称矩阵,b与w是n维列向量,c是一个标量;f(w)的泰勒在

Figure BDA0003223100000000032
的泰勒展开式为If f:B(w,r)→R is defined as f(w)=w T Aw+w T b+c; where A is an n×n invertible symmetric matrix, b and w are n-dimensional column vectors, and c is a scalar; the Taylor of f(w) in
Figure BDA0003223100000000032
The Taylor expansion of , is

Figure BDA0003223100000000033
Figure BDA0003223100000000033

H的第(i,j)元素由

Figure BDA0003223100000000034
定义;The (i,j)th element of H is given by
Figure BDA0003223100000000034
definition;

机器学习中的的线性回归模型的一般形式为The general form of a linear regression model in machine learning is

Figure BDA0003223100000000035
Figure BDA0003223100000000035

其中φi(x)为输入变量的非线性基函数,w0为偏差参数,x为图像数据向量;where φ i (x) is the nonlinear basis function of the input variable, w 0 is the deviation parameter, and x is the image data vector;

定义φ0(x)=1,从而(1)式可重写为Define φ 0 (x)=1, so equation (1) can be rewritten as

Figure BDA0003223100000000036
Figure BDA0003223100000000036

其中w=(w0,…,wM-1)T与φ(x)=(φ0(x),…,φM-1(x))Twhere w=(w 0 ,...,w M-1 ) T and φ(x)=(φ 0 (x),...,φ M-1 (x)) T ;

目标函数是带有加性高斯噪声的确定性函数y(x,w),即The objective function is a deterministic function y(x,w) with additive Gaussian noise, i.e.

t=y(x,w)+ε (3)t=y(x,w)+ε (3)

其中ε是0均值,精度为β的正态随机变量,从而获得where ε is a normal random variable with 0 mean and precision β, thus obtaining

p(t|x,w,β)=N(t|y(x,w),β-1) (4)。p(t|x,w,β)=N(t|y(x,w),β −1 ) (4).

进一步的,所述步骤1中权参数先验取如下的形式Further, in the step 1, the weight parameter prior takes the following form

Figure BDA0003223100000000037
Figure BDA0003223100000000037

其中α为精度(方差的逆)向量,α=(α1,…,αM)T与γ为均值向量,γ=(γ1,…,γM)Twhere α is the precision (inverse of variance) vector, α=(α 1 ,...,α M ) T and γ are the mean vector, γ=(γ 1 ,...,γ M ) T .

利用公式(4),获得似然函数Using formula (4), the likelihood function is obtained

Figure BDA0003223100000000041
Figure BDA0003223100000000041

其中,

Figure BDA0003223100000000042
M为所要确定的参数的个数。in,
Figure BDA0003223100000000042
M is the number of parameters to be determined.

相似地,Similarly,

Figure BDA0003223100000000043
Figure BDA0003223100000000043

其中α=diag(αi);where α=diag(α i );

获得权参数的后验分布p(w|t,X,α,β,γ)=N(w|m,Σ)也是正态分布,其中The posterior distribution p(w|t,X,α,β,γ)=N(w|m,Σ) for obtaining the weight parameters is also a normal distribution, where

m=(A+βΦTΦ)-1(Aγ+βΦTt) (8)m=(A+βΦ T Φ) -1 (Aγ+βΦ T t) (8)

Σ=(A+βΦTΦ)-1 (9)Σ=(A+βΦ T Φ) -1 (9)

其中

Figure BDA0003223100000000044
in
Figure BDA0003223100000000044

进一步的,所述步骤2具体为,Further, the step 2 is specifically:

p(t|X,α,β,γ)=∫p(t|X,w,β)p(w|α,γ)dw (10)p(t|X,α,β,γ)=∫p(t|X,w,β)p(w|α,γ)dw (10)

利用公式(6)和公式(7),可得Using formula (6) and formula (7), we can get

Figure BDA0003223100000000045
Figure BDA0003223100000000045

其中in

Figure BDA0003223100000000051
Figure BDA0003223100000000051

Figure BDA0003223100000000052
得到w=(A+βΦTΦ)-1(βΦTt+Aγ)=m,
Figure BDA0003223100000000053
可得make
Figure BDA0003223100000000052
Obtain w=(A+βΦ T Φ) -1 (βΦ T t+Aγ)=m,
Figure BDA0003223100000000053
Available

Figure BDA0003223100000000054
Figure BDA0003223100000000054

其中in

Figure BDA0003223100000000055
Figure BDA0003223100000000055

利用公式(11)与公式(12),可得Using formula (11) and formula (12), we can get

Figure BDA0003223100000000056
Figure BDA0003223100000000056

其中m=(A+βΦTΦ)-1(βΦTt+Aγ),

Figure BDA0003223100000000057
X X=(x1,x2,…,xN)。where m=(A+βΦ T Φ) -1 (βΦ T t+Aγ),
Figure BDA0003223100000000057
XX=(x 1 ,x 2 ,...,x N ).

进一步的,所述步骤3具体为,Further, the step 3 is specifically,

对公式(13)式取对数可得Taking the logarithm of formula (13), we can get

Figure BDA0003223100000000058
Figure BDA0003223100000000058

利用公式(9),公式(14)以及

Figure BDA0003223100000000059
可得Using Equation (9), Equation (14) and
Figure BDA0003223100000000059
Available

Figure BDA00032231000000000510
Figure BDA00032231000000000510

因为

Figure BDA0003223100000000061
because
Figure BDA0003223100000000061

利用

Figure BDA0003223100000000062
以及公式(15),可得use
Figure BDA0003223100000000062
and formula (15), we can get

Figure BDA0003223100000000063
Figure BDA0003223100000000063

由公式(16)可得According to formula (16), we can get

Figure BDA0003223100000000064
Figure BDA0003223100000000064

其中Σii是后验协方差Σ的主对角线的第i个元素;where Σ ii is the ith element of the main diagonal of the posterior covariance Σ;

由公式(12),可得From formula (12), we can get

Figure BDA0003223100000000065
Figure BDA0003223100000000065

其中mi是后验均值m的第i个分量;where m i is the ith component of the posterior mean m;

由公式(17)与公式(18),可得From formula (17) and formula (18), we can get

Figure BDA0003223100000000066
Figure BDA0003223100000000066

由(19)可得

Figure BDA0003223100000000067
从而可得It can be obtained from (19)
Figure BDA0003223100000000067
thus obtainable

Figure BDA0003223100000000068
Figure BDA0003223100000000068

其中λi=1-αiΣiiwhere λ i =1-α i Σ ii ;

根据公式(9)的Σ的定义以及

Figure BDA0003223100000000069
可得The definition of Σ according to equation (9) and
Figure BDA0003223100000000069
Available

Figure BDA00032231000000000610
Figure BDA00032231000000000610

根据

Figure BDA00032231000000000611
与Tr(xyT)=xTy,可得according to
Figure BDA00032231000000000611
With Tr(xy T )=x T y, we can get

Figure BDA00032231000000000612
Figure BDA00032231000000000612

因为(A+βΦTΦ)(A+βΦTΦ)-1=IM,可得Because (A+βΦ T Φ)(A+βΦ T Φ) -1 = IM , we can get

ΦTΦΣ=β-1(IM-AΣ) (23)Φ T ΦΣ=β -1 (I M -AΣ) (23)

根据公式(22)与公式(23),可以得到According to formula (22) and formula (23), we can get

Figure BDA0003223100000000071
Figure BDA0003223100000000071

从公式(12)式,可得From formula (12), we can get

Figure BDA0003223100000000072
Figure BDA0003223100000000072

从公式(24)与公式(25),可得From formula (24) and formula (25), we can get

Figure BDA0003223100000000073
Figure BDA0003223100000000073

从而得到thereby getting

Figure BDA0003223100000000074
Figure BDA0003223100000000074

根据公式(12),对γ求导可得

Figure BDA0003223100000000075
Figure BDA0003223100000000076
可得γ=m,所以According to formula (12), the derivative of γ can be obtained
Figure BDA0003223100000000075
make
Figure BDA0003223100000000076
γ=m can be obtained, so

γi=mi (27)γ i =m i (27)

其中γi为γ的第i个分量,mi为m的第i个分量。where γ i is the ith component of γ, and m i is the ith component of m.

本发明的有益效果是:The beneficial effects of the present invention are:

本发明采用更加一般的权参数先验形式,而不是传统的每个权参数服从均值为零的正态分布,参数具有更大取值范围,再由图像数据根据矩阵微积分、矩阵代数以及最优化的方法最大化含有超参数的证据函数,从而有利于提高对图像数据的分辨率。The present invention adopts a more general a priori form of weight parameters, instead of the traditional normal distribution with zero mean for each weight parameter. The optimized method maximizes the evidence function with hyperparameters, which is beneficial to improve the resolution of the image data.

附图说明Description of drawings

图1本发明的流程示意图。Fig. 1 is a schematic flow chart of the present invention.

具体实施方式Detailed ways

下面将结合本发明实施例中的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。基于本发明中的实施例,本领域普通技术人员在没有做出创造性劳动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。The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, but not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

一种基于相关向量机的图像处理证据函数估计方法,所述证据函数估计方法包括以下步骤:A correlation vector machine-based image processing evidence function estimation method, the evidence function estimation method comprising the following steps:

步骤1:将图像内的数据根据修正后的权参数先验形式,利用正态分布中均值与协方差,证明权参数的后验分布是正态分布;Step 1: Prove that the posterior distribution of the weight parameters is a normal distribution by using the mean and covariance in the normal distribution according to the modified prior form of the weight parameters in the data in the image;

步骤2:将图像内的数据根据多变量泰勒公式、似然函数与权重的先验分布的乘积对权参数做积分得到证据函数的具体表达式,即边缘似然函数;Step 2: Integrate the data in the image with the weight parameter according to the multivariate Taylor formula, the product of the likelihood function and the prior distribution of the weight to obtain the specific expression of the evidence function, that is, the edge likelihood function;

步骤3:基于步骤2图像内的数据的边缘似然函数,利用矩阵微积分、矩阵代数以及最优化的方法最大化含有超参数的证据函数,从而得到图像的各个超参数的进行优化迭代算法。Step 3: Based on the edge likelihood function of the data in the image in step 2, the method of matrix calculus, matrix algebra and optimization is used to maximize the evidence function containing hyperparameters, so as to obtain the optimization iterative algorithm of each hyperparameter of the image.

进一步的,所述步骤1具体为,当

Figure BDA0003223100000000081
其中x为标量,A为n×n可逆对称矩阵,Tr(×)为矩阵的迹、
Figure BDA0003223100000000082
Tr(xyT)=xTy,其中Tr(·)为矩阵的迹,x为向量,y为向量;算子
Figure BDA0003223100000000083
定义为
Figure BDA0003223100000000084
Figure BDA0003223100000000085
其中k∈N;Further, the step 1 is specifically, when
Figure BDA0003223100000000081
where x is a scalar, A is an n×n invertible symmetric matrix, Tr(×) is the trace of the matrix,
Figure BDA0003223100000000082
Tr(xy T )=x T y, where Tr( ) is the trace of the matrix, x is the vector, and y is the vector; the operator
Figure BDA0003223100000000083
defined as
Figure BDA0003223100000000084
and
Figure BDA0003223100000000085
where k∈N;

令h=(h1,h2)T

Figure BDA0003223100000000086
其中h1为向量h的第一个分量,h2为向量h的第二个分量,x1为向量x的第一个分量,x2为向量x的第二个分量,Let h=(h 1 , h 2 ) T and
Figure BDA0003223100000000086
where h 1 is the first component of the vector h, h 2 is the second component of the vector h, x 1 is the first component of the vector x, and x 2 is the second component of the vector x,

可得

Figure BDA0003223100000000087
Figure BDA0003223100000000088
Available
Figure BDA0003223100000000087
and
Figure BDA0003223100000000088

当算子

Figure BDA0003223100000000089
作用在x,得到when operator
Figure BDA0003223100000000089
acting on x, we get

Figure BDA00032231000000000810
Figure BDA00032231000000000810

Figure BDA00032231000000000811
Figure BDA00032231000000000811

则依据算子

Figure BDA00032231000000000812
对x作用的公式x(这个x去掉),可得then according to the operator
Figure BDA00032231000000000812
The formula x acting on x (this x is removed), we can get

Figure BDA0003223100000000091
Figure BDA0003223100000000091

Figure BDA0003223100000000092
Figure BDA0003223100000000092

其中h=(h1,h2)T

Figure BDA0003223100000000093
where h=(h 1 , h 2 ) T and
Figure BDA0003223100000000093

利用

Figure BDA0003223100000000094
可得use
Figure BDA0003223100000000094
Available

Figure BDA0003223100000000095
Figure BDA0003223100000000095

Figure BDA0003223100000000096
Figure BDA0003223100000000096

Figure BDA0003223100000000097
Figure BDA0003223100000000097

若f:B(w,r)→R被定义为f(w)=wTAw+wTb+c;其中A是n×n可逆对称矩阵,b与w是n维列向量,c是一个标量;f(w)的泰勒在

Figure BDA0003223100000000098
的泰勒展开式为If f:B(w,r)→R is defined as f(w)=w T Aw+w T b+c; where A is an n×n invertible symmetric matrix, b and w are n-dimensional column vectors, and c is a scalar; the Taylor of f(w) in
Figure BDA0003223100000000098
The Taylor expansion of , is

Figure BDA0003223100000000099
Figure BDA0003223100000000099

H的第(i,j)元素由

Figure BDA00032231000000000910
定义;The (i,j)th element of H is given by
Figure BDA00032231000000000910
definition;

机器学习中的的线性回归模型的一般形式为The general form of a linear regression model in machine learning is

Figure BDA00032231000000000911
Figure BDA00032231000000000911

其中φi(x)为输入变量的非线性基函数,w0为偏差参数,x为图像数据向量;where φ i (x) is the nonlinear basis function of the input variable, w 0 is the deviation parameter, and x is the image data vector;

定义φ0(x)=1,从而(1)式可重写为Define φ 0 (x)=1, so equation (1) can be rewritten as

Figure BDA00032231000000000912
Figure BDA00032231000000000912

其中w=(w0,…,wM-1)T与φ(x)=(φ0(x),…,φM-1(x))Twhere w=(w 0 ,...,w M-1 ) T and φ(x)=(φ 0 (x),...,φ M-1 (x)) T ;

目标函数是带有加性高斯噪声的确定性函数y(x,w),即The objective function is a deterministic function y(x,w) with additive Gaussian noise, i.e.

t=y(x,w)+ε (3)t=y(x,w)+ε (3)

其中ε是0均值,精度为β的正态随机变量,从而获得where ε is a normal random variable with 0 mean and precision β, thus obtaining

p(t|x,w,β)=N(t|y(x,w),β-1) (4)。p(t|x,w,β)=N(t|y(x,w),β −1 ) (4).

进一步的,所述步骤1中权参数先验取如下的形式Further, in the step 1, the weight parameter prior takes the following form

Figure BDA0003223100000000101
Figure BDA0003223100000000101

其中α为精度(方差的逆)向量,α=(α1,…,αM)T与γ为均值向量,γ=(γ1,…,γM)Twhere α is the precision (inverse of variance) vector, α=(α 1 ,...,α M ) T and γ are the mean vector, γ=(γ 1 ,...,γ M ) T .

利用公式(4),获得似然函数Using formula (4), the likelihood function is obtained

Figure BDA0003223100000000102
Figure BDA0003223100000000102

其中,

Figure BDA0003223100000000103
M为所要确定的参数的个数。in,
Figure BDA0003223100000000103
M is the number of parameters to be determined.

相似地,Similarly,

Figure BDA0003223100000000104
Figure BDA0003223100000000104

其中α=diag(αi);where α=diag(α i );

获得权参数的后验分布p(w|t,X,α,β,γ)=N(w|m,Σ)也是正态分布,其中The posterior distribution p(w|t,X,α,β,γ)=N(w|m,Σ) for obtaining the weight parameters is also a normal distribution, where

m=(A+βΦTΦ)-1(Aγ+βΦTt) (8)m=(A+βΦ T Φ) -1 (Aγ+βΦ T t) (8)

Σ=(A+βΦTΦ)-1 (9)Σ=(A+βΦ T Φ) -1 (9)

其中

Figure BDA0003223100000000111
in
Figure BDA0003223100000000111

在证明之前,先看如下内容:Before proving it, look at the following:

正态分布

Figure BDA0003223100000000112
取正态分布的负指数为normal distribution
Figure BDA0003223100000000112
Take the negative exponent of the normal distribution as

Figure BDA0003223100000000113
Figure BDA0003223100000000113

Figure BDA0003223100000000114
可以获得x=μ,这暗示f(x)的驻点是该正态分布的均值,同时
Figure BDA0003223100000000115
f(x)的二阶梯度就是协方差的逆。make
Figure BDA0003223100000000114
It can be obtained that x = μ, which implies that the stagnation point of f(x) is the mean of the normal distribution, while
Figure BDA0003223100000000115
The second-order gradient of f(x) is the inverse of the covariance.

下面证明p(w|t,X,α,β,γ)是正态分布;The following proves that p(w|t,X,α,β,γ) is a normal distribution;

由公式(6)与公式(7),得到p(t|X,w,β)p(w|α,γ)乘积的负指数为From formula (6) and formula (7), the negative exponent of the product of p(t|X,w,β)p(w|α,γ) is

Figure BDA0003223100000000116
Figure BDA0003223100000000116

Figure BDA0003223100000000117
因此,make
Figure BDA0003223100000000117
therefore,

m=w=(A+βΦTΦ)-1(Aγ+βΦTt),m=w=(A+βΦ T Φ) -1 (Aγ+βΦ T t),

又因为

Figure BDA0003223100000000118
所以Σ=(A+βΦTΦ)-1,得证。also because
Figure BDA0003223100000000118
So Σ=(A+βΦ T Φ) -1 , the proof is obtained.

进一步的,所述步骤2具体为,Further, the step 2 is specifically:

p(t|X,α,β,γ)=∫p(t|X,w,β)p(w|α,γ)dw (10)p(t|X,α,β,γ)=∫p(t|X,w,β)p(w|α,γ)dw (10)

利用公式(6)和公式(7),可得Using formula (6) and formula (7), we can get

Figure BDA0003223100000000119
Figure BDA0003223100000000119

其中in

Figure BDA00032231000000001110
Figure BDA00032231000000001110

Figure BDA0003223100000000121
Figure BDA0003223100000000121

Figure BDA0003223100000000122
得到w=(A+βΦTΦ)-1(βΦTt+Aγ)=m,
Figure BDA0003223100000000123
可得make
Figure BDA0003223100000000122
Obtain w=(A+βΦ T Φ) -1 (βΦ T t+Aγ)=m,
Figure BDA0003223100000000123
Available

Figure BDA0003223100000000124
Figure BDA0003223100000000124

其中in

Figure BDA0003223100000000125
Figure BDA0003223100000000125

利用公式(11)与公式(12),可得Using formula (11) and formula (12), we can get

Figure BDA0003223100000000126
Figure BDA0003223100000000126

其中m=(A+βΦTΦ)-1(βΦTt+Aγ),

Figure BDA0003223100000000127
X=(x1,x2,…,xN)。where m=(A+βΦ T Φ) -1 (βΦ T t+Aγ),
Figure BDA0003223100000000127
X=(x 1 , x 2 , . . . , x N ).

进一步的,所述步骤3具体为。Further, the step 3 is specifically:

对公式(13)式取对数可得Taking the logarithm of formula (13), we can get

Figure BDA0003223100000000128
Figure BDA0003223100000000128

利用公式(9),公式(14)以及

Figure BDA0003223100000000129
可得Using Equation (9), Equation (14) and
Figure BDA0003223100000000129
Available

Figure BDA00032231000000001210
Figure BDA00032231000000001210

因为

Figure BDA00032231000000001211
because
Figure BDA00032231000000001211

利用

Figure BDA0003223100000000131
以及公式(15),可得use
Figure BDA0003223100000000131
and formula (15), we can get

Figure BDA0003223100000000132
Figure BDA0003223100000000132

由公式(16)可得According to formula (16), we can get

Figure BDA0003223100000000133
Figure BDA0003223100000000133

其中Σii是后验协方差Σ的主对角线的第i个元素;where Σ ii is the ith element of the main diagonal of the posterior covariance Σ;

由公式(12),可得From formula (12), we can get

Figure BDA0003223100000000134
Figure BDA0003223100000000134

其中mi是后验均值m的第i个分量;where m i is the ith component of the posterior mean m;

由公式(17)与公式(18),可得From formula (17) and formula (18), we can get

Figure BDA0003223100000000135
Figure BDA0003223100000000135

由(19)可得

Figure BDA0003223100000000136
从而可得It can be obtained from (19)
Figure BDA0003223100000000136
thus obtainable

Figure BDA0003223100000000137
Figure BDA0003223100000000137

其中λi=1-αiΣii,;where λ i = 1-α i Σ ii ,;

根据公式(9)的Σ的定义以及

Figure BDA0003223100000000138
可得The definition of Σ according to equation (9) and
Figure BDA0003223100000000138
Available

Figure BDA0003223100000000139
Figure BDA0003223100000000139

根据

Figure BDA00032231000000001310
与Tr(xyT)=xTy,可得according to
Figure BDA00032231000000001310
With Tr(xy T )=x T y, we can get

Figure BDA00032231000000001311
Figure BDA00032231000000001311

因为(A+βΦTΦ)(A+βΦTΦ)-1=IM,可得Because (A+βΦ T Φ)(A+βΦ T Φ) -1 = IM , we can get

ΦTΦΣ=β-1(IM-AΣ) (23)Φ T ΦΣ=β -1 (I M -AΣ) (23)

根据公式(22)与公式(23),可以得到According to formula (22) and formula (23), we can get

Figure BDA0003223100000000141
Figure BDA0003223100000000141

从公式(12)式,可得From formula (12), we can get

Figure BDA0003223100000000142
Figure BDA0003223100000000142

从公式(24)与公式(25),可得From formula (24) and formula (25), we can get

Figure BDA0003223100000000143
Figure BDA0003223100000000143

从而得到thereby getting

Figure BDA0003223100000000144
Figure BDA0003223100000000144

根据公式(12),对γ求导可得

Figure BDA0003223100000000145
Figure BDA0003223100000000146
可得γ=m,所以According to formula (12), the derivative of γ can be obtained
Figure BDA0003223100000000145
make
Figure BDA0003223100000000146
γ=m can be obtained, so

γi=mi (27)γ i =m i (27)

其中γi为γ的第i个分量,mi为m的第i个分量。where γ i is the ith component of γ, and m i is the ith component of m.

Claims (5)

1. An image processing evidence function estimation method based on a correlation vector machine is characterized by comprising the following steps:
step 1: the data in the image is proved to be normal distribution by using the mean value and covariance in the normal distribution according to the corrected weight parameter prior form;
step 2: integrating the weight parameters of the data in the image according to a multivariate Taylor formula and a product of a likelihood function and the prior distribution of the weight to obtain a specific expression of an evidence function, namely an edge likelihood function;
and step 3: and (3) based on the edge likelihood function of the data in the image in the step (2), maximizing the evidence function containing the hyper-parameters by utilizing a matrix calculus, a matrix algebra and an optimization method, thereby obtaining the optimization iterative algorithm of each hyper-parameter of the image.
2. The method according to claim 1, wherein the step 1 is specifically when the method comprises
Figure FDA0003223099990000011
Wherein x is a scalar, A is an nxn invertible symmetric matrix, Tr (-) is a trace of the matrix,
Figure FDA0003223099990000012
Wherein Tr (·) is the trace of the matrix, and x is the vector and is the vector; operator
Figure FDA00032230999900000112
Is defined as
Figure FDA0003223099990000013
And is
Figure FDA0003223099990000014
wherein k∈N;
let h be (h)1,h2)TAnd
Figure FDA0003223099990000015
wherein h1Is the first component of the vector h, h2Is the second component of the vector h, x1Is the first component of the vector x, x2As the second component of the vector x, one obtains
Figure FDA0003223099990000016
And
Figure FDA0003223099990000017
when operator
Figure FDA0003223099990000018
Acting on x to obtain
Figure FDA0003223099990000019
Figure FDA00032230999900000110
Then the operator of the basis
Figure FDA00032230999900000111
Formula for the action of x, can be derived
Figure FDA0003223099990000021
Figure FDA0003223099990000022
wherein h=(h1,h2)TAnd
Figure FDA0003223099990000023
by using
Figure FDA0003223099990000024
Can obtain the product
Figure FDA0003223099990000025
Figure FDA0003223099990000026
Figure FDA0003223099990000027
If f is defined as f (w) → RTAw+wTb + c; wherein A is an nxn reversible symmetric matrix, b and w are n-dimensional column vectors, and c is a scalar; taylor of f (w)
Figure FDA0003223099990000028
Is of Taylor expansion type
Figure FDA0003223099990000029
The (i, j) th element of H is composed of
Figure FDA00032230999900000210
Defining;
the general form of a linear regression model in machine learning is
Figure FDA00032230999900000211
wherein φi(x) As a non-linear basis function of the input variable, w0Is a deviation parameter, x is an image data vector;
definition of phi0(x) 1, so that formula (1) can be rewritten as
Figure FDA00032230999900000212
wherein w=(w0,…,wM-1)TAnd phi (x) to (phi)0(x),…,φM-1(x))T
The objective function is a deterministic function y (x, w) with additive Gaussian noise, i.e.
t=y(x,w)+ε (3)
Where ε is the mean of 0 and the accuracy is the normal random variable of β, thereby obtaining
p(t|x,w,β)=N(t|y(x,w),β-1) (4)。
3. The method for estimating an image processing evidence function based on a relevance vector machine according to claim 1, wherein the weight parameter prior in the step 1 is in the form of
Figure FDA0003223099990000031
Where α is the precision vector, α ═ α1,…,αM)TAnd γ is the mean vector, γ ═ γ1,…,γM)T
Using equation (4), a likelihood function is obtained
Figure FDA0003223099990000032
wherein ,
Figure FDA0003223099990000033
m is the number of parameters to be determined.
In a similar manner to that described above,
Figure FDA0003223099990000034
wherein α ═ diag (α)i);
The posterior distribution p (w | t, X, α, β, γ) of the obtained weight parameters is also a normal distribution, where N (w | m, Σ) is
m=(A+βΦTΦ)-1(Aγ+βΦTt) (8)
Σ=(A+βΦTΦ)-1 (9)
wherein
Figure FDA0003223099990000041
4. The method for estimating an image processing evidence function based on a correlation vector machine according to claim 1, wherein the step 2 is specifically,
p(t|X,α,β,γ)=∫p(t|X,w,β)p(w|α,γ)dw (10)
using the formula (6) and the formula (7), it is possible to obtain
Figure FDA0003223099990000042
wherein
Figure FDA0003223099990000043
Order to
Figure FDA0003223099990000044
To obtain
Figure FDA0003223099990000045
Can obtain the product
Figure FDA0003223099990000046
wherein
Figure FDA0003223099990000047
Using the formula (11) and the formula (12), it is possible to obtain
Figure FDA0003223099990000048
Figure FDA0003223099990000051
Wherein m ═ a + β ΦTΦ)-1(βΦTt+Aγ),
Figure FDA0003223099990000052
X=(x1,x2,…,xN)。
5. The method for estimating an image processing evidence function based on a correlation vector machine according to claim 4, wherein the step 3 is specifically,
logarithm of the formula (13) is obtained
Figure FDA0003223099990000053
Using equations (9), (14) and
Figure FDA0003223099990000054
can obtain the product
Figure FDA0003223099990000055
Because of the fact that
Figure FDA0003223099990000056
By using
Figure FDA0003223099990000057
And equation (15), can be derived
Figure FDA0003223099990000058
From the formula (16), it can be found
Figure FDA0003223099990000059
wherein ΣiiIs the ith element of the main diagonal of the a posteriori covariance Σ;
from the formula (12), it can be found
Figure FDA00032230999900000510
wherein miIs the ith component of the posterior mean m;
from the formula (17) and the formula (18), it can be obtained
Figure FDA00032230999900000511
From (19) to
Figure FDA0003223099990000061
Thereby obtaining
Figure FDA0003223099990000062
wherein λi=1-αiΣii
Definition of Σ according to equation (9) and
Figure FDA0003223099990000063
can obtain the product
Figure FDA0003223099990000064
According to
Figure FDA0003223099990000065
And Tr (xy)T)=xTy, is obtained
Figure FDA0003223099990000066
Because (A + beta. phi)TΦ)(A+βΦTΦ)-1=IMIs obtained by
ΦTΦΣ=β-1(IM-AΣ) (23)
From the formula (22) and the formula (23), it is possible to obtain
Figure FDA0003223099990000067
From the formula (12), it can be obtained
Figure FDA0003223099990000068
From the formula (24) and the formula (25), it can be obtained
Figure FDA0003223099990000069
Thereby obtaining
Figure FDA00032230999900000610
According to equation (12), the derivative of γ can be obtained
Figure FDA00032230999900000611
Order to
Figure FDA00032230999900000612
Can get gamma as m, so
γi=mi (27)
wherein γiIs the i-th component of γ, miIs the ith component of m.
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Citations (10)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US20100070435A1 (en) * 2008-09-12 2010-03-18 Microsoft Corporation Computationally Efficient Probabilistic Linear Regression
CN102254193A (en) * 2011-07-16 2011-11-23 西安电子科技大学 Relevance vector machine-based multi-class data classifying method
CN103258213A (en) * 2013-04-22 2013-08-21 中国石油大学(华东) Vehicle model dynamic identification method used in intelligent transportation system
CN104732215A (en) * 2015-03-25 2015-06-24 广西大学 Remote-sensing image coastline extracting method based on information vector machine
CN106709918A (en) * 2017-01-20 2017-05-24 成都信息工程大学 Method for segmenting images of multi-element student t distribution mixed model based on spatial smoothing
CN108197435A (en) * 2018-01-29 2018-06-22 绥化学院 Localization method between a kind of multiple characters multi-region for containing error based on marker site genotype
CN108228535A (en) * 2018-01-02 2018-06-29 佛山科学技术学院 A kind of optimal weighting parameter evaluation method of unequal precision measurement data fusion
CN111914865A (en) * 2019-05-08 2020-11-10 天津科技大学 Probability main component analysis method based on random core
CN112053307A (en) * 2020-08-14 2020-12-08 河海大学常州校区 A method for linear reconstruction of X-ray images
US10867171B1 (en) * 2018-10-22 2020-12-15 Omniscience Corporation Systems and methods for machine learning based content extraction from document images

Patent Citations (10)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US20100070435A1 (en) * 2008-09-12 2010-03-18 Microsoft Corporation Computationally Efficient Probabilistic Linear Regression
CN102254193A (en) * 2011-07-16 2011-11-23 西安电子科技大学 Relevance vector machine-based multi-class data classifying method
CN103258213A (en) * 2013-04-22 2013-08-21 中国石油大学(华东) Vehicle model dynamic identification method used in intelligent transportation system
CN104732215A (en) * 2015-03-25 2015-06-24 广西大学 Remote-sensing image coastline extracting method based on information vector machine
CN106709918A (en) * 2017-01-20 2017-05-24 成都信息工程大学 Method for segmenting images of multi-element student t distribution mixed model based on spatial smoothing
CN108228535A (en) * 2018-01-02 2018-06-29 佛山科学技术学院 A kind of optimal weighting parameter evaluation method of unequal precision measurement data fusion
CN108197435A (en) * 2018-01-29 2018-06-22 绥化学院 Localization method between a kind of multiple characters multi-region for containing error based on marker site genotype
US10867171B1 (en) * 2018-10-22 2020-12-15 Omniscience Corporation Systems and methods for machine learning based content extraction from document images
CN111914865A (en) * 2019-05-08 2020-11-10 天津科技大学 Probability main component analysis method based on random core
CN112053307A (en) * 2020-08-14 2020-12-08 河海大学常州校区 A method for linear reconstruction of X-ray images

Non-Patent Citations (5)

* Cited by examiner, † Cited by third party
Title
CARL EDWARD RASMUSSEN等: "Healing the relevance vector machine by augmentation", PROCEEDINGS OF THE 22ND INTERNATIONAL CONFERENECE ON MACHINE LEARNING, pages 689 *
DAWEI ZOU等: "A Logical Framework of the Evidence Function Approximation Associated with Relevance Vector Machine", MATHEMATICAL PROBLEMS IN ENGINEERING, vol. 2020, pages 1 *
俞炯奇;梁国钱;: "地基沉降双曲线拟合的Bayes估计", 水力发电, vol. 34, no. 02, pages 26 *
张仕山等: "集成DS证据理论和模糊集的建筑物检测方法", 遥感信息, vol. 35, no. 5, pages 93 *
李鑫等: "基于相关向量机算法的研究与应用综述", 信息工程大学学报, vol. 21, no. 4, pages 433 *

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