CN113779502A - A Correlation Vector Machine-Based Evidence Function Estimation Method for Image Processing - Google Patents
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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。
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.
Description
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,当其中x为标量,A为n×n可逆对称矩阵,Tr(·)为矩阵的迹、Tr(xyT)=xTy,其中Tr(·)为矩阵的迹,x为向量,y为向量;算子定义为且其中k∈N;Further, the step 1 is specifically, when where x is a scalar, A is an n×n invertible symmetric matrix, Tr( ) is the trace of the matrix, 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 defined as and where k∈N;
令h=(h1,h2)T与其中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 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,
可得与 Available and
当算子作用在x,得到when operator acting on x, we get
则依据算子对x作用的公式,可得then according to the operator The formula acting on x can be obtained
其中h=(h1,h2)T与 where h=(h 1 , h 2 ) T and
利用可得use Available
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 The Taylor expansion of , is
H的第(i,j)元素由定义;The (i,j)th element of H is given by 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
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
其中w=(w0,…,wM-1)T与φ(x)=(φ0(x),…,φM-1(x))T;where 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
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)T。where 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
其中,M为所要确定的参数的个数。in, M is the number of parameters to be determined.
相似地,Similarly,
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 in
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
其中in
令得到w=(A+βΦTΦ)-1(βΦTt+Aγ)=m,可得make Obtain w=(A+βΦ T Φ) -1 (βΦ T t+Aγ)=m, Available
其中in
利用公式(11)与公式(12),可得Using formula (11) and formula (12), we can get
其中m=(A+βΦTΦ)-1(βΦTt+Aγ),X X=(x1,x2,…,xN)。where m=(A+βΦ T Φ) -1 (βΦ T t+Aγ), 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
利用公式(9),公式(14)以及可得Using Equation (9), Equation (14) and Available
因为 because
利用以及公式(15),可得use and formula (15), we can get
由公式(16)可得According to formula (16), we can get
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
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
由(19)可得从而可得It can be obtained from (19) thus obtainable
其中λi=1-αiΣii;where λ i =1-α i Σ ii ;
根据公式(9)的Σ的定义以及可得The definition of Σ according to equation (9) and Available
根据与Tr(xyT)=xTy,可得according to With Tr(xy T )=x T y, we can get
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
从公式(12)式,可得From formula (12), we can get
从公式(24)与公式(25),可得From formula (24) and formula (25), we can get
从而得到thereby getting
根据公式(12),对γ求导可得令可得γ=m,所以According to formula (12), the derivative of γ can be obtained make 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,当其中x为标量,A为n×n可逆对称矩阵,Tr(×)为矩阵的迹、Tr(xyT)=xTy,其中Tr(·)为矩阵的迹,x为向量,y为向量;算子定义为且其中k∈N;Further, the step 1 is specifically, when where x is a scalar, A is an n×n invertible symmetric matrix, Tr(×) is the trace of the matrix, 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 defined as and where k∈N;
令h=(h1,h2)T与其中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 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,
可得与 Available and
当算子作用在x,得到when operator acting on x, we get
则依据算子对x作用的公式x(这个x去掉),可得then according to the operator The formula x acting on x (this x is removed), we can get
其中h=(h1,h2)T与 where h=(h 1 , h 2 ) T and
利用可得use Available
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 The Taylor expansion of , is
H的第(i,j)元素由定义;The (i,j)th element of H is given by 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
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
其中w=(w0,…,wM-1)T与φ(x)=(φ0(x),…,φM-1(x))T;where 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
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)T。where 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
其中,M为所要确定的参数的个数。in, M is the number of parameters to be determined.
相似地,Similarly,
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 in
在证明之前,先看如下内容:Before proving it, look at the following:
正态分布取正态分布的负指数为normal distribution Take the negative exponent of the normal distribution as
令可以获得x=μ,这暗示f(x)的驻点是该正态分布的均值,同时f(x)的二阶梯度就是协方差的逆。make It can be obtained that x = μ, which implies that the stagnation point of f(x) is the mean of the normal distribution, while 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
令因此,make therefore,
m=w=(A+βΦTΦ)-1(Aγ+βΦTt),m=w=(A+βΦ T Φ) -1 (Aγ+βΦ T t),
又因为所以Σ=(A+βΦTΦ)-1,得证。also because 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
其中in
令得到w=(A+βΦTΦ)-1(βΦTt+Aγ)=m,可得make Obtain w=(A+βΦ T Φ) -1 (βΦ T t+Aγ)=m, Available
其中in
利用公式(11)与公式(12),可得Using formula (11) and formula (12), we can get
其中m=(A+βΦTΦ)-1(βΦTt+Aγ),X=(x1,x2,…,xN)。where m=(A+βΦ T Φ) -1 (βΦ T t+Aγ), 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
利用公式(9),公式(14)以及可得Using Equation (9), Equation (14) and Available
因为 because
利用以及公式(15),可得use and formula (15), we can get
由公式(16)可得According to formula (16), we can get
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
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
由(19)可得从而可得It can be obtained from (19) thus obtainable
其中λi=1-αiΣii,;where λ i = 1-α i Σ ii ,;
根据公式(9)的Σ的定义以及可得The definition of Σ according to equation (9) and Available
根据与Tr(xyT)=xTy,可得according to With Tr(xy T )=x T y, we can get
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
从公式(12)式,可得From formula (12), we can get
从公式(24)与公式(25),可得From formula (24) and formula (25), we can get
从而得到thereby getting
根据公式(12),对γ求导可得令可得γ=m,所以According to formula (12), the derivative of γ can be obtained make 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.
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