CN113779502B - Image processing evidence function estimation method based on correlation vector machine - Google Patents

Image processing evidence function estimation method based on correlation vector machine Download PDF

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CN113779502B
CN113779502B CN202110963746.2A CN202110963746A CN113779502B CN 113779502 B CN113779502 B CN 113779502B CN 202110963746 A CN202110963746 A CN 202110963746A CN 113779502 B CN113779502 B CN 113779502B
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邹大伟
马春华
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Abstract

The invention discloses an image processing evidence function estimation method based on a correlation vector machine. Step 1: the data in the image is proved to be normal distribution according to the prior form of the corrected weight parameters by using the mean value and covariance in the normal distribution; step 2: integrating the weight parameters of the data in the image according to the product of the multivariate Taylor formula, the likelihood function and the prior distribution of the weights to obtain a specific expression of the evidence function, namely an edge likelihood function; step 3: based on the edge likelihood function of the data in the image in the step 2, the evidence function containing the super parameters is maximized by utilizing matrix calculus, matrix algebra and optimization methods, so that the optimization iterative algorithm of each super parameter of the image is obtained. The invention is used for solving the problems that complex integration is faced and calculation is difficult in the process of integrating the weight parameters by the product of the likelihood function and the prior distribution of the weight in image processing to obtain the evidence function.

Description

Image processing evidence function estimation method based on correlation vector machine
Technical Field
The invention belongs to the field of image processing, and particularly relates to an image processing evidence function estimation method based on a correlation vector machine.
Background
In the estimation process of an evidence function related to a related vector machine in the image processing field, the posterior distribution needs to be proved to be normal distribution, and the mean value and covariance of the normal distribution are solved, so that the traditional method has to search for a complete square term, and the method is difficult to operate and lacks logic. Meanwhile, in the evidence function principle related to a related vector machine in the image processing field at present, the prior distribution of the weight parameters is the product of normal distribution with zero mean value, which lacks generality.
In the process of integrating the weight parameters by the product of the likelihood function and the prior distribution of the weights to obtain the evidence function, complex integration is faced, and the calculation is difficult. How to find a new method and logic framework for image processing to more simply and logically integrate the correlation of the image processing and more effectively maximize the evidence function is less studied.
Disclosure of Invention
The invention provides an image processing evidence function estimation method based on a correlation vector machine, which is used for solving the problems that complex integration is faced and calculation is difficult in the process of integrating a weight parameter by the product of a likelihood function and prior distribution of weight in image processing to obtain an evidence function.
The invention is realized by the following technical scheme:
an image processing evidence function estimation method based on a correlation vector machine, the evidence function estimation method comprising the following steps:
step 1: the data in the image is proved to be normal distribution according to the prior form of the corrected weight parameters by using the mean value and covariance in the normal distribution;
step 2: integrating the weight parameters of the data in the image according to the product of the multivariate Taylor formula, the likelihood function and the prior distribution of the weights to obtain a specific expression of the evidence function, namely an edge likelihood function;
step 3: based on the edge likelihood function of the data in the image in the step 2, the evidence function containing the super parameters is maximized by utilizing matrix calculus, matrix algebra and optimization methods, so that the optimization iterative algorithm of each super parameter of the image is obtained.
Further, the step 1 is specifically that, whenWherein x is a scalar, A is an n x n reversible 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, y is the vector; operator->Defined as->And-> wherein k∈N；
let h= (h 1 ,h 2 ) T And (3) with wherein h1 As the first component of the vector h, h 2 As the second component of the vector h, x 1 As the first component of vector x, x 2 As a second component of the vector x,
is available in the form ofAnd->
When an operatorActing on x to obtain
Then according to the operatorThe formula acting on x can be obtained
wherein h＝(h1 ,h 2 ) T And (3) with
By means ofIs available in the form of
If f: B (w, R) →R is defined as f (w) =w T Aw+w T b+c; wherein A is an n reversible symmetric matrix, b and w are n-dimensional column vectors, and c is a scalar; taylor at f (w)Taylor expansion of (2) is
The (i, j) th element of H is composed ofDefinition;
the general form of a linear regression model in machine learning is
wherein φi (x) As a nonlinear basis function of the input variable, w 0 As a deviation parameter, x is an image data vector;
definition phi 0 (x) =1, whereby (1) is rewritable as
wherein w＝(w0 ,…,w M-1 ) T And phi (x) = (phi) 0 (x),…,φ M-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 0 mean and the precision is β, to obtain a normal random variable
p(t|x,w,β)＝N(t|y(x,w),β -1 ) (4)。
Further, the weight parameters in the step 1 are obtained in the following form a priori
Where α is the precision (inverse of variance) vector, α= (α) 1 ,…,α M ) T And gamma is the mean vector, gamma= (gamma) 1 ,…,γ M ) T 。
Obtaining likelihood functions using equation (4)
wherein ,m is the number of parameters to be determined.
Similarly, the number of the devices to be used in the system,
where α=diag (α i )；
The posterior distribution p (w|t, X, α, β, γ) =n (w|m, Σ) of the acquisition weight parameters is also a normal distribution in which
m＝(A+βΦ T Φ) -1 (Aγ+βΦ T t) (8)
Σ＝(A+βΦ T Φ) -1 (9)
wherein
Further, the step 2 is specifically that,
p(t|X,α,β,γ)＝∫p(t|X,w,β)p(w|α,γ)dw (10)
using equation (6) and equation (7), one can obtain
wherein
Order theTo obtain w= (A+beta phi) T Φ) -1 (βΦ T t+Aγ)＝m，Is available in the form of
wherein
Using equation (11) and equation (12), one can obtain
Wherein m= (a+βΦ) T Φ) -1 (βΦ T t+Aγ)，X X＝(x 1 ,x 2 ,…,x N )。
Further, the step 3 is specifically that,
taking the logarithm of equation (13) to obtain
Using equation (9), equation (14) andis available in the form of
Because of
By means ofAnd equation (15) to obtain
From equation (16)
wherein Σii Is the i-th element of the main diagonal of the posterior covariance Σ;
from equation (12), it can be obtained
wherein mi Is the i-th component of the posterior mean m;
from equation (17) and equation (18), we can obtain
From (19)Thereby obtaining
wherein λi ＝1-α i Σ ii ；
Definition of Sigma and according to equation (9)Is available in the form of
According toAnd Tr (xy) T )＝x T y, can obtain
Because of (A+βΦ) T Φ)(A+βΦ T Φ) -1 ＝I M Can be obtained
Φ T ΦΣ＝β -1 (I M -AΣ) (23)
From the equation (22) and the equation (23), it is possible to obtain
From equation (12), it can be obtained
From equation (24) and equation (25), we can obtain
Thereby obtaining
Deriving gamma according to equation (12)Let->Can obtain gamma=m, so
γ i ＝m i (27)
wherein γi Is the ith component of gamma, m i Is the ith component of m.
The beneficial effects of the invention are as follows:
the invention adopts a more general weight parameter prior form instead of the traditional normal distribution that each weight parameter obeys the mean value to be zero, the parameters have a larger value range, and the image data maximizes the evidence function containing the super parameters according to matrix calculus, matrix algebra and optimization methods, thereby being beneficial to improving the resolution of the image data.
Drawings
FIG. 1 is a schematic flow chart of the present invention.
Detailed Description
The following description of the embodiments of the present invention will be made clearly and completely with reference to the accompanying drawings, in which it is apparent that the embodiments described are only some embodiments of the present invention, but not all embodiments. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
An image processing evidence function estimation method based on a correlation vector machine, the evidence function estimation method comprising the following steps:
step 1: the data in the image is proved to be normal distribution according to the prior form of the corrected weight parameters by using the mean value and covariance in the normal distribution;
step 2: integrating the weight parameters of the data in the image according to the product of the multivariate Taylor formula, the likelihood function and the prior distribution of the weights to obtain a specific expression of the evidence function, namely an edge likelihood function;
step 3: and (3) maximizing the evidence function containing the super parameters by utilizing matrix calculus, matrix algebra and optimization methods based on the edge likelihood function of the data in the image in the step (2), thereby obtaining the optimization iterative algorithm of each super parameter of the image.
Further, the step 1 is specifically that, whenWhere x is a scalar, A is an n x n reversible symmetric matrix, tr (X) is the trace of the matrix, ">Tr(xy T )＝x T y, where Tr (·) is the trace of the matrix, x is the vector, y is the vector; operator->Defined as->And-> wherein k∈N；
let h= (h 1 ,h 2 ) T And (3) with wherein h1 As the first component of the vector h, h 2 As the second component of the vector h, x 1 As the first component of vector x, x 2 As a second component of the vector x,
is available in the form ofAnd->
When an operatorActing on x to obtain
Then according to the operatorThe formula x acting on x (this x is removed) can be obtained
wherein h＝(h1 ,h 2 ) T And (3) with
By means ofIs available in the form of
If f: B (w, R) →R is defined as f (w) =w T Aw+w T b+c; wherein A is an n reversible symmetric matrix, b and w are n-dimensional column vectors, and c is a scalar; taylor at f (w)Taylor expansion of (2) is
The (i, j) th element of H is composed ofDefinition;
the general form of a linear regression model in machine learning is
wherein φi (x) As a nonlinear basis function of the input variable, w 0 As a deviation parameter, x is an image data vector;
definition phi 0 (x) =1, whereby (1) is rewritable as
wherein w＝(w0 ,…,w M-1 ) T And phi (x) = (phi) 0 (x),…,φ M-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 0 mean and the precision is β, to obtain a normal random variable
p(t|x,w,β)＝N(t|y(x,w),β -1 ) (4)。
Further, the weight parameters in the step 1 are obtained in the following form a priori
Where α is the precision (inverse of variance) vector, α= (α) 1 ,…,α M ) T And gamma is the mean vector, gamma= (gamma) 1 ,…,γ M ) T 。
Obtaining likelihood functions using equation (4)
wherein ,m is the number of parameters to be determined.
Similarly, the number of the devices to be used in the system,
where α=diag (α i )；
The posterior distribution p (w|t, X, α, β, γ) =n (w|m, Σ) of the acquisition weight parameters is also a normal distribution in which
m＝(A+βΦ T Φ) -1 (Aγ+βΦ T t) (8)
Σ＝(A+βΦ T Φ) -1 (9)
wherein
Before the certification, the following is seen:
normal distributionTaking the negative index of the normal distribution as
Order theX=μ can be obtained, which implies that the dwell 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.
The following demonstrates that p (w|t, X, α, β, γ) is a normal distribution;
from equation (6) and equation (7), the negative index of the product of p (t|X, w, beta) p (w|alpha, gamma) is obtained
Order theThus, the first and second substrates are bonded together,
m＝w＝(A+βΦ T Φ) -1 (Aγ+βΦ T t)，
and because ofSo Σ= (a+βΦ) T Φ) -1 And obtaining the evidence.
Further, the step 2 is specifically that,
p(t|X,α,β,γ)＝∫p(t|X,w,β)p(w|α,γ)dw (10)
using equation (6) and equation (7), one can obtain
wherein
Order theTo obtain w= (A+beta phi) T Φ) -1 (βΦ T t+Aγ)＝m，Is available in the form of
wherein
Using equation (11) and equation (12), one can obtain
Wherein m= (a+βΦ) T Φ) -1 (βΦ T t+Aγ)，X＝(x 1 ,x 2 ,…,x N )。
Further, the step 3 is specifically described.
Taking the logarithm of equation (13) to obtain
Using equation (9), equation (14) andavailable->
Because of
By means ofAnd equation (15) to obtain
From equation (16)
wherein Σii Is the i-th element of the main diagonal of the posterior covariance Σ;
from equation (12), it can be obtained
wherein mi Is the i-th component of the posterior mean m;
from equation (17) and equation (18), we can obtain
From (19)Thereby obtaining
wherein λi ＝1-α i Σ ii ，；
Definition of Sigma and according to equation (9)Is available in the form of
According toAnd Tr (xy) T )＝x T y, can obtain
Because of (A+βΦ) T Φ)(A+βΦ T Φ) -1 ＝I M Can be obtained
Φ T ΦΣ＝β -1 (I M -AΣ) (23)
From the equation (22) and the equation (23), it is possible to obtain
From equation (12), it can be obtained
From equation (24) and equation (25), we can obtain
Thereby obtaining
Deriving gamma according to equation (12)Let->Can obtain gamma=m, so
γ i ＝m i (27)
wherein γi Is the ith component of gamma, m i Is the ith component of m.

Claims (2)

1. The image processing evidence function estimation method based on the correlation vector machine is characterized by comprising the following steps of:
step 1: the data in the image is proved to be normal distribution according to the prior form of the corrected weight parameters by using the mean value and covariance in the normal distribution;
step 2: integrating the weight parameters of the data in the image according to the product of the multivariate Taylor formula, the likelihood function and the prior distribution of the weights to obtain a specific expression of the evidence function, namely an edge likelihood function;
step 3: based on the edge likelihood function of the data in the image in the step 2, maximizing the evidence function containing the super parameters by utilizing matrix calculus, matrix algebra and optimization methods, thereby obtaining the optimization iterative algorithm of each super parameter of the image;
the step 1 is specifically that whenWherein x is a scalar, A is an n x n reversible symmetric matrix, tr (·) is the trace of the matrix,>Tr(xy T )＝x T y, where Tr (·) is the trace of the matrix and x is the vector; operatorDefined as->And-> wherein k∈N；
let h= (h 1 ,h 2 ) T And (3) with wherein h1 As the first component of the vector h, h 2 As the second component of the vector h, x 1 As the first component of vector x, x 2 As the second component of the vector x, can be made +.>And (3) with
When an operatorActing on x to obtain
Then according to the operatorThe formula acting on x can be obtained
wherein h＝(h1 ,h 2 ) T And (3) with
By means ofIs available in the form of
If f: B (w, R) →R is defined as f (w) =w T Aw+w T b+c; wherein A is an n reversible symmetric matrix, b and w are n-dimensional column vectors, and c is a scalar; taylor at f (w)Taylor expansion of (2) is
The (i, j) th element of H is composed ofDefinition;
the general form of a linear regression model in machine learning is
wherein φi (x) As a nonlinear basis function of the input variable, w 0 As a deviation parameter, x is an image data vector;
definition phi 0 (x) =1, whereby (1) is rewritable as
wherein w＝(w0 ,…,w M-1 ) T And phi (x) = (phi) 0 (x),…,φ M-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 0 mean and the precision is β, to obtain a normal random variable
p(t|x,w,β)＝N(t|y(x,w),β -1 ) (4)；
The weight parameters in the step 1 are obtained in the following form in a priori manner
Where α is the precision vector, α= (α) 1 ,…,α M ) T And gamma is the mean vector, gamma= (gamma) 1 ,…,γ M ) T ；
Obtaining likelihood functions using equation (4)
wherein ,m is the number of parameters to be determined;
similarly, the number of the devices to be used in the system,
where α=diag (α i )；
The posterior distribution p (w|t, X, α, β, γ) =n (w|m, Σ) of the acquisition weight parameters is also a normal distribution in which
m＝(A+βΦ T Φ) -1 (Aγ+βΦ T t) (8)
Σ＝(A+βΦ T Φ) -1 (9)
wherein
The step 2 is specifically that,
p(t|X,α,β,γ)＝∫p(t|X,w,β)p(w|α,γ)dw (10)
using equation (6) and equation (7), one can obtain
wherein
Order theObtaining
w＝(A+βΦ T Φ) -1 (βΦ T t+Aγ)＝m，Is available in the form of
wherein
Using equation (11) and equation (12), one can obtain
Wherein m= (a+βΦ) T Φ) -1 (βΦ T t+Aγ)，X＝(x 1 ,x 2 ,…,x N )。
2. The method for estimating an image processing evidence function based on a relevance vector machine according to claim 1, wherein the step 3 is specifically,
taking the logarithm of equation (13) to obtain
Using equation (9), equation (14) andis available in the form of
Because of
By means ofAnd equation (15) to obtain
From equation (16)
wherein Σii Is the i-th element of the main diagonal of the posterior covariance Σ;
from equation (12), it can be obtained
wherein mi Is the i-th component of the posterior mean m;
from equation (17) and equation (18), we can obtain
From (19)Thereby obtaining
wherein λi ＝1-α i Σ ii ；
Definition of Sigma and according to equation (9)Is available in the form of
According toAnd Tr (xy) T )＝x T y, can obtain
Because of (A+βΦ) T Φ)(A+βΦ T Φ) -1 ＝I M Can be obtained
Φ T ΦΣ＝β -1 (I M -AΣ) (23)
From the equation (22) and the equation (23), it is possible to obtain
From equation (12), it can be obtained
From equation (24) and equation (25), we can obtain
Thereby obtaining
Deriving gamma according to equation (12)Let->Can obtain gamma=m, so
γ i ＝m i (27)
wherein γi Is the ith component of gamma, m i Is the ith component of m.
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