CN104537377B - A kind of view data dimension reduction method based on two-dimentional nuclear entropy constituent analysis - Google Patents
A kind of view data dimension reduction method based on two-dimentional nuclear entropy constituent analysis Download PDFInfo
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Abstract
The invention discloses a kind of view data dimension reduction method based on two-dimentional nuclear entropy constituent analysis, its step are as follows：(1) view data is read in；(2) using Parzen windows estimation kernel function；(3) nuclear matrix by all view data of column count is set up；(4) characteristic value and characteristic vector of the correlation matrix of view data are calculated；(5) the Renyi entropys of view data are calculated；(6) characteristic vector of the correlation matrix of view data is mapped using two-dimentional nuclear entropy component analyzing method, realizes the dimensionality reduction of view data.This method utilizes two-dimension analysis method, directly carries out kernel mapping to the row or column of image, and the entropy estimated the nuclear matrix of view data is ranked up, and obtains the intrinsic dimension of the view data after dimensionality reduction, moreover it is possible to keep the spatial structural form of view data；This method, without two-dimensional image data is converted into a n dimensional vector n, when progress kernel mapping tries to achieve correlation matrix, reduces the complexity of calculating due to directly pressing the nuclear matrix gone or by column count view data.
Description
Technical field
The present invention relates to a kind of two-dimentional nuclear entropy constituent analysis（KECA）View data dimension reduction method, belong to dimensional images number
According to processing method and applied technical field, suitable for the theoretical research with application technology of dimensionality reduction of high dimensional image.
Background technology
In the applications such as recognition of face, digital identification, medical image recognition, due to the higher-dimension of view data, usually need
First to carry out dimension-reduction treatment.View data is each grey scale pixel value represented with numerical value, and it can effectively represent the information of image,
And the spatial structural form of view data can be retained, but the dimension of view data is higher and data volume is big, therefore how to have
Effect obtains important information, and dimensionality reduction is carried out to view data, and reduces the complexity of calculating, turns into the pass of image real time transfer
Key link.
Many methods are proposed currently for the dimensionality reduction of view data, the dimension reduction method of view data mainly has principal component
Analysis method, core principle component analysis method, nuclear entropy component analyzing method, then, there is two-dimensional principal component analysis method.Principal component point
Analysis method is a kind of classical image data converting method, and it is a kind of linear transformation method, and core principle component analysis is then main
The nonlinear extensions of constituent analysis.Image data converting method using principal component analysis as representative, tries to achieve view data first
Covariance matrix, and the characteristic value and characteristic vector of this covariance matrix are obtained, then corresponding to maximum several characteristic values
Characteristic vector structure coordinate system, finally sample image data is projected on this coordinate system, obtains the view data after dimensionality reduction.
Nuclear entropy constituent analysis（Kernel Entropy Component Analysis, KECA）Method is a kind of based on the new of information theory
Image data converting method.This method, it regard the reference axis of the secondary Renyi entropy of original spatial image data as projection side
To, this is different from traditional data conversion spectrum transform method, the feature space of nuclear entropy constituent analysis (KECA) method choice dimensionality reduction,
View data after conversion has obvious angled arrangement attribute, so as to beneficial to further processing.But also exist as follows not
Foot：When above-mentioned principal component analytical method, core principle component analysis method, nuclear entropy component analyzing method conversion dimensionality reduction, first by two dimension
Picture element matrix is converted into one-dimensional characteristic vector, and this data transfer device is not tied effectively not only using the space of view data
Structure information, and when subsequently calculating covariance or calculating the nuclear matrix of view data, add the complexity of calculating；Next to that
Although the above-mentioned dimension reduction method based on two-dimensional principal component analysis make use of the space structure spatial information of view data, still, this
Kind linear processing methods still have limitation in the application.
In summary, the problem of dimension reduction method of current image data is primarily present be：View data can not effectively be utilized
Spatial structural form, and computation complexity is high.
The content of the invention
The purpose of the present invention is that the dimension reduction method for being directed to conventional images data can not be tied effectively using the space of view data
Structure information, the deficiencies of computation complexity is high, propose a kind of view data dimension reduction method based on two-dimentional nuclear entropy constituent analysis.
The present invention technical solution be：A kind of view data dimensionality reduction side based on two-dimentional nuclear entropy constituent analysis of the present invention
Method, in particular to a kind of space structure letter that data conversion directly is carried out to two-dimensional image data, view data can be remained
Breath, improve the dimensionality reduction performance of two-dimensional image data.This method is mainly directly to carry out kernel mapping by the row or column of image, and
The form of vector need not be converted images into, the characteristic value for the nuclear matrix for trying to achieve view data and characteristic vector are brought into entropy and estimated
In meter, selective entropy composition is mapped, and realizes the dimensionality reduction of view data, so as to improve the computation complexity for reducing data conversion.
A kind of view data dimension reduction method based on two-dimentional nuclear entropy constituent analysis of the present invention, its step are as follows：
(1) reads in view data；
(2) is using Parzen windows estimation kernel function；
(3) sets up the nuclear matrix by all view data of column count；
(4) calculates the characteristic value and characteristic vector of the correlation matrix of view data；
(5) calculates the Renyi entropys of view data；
(6) is mapped the characteristic vector of the correlation matrix of view data using two-dimentional nuclear entropy component analyzing method,
Realize the dimensionality reduction of view data；
Wherein, kernel function is estimated using Parzen windows described in step (2), be designated as, wherein, quadratic Renyi entropy
Expression formula：
(1)
In formula,It is M mN image data matrix；It is image data matrix's
Probability density function；It is monotonic function, only need to analyzes the quadratic Renyi entropy for removing negative sign, it is represented by, in order to estimate, Parzen window density estimators are introduced, it estimates expression formula：
(2)
In formula,It is Parzen windows pairEstimated obtained estimate；M is all view data squares
The number of battle array；I is the sequence number of M, and span arrives M for 1；It is the kernel function of Parzen windows estimation,It is
The width of window function；
Wherein, the nuclear matrix set up by all view data of column count described in step (3), its computational methods are as follows：
First, kernel mapping is carried out by column vector to all view data, obtains nuclear matrix, be designated as, its matrix is：
(3)
In formula,It is M mThe matrix of n view data, subscript n are total columns of image data matrix；Subscript M is
The total number of image data matrix；It is the n-component column vector of the data of M sub-pictures,It is M auxiliary image datas
The nuclear matrix arranged by the M auxiliary image datas n-th obtained by the n-th row progress kernel mapping；
Then, the nuclear matrix of view dataWith the nuclear matrix of the view data obtained by its transpositionIt is multiplied, gained
Product is that nuclear matrix is correlation matrix, is designated as：
(4)
In formula,It is nuclear matrixTransposition obtained by view data nuclear matrix；Subscript T represents transposition.
Wherein, the characteristic value and characteristic vector of the correlation matrix of the calculating view data described in step (4), its computational methods
It is as follows：
First, if the characteristic value of the correlation matrix of view dataWith the projection vector of the correlation matrix of view data, it is full
The following relational expression of foot：
Or, (5)
Then, it is assumed that the related nuclear matrix of M view data, be designated as, its expression formula is：
(6)
In formula,It isNuclear matrix corresponding to view data,It is image data matrixIn m image
The mean eigenvalue of the row vector of data；
If, then above-mentioned formula (5) is converted into following relationship：
(7)
Solved by above-mentioned relation formula (7), obtain the characteristic value of the related nuclear matrix of view dataWith corresponding image
The characteristic vector of the related nuclear matrix of data, its expression formula is respectively：
(8)
(9)
In formula,It is the m-th characteristic value of the related nuclear matrix of view data；It is the m-th picture number of formula (7)
According to related nuclear matrix characteristic vector；
If, then the characteristic vector of the related nuclear matrix of view data is obtained, its expression formula is：
(10)
In formula,It is the m-th characteristic vector of the related nuclear matrix of view data；
Wherein, the Renyi entropys of the calculating view data described in step (5), are designated asIts computational methods is as follows：
(11)
In formula,It is Parzen windows pairEstimate, i.e. with Parzen windows to original spatial image data two
The estimate in the direction of the reference axis of secondary Renyi entropys,
Formula (2) is updated in formula (11), obtains the Parzen window estimates of quadratic Renyi entropy, it is estimated
Expression formula：
(12)
In formula,WithA i-th of image data matrix and j-th of image data matrix is represented, by step (4)
The characteristic value and characteristic vector of related nuclear matrix are brought into formula (12), you can are obtainedEquivalence formula：
(13)
In formula,It is the related nuclear matrix m of view data1 unit vector；It is the related nuclear matrix m of view dataThe transposition of 1 unit vector；M is the number of image data matrix；It is the related nuclear moment for the view data that E transposition obtain
The characteristic vector of battle array；It is the transposition of the related nuclear matrix ith feature vector of view data；
Wherein, step (6) is entered using two-dimentional nuclear entropy component analyzing method to the characteristic vector of the correlation matrix of view data
Row mapping, realizes the dimensionality reduction of view data, its is specific as follows：
(14)
First, according to the Renyi entropy for the view data being calculated in calculating formula (13), dropped by its entropy size
Sequence sorts, and the Renyi entropy vector of d view data, is designated as before selection, its expression formula is：
(15)
Then, the entropy vector is mapped, obtains the nuclear matrix of view dataMap vector, be designated as；, the intrinsic dimension of the view data after dimensionality reduction is obtained using projective transformation, it is achieved thereby that the dimensionality reduction of view data.
This discovery compared with prior art the advantages of be：This method employs two-dimentional nuclear entropy component analyzing method, to figure
Nuclear matrix conversion is carried out by row or by row as data, Renyi entropys are estimated with the nuclear matrix of view data, after having obtained dimensionality reduction
The assertive evidence dimension of view data, realize the dimensionality reduction of view data.It has the following advantages that：
(1) this method utilizes two-dimension analysis method, directly kernel mapping is carried out to the row or column of image, to view data
Nuclear matrix estimate entropy is ranked up, obtain the intrinsic dimension of the view data after dimensionality reduction, moreover it is possible to keep the sky of view data
Between structural information；
(2) this method is due to directly pressing row or nuclear matrix by column count view data, without by two-dimensional image data
A n dimensional vector n is converted into, when progress kernel mapping tries to achieve the correlation matrix of view data, reduces the complexity of calculating.
Brief description of the drawings
Fig. 1 is a kind of realization stream of the view data dimension reduction method based on two-dimentional nuclear entropy constituent analysis of the present invention
Journey；
Fig. 2 is the dimension reduction method and the ratio of the nicety of grading of the dimension reduction method of existing view data of view data of the present invention
Compared with table.
Embodiment
In order to better illustrate a kind of view data dimension reduction method based on two-dimentional nuclear entropy constituent analysis of the present invention,
Carry out analyzing dimensionality reduction and classify using the forehead image of two kinds of different expressions of FERET face databases.
A kind of view data dimension reduction method based on two-dimentional nuclear entropy constituent analysis of the present invention, implementation process figure such as Fig. 1 institutes
Show, specific implementation step is as follows：
(1) reads in view data：FERET face database view data is read in, raw image data size is 8080,
In the present embodiment, view data is cut into size as 6060 view data；
(2) is designated as using Parzen windows estimation kernel function, wherein, quadratic Renyi entropy expression formula：
(1)
In formula,It is 200 6060 image data matrix；It is image data matrix
Probability density function, analysis remove the quadratic Renyi entropy after negative sign, it is represented by, in order to
Estimation, Parzen window density estimators are introduced, it estimates expression formula：
(2)
In formula,It is Parzen windows pairEstimated obtained estimate；200 be all view data squares
The number of battle array；It is the kernel function of Parzen windows estimation,It is the width of window function；
(3) sets up the nuclear matrix by all view data of column count, and its computational methods is as follows：
First, kernel mapping is carried out by column vector to all view data, obtains nuclear matrix, be designated as, its matrix is：
(3)
In formula,It is 200 6060 image data matrix, subscript 60 represent the matrix columns of view data；Subscript
200 represent the number of image data matrix；It is the 60th column vector of the data of the 200th image, column vector is 60
1；
Then, the nuclear matrix of view dataWith the image data matrix obtained by its transpositionIt is multiplied, the product of gained
It is related nuclear matrix for nuclear matrix, is designated as：
(4)
In formula,It is nuclear matrixTransposition obtained by view data nuclear matrix；Subscript T represents transposition；
(4) calculates the characteristic value and characteristic vector of the correlation matrix of view data, and its is specific as follows：
First, if the characteristic value of the correlation matrix of view dataWith the projection vector of the correlation matrix of view data, it is full
The following relational expression of foot：
Or, (5)
Then, it is assumed that the related nuclear matrix of 200 view data, be designated as, its expression formula is：
(6)
In formula,It isNuclear matrix corresponding to view data,It is image data matrixIn m picture number
According to row vector mean eigenvalue；
If, then above-mentioned formula (5) is converted into following relationship：
(7)
Solved by above-mentioned relation formula (7), obtain the characteristic value of the related nuclear matrix of view dataWith corresponding figure
As the characteristic vector of the related nuclear matrix of data, its expression formula is respectively：
(8)
(9)
In formula,It is the 200th characteristic value of related nuclear matrix of view data；It is the 200th image of formula (7)
The characteristic vector of the related nuclear matrix of data；
If, then the characteristic vector of the related nuclear matrix of view data is obtained, its expression formula is：
(10)
In formula,It is the m-th characteristic vector of the related nuclear matrix of view data；
(5) calculates the Renyi entropys of view data, is designated asIts computational methods is as follows：
(11)
In formula,It is Parzen windows pairEstimate, i.e. with Parzen windows to original spatial image data two
The estimate in the direction of the reference axis of secondary Renyi entropys,
Formula (3) is updated in formula (11), obtains the Parzen window estimates of quadratic Renyi entropy, it is estimated
Expression formula：
(12)
In formula,WithRepresent A i-th of image data matrix and j-th of image data matrix；By phase in step (4)
Close the characteristic value of nuclear matrix and characteristic vector is brought into formula (12) and can obtainEquivalence formula：
(13)
In formula,It is the related nuclear matrix 60 of view data1 unit vector；It is the related nuclear matrix m of view data
The transposition of 1 unit vector；
(6) is mapped the characteristic vector of the correlation matrix of view data using two-dimentional nuclear entropy component analyzing method, real
The dimensionality reduction of existing view data, its is specific as follows：
(14)
First according to the Renyi entropy for the view data being calculated in calculating formula (13), dropped by its entropy size
Sequence sorts, and the Renyi entropy vector of d view data, is designated as before selection, its expression formula is：
(15)
Then, the entropy vector is mapped, obtains the nuclear matrix of view dataMap vector, be designated as, the intrinsic dimension of the view data after dimensionality reduction is obtained using projective transformation, it is achieved thereby that the dimensionality reduction of view data.
In order to verify using a kind of view data dimension reduction method method based on two-dimentional nuclear entropy constituent analysis of the invention
Effect, in an experiment, the dimension reduction method of the dimension reduction method of the present invention and nuclear entropy component analyzing method of the prior art is made into ratio
Compared with as shown in Fig. 2 in the comparison sheet, often row represents the 10 different dimensions dropped to, at interval of 10 dimensions, is obtained final
To data characteristics drop to respectively 100 to 10 dimension；Each column expression is compared analysis with three kinds of methods, is respectively：By row nuclear entropy
Constituent analysis, by the constituent analysis of row nuclear entropy and nuclear entropy constituent analysis.It can be seen that from the table 1 in Fig. 2：Under same dimension, two dimension
The result of nuclear entropy constituent analysis is substantially better than the result of nuclear entropy constituent analysis；Under same dimension, it is better than by row nuclear entropy constituent analysis
By row nuclear entropy constituent analysis；Two-dimentional nuclear entropy constituent analysis has reached maximum when dimension is 60, and nuclear entropy constituent analysis is 100
Reach maximum during dimension.Two-dimentional nuclear entropy component analyzing method of the invention shown in the comparison sheet 1 of the nicety of grading is better than existing
Nuclear entropy component analyzing method in technology.
Claims (1)
1. a kind of view data dimension reduction method based on two-dimentional nuclear entropy constituent analysis, it is characterised in that its step is as follows：
(1) reads in view data, and these view data are to remain spatial structural form, it is not necessary to turns two-dimensional pixel matrix
Change the view data for being characterized vector into；
(2) is concretely comprised the following steps using Parzen windows estimation kernel function：
Described estimates kernel function using Parzen windows, is designated asWherein, quadratic Renyi entropy expression formula：
H (p)=- log ∫ p2(A)dA (1)
In formula, A is M m × n image data matrix；P (A) is image data matrix A=[A1... AM] probability density letter
Number；H (p) is monotonic function, need to analyze the quadratic Renyi entropy V (p) for removing negative sign, and it is represented by V (p)=∫ p2(A), to estimate
V (p) is counted, introduces Parzen window density estimators, it estimates expression formula：
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In formula,It is the estimate that Parzen windows are estimated to obtain to p (A)；M is of all image data matrixs
Number；I is the sequence number of M, and span arrives M for 1；Kσ(A, Ai) it is the kernel function that Parzen windows are estimated, σ is the width of window function；
(3) sets up the nuclear matrix by all view data of column count, concretely comprises the following steps：
First, kernel mapping is carried out by column vector to all view data, obtains nuclear matrix, be designated as Φ, its matrix is：
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In formula, AMIt is the matrix of M m × n view data, subscript n is total columns of image data matrix；Subscript M is picture number
According to the total number of matrix；It is the n-component column vector of the data of M sub-pictures,It is that M auxiliary image datas arrange by n-th
The nuclear matrix that M auxiliary image datas n-th obtained by progress kernel mapping arrange；
Then, the nuclear matrix Φ of the view data and nuclear matrix Φ of the view data obtained by its transpositionTIt is multiplied, the product of gained
For the related nuclear matrix of nuclear matrix, S is designated as：
S=Φ ΦT (4)
In formula, ΦTIt is the nuclear matrix of the view data obtained by nuclear matrix Φ transposition；Subscript T represents to turn；
(4) calculates the characteristic value and characteristic vector of the related nuclear matrix of the nuclear matrix of view data, concretely comprises the following steps：
First, it is full if the projection vector v of the eigenvalue λ of the related nuclear matrix of view data nuclear matrix related to view data
The following relational expression of foot：
λ v=Sv or λ v=Φ ΦTV, λ >=0 (5)
Then, it is assumed that the related nuclear matrix of M view data, be designated asIts expression formula is：
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In formula,It isNuclear matrix corresponding to view data,It is image data matrix AMIn m view data
The mean eigenvalue of row vector；
IfAbove-mentioned formula (5) is then converted into following relationship：
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Solved by above-mentioned relation formula (7), obtain the eigenvalue λ of the related nuclear matrix of view data and corresponding view data
Related nuclear matrix characteristic vector q, its expression formula is respectively：
λ=[λ1..., λM] (8)
Q=[q1..., qM] (9)
In formula, λMIt is the m-th characteristic value of the nuclear matrix of view data；qMIt is the related core of the m-th view data of formula (7)
The characteristic vector of matrix；
IfThe characteristic vector of the related nuclear matrix of view data is then obtained, its expression formula is：
V=[v1..., vM] (10)
In formula, vMIt is the m-th characteristic vector of the related nuclear matrix of view data；
(5) calculates the Renyi entropys of view data, concretely comprises the following steps：The Renyi entropys of described calculating view data, are designated asIts computational methods is as follows：
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In formula,It is estimate of the Parzen windows to V (p), i.e. secondary to original spatial image data with Parzen windows
The estimate in the direction of the reference axis of Renyi entropys,
Formula (2) is updated to the Parzen window estimates that quadratic Renyi entropy is obtained in formula (11)It estimates expression formula：
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<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>M</mi>
</msubsup>
<msubsup>
<mi>&Sigma;</mi>
<mrow>
<mi>j</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>M</mi>
</msubsup>
<msub>
<mi>K</mi>
<mi>&sigma;</mi>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>A</mi>
<mi>i</mi>
</msub>
<mo>,</mo>
<msub>
<mi>A</mi>
<mi>j</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>12</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula, AiAnd AjA i-th of image data matrix and j-th of image data matrix is represented, by related nuclear moment in step (4)
The characteristic value and characteristic vector of battle array are brought into formula (12) i.e. availableEquivalence formula：
Or
In formula, 1 is the unit vector of related nuclear matrix m × 1 of view data；1TIt is related nuclear matrix m × 1 of view data
The transposition of unit vector；M is the number of image data matrix；ETIt is the spy of the related nuclear matrix for the view data that E transposition obtain
Sign vector；It is the transposition of the related nuclear matrix ith feature vector of view data；
(6) is mapped the characteristic vector of the related nuclear matrix of view data using two-dimentional nuclear entropy component analyzing method, is realized
The dimensionality reduction of view data, is concretely comprised the following steps：
Y=[v1..., vd]TΦ(A) (14)
First, according to the Renyi entropy for the view data being calculated in calculating formula (13), descending row is carried out by its entropy size
Sequence, the Renyi entropy vector of d view data, is designated as Z, its expression formula is before selection：
Z=[z1..., zd] (15)
Then, the entropy vector is mapped, obtains the nuclear matrix Φ (A) of view data map vector, be designated as v1...,
vd, the intrinsic dimension of the view data after dimensionality reduction is obtained using projective transformation, it is achieved thereby that the dimensionality reduction of view data.
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