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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 PDF

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CN104537377B
CN104537377B CN201410791475.7A CN201410791475A CN104537377B CN 104537377 B CN104537377 B CN 104537377B CN 201410791475 A CN201410791475 A CN 201410791475A CN 104537377 B CN104537377 B CN 104537377B
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施俊
赵攀博
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University of Shanghai for Science and Technology
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Abstract

本发明公开了一种基于二维核熵成分分析的图像数据降维方法，其步骤如下：(1)读入图像数据；(2)采用Parzen窗估计核函数；(3)设立按列计算所有图像数据的核矩阵；(4)计算图像数据的相关矩阵的特征值和特征向量；(5)计算图像数据的Renyi熵；(6)采用二维核熵成分分析方法对图像数据的相关矩阵的特征向量进行映射，实现图像数据的降维。该方法利用二维分析方法，直接对图像的行或者列进行核变换，对图像数据的核矩阵估计的熵进行排序，得到降维后的图像数据的本征维数，还能保持图像数据的空间结构信息；该方法由于直接按行或者按列计算图像数据的核矩阵，无需将二维图像数据转换成一维矢量，进行核变换求得相关矩阵时，降低了计算的复杂度。

The invention discloses a dimensionality reduction method for image data based on two-dimensional kernel entropy component analysis. The steps are as follows: (1) read in image data; (2) use Parzen window to estimate the kernel function; The kernel matrix of the image data; (4) calculate the eigenvalue and the eigenvector of the correlation matrix of the image data; (5) calculate the Renyi entropy of the image data; The feature vector is mapped to realize the dimensionality reduction of the image data. This method uses a two-dimensional analysis method to directly perform kernel transformation on the rows or columns of the image, sort the entropy of the kernel matrix estimation of the image data, obtain the intrinsic dimension of the image data after dimensionality reduction, and maintain the image data. Spatial structure information; this method directly calculates the kernel matrix of the image data by row or column, without converting the two-dimensional image data into a one-dimensional vector, and reduces the computational complexity when performing kernel transformation to obtain the correlation matrix.

Description

A kind of view data dimension reduction method based on two-dimentional nuclear entropy constituent analysis
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)

1.一种基于二维核熵成分分析的图像数据降维方法，其特征在于，其步骤如下：1. an image data dimensionality reduction method based on two-dimensional nuclear entropy component analysis, is characterized in that, its steps are as follows: (1).读入图像数据，这些图像数据是保留了空间结构信息，不需要将二维像素矩阵转换成为特征向量的图像数据；(1). Read in the image data, which retains the spatial structure information and does not need to convert the two-dimensional pixel matrix into the image data of the feature vector; (2).采用Parzen窗估计核函数，具体步骤为：(2). Using the Parzen window to estimate the kernel function, the specific steps are: 所述的采用Parzen窗估计核函数，记为其中，二次Renyi熵表达式：The kernel function estimated using the Parzen window is denoted as Among them, the quadratic Renyi entropy expression: H(p)＝-log∫p2(A)dA (1)H(p)＝-log∫p 2 (A)dA (1) 式中，A是M个m×n的图像数据矩阵；p(A)是图像数据矩阵A＝[A1，...AM]的概率密度函数；H(p)是单调函数，需分析去掉负号的二次Renyi熵V(p)，其可表示为V(p)＝∫p2(A)，为估计V(p)，引入Parzen窗密度估计器，其估计表达式：In the formula, A is M m×n image data matrices; p(A) is the probability density function of image data matrix A=[A 1 ,...A M ]; H(p) is a monotone function, which needs to be analyzed The quadratic Renyi entropy V(p) with the minus sign removed can be expressed as V(p)=∫p 2 (A). To estimate V(p), a Parzen window density estimator is introduced, and its estimation expression is: <mrow> <mover> <mi>p</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>A</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mi>M</mi> </mfrac> <msubsup> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>M</mi> </msubsup> <msub> <mi>K</mi> <mi>&amp;sigma;</mi> </msub> <mrow> <mo>(</mo> <mi>A</mi> <mo>,</mo> <msub> <mi>A</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> <mrow><mover><mi>p</mi><mo>^</mo></mover><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mi>M</mi></mfrac><msubsup><mi>&amp;Sigma;</mi><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>M</mi></msubsup><msub><mi>K</mi><mi>&amp;sigma;</mi></msub><mrow><mo>(</mo><mi>A</mi><mo>,</mo><msub><mi>A</mi><mi>i</mi></msub><mo>)</mo></mrow><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow> 式中，是Parzen窗对p(A)进行估计得到的估计值；M是所有的图像数据矩阵的个数；i是M个的序号，取值范围为1到M；Kσ(A，Ai)是Parzen窗估计的核函数，σ是窗函数的宽度；In the formula, is the estimated value obtained by estimating p(A) by Parzen window; M is the number of all image data matrices; i is the serial number of M, and the value range is from 1 to M; K σ (A, A i ) is The kernel function estimated by the Parzen window, σ is the width of the window function; (3).设立按列计算所有图像数据的核矩阵，具体步骤为：(3). Set up a kernel matrix that calculates all image data by column. The specific steps are: 首先，对所有图像数据按列向量进行核变换，得到核矩阵，记为Φ，其矩阵为：First, perform kernel transformation on all image data according to column vectors to obtain a kernel matrix, which is denoted as Φ, and its matrix is: <mrow> <mi>&amp;Phi;</mi> <mo>=</mo> <mo>&amp;lsqb;</mo> <mi>&amp;phi;</mi> <mrow> <mo>(</mo> <msubsup> <mi>A</mi> <mn>1</mn> <mn>1</mn> </msubsup> <mo>)</mo> </mrow> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>&amp;phi;</mi> <mrow> <mo>(</mo> <msubsup> <mi>A</mi> <mn>1</mn> <mi>n</mi> </msubsup> <mo>)</mo> </mrow> <mo>,</mo> <mn>...</mn> <mi>&amp;phi;</mi> <mrow> <mo>(</mo> <msubsup> <mi>A</mi> <mi>M</mi> <mn>1</mn> </msubsup> <mo>)</mo> </mrow> <mo>,</mo> <mn>...</mn> <mi>&amp;phi;</mi> <mrow> <mo>(</mo> <msubsup> <mi>A</mi> <mi>M</mi> <mi>n</mi> </msubsup> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow> <mrow><mi>&amp;Phi;</mi><mo>=</mo><mo>&amp;lsqb;</mo><mi>&amp;phi;</mi><mrow><mo>(</mo><msubsup><mi>A</mi><mn>1</mn><mn>1</mn></msubsup><mo>)</mo></mrow><mo>,</mo><mn>...</mn><mo>,</mo><mi>&amp;phi;</mi><mrow><mo>(</mo><msubsup><mi>A</mi><mn>1</mn><mi>n</mi></msubsup><mo>)</mo></mrow><mo>,</mo><mn>...</mn><mi>&amp;phi;</mi><mrow><mo>(</mo><msubsup><mi>A</mi><mi>M</mi><mn>1</mn></msubsup><mo>)</mo></mrow><mo>,</mo><mn>...</mn><mi>&amp;phi;</mi><mrow><mo>(</mo><msubsup><mi>A</mi><mi>M</mi><mi>n</mi></msubsup><mo>)</mo></mrow><mo>&amp;rsqb;</mo><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></mrow> 式中，AM是M个m×n的图像数据的矩阵，上标n是图像数据矩阵的总列数；下标M是图像数据矩阵的总个数；是第M副图像的数据的第n个列向量，是第M副图像数据按第n列进行核变换所得到的第M副图像数据第n列的核矩阵；In the formula, AM is a matrix of M image data of m×n, superscript n is the total column number of image data matrix; subscript M is the total number of image data matrix; is the nth column vector of the data of the Mth image, is the kernel matrix of the nth column of the Mth secondary image data obtained by performing kernel transformation on the nth column of the Mth secondary image data; 然后，图像数据的核矩阵Φ与其转置所得到的图像数据的核矩阵ΦT相乘，所得的乘积为核矩阵的相关核矩阵，记为S：Then, the kernel matrix Φ of the image data is multiplied by the kernel matrix Φ T of the image data obtained by transposing it, and the resulting product is the correlation kernel matrix of the kernel matrix, which is denoted as S: S＝ΦΦT (4)S＝ ΦΦT (4) 式中，ΦT是核矩阵Φ的转置所得的图像数据的核矩阵；上标T表示转；In the formula, Φ T is the kernel matrix of the image data obtained by the transposition of the kernel matrix Φ; the superscript T means turn; (4).计算图像数据的核矩阵的相关核矩阵的特征值和特征向量，具体步骤为：(4). Calculate the eigenvalue and eigenvector of the correlation kernel matrix of the kernel matrix of image data, concrete steps are: 首先，设图像数据的相关核矩阵的特征值λ和图像数据的相关核矩阵的投影向量v，满足如下关系式：First, let the eigenvalue λ of the correlation kernel matrix of the image data and the projection vector v of the correlation kernel matrix of the image data satisfy the following relationship: λv＝Sv或λv＝ΦΦTv，λ≥0 (5)λv=Sv or λv=ΦΦ T v, λ≥0 (5) 然后，假定M个图像数据的相关核矩阵，记为其表达式为：Then, assuming the correlation kernel matrix of M image data, denoted as Its expression is: <mrow> <mover> <mi>&amp;Phi;</mi> <mo>~</mo> </mover> <mo>=</mo> <mo>&amp;lsqb;</mo> <mi>&amp;phi;</mi> <mrow> <mo>(</mo> <msub> <mover> <mi>A</mi> <mo>&amp;OverBar;</mo> </mover> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>&amp;phi;</mi> <mrow> <mo>(</mo> <msub> <mover> <mi>A</mi> <mo>&amp;OverBar;</mo> </mover> <mi>M</mi> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow> <mrow><mover><mi>&amp;Phi;</mi><mo>~</mo></mover><mo>=</mo><mo>&amp;lsqb;</mo><mi>&amp;phi;</mi><mrow><mo>(</mo><msub><mover><mi>A</mi><mo>&amp;OverBar;</mo></mover><mn>1</mn></msub><mo>)</mo></mrow><mo>,</mo><mn>...</mn><mo>,</mo><mi>&amp;phi;</mi><mrow><mo>(</mo><msub><mover><mi>A</mi><mo>&amp;OverBar;</mo></mover><mi>M</mi></msub><mo>)</mo></mrow><mo>&amp;rsqb;</mo><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>6</mn><mo>)</mo></mrow></mrow> 式中，是图像数据对应的核矩阵，是图像数据矩阵AM中的m个图像数据的行向量的平均特征值；In the formula, Yes The kernel matrix corresponding to the image data, is the average eigenvalue of the row vector of m image data in the image data matrix AM; 如果则将上述式子(5)转换成为下列关系式：if Then the above formula (5) is transformed into the following relational formula: <mrow> <mi>&amp;lambda;</mi> <mover> <mi>&amp;Phi;</mi> <mo>~</mo> </mover> <mi>q</mi> <mo>=</mo> <msup> <mi>&amp;Phi;&amp;Phi;</mi> <mi>T</mi> </msup> <mover> <mi>&amp;Phi;</mi> <mo>~</mo> </mover> <mi>q</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow> <mrow><mi>&amp;lambda;</mi><mover><mi>&amp;Phi;</mi><mo>~</mo></mover><mi>q</mi><mo>=</mo><msup><mi>&amp;Phi;&amp;Phi;</mi><mi>T</mi></msup><mover><mi>&amp;Phi;</mi><mo>~</mo></mover><mi>q</mi><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>7</mn><mo>)</mo></mrow></mrow> 由上述关系式(7)求解，得到图像数据的相关核矩阵的特征值λ和与其对应的图像数据的相关核矩阵的特征向量q，其表达式分别为：Solved by the above relational formula (7), the eigenvalue λ of the correlation kernel matrix of the image data and the eigenvector q of the correlation kernel matrix of the image data corresponding to it are obtained, and its expressions are respectively: λ＝[λ1，...，λM] (8)λ=[λ 1 ,...,λ M ] (8) q＝[q1，...，qM] (9)q=[q 1 , . . . , q M ] (9) 式中，λM是图像数据的核矩阵的第M个特征值；qM是式子(7)的第M个图像数据的相关核矩阵的特征向量；In the formula, λ M is the Mth eigenvalue of the kernel matrix of the image data; q M is the eigenvector of the correlation kernel matrix of the Mth image data of the formula (7); 如果则求出图像数据的相关核矩阵的特征向量，其表达式为：if Then find the eigenvector of the correlation kernel matrix of the image data, and its expression is: v＝[v1，...，vM] (10)v=[v 1 , . . . , v M ] (10) 式中，vM是图像数据的相关核矩阵的第M个特征向量；In the formula, v M is the Mth eigenvector of the correlation kernel matrix of the image data; (5).计算图像数据的Renyi熵，具体步骤为：所述的计算图像数据的Renyi熵，记为其计算方法如下：(5). Calculate the Renyi entropy of the image data, the specific steps are: the Renyi entropy of the described calculation image data, denoted as Its calculation method is as follows: <mrow> <mover> <mi>V</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mi>M</mi> </mfrac> <msubsup> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>M</mi> </msubsup> <mover> <mi>p</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow> <mrow><mover><mi>V</mi><mo>^</mo></mover><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mi>M</mi></mfrac><msubsup><mi>&amp;Sigma;</mi><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>M</mi></msubsup><mover><mi>p</mi><mo>^</mo></mover><mrow><mo>(</mo><msub><mi>A</mi><mi>i</mi></msub><mo>)</mo></mrow><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>11</mn><mo>)</mo></mrow></mrow> 式中，是Parzen窗对V(p)的估计值，即，用Parzen窗对原始空间图像数据二次Renyi熵的坐标轴的方向的估计值，In the formula, is the estimated value of V(p) by the Parzen window, that is, the estimated value of the direction of the coordinate axis of the quadratic Renyi entropy of the original spatial image data using the Parzen window, 将公式(2)代入到公式(11)中得到二次Renyi熵的Parzen窗估计值其估计表达式：Substituting formula (2) into formula (11) to get the estimated value of the Parzen window of the quadratic Renyi entropy Its estimated expression: <mrow> <mover> <mi>V</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <msup> <mi>M</mi> <mn>2</mn> </msup> </mfrac> <msubsup> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>M</mi> </msubsup> <msubsup> <mi>&amp;Sigma;</mi> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>M</mi> </msubsup> <msub> <mi>K</mi> <mi>&amp;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> <mrow><mover><mi>V</mi><mo>^</mo></mover><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><msup><mi>M</mi><mn>2</mn></msup></mfrac><msubsup><mi>&amp;Sigma;</mi><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>M</mi></msubsup><msubsup><mi>&amp;Sigma;</mi><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>M</mi></msubsup><msub><mi>K</mi><mi>&amp;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> 式中，Ai和Aj表示A的第i个图像数据矩阵和第j个图像数据矩阵，将步骤(4)中相关核矩阵的特征值和特征向量带入到公式(12)中即可得到的等价公式：In the formula, A i and A j represent the i-th image data matrix and the j-th image data matrix of A, and the eigenvalues and eigenvectors of the relevant kernel matrix in step (4) can be brought into formula (12) get The equivalent formula for is: 或 or 式中，1是图像数据的相关核矩阵m×1的单位向量；1T是图像数据的相关核矩阵m×1的单位向量的转置；M是图像数据矩阵的个数；ET是E转置得到的图像数据的相关核矩阵的特征向量；是图像数据的相关核矩阵第i个特征向量的转置；In the formula, 1 is the unit vector of the correlation kernel matrix m×1 of the image data; 1 T is the transpose of the unit vector of the correlation kernel matrix m×1 of the image data; M is the number of image data matrices; E T is E The eigenvectors of the correlation kernel matrix of the image data obtained by transposing; is the transpose of the ith eigenvector of the correlation kernel matrix of the image data; (6).采用二维核熵成分分析方法对图像数据的相关核矩阵的特征向量进行映射，实现图像数据的降维，具体步骤为：(6). The two-dimensional kernel entropy component analysis method is used to map the eigenvectors of the relevant kernel matrix of the image data to realize the dimensionality reduction of the image data. The specific steps are: Y＝[v1，...，vd]TΦ(A) (14)Y=[v 1 ,...,v d ] T Φ(A) (14) 首先，根据计算式(13)中计算得到的图像数据的Renyi熵值，按其熵值大小进行降序排序，选择前d个图像数据的Renyi熵值向量，记为Z，其表达式为：First, according to the Renyi entropy value of the image data calculated in formula (13), sort them in descending order according to their entropy value, select the Renyi entropy value vector of the first d image data, denote it as Z, and its expression is: Z＝[z1，...，zd] (15)Z=[z 1 , . . . , z d ] (15) 然后，对该熵值向量进行映射，得到图像数据的核矩阵Φ(A)的映射向量，记为v1，...，vd，利用投影变换得到降维后的图像数据的本征维数，从而实现了图像数据的降维。Then, map the entropy value vector to obtain the mapping vector of the kernel matrix Φ(A) of the image data, denoted as v 1 ,...,v d , and use projection transformation to obtain the eigendimensional dimension of the image data after dimensionality reduction number, thereby realizing the dimensionality reduction of the image data.
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