CN104537377A - Image data dimension reduction method based on two-dimensional kernel entropy component analysis - Google Patents
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Abstract
The invention provides an image data dimension reduction method based on two-dimensional kernel entropy component analysis. The method comprises the following steps that (1) image data are read in; (2) a kernel function is estimated through a Parzen window; (3) a kernel matrix for calculating all the image data in row is set; (4) the eigenvalue and eigenvector of a correlation matrix of the image data are calculated; (5) the Renyi entropy of the image data is calculated; (6) the eigenvector of the correlation matrix of the image data is mapped through a two-dimensional kernel entropy component analysis method, and data dimension reduction is achieved. According to the method, through the two-dimensional analysis method, kernel conversion is directly carried out on the rows or lines of an image, the entropy estimated by the kernel matrix of the image data is sorted, the intrinsic dimension of the image data obtained after dimension reduction is obtained, and the space structure information of the image data can be kept. According to the method, due to the fact that the kernel matrix directly calculates the image data in row or in line, two-dimensional image data do not need to be converted into one-dimensional vectors, and calculation complexity is reduced when the correlation matrix is obtained through kernel conversion.
Description
Technical field
The present invention relates to a kind of view data dimension reduction method of two-dimentional nuclear entropy constituent analysis (KECA), belong to high dimensional image disposal route and applied technical field, be applicable to the research of the dimensionality reduction Theory and applications technology of high dimensional image.
Background technology
In the application such as recognition of face, numeral 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 with numeric representation, it effectively can 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 large, therefore how effectively important information is obtained, dimensionality reduction is carried out to view data, and reduces the complexity calculated, become the key link of image real time transfer.
Dimensionality reduction at present for view data has proposed a lot of method, and the dimension reduction method of view data mainly contains principal component analytical method, core principle component analysis method, nuclear entropy component analyzing method, subsequently, has two-dimensional principal component analysis method.Principal component analytical method is a kind of image data converting method of classics, and it is a kind of linear transformation method, and core principle component analysis is then the nonlinear stretch of principal component analysis (PCA).Take principal component analysis (PCA) as the image data converting method of representative, first the covariance matrix of view data is tried to achieve, and obtain eigenwert and the proper vector of this covariance matrix, then coordinate system is built by the proper vector corresponding to maximum several eigenwerts, finally sample image data is projected on this coordinate system, obtain the view data after dimensionality reduction.Nuclear entropy constituent analysis (Kernel Entropy Component Analysis, KECA) method is a kind of based on information-theoretical new image data converting method.The method, using the coordinate axis of original spatial image data secondary Renyi entropy as projecting direction, these are different from traditional data transformation spectral transformation method, the feature space of nuclear entropy constituent analysis (KECA) method choice dimensionality reduction, view data after conversion has obvious angled arrangement attribute, thus is beneficial to further process.But also exist following not enough: during above-mentioned principal component analytical method, core principle component analysis method, nuclear entropy component analyzing method conversion dimensionality reduction, all first convert two-dimensional pixel matrix to one-dimensional characteristic vector, this data transfer device does not only effectively utilize the spatial structural form of view data, and when the nuclear matrix of subsequent calculations covariance or computed image data, add the complexity of calculating; Although next is the space structure spatial information that the above-mentioned dimension reduction method based on two-dimensional principal component analysis make use of view data, this linear processing methods still has limitation in the application.
In sum, the main Problems existing of dimension reduction method of current image data is: the spatial structural form that effectively can not utilize view data, and computation complexity is high.
Summary of the invention
The object of the invention is can not effectively utilize the high deficiency of the spatial structural form of view data, computation complexity to propose a kind of view data dimension reduction method based on two-dimentional nuclear entropy constituent analysis for the dimension reduction method of conventional images data.
Technical solution of the present invention is: a kind of view data dimension reduction method based on two-dimentional nuclear entropy constituent analysis of the present invention, specifically refer to that one directly carries out data transformation to two-dimensional image data, the spatial structural form of view data can be remained, improve the dimensionality reduction performance of two-dimensional image data.The method mainly direct row by image or row carries out kernel mapping, and without the need to image being converted to the form of vector, the eigenwert and proper vector of trying to achieve the nuclear matrix of view data are brought in entropy estimate, selective entropy composition maps, realize the dimensionality reduction of view data, thus improve the computation complexity reducing data transformation.A kind of view data dimension reduction method based on two-dimentional nuclear entropy constituent analysis of the present invention, its step is as follows:
(1). read in view data;
(2). adopt Parzen window to estimate kernel function;
(3). set up the nuclear matrix by all view data of column count;
(4). the eigenwert of the correlation matrix of computed image data and proper vector;
(5). the Renyi entropy of computed image data;
(6). adopt the proper vector of two-dimentional nuclear entropy component analyzing method to the correlation matrix of view data to map, realize the dimensionality reduction of view data;
Wherein, the employing Parzen window described in step (2) estimates kernel function, is designated as
, wherein, quadratic Renyi entropy expression formula:
(1)
In formula,
m m
the image data matrix of n;
it is image data matrix
probability density function;
be monotonic quantity, only need analyze the quadratic Renyi entropy removing negative sign
, it can be expressed as
, in order to estimate
, introduce Parzen window density estimator, it estimates expression formula:
(2)
In formula,
it is Parzen window pair
carry out the estimated value estimating to obtain; M is the number of all image data matrixs; I is the sequence number of M, and span is 1 to M;
the kernel function that Parzen window is estimated,
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 computing method are as follows:
First, by column vector, kernel mapping is carried out to all view data, obtains nuclear matrix, be designated as
, its matrix is:
(3)
In formula,
m m
the matrix of the view data of n, subscript n is total columns of image data matrix; Subscript M is total number of image data matrix;
the n-component column vector of the data of M sub-picture,
that M auxiliary image data carries out the nuclear matrix of M auxiliary image data n-th row that kernel mapping obtains by the n-th row;
Then, the nuclear matrix of view data
the nuclear matrix of the view data obtained with its transposition
be multiplied, the product of gained is nuclear matrix is correlation matrix, is designated as
:
(4)
In formula,
it is nuclear matrix
the nuclear matrix of view data of transposition gained; Subscript T represents transposition.
Wherein, the eigenwert of the correlation matrix of the computed image data described in step (4) and proper vector, its computing method are as follows:
First, if the eigenwert of the correlation matrix of view data
with the projection vector of the correlation matrix of view data
, meet following relational expression:
or
,
(5)
Then, assuming that the relevant nuclear matrix of M view data, be designated as
, its expression formula is:
(6)
In formula,
be
the nuclear matrix that view data is corresponding,
it is image data matrix
in the mean eigenvalue of row vector of m view data;
If
, then above-mentioned formula (5) is converted into following relationship:
(7)
Solved by above-mentioned relation formula (7), obtain the eigenwert of the relevant nuclear matrix of view data
and the proper vector of relevant nuclear matrix to its corresponding view data
, its expression formula is respectively:
(8)
(9)
In formula,
m eigenwert of the relevant nuclear matrix of view data;
it is the proper vector of the relevant nuclear matrix of M view data of formula (7);
If
, then obtain the proper vector of the relevant nuclear matrix of view data, its expression formula is:
(10)
In formula,
m proper vector of the relevant nuclear matrix of view data;
Wherein, the Renyi entropy of the computed image data described in step (5), is designated as
its computing method are as follows:
(11)
In formula,
it is Parzen window pair
estimated value, that is, by the estimated value of Parzen window to the direction of the coordinate axis of original spatial image data quadratic Renyi entropy,
Formula (2) is updated in formula (11), obtains the Parzen window estimated value of quadratic Renyi entropy
, it estimates expression formula:
(12)
In formula,
with
represent i-th image data matrix and a jth image data matrix of A, the eigenwert of nuclear matrix relevant in step (4) and proper vector are brought in formula (12), can obtain
equivalence formula:
(13)
In formula,
the relevant nuclear matrix m of view data
the vector of unit length of 1;
the relevant nuclear matrix m of view data
the transposition of the vector of unit length of 1; M is the number of image data matrix;
it is the proper vector of the relevant nuclear matrix of the view data that E transposition obtains;
it is the transposition of relevant nuclear matrix i-th proper vector of view data;
Wherein, step (6) adopts the proper vector of two-dimentional nuclear entropy component analyzing method to the correlation matrix of view data to map, and realize the dimensionality reduction of view data, it is specific as follows:
(14)
First, according to the Renyi entropy of the view data calculated in calculating formula (13), carry out descending sort by its entropy size, before selecting, the Renyi entropy vector of d view data, is designated as
, its expression formula is:
(15)
Then, this entropy vector is mapped, obtains the nuclear matrix of view data
mapping vector, be designated as
; , utilize projective transformation to obtain the intrinsic dimension of the view data after dimensionality reduction, thus achieve the dimensionality reduction of view data.
This discovery advantage is compared with prior art: this method have employed two-dimentional nuclear entropy component analyzing method, by row or by row nuclear matrix conversion is carried out to view data, Renyi entropy is estimated by the nuclear matrix of view data, obtain the assertive evidence dimension of the view data after dimensionality reduction, achieve the dimensionality reduction of view data.It has following advantage:
(1). the method utilizes two-dimension analysis method, the direct row to image or row carry out kernel mapping, estimate entropy sorts to the nuclear matrix of view data, obtain the intrinsic dimension of the view data after dimensionality reduction, the spatial structural form of view data can also be kept;
(2). the method, due to directly by row or by the nuclear matrix of column count view data, without the need to converting two-dimensional image data to a n dimensional vector n, being carried out kernel mapping when trying to achieve the correlation matrix of view data, being reduced the complexity of calculating.
Accompanying drawing explanation
Fig. 1 is the realization flow of a kind of view data dimension reduction method based on two-dimentional nuclear entropy constituent analysis that the present invention relates to;
Fig. 2 is the comparison sheet of the nicety of grading of the dimension reduction method of view data of the present invention and the dimension reduction method of existing view data.
Embodiment
In order to a kind of view data dimension reduction method based on two-dimentional nuclear entropy constituent analysis that the present invention relates to better is described, the forehead image of two of FERET face database kinds of different expressions is utilized to carry out analysis dimensionality reduction and classify.
A kind of view data dimension reduction method based on two-dimentional nuclear entropy constituent analysis of the present invention, as shown in Figure 1, specific implementation step is as follows for realization flow figure:
(1). read in view data: read in FERET face database view data, raw image data size is 80
80, in the present embodiment, view data being cut into size is 60
the view data of 60;
(2). adopt Parzen window to estimate kernel function, be designated as
, wherein, quadratic Renyi entropy expression formula:
(1)
In formula,
it is 200 60
the image data matrix of 60;
it is image data matrix
probability density function, analyze the quadratic Renyi entropy removed after negative sign
, it can be expressed as
, in order to estimate
, introduce Parzen window density estimator, it estimates expression formula:
(2)
In formula,
it is Parzen window pair
carry out the estimated value estimating to obtain; 200 is numbers of all image data matrixs;
the kernel function that Parzen window is estimated,
it is the width of window function;
(3). set up the nuclear matrix by all view data of column count, its computing method are as follows:
First, by column vector, kernel mapping is carried out to all view data, obtains nuclear matrix, be designated as
, its matrix is:
(3)
In formula,
it is 200 60
the image data matrix of 60, subscript 60 represents the matrix columns of view data; Subscript 200 represents the number of image data matrix;
be the 60th column vector of the data of the 200th image, column vector is 60
1;
Then, the nuclear matrix of view data
the image data matrix obtained with its transposition
be multiplied, the product of gained is nuclear matrix is relevant nuclear matrix, is designated as
:
(4)
In formula,
it is nuclear matrix
the nuclear matrix of view data of transposition gained; Subscript T represents transposition;
(4). the eigenwert of the correlation matrix of computed image data and proper vector, it is specific as follows:
First, if the eigenwert of the correlation matrix of view data
with the projection vector of the correlation matrix of view data
, meet following relational expression:
or
,
(5)
Then, assuming that the relevant nuclear matrix of 200 view data, be designated as
, its expression formula is:
(6)
In formula,
be
the nuclear matrix that view data is corresponding,
it is image data matrix
in the mean eigenvalue of row vector of m view data;
If
, then above-mentioned formula (5) is converted into following relationship:
(7)
Solved by above-mentioned relation formula (7), obtain the eigenwert of the relevant nuclear matrix of view data
and the proper vector of relevant nuclear matrix to its corresponding view data
, its expression formula is respectively:
(8)
(9)
In formula,
relevant nuclear matrix the 200th eigenwert of view data;
it is the proper vector of the relevant nuclear matrix of the 200th view data of formula (7);
If
, then obtain the proper vector of the relevant nuclear matrix of view data, its expression formula is:
(10)
In formula,
m proper vector of the relevant nuclear matrix of view data;
(5). the Renyi entropy of computed image data, is designated as
its computing method are as follows:
(11)
In formula,
it is Parzen window pair
estimated value, that is, by the estimated value of Parzen window to the direction of the coordinate axis of original spatial image data quadratic Renyi entropy,
Formula (3) is updated in formula (11), obtains the Parzen window estimated value of quadratic Renyi entropy
, it estimates expression formula:
(12)
In formula,
with
represent i-th image data matrix and a jth image data matrix of A; The eigenwert of nuclear matrix relevant in step (4) and proper vector are brought in formula (12) and can obtain
equivalence formula:
(13)
In formula,
it is the relevant nuclear matrix 60 of view data
the vector of unit length of 1;
the relevant nuclear matrix m of view data
the transposition of the vector of unit length of 1;
(6). adopt the proper vector of two-dimentional nuclear entropy component analyzing method to the correlation matrix of view data to map, realize the dimensionality reduction of view data, it is specific as follows:
(14)
First according to the Renyi entropy of the view data calculated in calculating formula (13), carry out descending sort by its entropy size, before selecting, the Renyi entropy vector of d view data, is designated as
, its expression formula is:
(15)
Then, this entropy vector is mapped, obtains the nuclear matrix of view data
mapping vector, be designated as
, utilize projective transformation to obtain the intrinsic dimension of the view data after dimensionality reduction, thus achieve the dimensionality reduction of view data.
In order to verify the effect using a kind of view data dimension reduction method method based on two-dimentional nuclear entropy constituent analysis of the present invention, in an experiment, the dimension reduction method of dimension reduction method of the present invention and nuclear entropy component analyzing method of the prior art is made comparisons, as shown in Figure 2, in this comparison sheet, often row represents drop to 10 different dimensions, at interval of 10 dimensions, the data characteristics finally obtained is dropped to respectively 100 to 10 dimensions; Every list is shown and is compared analysis by three kinds of methods, respectively: by the constituent analysis of row nuclear entropy, by row nuclear entropy constituent analysis and nuclear entropy constituent analysis.Table 1 as can be seen from Fig. 2: under same dimension, the result of two-dimentional nuclear entropy constituent analysis is obviously better than the result of nuclear entropy constituent analysis; Under same dimension, be better than nuclear entropy constituent analysis by row by the constituent analysis of row nuclear entropy; The constituent analysis of two dimension nuclear entropy reaches maximal value when dimension is 60, and nuclear entropy constituent analysis reaches maximal value when 100 dimension.Of the present invention two-dimentional nuclear entropy component analyzing method shown in the comparison sheet 1 of this nicety of grading is better than nuclear entropy component analyzing method of the prior art.
Claims (6)
1. based on a view data dimension reduction method for two-dimentional nuclear entropy constituent analysis, it is characterized in that, its step is as follows:
(1). read in view data;
(2). adopt Parzen window to estimate kernel function;
(3). set up the nuclear matrix by all view data of column count;
(4). the eigenwert of the correlation matrix of computed image data and proper vector;
(5). the Renyi entropy of computed image data;
(6). adopt the proper vector of two-dimentional nuclear entropy component analyzing method to the correlation matrix of view data to map, realize the dimensionality reduction of view data.
2. a kind of view data dimension reduction method based on two-dimentional nuclear entropy constituent analysis according to claim 1, is characterized in that: the employing Parzen window described in above-mentioned steps (2) estimates kernel function, is designated as
, wherein, quadratic Renyi entropy expression formula:
(1)
In formula,
m m
the image data matrix of n;
it is image data matrix
probability density function;
be monotonic quantity, the quadratic Renyi entropy removing negative sign need be analyzed
, it can be expressed as
, for estimating
, introduce Parzen window density estimator, it estimates expression formula:
(2)
In formula,
it is Parzen window pair
carry out the estimated value estimating to obtain; M is the number of all image data matrixs; I is the sequence number of M, and span is 1 to M;
the kernel function that Parzen window is estimated,
it is the width of window function.
3. a kind of view data dimension reduction method based on two-dimentional nuclear entropy constituent analysis according to claim 1, it is characterized in that: the nuclear matrix set up by all view data of column count described in above-mentioned steps (3), its computing method are as follows:
First, by column vector, kernel mapping is carried out to all view data, obtains nuclear matrix, be designated as
, its matrix is:
(3)
In formula,
m m
the matrix of the view data of n, subscript n is total columns of image data matrix; Subscript M is total number of image data matrix;
the n-component column vector of the data of M sub-picture,
that M auxiliary image data carries out the nuclear matrix of M auxiliary image data n-th row that kernel mapping obtains by the n-th row;
Then, the nuclear matrix of view data
the nuclear matrix of the view data obtained with its transposition
be multiplied, the product of gained is nuclear matrix is correlation matrix, is designated as
:
(4)
In formula,
it is nuclear matrix
the nuclear matrix of view data of transposition gained; Subscript T represents transposition.
4. a kind of view data dimension reduction method based on two-dimentional nuclear entropy constituent analysis according to claim 1, it is characterized in that: the eigenwert of the correlation matrix of the computed image data described in above-mentioned steps (4) and proper vector, its computing method are as follows:
First, if the eigenwert of the correlation matrix of view data
with the projection vector of the correlation matrix of view data
, meet following relational expression:
or
,
(5)
Then, assuming that the relevant nuclear matrix of M view data, be designated as
, its expression formula is:
(6)
In formula,
be
the nuclear matrix that view data is corresponding,
it is image data matrix
in the mean eigenvalue of row vector of m view data;
If
, then above-mentioned formula (5) is converted into following relationship:
(7)
Solved by above-mentioned relation formula (9), obtain the eigenwert of the relevant nuclear matrix of view data
and the proper vector of relevant nuclear matrix to its corresponding view data
, its expression formula is respectively:
(8)
(9)
In formula,
m eigenwert of the relevant nuclear matrix of view data;
it is the proper vector of the relevant nuclear matrix of M view data of formula (7);
If
, then obtain the proper vector of the relevant nuclear matrix of view data, its expression formula is:
(10)
In formula,
m proper vector of the relevant nuclear matrix of view data.
5. a kind of view data dimension reduction method based on two-dimentional nuclear entropy constituent analysis according to claim 1, is characterized in that: the Renyi entropy of the computed image data described in above-mentioned steps (5), is designated as
its computing method are as follows:
(11)
In formula,
it is Parzen window pair
estimated value, that is, by the estimated value of Parzen window to the direction of the coordinate axis of original spatial image data quadratic Renyi entropy,
Formula (2) is updated to the Parzen window estimated value obtaining quadratic Renyi entropy in formula (11)
, it estimates expression formula:
(12)
In formula,
with
represent i-th image data matrix and a jth image data matrix of A, the eigenwert of nuclear matrix relevant in step (4) and proper vector are brought in formula (12) and can obtain
equivalence formula:
(13)
In formula,
the relevant nuclear matrix m of view data
the vector of unit length of 1;
the relevant nuclear matrix m of view data
the transposition of the vector of unit length of 1; M is the number of image data matrix;
it is the proper vector of the relevant nuclear matrix of the view data that E transposition obtains;
it is the transposition of relevant nuclear matrix i-th proper vector of view data.
6. a kind of view data dimension reduction method based on two-dimentional nuclear entropy constituent analysis according to claim 1, it is characterized in that: the proper vector of employing two dimension nuclear entropy component analyzing method to the correlation matrix of view data described in above-mentioned steps (6) maps, realize the dimensionality reduction of view data, it is specific as follows:
(14)
First, according to the Renyi entropy of the view data calculated in calculating formula (13), carry out descending sort by its entropy size, before selecting, the Renyi entropy vector of d view data, is designated as
, its expression formula is:
(15)
Then, this entropy vector is mapped, obtains the nuclear matrix of view data
mapping vector, be designated as
, utilize projective transformation to obtain the intrinsic dimension of the view data after dimensionality reduction, thus achieve the dimensionality reduction of view data.
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