CN106960435A - A kind of double compression automatic testing methods of jpeg image - Google Patents

Abstract

The invention discloses a kind of double compression automatic testing methods of jpeg image, including jpeg image pretreatment；The poor feature of the first effective digital feature and adjacent coefficient of DCT coefficient of the jpeg image based on Markov model is extracted, fusion feature simultaneously carries out Feature Dimension Reduction；3 steps such as SVMs training and identification.The fusion feature proposed in the present invention is to various compression situations（Except general compression situation, be additionally included in that compression quality twice is equal and second of compression quality for 95 situation）Can effectively it use, and this method has done dimension-reduction treatment after fusion feature is extracted to eigenmatrix, and subsequent operation is carried out using the matrix after dimensionality reduction as characteristic vector, reduces computation complexity, improves efficiency of algorithm, it is easier to used in reality.

Description

A kind of double compression automatic testing methods of jpeg image

Technical field

The invention belongs to information security, pattern-recognition and digital image processing techniques field, more particularly to a kind of JPEG The double compression automatic testing methods of image.

Technical background

As image editing software is continued to develop, some basic skills of people GPRS just can be by distorted image " flawless ", light is difficult to differentiate with naked eyes.The life that the software of these simple easy to get started gives people brings facility, while New challenge has been welcome to digital evidence obtaining technology.Under the background in internet big epoch, image is easy to be carried out by criminal Malice is distorted, so determining the authenticity and integrity of a figure is extremely necessary.Joint Photographic Experts Group compressed format is extensive Ground is applied in digital camera and digital imaging processing software, therefore, and JPEG correlative study work has been received increasingly Many concerns.Image will pass through last step after being tampered --- preserves again.In most cases, operator is The image being tampered can be saved as into JPEG standard format to save memory space, ensure picture quality.Therefore, if detected Image passes through double squeeze operations, it can be deduced that a main conclusions, the image may be tampered.There are some to study work It is used to detect whether jpeg image have passed through double compressions and distort.In addition, some steganography patterns, such as F5 and OutGuess, If the original host image of input is also jpeg image, then it can also produce double compression images.So research JPEG figures Double compressed detecteds of picture contribute to the development of the steganalysis algorithm of complexity.

It is between Block DCT (DCT) grid according to the first second compression and the second jpeg image compressed Double compressed detecteds can be divided into two classes by no alignment：One class is the double compressed detecteds of jpeg image of alignment, in addition a class be block not The double compressed detecteds of the jpeg image of alignment.Whether identical according to first time and secondary quantization matrix for the first kind, it is also Two subclasses can be divided into, i.e., different with the quantization matrix of the second second compression for the first time, the quantization matrix of two second compressions is identical.It is right In detecting the different situation of the quantization matrix of two second compressions, there is researcher to propose some more successful methods.Luká^ S and Fridrich propose by explore be referred to as in each independent JPEG coefficients (i.e. quantization DCT coefficient) histogram it is " bimodal The statistics sexual deviation of value " or " missing values " detects double compression images.Popescu and Farid are proposed by measuring Fourier The potential periodic deviations of the JPEG coefficient histograms of leaf transformation detect double compressions.Dongdong Fu et al. are proposed will Benford is applied during image detection, and has put forward the broad sense Benford models for characteristics of image, afterwards, Bin Li et al. have selected the Benford rules that broad sense is used alone in the preceding 20 AC coefficients sorted according to zig-zag so that inspection Surveying result has a further raising, the Markov model of the first effective digitals of the Lisha Dong et al. based on DCT coefficient, Double compressed detecteds are carried out using second-order statisticses model, experimental precision is improved.Lanying Wu et al. are come using Benford laws Estimation compression number of times.Propose using the model histogram of low frequency JPEG coefficients as feature to differentiate with Fridrich Single, double compression image.In addition, Chen et al. is utilized along four direction (i.e. vertical, level, leading diagonal, counter-diagonal) JPEG 2 ties up the transition probability matrix of difference matrix to detect that JPEG weight contracts.

When detecting the compression of identical quantization matrix, all there is not satisfied performance to it above, because They can not be effectively described by with slight deviations caused by identical quantization matrix JPEG compression.Huang et al. passes through A new disturbance threshold method is proposed to pay close attention to this problem.In recent years, Lai andIt has studied and repeating JPEG compression matter The convergence of block's attribute in the case of the amount factor 100 (jpeg-100).Yang et al. is by calculating the IDCT systems with reconstructed image Error between number and pixel value, obtains error image, analysis rounding error block and truncated error block, proposes that series of features is come Characterize the statistical discrepancy of the error block between the mono- double compression images of JPEG.

The studies above achievement has all reached certain Detection results, but can be with improved place there are still some.One It is that most of detection methods are that the special circumstances Detection results such as 95 are poor for diagonal or second of compression ratio, it is suitable not find Feature accurately detect all compression of images situations.Two be that intrinsic dimensionality is too high, causes data redundancy, influence detection speed Rate and verification and measurement ratio.

The content of the invention

It is an object of the invention to provide a kind of double compression automatic testing methods of jpeg image, this method passes through based on Ma Er The fusion feature of the poor feature of the first effective digital feature and adjacent coefficient of the DCT coefficient of section's husband's model compresses and double to distinguish list JPEG image compression image, realizes the detection that the double compressions of jpeg image are distorted.The fusion feature of this method overcomes above-mentioned double compression inspections Survey method realizes efficiency, and various compression situations are used, and this method is done after fusion feature is extracted to eigenmatrix Dimension-reduction treatment, subsequent operation is carried out using the matrix after dimensionality reduction as characteristic vector, reduces computation complexity, improves detection effect Rate.

The technical solution adopted in the present invention is：A kind of double compression automatic testing methods of jpeg image, it is characterised in that bag Include following steps：

Step 1：Jpeg image is pre-processed；

Step 2：Extract the first effective digital feature of DCT coefficient of the jpeg image based on Markov model and adjacent system Number difference feature, fusion feature simultaneously carries out Feature Dimension Reduction；

Step 3：Characteristic vector is trained and recognized using SVMs.

The present invention general principle be：The respective advantage of two kinds of features is merged to recognize double compression images.When extracting feature, It is extracted the fusion feature of the poor feature of the first effective digital feature and adjacent coefficient of the DCT coefficient based on Markov model. Data Dimensionality Reduction is added during fusion feature, this make it that the invention has arrived good effect in big classification image classification.Its Various situations (including ordinary circumstance and cornerwise situation (first time compression quality etc. of JPEG compression can accurately be detected In the quality of the second second compression) and situation that second compression quality is 95).

It is of the invention compared with existing double compression recognizers, have the following advantages that and beneficial effect：

(1) fusion feature of the invention is compared with the single feature before fusion, in cornerwise situation (the first second compression matter Quality of the amount equal to the second second compression) and second compression quality have more preferable result for 95 situation, be jpeg image Double compressions distort evidence obtaining and provide a kind of algorithm of more popularity；

(2) present invention is simple for the feature extraction of image, and the dimension of the characteristic vector ultimately produced is low, for big During the data of scale, there can be more preferable time high efficiency；

(3) feature of present invention point is high for natural image stability, with very strong robustness.

Brief description of the drawings

Fig. 1 is the flow chart of the embodiment of the present invention；

Fig. 2 is the feature extraction flow chart of the embodiment of the present invention.

Embodiment

Understand for the ease of those of ordinary skill in the art and implement the present invention, below in conjunction with the accompanying drawings and embodiment is to this hair It is bright to be described in further detail, it will be appreciated that implementation example described herein is merely to illustrate and explain the present invention, not For limiting the present invention.

, it is necessary to be generated and handled for the image used by experiment before the experimental implementation of the present invention is carried out.This reality It is UCID (S.Milani, M.Tagliasacchi, and S.Tubaro, " Discriminating to test used natural image Multiple JPEG compression using first digit features, " in Proc.IEEE Int.Conf.Acoust., Speech, Signal Process., Mar.2012, pp.2253-2256.) in randomly select 500 figure images.For 500 uncompressed images, JPEG compression is carried out to image first in Photoshop, by even Continuous compression quality QF1 first, 500 pictures that second of compression quality QF2 is produced are designated as double compressed picture collection；Original graph Piece compresses 500 pictures produced by corresponding compression quality QF2, is designated as single compressed picture collection.In view of relatively low compression Quality has been able to recognize that, so Setup Experiments QF1 and QF2 scope are from 70 to 95, step-length is 5 by naked eyes.Using this The mode of kind, may finally obtain 36 groups of images, and every group is that QF2 single compression image and 500 press all comprising 500 compression qualities Contracting quality elder generation QF1 QF2 again double compression images.For different compression qualities and compression number of times, different lives are carried out to image Name and storage work.

Then, the experimental implementation of the present invention is carried out.The flow of the present invention is as shown in figure 1, whole detection process includes image Pretreatment, three processes of feature extraction and model training, most important of which part is exactly characteristic extraction procedure, in such as Fig. 2 The characteristic extraction procedure marked；The first place that extraction process extracts the DCT coefficient of the image based on Markov model first has Numerical characteristic and the poor feature of adjacent coefficient are imitated, then obtained characteristic vector is merged, carrying out Feature Dimension Reduction using PCA obtains To final characteristic vector.Characteristic vector is asked for after completing, it is necessary to SVM model training be carried out, using the side of cross validation Formula completes parameter Estimation, is classified finally for training image.

The double compression automatic testing methods of a kind of jpeg image that the present invention is provided, comprise the following steps：

Step 1：Jpeg image is pre-processed；

Step 1.1：Jpeg image gray value is extracted, and is rounded, gradation of image value matrix is obtained, is designated as I (i, j), wherein (i, j) represents the corresponding row and column of matrix；

Step 1.2：Gradation of image value matrix I (i, j) is divided into nonoverlapping N × N blocks (the present embodiment is 8 × 8)；

Step 2：Extract the first effective digital feature of DCT coefficient of the jpeg image based on Markov model and adjacent system Number difference feature, fusion feature simultaneously carries out Feature Dimension Reduction；

The first effective digital feature of DCT coefficient of the jpeg image based on Markov model is extracted described in step 2, its Implement including following sub-step：

Step A1：For ready-portioned sub-block in step 1.2, the DCT statistical natures of each sub-block are extracted respectively, are obtained To DCT coefficient matrix F (i, j)；

The process of implementing is each sub-block of order traversal image from top to bottom, to every height according to from left to right Block carries out dct transform；After obtaining DCT coefficient, it is carried out obtain after quantization operation, quantization image eigenmatrix F (i, j)；

The formula of each 8 × 8 pieces of dct transform is：

Wherein,F (i, j) represents the original picture block each divided, and wherein u is two-dimensional matrix Horizontal direction frequency, v is the vertical direction frequency of two-dimensional matrix.F (i, j) represents the eigenmatrix after dct transform.F (0,0) is DC coefficient, other are ac coefficient.In the present invention, ac coefficient is only taken.

Step A2：Preceding n coefficient (this reality of each N × N blocks in F (i, j) matrix is extracted according to Zig-Zag order Apply preceding 20 coefficients that example is each 8 × 8 pieces)；

Step A3：The first effective digital for the coefficient for obtaining extracting in step A2 according to broad sense Benford formula, and by number According to being stored in matrix；

Wherein, broad sense Benford formula are：

Wherein, N is normalized parameter, and s and q are according to different obtained by different images, different quantization quality Model parameter.

Step A4：The first effective digital matrix obtained with Markov Chain fit procedure A3, obtains jpeg image and is based on The characteristic vector of the first effective digital of the DCT coefficient of Markov model；

The process of implementing is：The first effective digital matrix based on DCT coefficient is fitted with Markov Chain, and with singly Walk transition probability matrix to characterize this process, finally draw Stationary Distribution vector, be required characteristic vector.

The method of Markov Chain fitting is as follows：

The scope of the first effective digital is 0-9 (be in order to obtain more information as far as possible) comprising 0, so Ma Erke Husband's chain is limited.F (i, j) represents DCT coefficient in the i-th row, the element of the first effective digital of jth row, then F is in the horizontal direction On a step transition probability matrix can represent following formula：

What m here, n were represented respectively is row and column.And there is following formula：

Because having level, vertical, leading diagonal, the transfer matrix of counter-diagonal four direction, but pass through test, four The transfer matrix in direction, which combines obtained result, will get well, also add dimension and calculating unlike singly taking horizontal direction to obtain result Difficulty, so for simplicity, the transfer matrix of horizontal direction is only taken here.

For the Markov chain of finite state, its Stationary Distribution is certainly existed, but not necessarily unique, only when the chain During for traversal and irreducible Markov chain, its Stationary Distribution is present and unique.Following equation is met in the presence of distribution π：

π=π p

Here p is transition probability matrix, and π is the vector of a non-negative element, and is 1.What's more, if Ma Er can Husband's chain is can not to simplify and travel through, and π is unique.According to theory of random processes, if a Markov Chain is stateful Space S and transfer matrix p, when following formula is met, Stationary Distribution is unique, is shown below：

In such case, only π can be calculated by following formula：

The characteristic vector of one 10 dimension is obtained according to formula, because being extracted preceding n characteristic point, the spy finally extracted It is 10n dimensions to levy vector.

The poor feature of adjacent coefficient of DCT coefficient of the jpeg image based on Markov model is extracted described in step 2, it has Body, which is realized, includes following sub-step：

Step B1：For ready-portioned sub-block in step 1.2, the DCT statistical natures of each sub-block are extracted respectively, are obtained To DCT coefficient matrix F (i, j)；

Step B2：According to Zig-Zag order in I (i, j) matrix each N × N blocks (and step A2 N × N blocks protect Hold consistent) preceding n coefficient carry out level, vertically, leading diagonal, the difference in the direction of counter-diagonal four, obtain four difference squares Battle array；

Whole picture, u ∈ [0, S are represented with F (u, v)_h-1],v∈[0,S_v- 1], S_hAnd S_vIt is ash to be detected respectively Spend image 2 dimension groups it is horizontal and vertical apart from size；

Four difference matrixs as shown under formula：

F_h(u, v)=F (u, v)-F (u+1, v),

F_v(u, v)=F (u, v)-F (u, v+1),

F_d(u, v)=F (u, v)-F (u+1, v+1),

F_m(u, v)=F (u+1, v)-F (u, v+1),

Wherein, F_h(u,v),F_v(u,v),F_d(u,v),F_mWhat (u, v) was represented respectively is vertical, level, leading diagonal, pair The difference matrix of diagonal four direction.In order to reduce computed losses, the scope of threshold value of difference value is set within (- 4-4), Value more than 4 is designated as 4, and the value less than -4 is designated as -4.

Step B3：The four difference matrix given thresholds obtained to step B2, with Markov one-step transition probability matrix Transfer vector is obtained, by this shifting science and technology in four directions addition of vectors, as adjacent coefficient difference of the jpeg image based on Markov model Characteristic vector；

Computational methods are as follows：

By taking horizontal direction as an example,：

Here m, n ∈ [- 4,4], and have

The other three direction asks the method similar with its.Finally, each direction can obtain the matrix of one 9 × 9, so One has 324 dimensional vectors.

Fusion feature described in step 2 simultaneously carries out Feature Dimension Reduction, by the jpeg image of extraction based on Markov model The first effective digital feature of DCT coefficient is directly added with the poor feature of adjacent coefficient, obtains 10n+324 (the present embodiment is 524) The eigen vector of dimension, PCA dimensionality reductions are carried out to obtained 10n+324 (the present embodiment is 524) dimension eigen vectors.

Characteristic vector 10n+324 (the present embodiment is 200+324=524) dimensions of each image extracted, dimension is still It is higher.Therefore, dimension-reduction treatment is carried out to eigenmatrix in this step and reduces data volume.The present invention uses main into composition Analytic approach (PCA) carries out Feature Dimension Reduction, this method be it is a kind of it is conventional information is compressed based on variable covariances matrix and The method of extraction, the data of higher-dimension are projected on lower dimensional space by linear transformation, and those dimensions fallen are dropped by PCA and are usually Picture noise either redundancy feature, because PCA methods ensure that data are not true as much as possible.

PCA dimensionality reductions wherein are carried out to obtained 10n+324 (the present embodiment is 524) dimension eigen vectors, it implements bag Include following sub-step：

Step C1：If n-dimensional vector w is a change in coordinate axis direction of target subspace, referred to as map vector, data are maximized Variance after mapping, is obtained：

Wherein：M is the number of data instance, X_iIt is that data instance i vector table reaches, X is being averaged for all data instances Vector；

Step C2：It is comprising the matrix that all map vectors are column vector, by linear algebraic transformation to define W：

Obtain following optimization object function：

The wherein track of tr representing matrixs, A is data covariance matrix；

Step C3：Output after PCA dimensionality reductions is Y=W'X, and k dimensions are reduced to by X original dimension.

Step 3：Characteristic vector is trained and recognized using SVMs.

Step 3.1：Classification design is carried out using SVM, the libsvm for being used herein as having designed is trained；

Step 3.2：Nu-SVM is selected as the model of classification；

Step 3.3：Data in training set are handed over according to natural image and tampered image data volume identical mode Fork checking and model training, then obtain optimal nu-SVM parameter, herein can be using the training optimal solution existed Algorithm；

The experimental result of the present invention, the experimental result obtained with the single feature before carrying out Fusion Features is contrasted, Resulting result is as shown in table 1 below：

(first place of the DCT coefficient based on Markov model has the fusion feature of the embodiment of the present invention of table 1 with single feature Feature extraction and the adjacent coefficient based on Markov model for imitating numeral are poor) testing result comparison diagram

From table 1 it follows that using this method and PCA dimensionality reductions, discrimination is higher than the independent feature of non-dimensionality reduction.Cause This, algorithm proposed by the invention is not only simple easily to be realized, efficiency of algorithm is high, and on accuracy of identification also than it is traditional based on The first effective digital feature of the DCT coefficient of Markov and the poor feature of adjacent coefficient will be high.Used in the double compression images of detection Distort, will there is higher efficiency, higher discrimination and the bigger scope of application.

It should be appreciated that the part that this specification is not elaborated belongs to prior art.

It should be appreciated that the above-mentioned description for preferred embodiment is more detailed, therefore it can not be considered to this The limitation of invention patent protection scope, one of ordinary skill in the art is not departing from power of the present invention under the enlightenment of the present invention Profit is required under protected ambit, can also be made replacement or be deformed, each fall within protection scope of the present invention, this hair It is bright scope is claimed to be determined by the appended claims.

Claims (10)

1. the double compression automatic testing methods of a kind of jpeg image, it is characterised in that comprise the following steps：

Step 1：Jpeg image is pre-processed；

Step 2：The first effective digital feature and adjacent coefficient for extracting DCT coefficient of the jpeg image based on Markov model are poor Feature, fusion feature simultaneously carries out Feature Dimension Reduction；

Step 3：Characteristic vector is trained and recognized using SVMs.

2. the double compression automatic testing methods of jpeg image according to claim 1, it is characterised in that the specific reality of step 1 Now include following sub-step：

Step 1.1：Jpeg image gray value is extracted, and is rounded, gradation of image value matrix is obtained, is designated as I (i, j), wherein (i, j) Represent the corresponding row and column of matrix；

Step 1.2：Gradation of image value matrix I (i, j) is divided into nonoverlapping N × N blocks.

3. the double compression automatic testing methods of jpeg image according to claim 2, it is characterised in that extracted described in step 2 The first effective digital feature of DCT coefficient of the jpeg image based on Markov model, it is implemented including following sub-step：

Step A1：For ready-portioned sub-block in step 1.2, the DCT statistical natures of each sub-block are extracted respectively, are obtained DCT coefficient matrix F (i, j)；

Step A2：The preceding n coefficient of each N × N blocks in F (i, j) matrix is extracted according to Zig-Zag order；

Step A3：The first effective digital for the coefficient for obtaining extracting in step A2 according to broad sense Benford formula, and data are protected Exist in matrix；

Step A4：The first effective digital matrix obtained with Markov Chain fit procedure A3, obtains jpeg image and is based on Ma Er The characteristic vector of the first effective digital of the DCT coefficient of section's husband's model.

4. the double compression automatic testing methods of jpeg image according to claim 3, it is characterised in that：In step A1, according to From left to right, each sub-block of order traversal image from top to bottom, dct transform is carried out to each sub-block；Obtain DCT coefficient it Afterwards, to it obtain the eigenmatrix F (i, j) of image after quantization operation, quantization；

Each the formula of the dct transform of N × N blocks is：

$F\left(u,v\right)=\frac{1}{4}C\left(u\right)C\left(v\right)\[\underset{i=0}{\overset{N-1}{\mathrm{\Σ}}}\underset{j=0}{\overset{N-1}{\mathrm{\Σ}}}f\left(i,j\right)\cos \frac{\left(2i+1u\mathrm{\π}\right)}{16}\cos \frac{\left(2j+1\right)v\mathrm{\π}}{16}\]$

Wherein,F (i, j) represents the original picture block each divided, and wherein u is the level of two-dimensional matrix Direction frequency, v is the vertical direction frequency of two-dimensional matrix；F (i, j) represents the eigenmatrix after dct transform, and F (0,0) is direct current Coefficient, other are ac coefficient.

5. the double compression automatic testing methods of jpeg image according to claim 3, it is characterised in that wide described in step A3 Adopted Benford formula are：

$p\left(x\right)=N{\log }_{10}(1+\frac{1}{s+x^q}),x=1,2,...,9;$

Wherein, N is normalized parameter, and s and q are according to the different models obtained by different images, different quantization quality Parameter.

6. the double compression automatic testing methods of jpeg image according to claim 3, it is characterised in that：In step A4, horse is used Er Kefu chains are fitted the first effective digital matrix based on DCT coefficient, and characterize with one-step transition probability matrix this mistake Journey, finally draws Stationary Distribution vector, is required characteristic vector.

7. the double compression automatic testing methods of jpeg image according to claim 2, it is characterised in that extracted described in step 2 The poor feature of the adjacent coefficient of DCT coefficient of the jpeg image based on Markov model, it is implemented including following sub-step：

Step B1：For ready-portioned sub-block in step 1.2, the DCT statistical natures of each sub-block are extracted respectively, are obtained DCT coefficient matrix F (i, j)；

Step B2：Level is carried out to the preceding n coefficient of each N × N blocks in I (i, j) matrix according to Zig-Zag order, hung down Directly, leading diagonal, the difference in the direction of counter-diagonal four, obtain four difference matrixs；

Step B3：The four difference matrix given thresholds obtained to step B2, are obtained with Markov one-step transition probability matrix Transfer vector, by the feature of this shifting science and technology in four directions addition of vectors, as adjacent coefficient difference of the jpeg image based on Markov model Vector.

8. the double compression automatic testing methods of jpeg image according to claim 7, it is characterised in that：In step B2, F is used (u, v) represents whole picture, u ∈ [0, S_h-1],v∈[0,S_v- 1], S_hAnd S_vIt is 2 dimensions of gray level image to be detected respectively Array it is horizontal and vertical apart from size；

Four difference matrixs as shown under formula：

F_h(u, v)=F (u, v)-F (u+1, v),

F_v(u, v)=F (u, v)-F (u, v+1),

F_d(u, v)=F (u, v)-F (u+1, v+1),

F_m(u, v)=F (u+1, v)-F (u, v+1),

Wherein, F_h(u,v),F_v(u,v),F_d(u,v),F_mWhat (u, v) was represented respectively is vertical, level, leading diagonal, counter-diagonal The difference matrix of four direction.

9. the double compression automatic testing methods of jpeg image according to claim 1, it is characterised in that：Melt described in step 2 Close feature and carry out Feature Dimension Reduction, by the first effective digital of DCT coefficient of the jpeg image of extraction based on Markov model The poor feature of feature and adjacent coefficient is directly added, and obtains the eigen vector of 324+10n dimensions, obtained 324+10n is tieed up characteristic to Amount carries out PCA dimensionality reductions.

10. the double compression automatic testing methods of jpeg image according to claim 9, it is characterised in that：Described pair obtains 324+10n dimension eigen vectors carry out PCA dimensionality reductions, and it is implemented including following sub-step：

Step C1：If n-dimensional vector w is a change in coordinate axis direction of target subspace, referred to as map vector, data mapping is maximized Variance afterwards, is obtained：

$\underset{w}{max}\frac{1}{m-1}\underset{i=1}{\overset{m}{\Σ}}{\left(w^T\right(X_i-\stackrel{\‾}{X}\left)\right)}^2$

Wherein：M is the number of data instance, X_iIt is that data instance i vector table reaches,It is the average vector of all data instances；

Step C2：It is comprising the matrix that all map vectors are column vector, by linear algebraic transformation to define W：

$\underset{W}{min}tr\left(W^{T}AW\right),s.t.W^{T}W=I$

Obtain following optimization object function：

$A=\frac{1}{m-1}\underset{i=1}{\overset{m}{\Σ}}(X_i-\stackrel{\‾}{X}){(X_i-\stackrel{\‾}{X})}^T$

The wherein track of tr representing matrixs, A is data covariance matrix；

Step C3：Output after PCA dimensionality reductions is Y=W'X, and k dimensions are reduced to by X original dimension.