CN106326935A - Image classification method based on sparse nonlinear subspace migration - Google Patents

Abstract

The invention discloses an image classification method based on sparse nonlinear subspace migration. The method comprises the steps of mapping data from an original space to a regenerated kernel Hilbert space via a kernel method, mapping target domain training data XT to a preset subspace via predefined fundamental transformation P in the kernel Hilbert space to obtain target data PXT, mapping source domain training data XS to the preset subspace via the fundamental transformation P to obtain P[XS, XT], transforming the source domain data P[XS, XT] via a spare matrix Z, and distributing the P[XS, XT] and the PXT in the preset subspace in a sharing mode. The method has the advantages of improving the migration accuracy of image data in the preset subspace and being applicable to transformation of nonlinear data.

Description

A kind of image classification method migrated based on sparse nonlinear subspace

Technical field

The invention belongs to a kind of image classification method, be specifically related to a kind of image migrated based on sparse nonlinear subspace Sorting technique.

Background technology

In general, the data collected when sensor does not occur drift are referred to as source domain, in use for some time The data that sensor obtains after drifting about are referred to as aiming field.At image domains, any one attitude of object can be referred to as Source domain, will be changed the another kind of attitude that (such as angle, light intensity, backcolor change etc.) obtain and be referred to as by external condition Aiming field, the basic task of image classification is exactly that the similar object that can these be under different conditions searches out.

In order to describe simplicity, present invention provide that:

" S " represents source domain, and " T " represents aiming field；Source domain training data is designated asAiming field training data is remembered ForWherein D represents dimension, N_SAnd N_TIt it is the number of training sample in two territories；WithRepresent minority labelling in aiming field and a large amount of unlabelled data；Represent source data and target The luv space of data is mapped to the basis conversion with dimension d of latent space；Z represents X_sAnd X_TBetween sparse reconstruction square Battle array；1_nRepresent the non-zero column vector of a length of n；I representation unit matrix；‖·‖₀Represent the number of a vectorial nonzero element Mesh；‖·‖₁Represent L₁-norm；‖·‖_FThe Frobenius norm of representing matrix；[X]_iRepresent i-th row of X.In the present invention, Matrix uses capitalization black matrix, and vector is small letter black matrix, and variable is italic.

M.Shao,D.Kit,and Y.Fu,“Generalized Transfer Subspace Learning Through Low-Rank Constraint, " Int.J.Comput.Vis., vol.109, pp.74-93,2014. (M.Shao, D.Kit and Y.Fu was published in the paper of volume 109 the 74-93 page in " international computer vision periodical " " under low-rank constraints in 2014 Broad sense transfer sub-space learning "), it is provided that a kind of subspace moving method (being called for short LTSL) represented based on low-rank, LTSL The data transfer principle figure of method is as shown in Figure 1:

Source domain training data X_SWith aiming field training data X_TOne conversion base W of trained acquisition, source domain data and aiming field Data converted base W be mapped to subspace (subspace refer to by projection, it is achieved high dimensional feature to the mapping of lower dimensional space, Ensureing not lose information, this is a kind of conventional dimensionality reduction thought, when this sub spaces training data, can effectively prevent as far as possible Over-fitting), in subspace, source data WX_SBy low-rank Z so that and aiming field data WX_TDistribution is consistent, the most similar warp Cross mapping closer to, thus realize classification.

(such as, principal component analysis (PCA), linear discriminant analysis (LDA), local keeps the sub-space learning method existed Projection (LPP) etc.), it is used to Learning Subspaces projection W, in subspace, source data WX_SRepresent that (LRR) is used for by low-rank Rebuild target data WX_T(because of target data WX_TIt is to exist, WX_SRepresented by low-rank and approach).

LTSL solves subspace branch problem, by minimizing object function as follows and realize (that is: find one group of W, Z, The value of E so that the value of this formula minimizes):

s.t.W^TX_T=W^TX_SZ+E,W^TU₂W=I

In formula, the order of rank (.) representing matrix, ‖ E ‖_2,1Represent l_2,1-norm, and λ₁And λ₂It it is regularization coefficient.Express Formula ‖ E ‖_2,1Being used to encourage the error of E to be classified as 0, therefore noise or abnormal source domain can be deleted during adapting to.But What the minima of rank (Z) was the most permissible finds a reconstructed coefficients matrix by inferior grade structure.F(W,X_s) it is a broad sense Sub-space learning function, it can be write as Tr (W^TU₁W), U₁And U₂Being selected by sub-space learning pattern, learning model is such as PCA or LDA etc..W is a kind of sub-space learning representation, and learning model has PCA or LDA；Z is that low-rank represents.

When sub-space learning pattern is selected, W fixes, use non-precision augmented vector approach solve above formula with Obtain Z, E.Based on low-rank method for expressing knowledge adaptability should be provided a more complete theory and spatial analysis, but only Under conditions of meeting subspace independence and having adequate data, the performance of LTSL is good.It practice, subspace is independent It is the most inappeasable with data volume abundance, so the accuracy of the subspace moving method represented based on low-rank is relatively low.

It addition, LTSL method is only applicable to the conversion of linear data, but nonlinear data can not be processed, and this Method invention is just being aimed at nonlinear data.

Summary of the invention

The technical problem to be solved is just to provide a kind of image migrated based on sparse nonlinear subspace and divides Class method, it can improve the accuracy that view data migrates in subspace, and can be suitably used for the conversion of nonlinear data.

The technical problem to be solved is realized by such technical scheme, and it includes following steps:

Step 1, input source domain sampleWith aiming field sample Calculate initial dataX_S ¹Be source domain first sample,It is the N of source domain_SIndividual sample Originally, X_T ¹Be aiming field first sample,It is the N of aiming field_TIndividual sample；

Step 2, calculates K_TAnd K

By formulaWithSolve KT and K respectively；It is that initial data X projects to The data of Hilbert space；

Step 3, obtains the Eigenvalues Decomposition of K

By formula K=VSV^TSolve the Eigenvalues Decomposition of K；V is characterized the matrix of vector composition, element on the diagonal of S It is characterized the eigenvalue corresponding to vector；

Step 4, initialization feature vector Φ

By formula Φ :=V (:, v) take the characteristic vector corresponding to front d eigenvalue of maximum of V, in formula v be the front d of V Big characteristic vector corresponding to eigenvalue；

Step 5, updates Z

Φ is fixed, uses ADMM method, update with the Z in following formula (8):

${\min }_Z\left|\right|Z|{|}_1+{\mathrm{\λ}}_1||{\mathrm{\Φ}}^T{K}_T-{\mathrm{\Φ}}^{T}KZ|{|}_F^2---\left(8\right)$

$s.t{.1}_{N_S+N_T}^{T}Z=1_{N_T}^T$

In formula (8), Z is sparse matrix, λ₁Represent balance parameter,Represent a length of N_S+N_TNon-zero column vector, Represent a length of N_TNon-zero column vector；

Step 6, updates Φ

Z is fixed, the method using Eigenvalues Decomposition, updates with the Φ in following formula (13):

${\min }_{\mathrm{\Φ}}{\mathrm{\λ}}_1\left|\right|{\mathrm{\Φ}}^T{K}_T-{\mathrm{\Φ}}^{T}KZ|{|}_F^2+{\mathrm{\λ}}_{2}Tr\left({\left(I-{\mathrm{\Φ\Φ}}^{T}K\right)}^{T}K\right(I-{\mathrm{\Φ\Φ}}^{T}K\left)\right)---\left(13\right)$

s.t.Φ^TK Φ=I

Step 7, checks convergence situation

If not restraining, then repeat step 5 and step 6；If convergence, then export the value of Φ and Z；

Step 8, byP must be mapped^*, calculate source domain data X_SIt is mapped to value M of default subspace_S,

Step 9, utilizes and maps P^*, calculate the target training data X of labelling_TlIt is mapped to value M of default subspace_Tl,

Represent non-linear X_S, project to Hilbert space and beThen by sparse square Battle array Z-direction aiming field conversion；

Step 10, calculates not labeled target detection data X_TuIt is mapped to value M of default subspace_Tu,

Step 11, training data [M based on labelling_S,M_Tl] and the labelling of they correspondences, utilize a two norm canonicals young waiter in a wineshop or an inn Method is taken advantage of to train a grader W；

Step 12, by calculating decision functionNot labeled target detection data are carried out classification differentiation.

The solution have the advantages that: improve the accuracy that view data migrates in default subspace；Due to the present invention First pass through kernel method and data are mapped to the core Hilbert space of regeneration from luv space, therefore can be suitably used for non-linear number According to conversion.

Accompanying drawing explanation

The accompanying drawing of the present invention is described as follows:

Fig. 1 is the data transfer principle figure of the LTSL of background technology；

Fig. 2 is the data transfer principle figure of the present invention；

Fig. 3 is source domain and the three-dimensional distribution map of aiming field of data verification；

Fig. 4 is that the data of Fig. 3 two dimension after the present invention changes presets Subspace Distribution figure；

Fig. 5 is the sample image of an experimenter；

Fig. 6 is that the present invention is applied to the pose adjustment result figure in expression of smiling in Fig. 5.

Detailed description of the invention

The invention will be further described with embodiment below in conjunction with the accompanying drawings:

The present invention first passes through kernel method and data is mapped to the core Hilbert space (RKHS) of regeneration from luv space, Again by conversion as shown in Figure 2, at core Hilbert space, by a predefined basis conversion P, aiming field is trained Data X_TIt is mapped to default subspace by basis conversion P and obtains target data PX_T, by source domain training data X_STurned by basis Change P to be mapped to default subspace and obtain PX_S, owing to the present invention is by source domain and aiming field data P [X_S,X_T] come target data Rebuild, therefore P [X_S,X_T] it is collectively referred to as again source domain data, then utilize source domain data P [X_S,X_T] pass through sparse matrix Z changes, with PX_TDistribution is shared in default subspace.

The principle of the present invention is:

LSDT (linearly) conversion and the difference changed of NLSDT (non-linear) are: due to can not be linear in lower dimensional space The point set of segmentation, during by the point set that is converted in higher dimensional space, it is possible to become linear separability, but higher dimensional space again can Cause dimension disaster, the method therefore using core, first initial data is projected on Hilbert space, reduce dimension and just can keep away Exempt from dimension disaster, i.e. X projects to(X is initial data,It is the data after X projects to Hilbert space).

First nuclear matrix is definedAndWherein κ It is kernel function, uses gaussian kernel function,x_i、x_jFor original number According to,It is respectively initial data x_i、x_jProject to the data of Hilbert space.

Source domain data P [X_S,X_T] changed by sparse matrix Z, thus with aiming field data PX_TIn default subspace Interior sharing is distributed, then sparse nonlinear subspace migrates the mathematical model optimized and is

In formula (1), T₀It it is sparse level；For the target data of default subspace,For in advance If the source domain data of subspace；Z is to make to preset source domain and the sparse matrix of aiming field data sharing distribution in subspace.

Because based on L₀The optimization of norm is non-convex problem, and model the most of the present invention employs the convex of this specification and approaches, i.e. L₁ Norm.

In order to make the P of acquisition be able to ensure that, projection does not make data be distorted, and can keep a lot of useful information, that There is following expression,

Formula (1) is combined with formula (2) formula, finally can be expressed as:

In formula (3), the row of P must be orthogonal, and necessary normalization so that PP^T=I, I are unit matrixs, with this Avoid degenerating.Furthermore, it is assumed that(Represent a length of N_S+N_TNon-zero column vector,Represent a length of N_T's Non-zero column vector) so that source domain data and aiming field data are in united affine default subspace, and non-linear default subspace (affine default subspace and the linear difference presetting subspace are whether there is selected initial point.In affine default subspace The status of any 2 is of equal value, and the initial point linearly presetting subspace is a special point), λ₁And λ₂Then represent balance parameter (i.e. The regularization coefficient λ of background technology₁And λ₂, illustrate the weight of each).

To put it more simply, orderSo object function (3) can be written as:

In object function (3) or (4), the solution of P has a lot, i.e. solution is the most unique.A kind of method for solving of basis conversion P is such as Under:

There is optimal solution P*, it can be expressed as intuitivelyIn the case of, initial data and number of targets According to linear combination, representation is as follows:

In formula (5), φ is characterized vector, φ^T、Be respectively φ andTransposed matrix.

Convolution (4) and formula (5), object function can be expressed as:

OrderSo previously described formula (3) can be expressed as follows:

${\min }_{Z,\mathrm{\Φ}}\left|\right|z|{|}_1+{\mathrm{\λ}}_1||{\mathrm{\Φ}}^T{K}_T-{\mathrm{\Φ}}^{T}KZ|{|}_F^2+{\mathrm{\λ}}_{2}Tr\left({\left(I-{\mathrm{\Φ\Φ}}^{T}K\right)}^{T}K\right(I-{\mathrm{\Φ\Φ}}^{T}K\left)\right)---\left(7\right)$

$s.t.{\mathrm{\Φ}}^{T}K\mathrm{\Φ}=I,1_{N_S+N_T}^{T}Z=1_{N_T}^T$

From formula (7) it can be seen that formula relates to two variable Z and Φ.In order to obtain the minimal solution of formula (7), can use Fix a variable, and change another variable to solve this problem.Therefore, including following 2 key steps:

The first, sparse matrix (Z) is updated

In order to obtain Z, Φ can be fixed, then formula (7) can be expressed as:

$\begin{array}{c}\underset{Z}{\min }\left|\right|Z|{|}_1+{\mathrm{\λ}}_1||{\mathrm{\Φ}}^T{K}_T-{\mathrm{\Φ}}^{T}KZ|{|}_F^2\\ s.t{.1}_{N_S+N_T}^{T}Z=1_{N_T}^T\end{array}---\left(8\right)$

This is typical sparse lasso trick (Lasso) optimization problem, and alternating direction multiplier method (ADMM) can be used to have Effect ground solves.

Introducing auxiliary variable L, formula (8) can be rewritten as:

$\begin{array}{c}\underset{Z,L}{\min }\left|\right|Z|{|}_1+{\mathrm{\λ}}_1||{\mathrm{\Φ}}^T{K}_T-{\mathrm{\Φ}}^{T}KL|{|}_F^2\\ s.t.L=Z,1_{N_S+N_T}^{T}L=1_{N_T}^T\end{array}---\left(9\right)$

The enhancing Lagrangian of formula (9) can be expressed as:

$\begin{array}{c}J(Z,L)=\left|\right|Z|{|}_1+\frac{{\mathrm{\&mu;}}_1}{2}||{\mathrm{\&Phi;}}^T{K}_T-{\mathrm{\&Phi;}}^{T}KL|{|}_F^2+<Y_A,L-Z>\\ +<Y_B,1_{N_S+N_T}^{T}L-1_{N_T}^T>+\frac{{\mathrm{\&mu;}}_1}{2}\left(\right||L-Z|{|}_F^2+\left|\right|1_{N_S+N_T}^{T}L-1_{N_T}^T\left|{|}_2^2\right)\end{array}---\left(10\right)$

In formula (10), Y_AAnd Y_BRepresent Lagrange multiplier, λ respectively₁=μ₁/2；λ₁And μ₁It is coefficient variation, Ke Yisui Meaning selects, and takes λ₁=μ₁/ 2 are intended merely to convenience when that derivation obtaining (11) formula, and (two norm derivations just have 2 and 1/2 the most about Fall)；μ₂Coefficient when being to solve with augmented vector approach.

For understanding the L of formula (9), now fix Z, and make the local derviation of formula (10)It it is 0, it is possible to obtain the expression formula of L As follows:

$\begin{array}{c}L={({\mathrm{\&mu;}}_1{K}^T{\mathrm{\&Phi;\&Phi;}}^{T}K+{\mathrm{\&mu;}}_{2}I+{\mathrm{\&mu;}}_2{1}_{N_S+N_T}{1}_{N_S+N_T}^T)}^{-1}\\ ({\mathrm{\&mu;}}_1{K}^T{\mathrm{\&Phi;\&Phi;}}^T{K}_T-Y_A-1_{N_S+N_T}{Y}_B+{\mathrm{\&mu;}}_{2}Z+{\mathrm{\&mu;}}_2{1}_{N_S+N_T}{1}_{N_T}^T)\end{array}---\left(11\right)$

Fixing L, then Z can be solved by following formula:

$\begin{array}{c}\underset{Z}{\min }\left|\right|Z|{|}_1+\frac{{\mathrm{\&mu;}}_2}{2}||L-Z|{|}_F^2+<Y_A,L-Z>\\ \&Proportional;\underset{Z}{\min }\left|\right|Z|{|}_1+\frac{{\mathrm{\&mu;}}_2}{2}\left(\right||L-Z|{|}_F^2+\frac{{\mathrm{\&mu;}}_2}{2}<Y_A,L-Z>)\\ \&Proportional;\underset{Z}{\min }\left|\right|Z|{|}_1+\frac{{\mathrm{\&mu;}}_2}{2}||Z-(L+\frac{Y_A}{{\mathrm{\&mu;}}_2})|{|}_F^2\end{array}---\left(12\right)$

Therefore, using ADMM method, the step solving Z according to formula (8) is as follows:

Step 1), initiation parameter

Make Z, Y_A,Y_BIt is 0, μ₂For λ₁, ρ=1.1 and max_μ=10⁶；Wherein Y_AAnd Y_BRepresent Lagrange multiplier respectively, λ₁And λ₂Then represent balance parameter；

Step 2), update L

Z is fixed, by utilizing following formula that L is solved when not restraining when:

$L={({\mathrm{\&mu;K}}^T{\mathrm{\&Phi;\&Phi;}}^{T}K+{\mathrm{\&mu;}}_{2}I+{\mathrm{\&mu;}}_2{1}_{N_S+N_T}{1}_{N_S+N_T}^T)}^{-1}({\mathrm{\&mu;K}}^T{\mathrm{\&Phi;\&Phi;}}^T{K}_T-Y_A-1_{N_S+N_T}{Y}_B+{\mathrm{\&mu;}}_{2}Z+{\mathrm{\&mu;}}_2{1}_{N_S+N_T}{1}_{N_T}^T)$

In formula, L is auxiliary variable, λ₁=μ₁/ 2, μ₂Coefficient when being to solve with augmented vector approach；

Step 3), update Z

When not restraining when, fix L, by following formula, Z solved:

$\underset{Z}{min}\left|\right|Z|{|}_1+\frac{{\mathrm{\&mu;}}_2}{2}||Z-(L+\frac{Y_A}{{\mathrm{\&mu;}}_2})|{|}_F^2$

Step 4), update Lagrange multiplier

According to step 2) and step 3) more newly obtained L and Z, substitute into following formula,

Y_A=Y_A+μ₂(L-Z),Y_B=Y_B+μ₂(1^TL-1^T)

Multiplier after being updated；

In above-mentioned formula (10), Y_AAnd Y_BRepresent Lagrange multiplier respectively, owing to this is a typical sparse lasso trick optimization Problem, therefore uses alternating direction multiplier method (ADMM), because formula (10) has Lagrange multiplier, therefore title is called again multiplication Device；

Step 5), undated parameter μ₂

μ₂Take μ₂ρ and max_μIn minima (μ₂ρ and max_μBeing the value set, in step 1, initiation parameter part is the most supplementary Their setting value)；

Step 6), check convergence situation

If not restraining, then repeat step 2) to step 5)；

If convergence, then export Z.

The second, Φ is updated

In order to solve Φ, by fixing Z, formula (7) can be written as:

The object function of formula (13) is expanded as follows:

In formula (14),

Make Φ^TK Φ=I, then Φ Φ^TKΦΦ^T=QKQ^T=Φ Φ^T=Q, then object function can be reduced to:

Tr((λ₁(K^-1K_T-Z)(K^-1K_T-Z)^T-λ₂I)K^TQ^TK) (15)

According to formula K=VSV^TEigenvalues Decomposition, wherein V is characterized vector, on S diagonal element be characterized vector right The eigenvalue answered:

$K^T{Q}^{T}K={\mathrm{VS}}^{\frac{1}{2}}{\mathrm{\&Omega;\&Omega;}}^T{S}^{\frac{1}{2}}{V}^T$

In formula,Characteristic function (15) dissolve for:

$\begin{array}{c}Tr\left({\mathrm{\&Omega;}}^T{S}^{\frac{1}{2}}{V}^T\right({\mathrm{\&lambda;}}_1\left(K^{-1}{K}_T-Z\right){\left(K^{-1}{K}_T-Z\right)}^T-{\mathrm{\&lambda;}}_{2}I\left){\mathrm{VS}}^{\frac{1}{2}}\mathrm{\&Omega;}\right)\\ =Tr\left({\mathrm{\&Omega;}}^T\mathrm{\&Theta;}\mathrm{\&Omega;}\right)\end{array}---\left(16\right)$

In formula (16),

And Ω^TΩ=Φ^TVSV^TΦ=Φ^TK Φ=I

Finally, formula (13) is rewritten as:

${\mathrm{\&Omega;}}^{*}=\arg \underset{\mathrm{\&Omega;}}{\min }Tr\left({\mathrm{\&Omega;}}^T\mathrm{\&Theta;}\mathrm{\&Omega;}\right),s.t.{\mathrm{\&Omega;}}^T\mathrm{\&Omega;}=I---\left(17\right)$

Optimal solution Ω^*It is to be obtained by characteristic vector l of the minimal eigenvalue about Θ.

Ω^*It is to pass throughAnd VV^T=I is tried to achieve, Φ^*Optimum can be represented as:

${\mathrm{\&Phi;}}^{*}={\mathrm{VS}}^{-\frac{1}{2}}{\mathrm{\&Omega;}}^{*}---\left(18\right)$

Therefore, the step of Φ is solved according to formula (13) as follows:

Step (1), seeks the Eigenvalues Decomposition of K

By K=VSV^TObtain the Eigenvalues Decomposition of K；Wherein K=X^TX, V are characterized vector, and on S diagonal, element is characterized Vector characteristic of correspondence value；

Step (2), is solved Θ by following formula:

$\mathrm{\&Theta;}=S^{\frac{1}{2}}{V}^T\left({\mathrm{\&lambda;}}_1\right(K^{-1}{K}_T-Z){\left(K^{-1}{K}_T-Z\right)}^T-{\mathrm{\&lambda;}}_{2}I){\mathrm{VS}}^{\frac{1}{2}}$

Step (3), obtains the Eigenvalues Decomposition of Θ

By formula Θ=U Σ U^TSolving the Eigenvalues Decomposition of Θ, U is the characteristic vector of Θ, and Σ is a diagonal matrix, and it is right Element on angle is characterized vector characteristic of correspondence value；

Step (4), solves Ω

Solve Ω by Ω=U (:, U), this formula be Θ is carried out feature decomposition after, take front d the minimal eigenvalue of U Characteristic of correspondence vector, in formula, U is the characteristic vector corresponding to front d minimal eigenvalue of U；

The scope of d is all eigenvalue numbers of 1～U, circulates successively with eigenvalues all in U, and treating excess syndrome is tested effective Those values；

Step (5), solves Φ

By formulaSolve Φ.

Summary content, obtains the image classification method migrated based on sparse nonlinear subspace of the present invention, including with Lower step:

Step 1, input source domain sampleWith aiming field sample CalculateX_S ¹Be source domain first sample,It is the N of source domain_SIndividual sample, X_T ¹It it is mesh Mark territory first sample,It is the N of aiming field_TIndividual sample；

Step 2, calculates K_TAnd K

By formulaWithSolve KT and K respectively；It is that initial data X projects to The data of Hilbert space；

Step 3, obtains the Eigenvalues Decomposition of K

By formula K=VSV^TSolve the Eigenvalues Decomposition of K；V is characterized the matrix that vector is formed, element on S diagonal It is characterized vector characteristic of correspondence value；

Step 4, initializes Φ

By formula Φ :=V (:, v) take the characteristic vector corresponding to front d eigenvalue of maximum of V, in formula v be the front d of V Big characteristic vector corresponding to eigenvalue；

The scope of d is 1～(all eigenvalue numbers of V), circulates successively with eigenvalues all in V, and treating excess syndrome is tested effective Those values；

Step 5, updates Z

Φ is fixed, uses ADMM method, update with the Z in following formula (8):

$\underset{Z}{min}\left|\right|Z|{|}_1+{\mathrm{\&lambda;}}_1||{\mathrm{\&Phi;}}^T{K}_T-{\mathrm{\&Phi;}}^{T}KZ|{|}_F^2---\left(8\right)$

$s.t{.1}_{N_S+N_T}^{T}Z=1_{N_T}^T$

Step 6, updates Φ

Z is fixed, the method using Eigenvalues Decomposition, updates with the Φ in following formula (13):

${\min }_{\mathrm{\&Phi;}}{\mathrm{\&lambda;}}_1\left|\right|{\mathrm{\&Phi;}}^T{K}_T-{\mathrm{\&Phi;}}^{T}KZ|{|}_F^2+{\mathrm{\&lambda;}}_{2}Tr\left({\left(I-{\mathrm{\&Phi;\&Phi;}}^{T}K\right)}^{T}K\right(I-{\mathrm{\&Phi;\&Phi;}}^{T}K\left)\right)---\left(13\right)$

s.t.Φ^TK Φ=I

Step 7, checks convergence situation

If not restraining, then repeat step 5 and step 6；If convergence, then export the value of Φ and Z；

Step 8, byP must be mapped^*, calculate source domain data X_SIt is mapped to value M of default subspace_S,

Step 9, utilizes and maps P^*, calculate the target training data X of labelling_TlIt is mapped to value M of default subspace_Tl,

Represent non-linear X_S, project to Hilbert space and beBy sparse after P Matrix Z-direction aiming field is changed；

Step 10, calculates not labeled target detection data X_TuIt is mapped to value M of default subspace_Tu,

The target training data X of labelling is set_TlNot labeled target detection data X_Tu, it is because source domain to be passed through Data X_SAiming field data X with labelling_TlTrain a grader W, then use not labeled aiming field data X_TuEnter Row test, it is judged that the quality of the grader W that step 11 trains, to check the quality of the present invention；

Step 11, training data [M based on labelling_S,M_Tl] and the labelling of they correspondences, utilize a two norm canonicals young waiter in a wineshop or an inn Method is taken advantage of to train a grader W；

Step 12, by calculating decision functionNot labeled target detection data are carried out classification differentiation.

Experimental verification

One, data verification

As it is shown on figure 3, for data produced by the array of default subspace, by the Gauss distribution of different modes and association side Difference matrix, it is thus achieved that (target data is random producing for the target data of minority labelling and majority unlabelled target data distributed in three dimensions The raw data meeting Gauss distribution), source domain has two class data, every class packet contains 50 samples, and easily in source domain Find the decision boundary of two class data；In aiming field, also have two class data, every class have the sample of 5 labellings and 50 unmarked Sample.From Fig. 3 it will be clear that: 1, of a sort data point has different distributions in source domain and aiming field；2、 The Optimal Separating Hyperplane of source domain is not suitable for the decision boundary (decision boundary refers to separate for inhomogeneity data line or plane) of aiming field.

The sparse Z subspace utilizing the present invention migrates, and after projection and rebuilding, obtains the source data of two-dimentional default subspace And target data, as shown in Figure 4, the spatial offset rate between source data and target data decreases, and between this two class Decision boundary clear, it is easy to finding a kind of grader to realize classification, so using present method invention to carry out data classification being Feasible, effective.

Two, object identification

CMU Multi-PIE human face data collection is a public data collection, comprises the comprehensive face number of 337 experimenters altogether According to collection, image therein passes through 15 postures, 20 kinds of illuminations, 6 kinds of expression, 4 realization captures of different stages.In order to be thought The result wanted, in an experiment 60 experimenters before stage 1 and stage 2 have selected respectively, 7 kinds of expression figures of each experimenter Picture.As shown in Figure 5: the stage 1 is the first row, face is with general expression；Stage 2 is the second row, and experimenter is also with smile.? In this experiment, four kinds of Setup Experiments are as follows:

Arrange 1: the red rectangle front expression of each experimenter and the extreme attitude of blue rectangle (60 ° of angles) are respectively intended to As source domain and the training data of aiming field, remaining expression is i.e. used as probe face.

Arrange 2: with reference to arranging 1, similarly configure.

1+2 is set: express one's feelings the two kinds of front expressions arranged in 1 and 2 and two kinds of extreme 60 ° of attitudes as training number According to, remaining expression is used as probe face.

Arranged in a crossed manner: typically to express one's feelings arrange each experimenter in 1 as source domain, will arrange 2 each experimenters' Expression of smiling, as aiming field, adapts to the change of expression with this.

Nonlinearity due to three-dimensional rotation face so that attitude tracking is extremely challenging.Fig. 6 is that the present invention is applied to Stage 2 smile expression on pose adjustment result figure, it using in Fig. 5 front express one's feelings as source data, the extreme attitude in back (60 ° of angles) is used as target data.We can see that from Fig. 6: owing to remaining the removal of (noise), target face alignment appearance State is good.Using the invention of this nonlinear method to carry out recognition of face, its recognition accuracy compares with LTSL, and it the results are shown in Table 1.

Table 1. is for the recognition accuracy ratio of human face posture identification

Task	Source domain	Aiming field	LTSL-PCA	LTSL-LDA	The present invention
						Arrange 1	Front	The angle of 60 degree	55.7	56.0	63.7
Arrange 2	Front	The angle of 60 degree	58.7	60.7	70.7
						1+2 is set	Front	The angle of 60 degree	57.8	60.7	67.5
Arranged in a crossed manner	Arrange 1	Arrange 2	96.7	96.7	99.4

This method is nonlinear method, owing to initial data is non-linear, is projected on Hilbert space carrying out again Process, thus greatly improve recognition accuracy.By contrast linear method LTSL, as can be seen from Table 1: present method invention ratio The recognition effect of linear method LTSL is good.

Claims (3)

1. the image classification method migrated based on sparse nonlinear subspace, is characterized in that, comprises the steps of

Step 1, input source domain sampleWith aiming field sample Calculate initial dataX_S ¹Be source domain first sample,It is the N of source domain_SIndividual sample Originally, X_T ¹Be aiming field first sample,It is the N of aiming field_TIndividual sample；

Step 2, calculates K_TAnd K

By formulaWithSolve KT and K respectively；It is that initial data X projects to Xi Er The data in Bert space；

Step 3, obtains the Eigenvalues Decomposition of K

By formula K=VSV^TSolve the Eigenvalues Decomposition of K；V is characterized the matrix of vector composition, and on the diagonal of S, element is characterized Eigenvalue corresponding to vector；

Step 4, initialization feature vector Φ

By formulaTaking the characteristic vector corresponding to front d eigenvalue of maximum of V, in formula, v is front d the maximum feature of V Characteristic vector corresponding to value；

Step 5, updates Z

Φ is fixed, uses ADMM method, update with the Z in following formula (8):

${\min }_Z\left|\right|Z|{|}_1+{\mathrm{\&lambda;}}_1||{\mathrm{\&Phi;}}^T{K}_T-{\mathrm{\&Phi;}}^{T}KZ|{|}_F^2---\left(8\right)$

$s.t{.1}_{N_S+N_T}^{T}Z=1_{N_T}^T$

In formula (8), Z is sparse matrix, λ₁Represent balance parameter,Represent a length of N_S+N_TNon-zero column vector,Represent A length of N_TNon-zero column vector；

Step 6, updates Φ

Z is fixed, the method using Eigenvalues Decomposition, updates with the Φ in following formula (13):

${\min }_{\mathrm{\&Phi;}}{\mathrm{\&lambda;}}_1\left|\right|{\mathrm{\&Phi;}}^T{K}_T-{\mathrm{\&Phi;}}^{T}KZ|{|}_F^2+{\mathrm{\&lambda;}}_{2}Tr\left({\left(I-{\mathrm{\&Phi;\&Phi;}}^{T}K\right)}^{T}K\right(I-{\mathrm{\&Phi;\&Phi;}}^{T}K\left)\right)---\left(13\right)$

s.t.Φ^TK Φ=I

Step 7, checks convergence situation

If not restraining, then repeat step 5 and step 6；If convergence, then export the value of Φ and Z；

Step 8, byP must be mapped^*, calculate source domain data X_SIt is mapped to value M of default subspace_S,

Step 9, utilizes and maps P^*, calculate the target training data X of labelling_TlIt is mapped to value M of default subspace_Tl,

Represent non-linear X_S, project to Hilbert space and beThen by sparse matrix Z Change to aiming field；

Step 10, calculates not labeled target detection data X_TuIt is mapped to value M of default subspace_Tu,

Step 11, training data [M based on labelling_S,M_Tl] and the labelling of they correspondences, utilize two norm canonical least squares Method one grader W of training；

Step 12, by calculating decision functionNot labeled target detection data are carried out classification differentiation.

The image classification method migrated based on sparse nonlinear subspace the most according to claim 1, is characterized in that, in step Rapid 5 steps updating Z include:

Step 1), initiation parameter

Make Z, Y_A,Y_BIt is 0, μ₂For λ₁, ρ=1.1 and max_μ=10⁶；Wherein Y_AAnd Y_BRepresent Lagrange multiplier, λ respectively₁And λ₂ Then represent balance parameter；

Step 2), update L

Z is fixed, by utilizing following formula that L is solved when not restraining when:

$L={({\mathrm{\&mu;}}_1{K}^T{\mathrm{\&Phi;\&Phi;}}^{T}K+{\mathrm{\&mu;}}_{2}I+{\mathrm{\&mu;}}_2{1}_{N_S+N_T}{1}_{N_S+N_T}^T)}^{-1}({\mathrm{\&mu;}}_1{K}^T{\mathrm{\&Phi;\&Phi;}}^T{K}_T-Y_A-1_{N_S+N_T}{Y}_B+{\mathrm{\&mu;}}_{2}Z+{\mathrm{\&mu;}}_2{1}_{N_S+N_T}{1}_{N_T}^T)$

In formula, L is auxiliary variable, λ₁=μ₁/ 2, μ₂Coefficient when being to solve with augmented vector approach；

Step 3), update Z

When not restraining when, fix L, by following formula, Z solved:

$\underset{Z}{min}\left|\right|Z|{|}_1+\frac{{\mathrm{\&mu;}}_2}{2}||Z-(L+\frac{Y_A}{{\mathrm{\&mu;}}_2})|{|}_F^2$

Step 4), update Lagrange multiplier

According to step 2) and step 3) more newly obtained L and Z, substitute into following formula,

Y_A=Y_A+μ₂(L-Z),Y_B=Y_B+μ₂(1^TL-1^T)

Multiplier after being updated；

Step 5), undated parameter μ₂

μ₂Take μ₂ρ and max_μIn minima；

Step 6), check convergence situation

If not restraining, then repeat step 2) to step 5)；

If convergence, then export Z.

The image classification method migrated based on sparse nonlinear subspace the most according to claim 1 and 2, is characterized in that, The step updating Φ in step 6 includes:

Step (1), seeks the Eigenvalues Decomposition of K

By K=VSV^TObtain the Eigenvalues Decomposition of K；Wherein K=X^TX, V are characterized vector, and on S diagonal, element is characterized vector Characteristic of correspondence value；

Step (2), is solved Θ by following formula:

$\mathrm{\&Theta;}=S^{\frac{1}{2}}{V}^T\left({\mathrm{\&lambda;}}_1\right(K^{-1}{K}_T-Z){\left(K^{-1}{K}_T-Z\right)}^T-{\mathrm{\&lambda;}}_{2}I){\mathrm{VS}}^{\frac{1}{2}}$

Step (3), obtains the Eigenvalues Decomposition of Θ

By formula Θ=U Σ U^TSolving the Eigenvalues Decomposition of Θ, U is the characteristic vector of Θ, and Σ is a diagonal matrix, on its diagonal angle Element be characterized vector characteristic of correspondence value；

Step (4), solves Ω

Solve Ω by Ω=U (:, U), this formula be Θ is carried out feature decomposition after, front d the minimal eigenvalue taking U is corresponding Characteristic vector, in formula, U is the characteristic vector corresponding to front d minimal eigenvalue of U；

Step (5), solves Φ

By formulaSolve Φ.