CN104318261B - A kind of sparse representation face identification method representing recovery based on figure embedding low-rank sparse - Google Patents

Abstract

The invention discloses a kind of figure embedding low-rank sparse and represent recovery sparse representation face identification method, belong to computer vision and mode identification technology.The present invention includes following steps: first, propose a kind of figure embedding low-rank sparse and represent restoration methods, the strong clean training sample data matrix of judgement index can be recovered from training sample data matrix, obtain training sample data error matrix simultaneously; Then, with clean training sample data matrix for dictionary, with training sample data error matrix for error dictionary, adopt norm optimization technique solve the rarefaction representation coefficient of human face data to be identified; Further, utilize the rarefaction representation coefficient of human face data to be identified, class association reconstruct is carried out to human face data to be identified; Finally, based on the class association reconstructed error of human face data to be identified, the identification of facial image to be identified is completed.The present invention can solve the recognition of face problem that training sample image and image to be identified are all blocked in situation by noise pollution or local.

Description

A kind of sparse representation face identification method representing recovery based on figure embedding low-rank sparse

Technical field

The present invention relates to computer vision and mode identification technology, be specifically related to a kind of sparse representation face identification method representing recovery based on figure embedding low-rank sparse.

Background technology

At present, face recognition technology due to its widespread use in identification, video frequency searching, security monitoring etc., one of hot research problem becoming computer vision and mode identification technology.

In recent years, along with compressive sensing theory and l ₁the development of norm optimization technique, rarefaction representation is subject to lot of domestic and foreign focus of attention.Under rarefaction representation, signal can be expressed as the most sparse linear combination of given dictionary atom.Research shows, sparse representation model and human visual system's principle closely similar.Therefore, rarefaction representation is widely used in the computer vision fields such as image filtering, Image Reconstruction, compression of images.

2009, the people such as JohnWright proposed a kind of face identification method based on rarefaction representation sorter (SparseRepresentationclassification, SRC).The method first respectively with original training sample data matrix and unit matrix for dictionary and error dictionary, adopt l ₁norm minimum technology solves the rarefaction representation coefficient of facial image to be identified; Then, utilize the rarefaction representation coefficient of facial image to be identified, class association reconstruct is carried out to facial image to be identified; Finally, based on the class association reconstructed error of facial image to be identified, the identification of facial image to be identified is completed.Thus, the face identification method based on rarefaction representation receives numerous concern, and many scholars have done many research work in this respect.

Under the precondition that the training sample image as dictionary is clean, the face identification method based on rarefaction representation achieves good recognition effect.But when training sample image is blocked by noise pollution or local, the recognition effect based on the face identification method of rarefaction representation is poor.In addition, owing to adopting the unit matrix of higher-dimension to make error dictionary based on the face identification method of rarefaction representation, process the noise in facial image to be identified and block, its computational complexity is very high.

Summary of the invention

The object of this invention is to provide and a kind ofly embed low-rank sparse based on figure and represent that the sparse representation face identification method of recovery, object solve training sample image and image to be identified all by the recognition of face problem under noise pollution or the situation that is locally blocked.

For above-mentioned based on Problems existing in the face identification method of rarefaction representation sorter, the present invention proposes a kind of sparse representation face identification method representing recovery based on figure embedding low-rank sparse.

The present invention adopts following technical proposals: a kind of sparse representation face identification method representing recovery based on figure embedding low-rank sparse, comprises the following steps:

S01: suppose there is K class training sample image, every class has n to open training sample image, and N=K × n opens training sample image, if often opening training sample image resolution is r × c altogether, be converted to M=r × c dimensional vector by often opening training sample image, then training sample data matrix is designated as

S02: the class label utilizing training sample data matrix X and correspondence thereof, builds a undirected neighbour of supervision that has comprising N number of node and schemes G; Wherein, node annexation and the weight thereof of scheming G are: if data x _iwith data x _jbelong to same class, and x _ifor x _jk neighbour or x _jfor x _ik neighbour, k is positive integer, span 3≤k≤n-1, then the node i scheming G is connected with node j, and its connection weight W _ijbe 1; Otherwise the node i of figure G is not connected with node j, its weights W _ijbe 0, definition data x _idegree be:

$d_i={\mathrm{\Σ}}_{j=1}^N{W}_{ij},i=1,2,\mathrm{...},N;$

Definition data read matrix D is: D=diag (d ₁, d ₂..., d _n), wherein diag (d ₁, d ₂..., d _n) represent with d ₁, d ₂..., d _nfor the diagonal matrix of diagonal element;

S03: the weight matrix W tried to achieve according to step S02 and data read matrix D, definition Laplacian Matrix L: suppose Z=[z ₁, z ₂..., z _n] be the expression matrix of coefficients of X, in order to enable the expression coefficient of training sample data keep the local geometry of data space and have strong judgement index, based on Laplacian Matrix L, defining one has the figure of supervision to embed regular terms:

$\frac{1}{2}\underset{i,j=1}{\overset{N}{\Σ}}\left|\right|z_i-z_j|{|}_2^2{W}_{ij}=Tr\left({\mathrm{ZLZ}}^T\right),$

In formula, the mark of Tr () representing matrix;

S04: make E be training sample data error matrixes, α > 0, β > 0, γ > 0 are iotazation constant, definition figure embeds low-rank sparse and represents that restoration methods objective function is:

$\begin{array}{ccc}\underset{Z,E}{\min }\left|\right|Z|{|}_{*}+\mathrm{\α}|\left|E\right|{|}_1+\mathrm{\β}\left|\right|Z|{|}_1+\frac{\mathrm{\γ}}{2}Tr\left({\mathrm{ZLZ}}^T\right)& s.t.& X=XZ+E\end{array};$

In formula || || _*the nuclear norm of representing matrix, i.e. singular values of a matrix sum, || || ₁the l of representing matrix ₁norm, the mark of Tr () representing matrix;

Embed low-rank sparse according to figure and represent that restoration methods objective function tries to achieve expression matrix of coefficients Z and the training sample data error matrix E of training sample data matrix;

S05: according to the expression matrix of coefficients Z of the training sample data matrix X that step S04 tries to achieve, adopts following formula to recover clean training sample data Matrix C:

C＝XZ；

S06: for arbitrary facial image to be identified, be translated into M dimensional vector, be designated as face image data y to be identified,

S07: take C as dictionary, E is error dictionary, adopts l ₁norm optimization technique, solve the rarefaction representation coefficient of face image data y to be identified by following formula:

${\min }_{{\[{\mathrm{\α}}^T,{\mathrm{\β}}^T\]}^T}\left|\right|y-C\mathrm{\α}-E\mathrm{\β}|{|}_2^2+\mathrm{\λ}|\left|{\[{\mathrm{\α}}^T,{\mathrm{\β}}^T\]}^T\right|{|}_1;$

In formula, iotazation constant λ=0.05;

S08: calculate the class reconstructed error of all kinds of training sample to face image data y to be identified:

$e\left(i\right)=\left|\right|y-{\mathrm{C\δ}}_i\left(\mathrm{\α}\right)-E\mathrm{\β}|{|}_2^2,i=1,2,\mathrm{...},K;$

In formula, δ _i(α) be only retain with the i-th class coefficient of correspondence, be set to the coefficient vector of 0 with other class coefficient of correspondence;

S09: according to class reconstructed error e (i), i=1,2 ..., K, calculates the class label of face image data y to be identified:

Class (y)=argmin _ie (i); In formula, Class (y) represents the class label of face image data y to be identified, argmin _ithe i of e (i) correspondence that e (i) representative value is minimum;

S10: the recognition result exporting facial image to be identified.

Figure described in described step S04 embeds low-rank sparse and represents that solving of restoration methods objective function is an iterative process, and algorithm flow is:

A), introduce auxiliary variable J, figure embedded low-rank sparse and represents that restoration methods objective function equivalence is converted into:

$\begin{array}{ccc}\underset{Z,E}{\min }\left|\right|Z|{|}_{*}+\mathrm{\α}|\left|E\right|{|}_1+\mathrm{\β}\left|\right|J|{|}_1+\frac{\mathrm{\γ}}{2}Tr\left({\mathrm{ZLZ}}^T\right)& s.t.& X=XZ+E,Z=J\end{array};$

B), constitution step a) in the Augmented Lagrangian Functions of objective function:

L(Z,J,E,Y ₁,Y ₂,μ)＝||Z|| _*+α||E|| ₁+β||J|| ₁+f(Z,J,E,Y ₁,Y ₂,μ)；

Wherein, Y ₁and Y ₂for Lagrange multiplier matrix, μ > 0 is punishment parameter; F (Z, J, E, Y ₁, Y ₂, μ) be:

$f(Z,J,E,Y_1,Y_2,\mathrm{\μ})=\frac{\mathrm{\γ}}{2}Tr\left({\mathrm{ZLZ}}^T\right)+\frac{\mathrm{\μ}}{2}\left(\right||X-XZ-E+\frac{Y_1}{\mathrm{\μ}}|{|}_F^2+\left|\right|Z-J+\frac{Y_2}{\mathrm{\μ}}\left|{|}_F^2\right);$

Wherein || || _ffor the F-norm of matrix;

C), initialization $Z^0=J^0=E^0=Y_1^0=Y_2^0=\mathrm{\θ},{\mathrm{\μ}}^0=0.1,\mathrm{\ρ}=1.1,{\mathrm{\μ}}_{max}={10}^{30},$ Given maximum iteration time maxiter=500, iteration error ε=0.001; Given parameters α=0.5, β=0.2, γ=500; Initialization iterative steps k=0;

D), fixing J ^k, E ^k, Y ₁ ^k, μ ^k, adopt speedup gradient method to solve Z by following formula ^k+1:

$Z^{k+1}=\underset{Z}{\arg \min }\left|\right|Z|{|}_{*}+f(Z,J^k,E^k,Y_1^k,Y_2^k,{\mathrm{\μ}}^k);$

E), fixing Z ^k+1, Y ₁ ^k, μ ^k, solve J by following formula ^k+1:

$J^{k+1}=\underset{J}{\arg \min }\frac{\mathrm{\β}}{{\mathrm{\μ}}^k}\left|\right|J|{|}_1+\frac{1}{2}||Z^{k+1}-J+\frac{Y_2^k}{{\mathrm{\μ}}^k}|{|}_F^2=S_{\mathrm{\β}/{\mathrm{\μ}}^k}(Z^{k+1}+Y_2^k/{\mathrm{\μ}}^k);$

Wherein S _bx () is contracting function, be defined as $S_b\left(x\right)\stackrel{\mathrm{\&Delta;}}{=}\left\{\begin{array}{cc}x-b,& x>b\\ 0,& -b\&le;x\&le;b\\ x+b,& x<-b\end{array}\right.;$

F), fixing Z ^k+1, Y ₁ ^k, μ ^k, solve E by following formula ^k+1:

$E^{k+1}=\underset{E}{\mathrm{argmin}}\frac{\mathrm{\&alpha;}}{{\mathrm{\&mu;}}^k}\left|\right|E|{|}_1+\frac{1}{2}||X-{\mathrm{XZ}}^{k+1}-E+\frac{Y_1^k}{{\mathrm{\&mu;}}^k}|{|}_F^2=S_{\mathrm{\&alpha;}/{\mathrm{\&mu;}}^k}(X-{\mathrm{XZ}}^{k+1}+Y_1^k/{\mathrm{\&mu;}}^k);$

Wherein S _bx () is contracting function, its definition and step e) identical;

G), Lagrange multiplier matrix is upgraded:

$Y_1^{k+1}=Y_1^k+{\mathrm{\&mu;}}^k(X-{\mathrm{XZ}}^{k+1}-E^{k+1});$

$Y_2^{k+1}=Y_2^k+{\mathrm{\&mu;}}^k(Z^{k+1}-J^{k+1});$

H), undated parameter μ: μ ^k+1=min (ρ μ ^k, μ _max);

I), convergence conditions is checked, if met

|| X-XZ ^k+1-E ^k+1|| _∞< ε and || Z ^k+1-J ^k+1|| _∞< ε, jump procedure j);

Otherwise k=k+1, returns steps d);

J) expression matrix of coefficients Z and data error matrix E, is exported.

Compared with prior art, beneficial effect of the present invention is:

1, the present invention is by addition sparsity constraints and the low-rank constraint of the representing matrix of training sample, and introducing figure embeds data space local geometry hold facility and judgement index that regular terms emphasizes data representation coefficient, effectively the strong clean training sample data matrix of judgement index can be recovered from the face training sample image data matrix be blocked by noise pollution or local;

2, the figure utilizing the present invention to propose embeds low-rank sparse and represents the data error matrix that restoration methods is tried to achieve, and can describe the noise in facial image and partial occlusion exactly.Therefore, make error dictionary with this data error matrix, effectively can process the noise in facial image to be identified and block, and computational complexity is low;

3, with clean training sample data matrix for dictionary, with data error matrix for error dictionary, effectively can solve the recognition of face problem that training sample image and image to be identified are all blocked in situation by noise pollution or local, increase substantially face recognition accuracy rate and operation efficiency.

Accompanying drawing explanation

Fig. 1 is a kind of sparse representation face identification method process flow diagram representing recovery based on figure embedding low-rank sparse.

Embodiment

The present invention proposes a kind of sparse representation face identification method representing recovery based on figure embedding low-rank sparse, object solves training sample image and image to be identified all by the recognition of face problem under noise pollution or the situation that is locally blocked, as shown in Figure 1, specifically comprise the following steps:

S01: suppose there is K class training sample image, every class has n to open training sample image, N=K × n opens training sample image altogether.If often opening training sample image resolution is r × c, be converted to M=r × c dimensional vector by often opening training sample image, then training sample data matrix is designated as

S02: the class label utilizing training sample data matrix X and correspondence thereof, builds a undirected neighbour of supervision that has comprising N number of node and schemes G; Wherein, node annexation and the weight thereof of scheming G are: if data x _iwith data x _jbelong to same class, and x _ifor x _jk neighbour or x _jfor x _ik neighbour (k is positive integer, span 3≤k≤n-1), then the node i scheming G is connected with node j, and its connection weight W _ijbe 1; Otherwise the node i of figure G is not connected with node j, its weights W _ijbe 0.Definition data x _idegree be:

$d_i={\mathrm{\&Sigma;}}_{j=1}^N{W}_{ij},i=1,2,\mathrm{...},N;$

Definition data read matrix D is: D=diag (d ₁, d ₂..., d _n), wherein diag (d ₁, d ₂..., d _n) represent with d ₁, d ₂..., d _nfor the diagonal matrix of diagonal element;

S03: the weight matrix W tried to achieve according to step S02 and data read matrix D, definition Laplacian Matrix suppose Z=[z ₁, z ₂..., z _n] be the expression matrix of coefficients of X, in order to enable the expression coefficient of training sample data keep the local geometry of data space and have strong judgement index, based on Laplacian Matrix L, defining one has the figure of supervision to embed regular terms:

$\frac{1}{2}\underset{i,j=1}{\overset{N}{\&Sigma;}}\left|\right|z_i-z_j|{|}_2^2{W}_{ij}=Tr\left({\mathrm{ZLZ}}^T\right),$

In formula, the mark of Tr () representing matrix;

S04: make E be training sample data error matrixes, α > 0, β > 0, γ > 0 are iotazation constant, definition figure embeds low-rank sparse and represents that restoration methods objective function is:

$\begin{array}{ccc}\underset{Z,E}{\min }\left|\right|Z|{|}_{*}+\mathrm{\&alpha;}|\left|E\right|{|}_1+\mathrm{\&beta;}\left|\right|Z|{|}_1+\frac{\mathrm{\&gamma;}}{2}Tr\left({\mathrm{ZLZ}}^T\right)& s.t.& X=XZ+E\end{array};$

In formula || || _*the nuclear norm of representing matrix, i.e. singular values of a matrix sum, || || ₁the l of representing matrix ₁norm, the mark of Tr () representing matrix;

Embed low-rank sparse according to figure and represent that restoration methods objective function tries to achieve expression matrix of coefficients Z and the training sample data error matrix E of training sample data matrix; Wherein figure embedding low-rank sparse represents that solving of restoration methods objective function is an iterative process, and algorithm flow is:

A) introduce auxiliary variable J, figure embedded low-rank sparse and represents that restoration methods objective function equivalence is converted into:

$\begin{array}{ccc}\underset{Z,E}{\min }\left|\right|Z|{|}_{*}+\mathrm{\&alpha;}|\left|E\right|{|}_1+\mathrm{\&beta;}\left|\right|J|{|}_1+\frac{\mathrm{\&gamma;}}{2}Tr\left({\mathrm{ZLZ}}^T\right)& s.t.& X=XZ+E,Z=J\end{array};$

B) constitution step a) in the Augmented Lagrangian Functions of objective function:

L(Z,J,E,Y ₁,Y ₂,μ)＝||Z|| _*+α||E|| ₁+β||J|| ₁+f(Z,J,E,Y ₁,Y ₂,μ)；

Wherein, Y ₁and Y ₂for Lagrange multiplier matrix, μ > 0 is punishment parameter; F (Z, J, E, Y ₁, Y ₂, μ) be:

$f(Z,J,E,Y_1,Y_2,\mathrm{\&mu;})=\frac{\mathrm{\&gamma;}}{2}Tr\left({\mathrm{ZLZ}}^T\right)+\frac{\mathrm{\&mu;}}{2}\left(\right||X-XZ-E+\frac{Y_1}{\mathrm{\&mu;}}|{|}_F^2+\left|\right|Z-J+\frac{Y_2}{\mathrm{\&mu;}}\left|{|}_F^2\right);$

Wherein || || _ffor the F-norm of matrix;

C) initialization $Z^0=J^0=E^0=Y_1^0=Y_2^0=\mathrm{\&theta;},{\mathrm{\&mu;}}^0=0.1,\mathrm{\&rho;}=1.1,{\mathrm{\&mu;}}_{max}={10}^{30},$ Given maximum iteration time maxiter=500, iteration error ε=0.001; Given parameters α=0.5, β=0.2, γ=500; Initialization iterative steps k=0;

D) fixing J ^k, E ^k, Y ₁ ^k, μ ^k, adopt speedup gradient method (acceleratedgradientmethod) to solve Z by following formula ^k+1:

$Z^{k+1}=\underset{Z}{\arg \min }\left|\right|Z|{|}_{*}+f(Z,J^k,E^k,Y_1^k,Y_2^k,{\mathrm{\&mu;}}^k);$

E) fixing Z ^k+1, Y ₁ ^k, μ ^k, solve J by following formula ^k+1:

$J^{k+1}=\underset{J}{\arg \min }\frac{\mathrm{\&beta;}}{{\mathrm{\&mu;}}^k}\left|\right|J|{|}_1+\frac{1}{2}||Z^{k+1}-J+\frac{Y_2^k}{{\mathrm{\&mu;}}^k}|{|}_F^2=S_{\mathrm{\&beta;}/{\mathrm{\&mu;}}^k}(Z^{k+1}+Y_2^k/{\mathrm{\&mu;}}^k);$

Wherein S _bx () is contracting function, be defined as $S_b\left(x\right)\stackrel{\mathrm{\&Delta;}}{=}\left\{\begin{array}{cc}x-b,& x>b\\ 0,& -b\&le;x\&le;b\\ x+b,& x<-b\end{array}\right.;$

F) fixing Z ^k+1, Y ₁ ^k, Y ₂ ^k, μ ^k, solve E by following formula ^k+1:

$E^{k+1}=\underset{E}{\arg \min }\frac{\mathrm{\&alpha;}}{{\mathrm{\&mu;}}^k}\left|\right|E|{|}_1+\frac{1}{2}||X-{\mathrm{XZ}}^{k+1}-E+\frac{Y_1^k}{{\mathrm{\&mu;}}^k}|{|}_F^2=S_{\mathrm{\&alpha;}/{\mathrm{\&mu;}}^k}(X-{\mathrm{XZ}}^{k+1}+Y_1^k/{\mathrm{\&mu;}}^k);$

Wherein S _bx () is contracting function, its definition and step e) identical;

G) Lagrange multiplier matrix is upgraded:

$Y_1^{k+1}=Y_1^k+{\mathrm{\&mu;}}^k(X-{\mathrm{XZ}}^{k+1}-E^{k+1});$

$Y_2^{k+1}=Y_2^k+{\mathrm{\&mu;}}^k(Z^{k+1}-J^{k+1});$

H) undated parameter μ: μ ^k+1=min (ρ μ ^k, μ _max);

I) convergence conditions is checked, if met

|| X-XZ ^k+1-E ^k+1|| _∞< ε and || Z ^k+1-J ^k+1|| _∞< ε, jump procedure j);

Otherwise k=k+1, returns steps d);

J) expression matrix of coefficients Z and data error matrix E is exported.

S05: according to the expression matrix of coefficients Z of the training sample data matrix X that step S04 tries to achieve, adopts following formula to recover clean training sample data Matrix C:

C＝XZ；

S06: for arbitrary facial image to be identified, be translated into M dimensional vector, be designated as face image data y to be identified,

S07: take C as dictionary, E is error dictionary, adopts l ₁norm optimization technique, solve the rarefaction representation coefficient of face image data y to be identified by following formula:

${\min }_{{\&lsqb;{\mathrm{\&alpha;}}^T,{\mathrm{\&beta;}}^T\&rsqb;}^T}\left|\right|y-C\mathrm{\&alpha;}-E\mathrm{\&beta;}|{|}_2^2+\mathrm{\&lambda;}|\left|{\&lsqb;{\mathrm{\&alpha;}}^T,{\mathrm{\&beta;}}^T\&rsqb;}^T\right|{|}_1;$

In formula, iotazation constant λ=0.05;

S08: calculate the class reconstructed error of all kinds of training sample to face image data y to be identified:

$e\left(i\right)=\left|\right|y-{\mathrm{C\&delta;}}_i\left(\mathrm{\&alpha;}\right)-E\mathrm{\&beta;}|{|}_2^2,i=1,2,\mathrm{...},K;$

In formula, δ _i(α) be only retain with the i-th class coefficient of correspondence, be set to the coefficient vector of 0 with other class coefficient of correspondence;

S09: according to class reconstructed error e (i), i=1,2 ..., K, calculates the class label of face image data y to be identified:

Class(y)＝argmin _ie(i)；

In formula, Class (y) represents the class label of face image data y to be identified, argmin _ithe i of e (i) correspondence that e (i) representative value is minimum;

S10: the recognition result exporting facial image to be identified.

By said method, the present invention can solve the recognition of face problem that training sample image and image to be identified are all blocked in situation by noise pollution or local effectively, improves face recognition accuracy rate and operation efficiency.

Be more than the present invention's better embodiment, all changes made with technical solution of the present invention, when the function produced does not exceed the scope of technical solution of the present invention, all belong to protection scope of the present invention.

Claims (2)

1. represent a sparse representation face identification method for recovery based on figure embedding low-rank sparse, it is characterized in that: comprise the following steps:

S01: suppose there is K class training sample image, every class has n to open training sample image, and N=K × n opens training sample image, if often opening training sample image resolution is r × c altogether, be converted to M=r × c dimensional vector by often opening training sample image, then training sample data matrix is designated as

S02: the class label utilizing training sample data matrix X and correspondence thereof, builds a undirected neighbour of supervision that has comprising N number of node and schemes G; Wherein, node annexation and the weight thereof of scheming G are: if data x _iwith data x _jbelong to same class, and x _ifor x _jk neighbour or x _jfor x _ik neighbour, k is positive integer, span 3≤k≤n-1, then the node i scheming G is connected with node j, and its connection weight W _ijbe 1; Otherwise the node i of figure G is not connected with node j, its weights W _ijbe 0, definition data x _idegree be:

$d_i={\mathrm{\&Sigma;}}_{j=1}^N{W}_{ij},i=1,2,...,N;$

Definition data read matrix D is: D=diag (d ₁, d ₂..., d _n), wherein diag (d ₁, d ₂..., d _n) represent with d ₁, d ₂..., d _nfor the diagonal matrix of diagonal element;

S03: the weight matrix W tried to achieve according to step S02 and data read matrix D, definition Laplacian Matrix suppose Z=[z ₁, z ₂..., z _n] be the expression matrix of coefficients of X, in order to enable the expression coefficient of training sample data keep the local geometry of data space and have strong judgement index, based on Laplacian Matrix L, defining one has the figure of supervision to embed regular terms:

$\frac{1}{2}\underset{i,j=1}{\overset{N}{\&Sigma;}}\left|\right|z_i-z_j|{|}_2^2{W}_{ij}=Tr\left({\mathrm{ZLZ}}^T\right),$

In formula, the mark of Tr () representing matrix;

S04: make E be training sample data error matrixes, α > 0, β > 0, γ > 0 are iotazation constant, definition figure embeds low-rank sparse and represents that restoration methods objective function is:

$\begin{array}{ccc}\underset{Z,E}{min}\left|\right|Z|{|}_{*}+\mathrm{\&alpha;}|\left|E\right|{|}_1+\mathrm{\&beta;}\left|\right|Z|{|}_1+\frac{\mathrm{\&gamma;}}{2}Tr\left({\mathrm{ZLZ}}^T\right)& s.t.& X=XZ+E;\end{array}$

In formula || || _*the nuclear norm of representing matrix, i.e. singular values of a matrix sum, || || ₁the l of representing matrix ₁norm, the mark of Tr () representing matrix;

Embed low-rank sparse according to figure and represent that restoration methods objective function tries to achieve expression matrix of coefficients Z and the training sample data error matrix E of training sample data matrix;

S05: according to the expression matrix of coefficients Z of the training sample data matrix X that step S04 tries to achieve, adopts following formula to recover clean training sample data Matrix C:

C＝XZ；

S06: for arbitrary facial image to be identified, be translated into M dimensional vector, be designated as face image data y to be identified,

S07: take C as dictionary, E is error dictionary, adopts l ₁norm optimization technique, solve the rarefaction representation coefficient of face image data y to be identified by following formula:

${\min }_{{\&lsqb;{\mathrm{\&alpha;}}^T,{\mathrm{\&beta;}}^T\&rsqb;}^T}\left|\right|y-C\mathrm{\&alpha;}-E\mathrm{\&beta;}|{|}_2^2+\mathrm{\&lambda;}|\left|{\&lsqb;{\mathrm{\&alpha;}}^T,{\mathrm{\&beta;}}^T\&rsqb;}^T\right|{|}_1;$

In formula, iotazation constant λ=0.05;

S08: calculate the class reconstructed error of all kinds of training sample to face image data y to be identified:

$e\left(i\right)=\left|\right|y-{\mathrm{C\&delta;}}_i\left(\mathrm{\&alpha;}\right)-E\mathrm{\&beta;}|{|}_2^2,i=1,2,...,K;$

In formula, δ _i(α) be only retain with the i-th class coefficient of correspondence, be set to the coefficient vector of 0 with other class coefficient of correspondence;

S09: according to class reconstructed error e (i), i=1,2 ..., K, calculates the class label of face image data y to be identified:

Class(y)＝argmin _ie(i)；

In formula, Class (y) represents the class label of face image data y to be identified, argmin _ithe i of e (i) correspondence that e (i) representative value is minimum;

S10: the recognition result exporting facial image to be identified.

2. a kind of sparse representation face identification method representing recovery based on figure embedding low-rank sparse according to claim 1, it is characterized in that: the figure described in described step S04 embeds low-rank sparse and represents that solving of restoration methods objective function is an iterative process, and algorithm flow is:

A), introduce auxiliary variable J, figure embedded low-rank sparse and represents that restoration methods objective function equivalence is converted into:

$\begin{array}{ccc}\underset{Z,E}{min}\left|\right|Z|{|}_{*}+\mathrm{\&alpha;}|\left|E\right|{|}_1+\mathrm{\&beta;}\left|\right|J|{|}_1+\frac{\mathrm{\&gamma;}}{2}Tr\left({\mathrm{ZLZ}}^T\right)& s.t.& X=XZ+E,Z=J;\end{array}$

B), constitution step a) in the Augmented Lagrangian Functions of objective function:

L(Z,J,E,Y ₁,Y ₂,μ)＝||Z|| _*+α||E|| ₁+β||J|| ₁+f(Z,J,E,Y ₁,Y ₂,μ)；

Wherein, Y ₁and Y ₂for Lagrange multiplier matrix, μ > 0 is punishment parameter; F (Z, J, E, Y ₁, Y ₂, μ) be:

$f(Z,J,E,Y_1,Y_2,\mathrm{\&mu;})=\frac{\mathrm{\&gamma;}}{2}Tr\left({\mathrm{ZLZ}}^T\right)+\frac{\mathrm{\&mu;}}{2}\left(\right||X-XZ-E+\frac{Y_1}{\mathrm{\&mu;}}|{|}_F^2+\left|\right|Z-J+\frac{Y_2}{\mathrm{\&mu;}}\left|{|}_F^2\right);$

Wherein || || _ffor the F-norm of matrix;

C), initialization μ ⁰=0.1, ρ=1.1, μ _max=10 ³⁰, given maximum iteration time maxiter=500, iteration error ε=0.001; Given parameters α=0.5, β=0.2, γ=500; Initialization iterative steps k=0;

D), fixing J ^k, E ^k, μ ^k, adopt speedup gradient method to solve Z by following formula ^k+1:

$Z^{k+1}=\underset{Z}{\arg min}\left|\right|Z|{|}_{*}+f(Z,J^k,E^k,Y_1^k,Y_2^k,{\mathrm{\&mu;}}^k);$

E), fixing j is solved by following formula ^k+1:

$J^{k+1}=\underset{J}{\arg min}\frac{\mathrm{\&beta;}}{{\mathrm{\&mu;}}^k}\left|\right|J|{|}_1+\frac{1}{2}||Z^k+1-J+\frac{Y_2^k}{{\mathrm{\&mu;}}^k}|{|}_F^2=S_{\mathrm{\&beta;}/{\mathrm{\&mu;}}^k}(Z^{k+1}+Y_2^k/{\mathrm{\&mu;}}^k);$

Wherein S _bx () is contracting function, be defined as $S_b\left(x\right)\stackrel{\mathrm{\&Delta;}}{=}\left\{\begin{array}{cc}x-b,& x>b\\ 0,& -b\&le;x\&le;b\\ x+b,& x<-b\end{array}\right.;$

F), fixing Z ^k+1, μ ^k, solve E by following formula ^k+1:

$E^{k+1}=\underset{E}{\arg min}\frac{\mathrm{\&alpha;}}{{\mathrm{\&mu;}}^k}\left|\right|E|{|}_1+\frac{1}{2}||X-{\mathrm{XZ}}^{k+1}-E+\frac{Y_1^k}{{\mathrm{\&mu;}}^k}|{|}_F^2=S_{\mathrm{\&alpha;}/{\mathrm{\&mu;}}^k}(X-{\mathrm{XZ}}^{k+1}+Y_1^k/{\mathrm{\&mu;}}^k);$

Wherein S _bx () is contracting function, its definition and step e) identical;

G), Lagrange multiplier matrix is upgraded:

$Y_1^{k+1}=Y_1^k+{\mathrm{\&mu;}}^k(X-{\mathrm{XZ}}^{k+1}-E^{k+1});$

$Y_2^{k+1}=Y_2^k+{\mathrm{\&mu;}}^k(Z^{k+1}-J^{k+1});$

H), undated parameter μ: μ ^k+1=min (ρ μ ^k, μ _max);

I), convergence conditions is checked, if met

|| X-XZ ^k+1-E ^k+1|| _∞< ε and || Z ^k+1-J ^k+1|| _∞< ε, jump procedure j);

Otherwise k=k+1, returns steps d);

J) expression matrix of coefficients Z and data error matrix E, is exported.