CN106599787A - Single sample face recognition method based on semi-supervised block joint regression - Google Patents

Abstract

The invention discloses a single sample face recognition method based on semi-supervised block joint regression. The method comprises steps: a face is firstly divided into multiple blocks; a semi-supervised-based block joint regression model is then brought forward to make full use of face images with labels and with no labels to learn various change information of the face images, isometric constraints with each class of the face images with no labels in class label coordinate space are added to avoid influenced model discrimination by the data with no labels, a non-strict augmented Lagrange multiplier is used for acquiring a mapping matrix corresponding to each block; on the basis, regression classification on a test image block can be realized through the mapping matrix; and finally, all test image blocks are voted to finally determine a classification result. The single sample face recognition method has good robustness towards expressions, illumination variation and occlusion, and the recognition precision is high.

Description

Single sample face recognition method based on semi-supervised sub-block joint regression

Technical field

The present invention relates to single sample face recognition method, more particularly to each object to be identified only have a width training image The single sample face recognition method based on semi-supervised sub-block joint regression, belong to technical field of face recognition.

Background technology

In past 40 years, face recognition technology makes remarkable progress, and the face recognition technology under controlled condition is It is mature on the whole.But under the conditions of non-controllable, due to by illumination, expression, attitude, noise, the factor such as blocking and affected, face is known Studying a question for a very challenging property is not remained.The most direct method for solving these problems is exactly to increase training sample, but In actual applications such as in many practical applications such as identity card identification, passport identification, judicial confirmation, access control, generally only One training sample can be acquired, and recognition of face problem in this case is referred to as single sample recognition of face problem (Single Sample per Person, SSPP), its be further exacerbated by it is non-controllable under the conditions of recognition of face difficulty.

Technically, single sample recognition of face has more highlighted training sample less and the lance between view data dimension height Shield so that or serious hydraulic performance decline occur in many existing face recognition technologies, or cannot work at all, thus it is public It is considered one of most challenging research topic.But say from practical standpoint, everyone single image itself but have collection easily, Many outstanding advantages such as storage cost is little, be easy to promote on a large scale.Therefore, single sample recognition of face problem not only has very strong Research value, and be also face identification system move towards it is practical during the problem that can not avoid, must solve, solution This problem is of great immediate significance.

Change in class during single specimen discerning inestimable problem to overcome, general learning method (generic Learning a general training set containing everyone multiple faces (generic set)) has been additionally introduced.As face has Change in similar class, it is therefore possible to the facial image Changing Pattern learnt on generic set is used to analyze single instruction Practice the change of facial image in sample set (gallery set).Based on it is above-mentioned it is assumed that Wang et al. first proposed one it is general Framework (J.Wang, K.N.Plataniotis, J.Lu, A.N.Venetsanopoulos, " the On solving the of study face recognition problem with one training sample per subject”.Pattern Recognition,2006,39(9):1746-1762), then Su (S.Yu, S.Shan, X.Chen, W.Gao, " Adaptive Generic Learning for Face Recognition from a Single Sample per Person”IEEE Computer Vision and Pattern Recognition.San Francisco,CA,June 2010:2699-2706) Adaptive universal study (Adaptive Generic Learning, AGL) method of proposition attempts to one from general training The forecast model acquired on collection further improves the hypothesis.Kan et al. (M.Kan, S.Shan, Y.Su, et al. " Adaptive discriminant learning for face recognition”.Pattern Recognition,2013,46(9): Self adaptation discriminant analysiss (the Adaptive Discriminant Analysis) method for 2497-2509) proposing is then based on above-mentioned The thought of adaptive universal study, goes to estimate test sample using some training samples in the generic set neighbouring with test sample Class in Scatter Matrix.Recently, it is to solve the problems, such as single specimen discerning, document (B.Wang, F.Zhou, W.Li, Z.Li, Q.Liao “Combining Specific Learning and Generic Learning for Single-Sample Face Recognition,”5th International Congress on Image and Signal Processing, Chongqing,Oct.2012:1219-1223) will specifically learn (Specific Learning) and general study (Generic Learning) combine, and Deng et al. (W.Deng, J.Hu, X.Zhou, et al. " Equidistant prototypes embedding for single sample based face recognition with generic learning and incremental learning,”Pattern Recognition,2014,47(12):3738-3749) then based on " equidistant former Type is embedded in " principle proposes a kind of linear regression analyses (linear regression Analysis, LRA) method, and introduce The general generalization ability learnt to lift the method.

In recent years, although in class, change is unable to estimate when general learning method can alleviate single specimen discerning to a certain extent Predicament, but generally assuming that for its foundation be not all suitable in all cases, such as when the Genus Homo in two set is not in With race or with the different colours of skin or during the age, and when in the class in generic set change all cannot cover In class in gallery set during change.Therefore how to find one and can more preferably cover change in class in gallery set Generic set, and how effectively to excavate Changing Pattern in the class in two set between sample be all that further research is needed Problem to be considered.

The content of the invention

The technical problem to be solved is：Single sample recognition of face based on semi-supervised sub-block joint regression is provided Method, provides a kind of simple and effective solution for single sample recognition of face problem.

The present invention is employed the following technical solutions to solve above-mentioned technical problem：

Based on single sample face recognition method of semi-supervised sub-block joint regression, comprise the steps：

Step 1, is carried out to the training facial image of all classes and test facial image with equal-sized square window Piecemeal, square window slip over each pixel of image, obtain at each pixel of image corresponding one it is square Shape block, k-th class training facial image pixel i at square block corresponding to vector representation beK=1 ..., K, K table Show the total number or total classification number of training facial image；

Step 2, using corresponding square block structure block training sample set at training facial image pixel i of all classesClass label matrix corresponding to which isUsing all tests Corresponding square block structure block test sample collection Z at facial image pixel iⁱIf the class label matrix corresponding to which isN represents the total number of test facial image；Setting block training sample set XⁱTest with block Sample set ZⁱTo its class label matrix Y and Y_unlabeledMapping matrix be W_i, corresponding piece of training sample at joint all pixels i This collection and block test sample collection, so as to obtain based on semi-supervised joint regression model；

Step 3, by the augmentation Lagrange multiplier solution procedure 2 of non-critical based on semi-supervised joint regression model, Obtain the corresponding mapping matrix W of block training sample set and block test sample collection at all pixels i_i；

Step 4, according to mapping matrix W_i, each square block of each test facial image is classified；

Step 5, after recognizing the classification of each square block of each test facial image, determines test by way of ballot Classification belonging to facial image.

Used as a preferred embodiment of the present invention, the detailed process of the step 1 is：

1.1 define in images centered on any one pixel i, radius be r, size for (2r+1) × (2r+1) square Block is used as the corresponding square block of pixel i；

1.2 edge pixels that image is processed using the method for edge pixel border picture, in obtaining with the edge pixel of image being The square block of the heart.

As a preferred embodiment of the present invention, described in step 2 based on semi-supervised joint regression model it is：

Wherein,Represent block test sample collection ZⁱBe mapped to [1, 1,…,1]^T∈R^KWhen the error that produces, E=[E¹；E²；…；E^M], EⁱRepresent error, W=[W₁；W₂；…；W_M], α is representedError term coefficient, i=1 ..., M, M represent each training facial image or the test square that is divided into of facial image The total block data of block, λ represent | | W | |_2,1Regularization coefficient, | | | |_2,12,1 norm is represented, T represents transposition.

Used as a preferred embodiment of the present invention, the detailed process of the step 3 is：

3.1 by step 2 based on semi-supervised joint regression model conversation be following form：

Wherein,W=[W₁；W₂；…；W_M], Y_α=[Y, α Y_u],I= 1 ..., M, M represent the total block data of the square block that each training facial image or test facial image are divided into, and λ represents | | W | |_2,1's Regularization coefficient, | | | |_2,12,1 norm is represented, T represents transposition；

3.2 is that 3.1 object function introduces auxiliary variable J, is obtained：

3.2 object function is converted to Augmented Lagrangian Functions L by 3.3：

Wherein, J=[J₁；J₂；…；J_M], G, H represent Lagrange multiplier, G=[G₁；G₂；…；G_M], H=[H₁； H₂；…；H_M], the mark of Tr () representing matrix, | | | |_FF norms are represented, μ is the punishment parameter more than 0；

3.4E_α, J, W, G, H be initialized as 0, μ=0.5, fixes other unknown numbers, updates W, then the object function of model turns Turn to the object function for seeking W as follows：

The more new formula that the object function of 3.5 solutions 3.4 obtains W is as follows：

Wherein, t represents iterationses, and I is unit battle array；

3.6 fix other unknown numbers, update J, obtain following Lagrangian subproblem：

The object function is changed intoForm, wherein, S=J； Which has following closed solution：

3.7 fix other unknown numbers, update E in the following way_α：

In 3.8 above-mentioned 3.7, object function is decomposed into M following subproblem：

The object function is changed intoForm, wherein,Which has following closed solution：

3.9 update Lagrange multiplier：

3.10 undated parameter μ：μ=min (ρ μ, μ_max), wherein ρ=1.1, μ_max=10¹⁰；

3.11 check whether and meet the following condition of convergence：

||J^t+1-W^t+1||_∞<ε

Wherein, ε=10^-8, when the above-mentioned condition of convergence is unsatisfactory for, repeat 3.4-3.11, until meeting the condition of convergence.

Used as a preferred embodiment of the present invention, the formula classified described in step 4 is：

Wherein, z represents a test facial image, zⁱRepresent the test corresponding square block of facial image pixel i to Amount expression, label (zⁱ) represent zⁱAffiliated class label,Represent using corresponding mapping square at pixel i Battle array W_iBy zⁱThe categorization vector being mapped to corresponding to which, and a maximum class label the most final of categorization vector intermediate value is taken, K=1,2 ..., K, K represent the total number or total classification number of training facial image, and T represents transposition.

The present invention adopts above technical scheme compared with prior art, with following technique effect：

1st, present invention list sample face recognition method, on the one hand due to introducing more unlabeled exemplars portraying in class Change, to expression, illumination variation and blocks etc. and to have good robustness, thus with higher accuracy of identification；On the other hand Be mapped to due to the sample without label [1,1 ..., 1]^T∈R^K, the distance of its each class of distance in Tag Coordinate Space is equal, The discriminating power of model can't be affected.

2nd, present invention list sample face recognition method, without the need for feature extraction, classifies simple；And taken full advantage of mark Sign the information with unlabeled exemplars.

Description of the drawings

Fig. 1 is flow chart of the present invention based on single sample face recognition method of semi-supervised sub-block joint regression.

Fig. 2 is image block signal in single sample face recognition method of the present invention based on semi-supervised sub-block joint regression Figure.

Fig. 3 is the test specimens without label in single sample face recognition method of the present invention based on semi-supervised sub-block joint regression This geometric interpretation schematic diagram in Tag Coordinate Space.

Specific embodiment

Embodiments of the present invention are described below in detail, the example of the embodiment is shown in the drawings.Below by The embodiment being described with reference to the drawings is exemplary, is only used for explaining the present invention, and is not construed as limiting the claims.

Face change information is actually contained in the test facial image without label, therefore in single sample recognition of face The test facial image without label is taken into full account, can effectively learn facial image change, so as to avoid general learning method pair The dependence and restriction of general training sample set.Based on this idea, the present invention proposes that one kind is combined back based on semi-supervised sub-block The single sample face recognition method returned.

As shown in figure 1, single sample face recognition method of the present invention based on semi-supervised sub-block joint regression, including following step Suddenly：

1st, piecemeal is carried out to the training facial image of all classes and test facial image with equal-sized square window, Square window slips over each pixel of image, and a corresponding square block is obtained at each pixel of image, It is specific as follows：

(1) facial image of all one training samples and test sample is divided into the block of multiple overlaps, each block center One pixel of correspondence image, centered on the pixel, square window of the radius as r carry out piecemeal.That is scheme As any one pixel i, the square block of size for (2r+1) × (2r+1) centered on the pixel of correspondence one, be expressed as to Amount form isWherein k represents the sample of kth class.Above-mentioned fixation is used to all training facial images and test facial image The square block of size carries out piecemeal, the pixel in the correspondence image of block center, as shown in Figure 2.

(2) for image edge pixels, processed using the method for edge pixel border picture because bulk is beyond image border Cause the phenomenon of its interior section pixel value disappearance, obtain the bulk centered on image edge pixels.

2nd, using corresponding square block structure block training sample set X at training facial image pixel i of all classesⁱ, its Corresponding class label isY∈R^K×K, R^K×KRepresent the space of K × K, it is assumed that Block training set XⁱMapping matrix to its class label matrix Y is W_i, reflect by all sub-blocks of regression model joint solution are corresponding Penetrate matrix W_i, comprise the following steps that：

(1) block training sample setWhereinRepresent k-th or class trains facial image in pixel Corresponding square block at i, k=1,2 ..., K, K represent the total number or total classification number of training facial image；

(2) by block training sample setCorresponding class label is defined as Y∈R^K×K, it is assumed that block training set XⁱMapping matrix to its class label matrix Y is W_i, the corresponding regression model of the sub-block can be with It is expressed as：Y^T=(Xⁱ)^TW_i+Eⁱ, wherein EⁱRepresent error；

(3) combine the mapping matrix W for solving all sub-blocks using following object function_i：

Wherein, E=[E¹；E²；…；E^M], W=[W₁；W₂；…；W_M], i=1,2 ..., M, M represent sub-block number, and λ is represented | |W||_2,1Regularization coefficient, | | | |_2,12,1 norm is represented, can effectively suppress noise and outlier.

3rd, the sub-block of test sample formed objects at pixel i and training sample of whether there is label is taken, block is constituted Test sample collection Zⁱ, the class label of these test samples be it is unknown, but due in step 2 regression model only used uniquely One training sample, it is impossible to fully change information of the study to face, therefore the present invention is by block test set ZⁱIt is mapped toSo as to obtain based on semi-supervised joint regression model, the model can make full use of people The change information of face, while all test samples are mapped to [1,1 ..., 1]^T∈R^KCan ensure that the test sample without label exists In Tag Coordinate Space, the distance of each class of distance is equal, so can be protected while face change information is made full use of Demonstrate,prove these samples without tag attributes, its geometric interpretation in Tag Coordinate Space is as shown in Figure 3.Should be based on semi-supervised son Block joint regression model is as follows：

Wherein,N represents test sample number, Represent block test set ZⁱBe mapped to [1,1 ..., 1]^T∈R^KWhen the error that produces, α represents | | E_unlabeled||_2,1Error term system Number.

4th, by the semi-supervised joint regression model of the augmentation Lagrange multiplier solution procedure 3 of non-critical, owned The corresponding mapping matrix W of sub-block_i, comprise the following steps that：

(1) by step 3 based on semi-supervised joint regression model conversation be following form：

Wherein, Y_α=[Y, α Y_unlabeled],W=[W₁；W₂；…； W_M], λ is represented | | W | |_2,1Regularization coefficient, | | | |_2,1Represent 2,1 norms；

(2) object function for (1) introduces auxiliary variable J：

(3) object function of (2) is converted to into Augmented Lagrangian Functions L：

Wherein, the mark of Tr () representing matrix, J=[J₁；J₂；…；J_M], G, H represent Lagrange multiplier, G=[G₁； G₂；…；G_M], H=[H₁；H₂；…；H_M], T represents transposition, and μ is the punishment parameter more than 0；

(4) first by E_α, J, W, G, H are initialized as 0, μ=0.5, μ_max=10¹⁰, ρ=1.1, ε=10-⁸；

(5) using alternating minimization updating E_α, J, W, G, H fix other unknown numbers, update W, then the target letter of model Number can be converted into the object function for seeking W as follows：

(6) solve (5) object function can obtain W more new formula it is as follows：

Wherein I is unit battle array, and t represents iterationses；

(7) other unknown numbers are fixed, it is possible to obtain following Lagrangian subproblem：

The object function can be changed intoForm, wherein, S=J；Which has following closed solution：

J is updated by above-mentioned formula；

(8) other unknown numbers are fixed, E is updated in the following way_α：

(9) in above-mentioned (8), object function can be decomposed into M following subproblem：

The object function can also be converted intoForm, wherein,There is following closed solution：

(10) update Lagrange multiplier：

G^t+1=G^t+μ(J^t+1-W^t+1)

(11) undated parameter μ：μ=min (ρ μ, μ_max)；

(12) check whether and meet the following condition of convergence：

||J^t+1-W^t+1||_∞<ε

Wherein, ε=10^-8, when the above-mentioned condition of convergence is unsatisfactory for, repeat (5)-(12), until meeting the condition of convergence.

5th, the mapping matrix W according to each sub-block locations_i, each square block of each test facial image is carried out point Class, classification formula are as follows：

Wherein, z represents a test sample, zⁱRepresent the vector of the corresponding square block of pixel i of test facial image Represent, label (zⁱ) represent zⁱAffiliated class label,Represent using corresponding mapping matrix at pixel i W_iBy zⁱMapping is with the categorization vector corresponding to which, and takes a maximum class label the most final of categorization vector intermediate value.

6th, after recognizing the classification of each test face block, finally determined belonging to test facial image by the method voted Classification.

Single sample face recognition method of the present invention based on semi-supervised sub-block joint regression, the classification per block can be grasped parallel Make, the inventive method has very strong robustness to illumination variation, can be under conditions of extra training sample set is not relied on Still obtain preferable performance, at the same to illumination, express one's feelings, block, time change all has good robustness.

Above example technological thought only to illustrate the invention, it is impossible to which protection scope of the present invention is limited with this, it is every According to technological thought proposed by the present invention, any change done on the basis of technical scheme, the scope of the present invention is each fallen within Within.

Claims (5)

1. the single sample face recognition method based on semi-supervised sub-block joint regression, it is characterised in that comprise the steps：

Step 1, carries out piecemeal to the training facial image of all classes and test facial image with equal-sized square window, Square window slips over each pixel of image, and a corresponding square block is obtained at each pixel of image, K-th class training facial image pixel i at square block corresponding to vector representation beK=1 ..., K, K represent instruction Practice the total number or total classification number of facial image；

Step 2, using corresponding square block structure block training sample set at training facial image pixel i of all classesClass label matrix corresponding to which isY∈R^K×K；Using all testers Corresponding square block structure block test sample collection Z at face image pixel iⁱIf the class label matrix corresponding to which isN represents the total number of test facial image；Setting block training sample set XⁱTest with block Sample set ZⁱTo its class label matrix Y and Y_unlabeledMapping matrix be W_i, corresponding piece of training sample at joint all pixels i This collection and block test sample collection, so as to obtain based on semi-supervised joint regression model；

Step 3, by the augmentation Lagrange multiplier solution procedure 2 of non-critical based on semi-supervised joint regression model, obtains The corresponding mapping matrix W of block training sample set and block test sample collection at all pixels i_i；

Step 4, according to mapping matrix W_i, each square block of each test facial image is classified；

Step 5, after recognizing the classification of each square block of each test facial image, determines test face by way of ballot Classification belonging to image.

2. single sample face recognition method according to claim 1 based on semi-supervised sub-block joint regression, it is characterised in that The detailed process of the step 1 is：

1.1 define in images centered on any one pixel i, radius is that r, size are made for the square block of (2r+1) × (2r+1) For the corresponding square block of pixel i；

1.2 edge pixels that image is processed using the method for edge pixel border picture, are obtained centered on the edge pixel of image Square block.

3. single sample face recognition method according to claim 1 based on semi-supervised sub-block joint regression, it is characterised in that Described in step 2 based on semi-supervised joint regression model it is：

$\begin{array}{ccc}{\min }_{E,E_{unlabeled},W}\left|\right|E|{|}_{2,1}+\mathrm{\&alpha;}|\left|E_{unlabeled}\right|{|}_{2,1}+\mathrm{\&lambda;}\left|\right|W|{|}_{2,1},& s.t.& Y^T={\left(X^i\right)}^T{W}_i+E^i;Y_{unlabeled}^T={\left(Z^i\right)}^T{W}_i+E_{unlabeled}^i\end{array}$

Wherein,Represent block test sample collection ZⁱBe mapped to [1,1 ..., 1]^T ∈R^KWhen the error that produces, E=[E¹；E²；…；E^M], EⁱRepresent error, W=[W₁；W₂；…；W_M], α is representedMistake Difference term coefficient, i=1 ..., M, M represent the total block data of the square block that each training facial image or test facial image are divided into, λ Represent | | W | |_2,1Regularization coefficient, | | | |_2,12,1 norm is represented, T represents transposition.

4. single sample face recognition method according to claim 3 based on semi-supervised sub-block joint regression, it is characterised in that The detailed process of the step 3 is：

3.1 by step 2 based on semi-supervised joint regression model conversation be following form：

$\begin{array}{ccc}{\min }_{E_{\mathrm{\&alpha;}},W}\left|\right|E_{\mathrm{\&alpha;}}|{|}_{2,1}+\mathrm{\&lambda;}|\left|W\right|{|}_{2,1},& s.t.& Y_{\mathrm{\&alpha;}}^T={\left(X_{\mathrm{\&alpha;}}^i\right)}^T{W}_i+E_{\mathrm{\&alpha;}}^i,\&ForAll;i\end{array}$

Wherein,W=[W₁；W₂；…；W_M], Y_α=[Y, α Y_u],I=1 ..., M, M represent the total block data of the square block that each training facial image or test facial image are divided into, and λ represents | | W | |_2,1Canonical Term coefficient, | | | |_2,12,1 norm is represented, T represents transposition；

3.2 is that 3.1 object function introduces auxiliary variable J, is obtained：

$\begin{array}{ccc}{\min }_{E_{\mathrm{\&alpha;}},W}\frac{1}{2}\left|\right|E_{\mathrm{\&alpha;}}|{|}_{2,1}+\mathrm{\&lambda;}|\left|J\right|{|}_{2,1},& s.t.& Y_{\mathrm{\&alpha;}}^T={\left(X_{\mathrm{\&alpha;}}^i\right)}^T{W}_i+E_{\mathrm{\&alpha;}}^i,J=W,\&ForAll;i\end{array}$

3.2 object function is converted to Augmented Lagrangian Functions L by 3.3：

$L(E_{\mathrm{\&alpha;}},J,W,G,H,\mathrm{\&mu;})=\left|\right|E_{\mathrm{\&alpha;}}|{|}_{2,1}+\mathrm{\&lambda;}|\left|J\right|{|}_{2,1}+{\mathrm{\&Sigma;}}_{i=1}^{M}Tr\left(H_i^T\right(Y_{\mathrm{\&alpha;}}^T-X_{\mathrm{\&alpha;}}^i{W}_i-E_{\mathrm{\&alpha;}}^i\left)\right)$

$\begin{array}{c}Tr\left(G^T\right(J-W\left)\right)+\frac{\mathrm{\&mu;}}{2}\left({\mathrm{\&Sigma;}}_{i=1}^M\right||Y_{\mathrm{\&alpha;}}^T-X_{\mathrm{\&alpha;}}^i{W}_i-E_{\mathrm{\&alpha;}}^i|{|}_F^2+\left|\right|J-W\left|{|}_F^2\right)={\mathrm{\&Sigma;}}_{i=1}^M\left|\right|E_{\mathrm{\&alpha;}}^i|{|}_{2,1}+\\ \mathrm{\&lambda;}\left|\right|J_i|{|}_{2,1}+\frac{\mathrm{\&mu;}}{2}||Y_{\mathrm{\&alpha;}}^T-X_{\mathrm{\&alpha;}}^i{W}_i-E_{\mathrm{\&alpha;}}^i+\frac{H_i}{\mathrm{\&mu;}}|{|}_F^2+\frac{\mathrm{\&mu;}}{2}\left|\right|J_i-W_i+\frac{G_i}{\mathrm{\&mu;}}|{|}_F^2-\frac{1}{2\mathrm{\&mu;}}|\left|H_i\right|{|}_F^2-\frac{1}{2\mathrm{\&mu;}}\left|\right|G_i|{|}_F^2\end{array}$

Wherein, J=[J₁；J₂；…；J_M], G, H represent Lagrange multiplier, G=[G₁；G₂；…；G_M], H=[H₁；H₂；…； H_M], the mark of Tr () representing matrix, | | | |_FF norms are represented, μ is the punishment parameter more than 0；

3.4E_α, J, W, G, H be initialized as 0, μ=0.5, fixes other unknown numbers, updates W, then the object function of model is converted into The object function of W is sought as follows：

$W=\arg {\min }_{W}L\left(W\right)=\arg {\min }_W{\mathrm{\&Sigma;}}_{i=1}^M\frac{\mathrm{\&mu;}}{2}\left|\right|Y_{\mathrm{\&alpha;}}^T-\left(X_{\mathrm{\&alpha;}}^i\right)W_i-E_{\mathrm{\&alpha;}}^i+\frac{H_i}{\mathrm{\&mu;}}|{|}_F^2+\frac{\mathrm{\&mu;}}{2}||J_i-W_i+\frac{G_i}{\mathrm{\&mu;}}|{|}_F^2$

The more new formula that the object function of 3.5 solutions 3.4 obtains W is as follows：

$W_i^{t+1}={(X_{\mathrm{\&alpha;}}^i{\left(X_{\mathrm{\&alpha;}}^i\right)}^T+I)}^{-1}(X_{\mathrm{\&alpha;}}^i{Y}_{\mathrm{\&alpha;}}^T-X_{\mathrm{\&alpha;}}^i{\left(X_{\mathrm{\&alpha;}}^i\right)}^t+J_i^t+\frac{(X_{\mathrm{\&alpha;}}^i{H}_i^t+G_i^t)}{\mathrm{\&mu;}}),\&ForAll;i$

Wherein, t represents iterationses, and I is unit battle array；

3.6 fix other unknown numbers, update J, obtain following Lagrangian subproblem：

$J^{t+1}=\arg {\min }_J\frac{\mathrm{\&lambda;}}{\mathrm{\&mu;}}\left|\right|J|{|}_{2,1}+\frac{1}{2}||J-W^{t+1}+\frac{G^t}{\mathrm{\&mu;}}|{|}_F^2$

The object function is changed intoForm, wherein, S=J； Which has following closed solution：

$S(j,:)=\left\{\begin{array}{cc}\frac{\left|\right|T(j,:)|{|}_2-\mathrm{\&gamma;}}{\left|\right|T(j,:)|{|}_2},& if\left|\right|T(j,:)|{|}_2>\mathrm{\&gamma;}\\ 0,& otherwise\end{array}\right.;$

3.7 fix other unknown numbers, update E in the following way_α：

$E_{\mathrm{\&alpha;}}^{t+1}=\arg \min L\left(E_{\mathrm{\&alpha;}}\right)=\arg {\min }_{E_{\mathrm{\&alpha;}}}{\mathrm{\&Sigma;}}_{i=1}^M\left|\right|E_{\mathrm{\&alpha;}}^i|{|}_{2,1}+\frac{\mathrm{\&mu;}}{2}||Y_{\mathrm{\&alpha;}}^T-{\left(X_{\mathrm{\&alpha;}}^i\right)}^T{W}_i^{t+1}-E_{\mathrm{\&alpha;}}^i+\frac{H_i^t}{\mathrm{\&mu;}}|{|}_F^2$

In 3.8 above-mentioned 3.7, object function is decomposed into M following subproblem：

${\left(E_{\mathrm{\&alpha;}}^i\right)}^{t+1}=\arg {\min }_{E_{\mathrm{\&alpha;}}^i}\left|\right|E_{\mathrm{\&alpha;}}^i|{|}_{2,1}+\frac{\mathrm{\&mu;}}{2}||Y_{\mathrm{\&alpha;}}^T-{\left(X_{\mathrm{\&alpha;}}^i\right)}^T{W}_i^{t+1}-E_{\mathrm{\&alpha;}}^i+\frac{H_i^t}{\mathrm{\&mu;}}|{|}_F^2,\&ForAll;i$

The object function is changed intoForm, wherein, Which has following closed solution：

$S(j,:)=\left\{\begin{array}{cc}\frac{\left|\right|T(j,:)|{|}_2-\mathrm{\&gamma;}}{\left|\right|T(j,:)|{|}_2},& if\left|\right|T(j,:)|{|}_2>\mathrm{\&gamma;}\\ 0,& otherwise\end{array}\right.;$

3.9 update Lagrange multiplier：

3.10 undated parameter μ：μ=min (ρ μ, μ_max), wherein ρ=1.1, μ_max=10¹⁰；

3.11 check whether and meet the following condition of convergence：

$\left|\right|Y_{\mathrm{\&alpha;}}^T-{\left(X_{\mathrm{\&alpha;}}^i\right)}^T{W}_i^{t+1}-{\left(E_{\mathrm{\&alpha;}}^i\right)}^{t+1}|{|}_{\mathrm{\&infin;}}<\mathrm{\&epsiv;}$

||J^t+1-W^t+1||_∞<ε

Wherein, ε=10^-8, when the above-mentioned condition of convergence is unsatisfactory for, repeat 3.4-3.11, until meeting the condition of convergence.

5. single sample face recognition method according to claim 1 based on semi-supervised sub-block joint regression, it is characterised in that The formula classified described in step 4 is：

$label\left(z^i\right)=\arg {\max }_k\left(W_i^T{z}^i\right)$

Wherein, z represents a test facial image, zⁱRepresent the vector table of the corresponding square block of test facial image pixel i Show, label (zⁱ) represent zⁱAffiliated class label,Represent using corresponding mapping matrix W at pixel i_i By zⁱThe categorization vector being mapped to corresponding to which, and take a maximum class label the most final of categorization vector intermediate value, k= 1,2 ..., K, K represent the total number or total classification number of training facial image, and T represents transposition.