CN109508697B - Face recognition method, system and storage medium based on semi-nonnegative matrix factorization of E auxiliary function - Google Patents

Face recognition method, system and storage medium based on semi-nonnegative matrix factorization of E auxiliary function Download PDF

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CN109508697B
CN109508697B CN201811536236.1A CN201811536236A CN109508697B CN 109508697 B CN109508697 B CN 109508697B CN 201811536236 A CN201811536236 A CN 201811536236A CN 109508697 B CN109508697 B CN 109508697B
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陈文胜
陈海涛
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Abstract

The invention provides a face recognition method, a system and a storage medium based on E auxiliary function semi-nonnegative matrix decomposition, wherein the face recognition method comprises the following steps: converting training sample images into a training sample matrix
Figure DEST_PATH_IMAGE002
Setting an error threshold
Figure DEST_PATH_IMAGE004
Maximum number of iterations
Figure DEST_PATH_IMAGE006
And inputting the training sample matrix
Figure DEST_PATH_IMAGE002A
Error threshold and maximum number of iterations
Figure DEST_PATH_IMAGE006A
(ii) a The second step is as follows: to the base image matrix and coefficient matrix
Figure DEST_PATH_IMAGE010
Carrying out initialization; the fourth step: updating the base image matrix according to equation (6)
Figure DEST_PATH_IMAGE012
Sum coefficient matrix
Figure DEST_PATH_IMAGE010A
(ii) a A sixth step: determining an objective function
Figure DEST_PATH_IMAGE015
Or whether the iteration number n reaches the maximum iteration number
Figure DEST_PATH_IMAGE006AA
And if so, outputting the base image matrix
Figure DEST_PATH_IMAGE012A
Sum coefficient matrix
Figure DEST_PATH_IMAGE010AA
Otherwise, executing the fourth step. The invention has the beneficial effects that: the face recognition method has the advantages of high recognition performance and low calculation complexity, and the results show that the method developed by the patent has certain superiority by experimental comparison with related algorithms in a public face database.

Description

Face recognition method, system and storage medium based on semi-nonnegative matrix factorization of E auxiliary function
Technical Field
The invention relates to the technical field of data processing, in particular to a face recognition method, a face recognition system and a storage medium based on E auxiliary function semi-nonnegative matrix decomposition.
Background
With the advent of the information age, biometric identification technology for identifying an individual's identity using physiological and behavioral characteristics inherent to a human body has become one of the most active research fields. Among the many branches of biometric technology, one of the most well accepted techniques is face recognition technology, because face recognition is non-invasive, non-mandatory, non-contact, and concurrent with respect to other biometric technologies.
The face recognition technology comprises two stages, wherein the first stage is feature extraction, namely extracting face feature information in a face image, and the first stage directly determines the quality of the face recognition technology; the second stage is identity authentication, and personal identity authentication is carried out according to the extracted characteristic information. Principal Component Analysis (PCA) and Singular Value Decomposition (SVD) are more classical feature extraction methods, but feature vectors proposed by the two methods usually contain negative elements, so that the methods are not reasonable and interpretable under the condition that an original sample is non-negative data. non-Negative Matrix Factorization (NMF) is a feature extraction method for processing non-negative data, and its application is very wide, such as hyperspectral data processing, face image recognition, etc. The NMF algorithm has non-negative limitation on the extracted features in the original sample non-negative data matrix decomposition process, namely all components after decomposition are non-negative, so that non-negative sparse features can be extracted. The essence of the NMF algorithm is that the non-negative matrix X is approximately decomposed as the product of the base image matrix W and the coefficient matrix H, i.e. X ≈ WH, and both W and H are non-negative matrices. Thus each column of the matrix X can be represented as a non-negative linear combination of the vectors of the columns of the matrix W, which also follows the construction basis of the NMF algorithm-the perception of the whole is made up of the perception of the parts that make up the whole (purely additive).
However, NMF requires the original sample to be non-negative data, which limits the range of application of the algorithm to some extent. In order to expand the range of NMF applications, in recent years, researchers have proposed many algorithms for NMF deformation, such as Semi-non-negative matrix factorization (Semi-NMF) and Convex non-negative matrix factorization (Convex-NMF). Both algorithms can be used for non-negative data as well as for other data. However, since they allow the original sample to contain negative numbers, it becomes a complicated problem to prove the convergence of the iterative formula. Meanwhile, the existing Semi-NMF iteration method has the problems of low convergence rate, low algorithm recognition rate and the like.
1. Auxiliary Function (Auxiliary Function)
The auxiliary function is a common tool for proving the convergence of the algorithm, and is defined and characterized as follows: definition 1: for arbitrary matrices H and H(t)If the condition is satisfied
G(H,H(t)) ≧ f (H), and G (H)(t),H(t))＝f(H(t))
Then called G (H, H)(t)) Is an auxiliary function of the function f (h).
Introduction 1: if G (H, H)(t)) Is an auxiliary function of f (H), then f (H) is as followsThe new rule is that the operation is monotonous and does not increase,
Figure BDA0001906953360000021
2. semi non-negative matrix factorization (Semi-NMF)
Semi-non-negative matrix factorization (Semi-NMF) does not limit the sign of the sample matrix, thereby expanding the application range of NMF. For matrix decomposition X ≈ WH, the optimization problem that Semi-NMF needs to solve is:
Figure BDA0001906953360000022
the updating iteration formula is as follows:
Figure BDA0001906953360000023
wherein
Figure BDA0001906953360000024
3. Convex non-negative matrix factorization (Convex-NMF)
The main idea of Convex non-negative matrix factorization (Convex-NMF) is to restrict the columns in the base image matrix to one Convex combination of columns in the original matrix. For matrix decomposition X ≈ XWH, the optimization problem that the Convex-NMF needs to solve is:
Figure BDA0001906953360000025
the updating iteration formula is as follows:
Figure BDA0001906953360000031
Figure BDA0001906953360000032
as can be seen from the above Semi-NMF and Convex-NMF iterative formulas, the square root of the open arithmetic appears therein, which greatly increases the computational complexity of the algorithm and reduces the computational efficiency of the algorithm.
To sum up:
1. the application range of the non-negative matrix factorization algorithm (NMF) is narrow, and the non-negative matrix factorization algorithm is only suitable for non-negative data.
2. Although the application range of the non-negative matrix factorization algorithm is expanded by the traditional Semi-non-negative matrix factorization algorithm (Semi-NMF), the effect and the convergence rate of the non-negative matrix factorization algorithm still need to be improved.
Disclosure of Invention
The invention provides a face recognition method based on E auxiliary function semi-nonnegative matrix decomposition, which comprises a training step, wherein the training step comprises the following steps:
the first step is as follows: converting the training sample image into a training sample matrix X, and setting an error threshold epsilon and a maximum iteration number ImaxAnd inputting a training sample matrix X, an error threshold epsilon and a maximum iteration number Imax
The second step is as follows: initializing a base image matrix W and a coefficient matrix H;
the third step: setting the iteration number n to be 0;
the fourth step: updating the base image matrix W and the coefficient matrix H according to formula (6);
the fifth step: n is n + 1;
a sixth step: judging whether the objective function F (W, H) is less than or equal to epsilon or the iteration number n reaches the maximum iteration number ImaxIf yes, outputting a base image matrix W and a coefficient matrix H, otherwise, executing a fourth step;
in the fourth step, equation (6) is as follows:
W←XHT(HHT)-1
Figure BDA0001906953360000033
in formula (6), W represents a base image matrix, H represents a coefficient matrix, and X represents a training sample matrix.
As a further improvement of the invention: the face recognition method further comprises a recognition step which is executed after the training step, wherein the recognition step comprises the following steps:
a seventh step of: calculating the average characteristic vector m of each class in the training samplej(j＝1,…,c)；
The eighth step is that the face image y to be recognized is input and the characteristic vector h of the face image y is calculatedy＝W-1y；
The ninth step is that the characteristic vector h of the face image to be recognized is calculatedyAverage feature vector m to each classjIf h isyAnd mjIf the distance is the minimum, the face image y to be recognized is classified into the P-th class;
a tenth step: and outputting the category P.
The invention also provides a face recognition system based on the semi-nonnegative matrix factorization of the E auxiliary function, which comprises a training module, wherein the training module comprises:
an input module: for converting training sample image into training sample matrix X, setting error threshold epsilon and maximum iteration number ImaxAnd inputting a training sample matrix X, an error threshold epsilon and a maximum iteration number Imax；
An initialization module: the method comprises the steps of initializing a base image matrix W and a coefficient matrix H;
an assignment module: setting the iteration number n to be 0;
an update module: for updating the base image matrix W and the coefficient matrix H according to equation (6); a counting module: n is n + 1;
a judging module: judging whether the objective function F (W, H) is less than or equal to epsilon or the iteration number n reaches the maximum iteration number ImaxIf yes, outputting a base image matrix W and a coefficient matrix H, otherwise, executing an updating module;
in the update module, equation (6) is as follows:
W←XHT(HHT)-1
Figure BDA0001906953360000041
in formula (6), W represents a base image matrix, H represents a coefficient matrix, and X represents a training sample matrix.
As a further improvement of the invention: the face recognition system further comprises a recognition module executed after the training module, the recognition module comprising:
an average feature vector calculation module: for calculating an average feature vector m for each class in a training samplej(j＝1,…,c)；
The characteristic vector calculation module is used for inputting the face image y to be recognized and calculating the characteristic vector h thereofy＝W-1y；
Distance calculating module for calculating characteristic vector h of face image to be recognizedyAverage feature vector m to each classjIf h isyAnd mjIf the distance is the minimum, the face image y to be recognized is classified into the P-th class;
an output module: for outputting the category P.
The invention also discloses a computer-readable storage medium, in which a computer program is stored which, when being invoked by a processor, is configured to carry out the steps of the method as claimed in the claims.
The invention has the beneficial effects that: the face recognition method has the advantages of high recognition performance and low calculation complexity, and the results show that the method developed by the patent has certain superiority by experimental comparison with related algorithms in a public face database.
Drawings
FIG. 1 is a flow chart of the algorithm construction process of the present invention;
FIG. 2 is a flow chart of a method of the present invention;
FIG. 3 shows the method and related algorithms (Convex-NMF and Semi-NMF) proposed by the present invention
Comparison of recognition rates on ORL face database, where FSNMF represents the method of the invention;
FIG. 4 is a graph comparing the convergence curves of the inventive method and the Semi-NMF algorithm, where FSNMF represents the inventive method.
Detailed Description
The invention firstly provides a new concept of the E auxiliary function of the objective function, and accordingly provides a new basic theory and framework for constructing the auxiliary function, which greatly expand the selection range of the auxiliary function and also provide a powerful tool for flexibly constructing the auxiliary function to design a new high-performance non-negative characteristic algorithm. According to the method for the E auxiliary function, provided by the invention, a new auxiliary function is constructed, and a new fast semi-nonnegative matrix factorization (FSNMF) algorithm is deduced according to the new auxiliary function. The FSNMF algorithm obtained by the invention is convergent according to the property of the auxiliary function. The experimental result shows that compared with other NMF-based algorithms, the FSNMF iteration method provided by the invention has better recognition rate and faster convergence rate. Meanwhile, the E auxiliary function method provided by the invention is also suitable for developing other biological feature identification iterative algorithms and proving the convergence of the algorithms.
Keyword interpretation:
1. description of the symbols:
x matrix
XTTranspose of X
Taking absolute value of every element of matrix X
XijThe ijth element of the matrix X
xjJ column of matrix X
Figure BDA0001906953360000061
Product of corresponding elements in matrices A and B
Figure BDA0001906953360000062
Quotient of corresponding elements in matrices A and B
AdD powers of each element in the matrix A
2. Non-negative Matrix Factorization (Non-negative Matrix Factorization, NMF)
The basic idea of NMF is to use a non-negative sample matrix
Figure BDA0001906953360000063
The approximate decomposition is the product of two non-negative matrices, namely:
X≈WH,
wherein the content of the first and second substances,
Figure BDA0001906953360000064
and
Figure BDA0001906953360000065
referred to as base image matrix and coefficient matrix, respectively. And, by constructing a loss function metric that measures the degree of approximation between X and WH, the loss function is typically defined based on the F-norm, written as:
F(W,H)＝||X-WH||2 (1)
where | · | | represents the F-norm.
3. Semi-Non-negative Matrix Factorization (Semi-Non-negative Matrix Factorization, Semi-NMF)
The Semi-NMF does not limit the sign of the sample matrix, thereby expanding the application range of the NMF. For sample matrix
Figure BDA0001906953360000071
Considering the decomposition:
X≈WH,
wherein the content of the first and second substances,
Figure BDA0001906953360000072
and
Figure BDA0001906953360000073
i.e., Semi-NMF only limits the coefficient matrix to non-negative.
The specific technical scheme is as follows:
1. the main objects of the present invention are:
(1) the application range of the NMF is expanded, so that the NMF still has effectiveness on data containing negative numbers;
(2) a new concept of the E-helper function is proposed, whereby a basic theoretical framework and method of constructing a new helper function is presented.
(3) According to the proposed E-helper function method, a new fast semi-nonnegative matrix factorization (FSNMF) face recognition algorithm with high recognition performance and low computational complexity is developed.
2. The technical scheme is as follows:
(1) definition and properties of the E-helper function:
for many objective functions, it is difficult to construct a suitable helper function. For example, the auxiliary function constructed in the conventional semi-nonnegative matrix factorization algorithm is extremely complex, and the convergence rate of the obtained iterative formula is slow. In order to overcome the defects, the patent firstly provides the concept of the E auxiliary function, and obtains a new fast semi-nonnegative matrix factorization (FSNMF) face recognition algorithm with high recognition performance and low computation complexity based on the E auxiliary function. The definition and nature of the E-helper function is as follows:
definition 2: for arbitrary matrices H and H(t)If E (H, H)(t)) Satisfies the conditions
E(H,H(t)) Not less than 0, and E (H)(t),H(t))＝0
Then called E (H, H)(t)) An E helper function for the function f (H).
Theorem 1: if the function G (H, H)(t)) Is an auxiliary function of the function f (H), if the function E (H, H)(t)) Is an E-helper function of the function f (h). Then there are:
Figure BDA0001906953360000074
is still an auxiliary function of f (H), where λ ≧ 0.
And (3) proving that: from the definition of the helper function and the E helper function, we have:
Figure BDA0001906953360000081
Figure BDA0001906953360000082
it can be known that
Figure BDA0001906953360000083
Is still an auxiliary function of f (H).
Theorem 1 provides a theoretical framework and a method for constructing a new auxiliary function based on an E auxiliary function, which greatly expand the selection range of the auxiliary function and also provide a powerful tool for flexibly constructing the auxiliary function to design a new high-performance non-negative characteristic algorithm.
(2) Proposal of new FSNMF:
construction of an objective function
For sample matrix
Figure BDA0001906953360000084
Consider Semi-NMF:
X≈WH
wherein the content of the first and second substances,
Figure BDA0001906953360000085
and
Figure BDA0001906953360000086
the objective function to obtain the fast semi-nonnegative matrix factorization algorithm (FSNMF) is:
F(W,H)＝||X-WH||2 (3)
to solve the two unknown matrices W and H in the objective function (3), we convert the objective function into two sub-objective functions, which are:
f1(W)＝||X-WH||2in which H is fixed
f2(H)＝||X-WH||2Where W is fixed, problem (3) also evolves into two sub-problems, respectively:
minf1(W) (4)
minf2(H)s.t.H≥0 (5)
learning of the base image matrix W
The H is fixed, and the reaction is carried out,due to f1(W) is convex, and the sign of W is not limited, so that the extreme point is the optimal solution. The gradient is calculated and made 0 with:
Figure BDA0001906953360000087
the closed form solution of W can be solved:
W＝XHT(HHT)-1
learning of coefficient matrix H
Fixing W, we construct the objective function f first2(H) An auxiliary function G (H, H)(t)). An objective function f2(H) The expansion is as follows:
f2(H)＝tr(XTX)-2tr(HT(WTX)+-HT(WTX)-)+tr((WTW)+-(WTW)-)HHT
for the last two terms on the right of the equation, we have:
2, leading: function(s)
Figure BDA0001906953360000091
Is tr ((W)TW)+HHT) An auxiliary function of (2).
And 3, introduction: function(s)
G2(H,H(t))＝-2tr((WTW)-H(t)HT)+tr((WTW)-H(t)H(t)T)
Is-tr ((W)TW)-HHT) An auxiliary function of (2).
From the above two lemmas, we can get theorem 2.
Theorem 2: function(s)
Figure BDA0001906953360000092
Is f2(H) An auxiliary function of (2).
And (3) proving that: as can be seen from lemons 2 and 3,
Figure BDA0001906953360000093
is tr ((W)TW)+HHT) The auxiliary function of (a) is selected,
Figure BDA0001906953360000094
is-tr ((W)TW)-HHT) The auxiliary function of (2).
Therefore, there is G (H, H)(t)) Is f2(H) An auxiliary function of (2).
Generally, updating the iterative formula can be derived directly from the auxiliary function. But here by an auxiliary function G (H, H)(t)) The derived formula does not satisfy the non-negativity, so that the method needs to reconstruct a new auxiliary function by means of the E auxiliary function.
Theorem 3: function(s)
Figure BDA0001906953360000095
Is f2(H) An E auxiliary function of.
And (3) proving that: it is clear that,
E(H,H(t)) Not less than 0, and E (H)(t),H(t))＝0
Thus, E (H, H) is known(t)) Is still f2(H) An E auxiliary function of.
Theorem 4: defining a function G (H, H)(t)) And E (H, H)(t)) The following were used:
Figure BDA0001906953360000096
Figure BDA0001906953360000097
then
Figure BDA0001906953360000098
Is f2(H) An auxiliary function of (2).
And (3) proving that: from theorems 2 and 3, it can be seen that: function G (H, H)(t)) Is a function f2(H) An auxiliary function of, and a function E (H, H)(t)) Is a function f2(H) An E auxiliary function of. With theorem 1, we can directly deduce that λ is 1
Figure BDA0001906953360000101
Is still f2(H) An auxiliary function of (2).
From definition 1 and lemma 1, we know the function
Figure BDA0001906953360000102
Is a function f2(H) An upper limit of (2), and
Figure BDA0001906953360000103
to obtain
Figure BDA0001906953360000104
We solve its derivative and make it 0, have
Figure BDA0001906953360000105
From this equation, the updated iterative equation for H can be solved:
Figure BDA0001906953360000106
through simple derivation, the iterative formula can be proved to meet the KKT condition.
To sum up, the updating iterative formula of FSNMF proposed by this patent is:
Figure BDA0001906953360000107
compared with the traditional Semi-NMF iterative formula (2), the updated iterative formula (6) provided by the patent not only greatly improves the convergence rate and the calculation rate, but also has a better recognition effect.
To sum up:
as shown in fig. 2, the present invention provides a face recognition method based on E-helper function semi-nonnegative matrix factorization, which includes a training step, where the training step includes the following steps:
the first step is as follows: converting the training sample image into a training sample matrix X, and setting an error threshold epsilon and a maximum iteration number ImaxAnd inputting a training sample matrix X, an error threshold epsilon and a maximum iteration number Imax；
The second step is as follows: initializing a base image matrix W and a coefficient matrix H;
the third step: setting the iteration number n to be 0;
the fourth step: updating the base image matrix W and the coefficient matrix H according to formula (6);
the fifth step: n is n + 1;
a sixth step: judging whether the objective function F (W, H) is less than or equal to epsilon or the iteration number n reaches the maximum iteration number ImaxIf yes, outputting a base image matrix W and a coefficient matrix H, otherwise, executing a fourth step;
in the fourth step, equation (6) is as follows:
W←XHT(HHT)-1
Figure BDA0001906953360000111
in formula (6), W represents a base image matrix, H represents a coefficient matrix, X represents a training sample matrix, and T represents a transpose of the matrix.
The face recognition method further comprises a recognition step which is executed after the training step, wherein the recognition step comprises the following steps:
a seventh step of: calculating each class in the training sampleAverage feature vector m ofj(j ═ 1, …, C), j denotes the jth class, C denotes a total of C different face classes;
the eighth step is that the face image y to be recognized is input and the characteristic vector h of the face image y is calculatedy＝W-1y, W represent the base image matrix;
the ninth step is that the characteristic vector h of the face image to be recognized is calculatedyAverage feature vector m to each classjIf h isyAnd mjIf the distance is the minimum, the face image y to be recognized is classified into the P-th class;
a tenth step: and outputting the category P.
And outputting the type P, namely the face image y to be recognized is recognized as the No. P personal face type, so that after the type P is output, the face recognition is finished.
The invention also provides a face recognition system based on the semi-nonnegative matrix factorization of the E auxiliary function, which comprises a training module, wherein the training module comprises:
an input module: for converting training sample image into training sample matrix X, setting error threshold epsilon and maximum iteration number ImaxAnd inputting a training sample matrix X, an error threshold epsilon and a maximum iteration number Imax；
An initialization module: the method comprises the steps of initializing a base image matrix W and a coefficient matrix H;
an assignment module: setting the iteration number n to be 0;
an update module: for updating the base image matrix W and the coefficient matrix H according to equation (6); a counting module: n is n + 1;
a judging module: judging whether the objective function F (W, H) is less than or equal to epsilon or the iteration number n reaches the maximum iteration number ImaxIf yes, outputting a base image matrix W and a coefficient matrix H, otherwise, executing an updating module;
in the update module, equation (6) is as follows:
W←XHT(HHT)-1
Figure BDA0001906953360000121
in formula (6), W represents a base image matrix, H represents a coefficient matrix, X represents a training sample matrix, and T represents a transpose of the matrix.
The face recognition system further comprises a recognition module executed after the training module, the recognition module comprising:
an average feature vector calculation module: for calculating an average feature vector m for each class in a training samplej(j ═ 1, …, C), j denotes the jth class, C denotes a total of C different face classes;
the characteristic vector calculation module is used for inputting the face image y to be recognized and calculating the characteristic vector h thereofy＝W-1y, W represent the base image matrix;
distance calculating module for calculating characteristic vector h of face image to be recognizedyAverage feature vector m to each classjIf h isyAnd mjIf the distance is the minimum, the face image y to be recognized is classified into the P-th class;
an output module: for outputting the category P.
The invention also discloses a computer-readable storage medium, in which a computer program is stored which, when being invoked by a processor, is configured to carry out the steps of the method as claimed in the claims.
Table 1 shows the recognition rate (%) of the Method proposed in this patent (Our Method) in ORL face database compared with Convex non-negative matrix factorization (Convex-NMF) and Semi-non-negative matrix factorization (Semi-NMF) (TN indicates the number of training samples per class)
TN 3 4 5 6 7 8 9
Convex-NMF 72.21 76.79 83.70 85.75 87.17 88.50 88.50
Semi-NMF 75.93 80.92 85.85 86.00 89.00 90.13 89.00
Our Method 78.29 82.92 88.05 88.56 91.83 91.63 91.75
Table 2 shows the comparison of the recognition speed(s) of the Method proposed by this patent (Our Method) with the Convex non-negative matrix factorization (Convex-NMF) and the Semi-non-negative matrix factorization (Semi-NMF) on the ORL face database (TN indicates the number of training samples per class)
TN 3 4 5 6 7 8 9
Convex-NMF 0.49 0.76 1.18 1.53 1.70 2.39 2.72
Semi-NMF 0.53 0.63 0.95 1.08 1.09 1.24 1.38
Our Method 0.46 0.56 0.88 0.89 0.90 1.09 1.21
The invention has the beneficial effects that:
1. the invention allows the input matrix to contain negative numbers, and expands the application range of NMF.
2. The patent proposes a new concept of the E-helper function, from which a basic theoretical framework and method of constructing a new helper function is presented.
3. The face recognition method has the advantages of high recognition performance and low calculation complexity.
4. An iterative formula with better recognition effect and faster convergence rate is obtained by constructing a new E auxiliary function.
5. The convergence of the algorithm provided by the patent is not only proved theoretically by using the auxiliary function, but also verified in experiments, and the algorithm has higher convergence.
6. The results of experiments and comparisons of the disclosed face database and related algorithms show that the method developed by the patent has certain superiority.
The foregoing is a more detailed description of the invention in connection with specific preferred embodiments and it is not intended that the invention be limited to these specific details. For those skilled in the art to which the invention pertains, several simple deductions or substitutions can be made without departing from the spirit of the invention, and all shall be considered as belonging to the protection scope of the invention.

Claims (3)

1. A face recognition method based on E auxiliary function semi-nonnegative matrix decomposition is characterized by comprising a training step, wherein the training step comprises the following steps:
the first step is as follows: converting the training sample image into a training sample matrix X, and setting an error threshold epsilon and a maximum iteration number ImaxAnd inputting a training sample matrix X, an error threshold epsilon and a maximum iteration number Imax；
The second step is as follows: initializing a base image matrix W and a coefficient matrix H;
the third step: setting the iteration number n to be 0;
the fourth step: updating the base image matrix W and the coefficient matrix H according to formula (6);
the fifth step: n is n + 1;
a sixth step: judging whether the objective function F (W, H) is less than or equal to epsilon or the iteration number n reaches the maximum iteration number ImaxIf yes, outputting a base image matrix W and a coefficient matrix H, otherwise, executing a fourth step; the face recognition method further comprises a recognition step which is executed after the training step, wherein the recognition step comprises the following steps:
a seventh step of: calculating the average characteristic vector m of each class in the training samplej(j ═ 1, …, c), j denotes the jth class, c denotes a total of c different face classes;
the eighth step is that the face image y to be recognized is input and the characteristic vector h of the face image y is calculatedy＝W-1y, W represent the base image matrix;
the ninth step is that the characteristic vector h of the face image to be recognized is calculatedyAverage feature vector m to each classjIf h isyAnd mjIf the distance is the minimum, the face image y to be recognized is classified into the P-th class;
a tenth step: outputting the class P, thereby completing face recognition;
in the fourth step, equation (6) is as follows:
W←XHT(HHT)-1
Figure FDA0003389904160000011
in formula (6), W represents a base image matrix, H represents a coefficient matrix, X represents a training sample matrix, and T represents a transpose of the matrix;
the definition and theorem of the E auxiliary function is as follows:
defining: for arbitrary matrices H and H(t)If E (H, H)(t)) Satisfies the conditions
E(H,H(t)) Not less than 0, and E (H)(t),H(t))＝0
Then called E (H, H)(t)) An E helper function that is a function f (H);
theorem: if the function G (H, H)(t)) Is an auxiliary function of the function f (H), if the function E (H, H)(t)) Is an E helper function for function f (H) then:
Figure FDA0003389904160000021
is still an auxiliary function of (f) (H), where λ ≧ 0;
the iterative formula derivation process for updating the coefficient matrix H is as follows:
sub-targeting function f2(H)＝||X-WH||2Wherein W is fixed;
function(s)
Figure FDA0003389904160000022
Is f2(H) An auxiliary function of
Function(s)
Figure FDA0003389904160000023
E(H,H(t)) Not less than 0, and E (H)(t),H(t))＝0
Thus, E (H, H) is known(t)) Is still f2(H) An E helper function of;
then
Figure FDA0003389904160000024
Is f2(H) An auxiliary function of, function G (H, H)(t)) Is a function f2(H) An auxiliary function of, and a function E (H, H)(t)) Is a function f2(H) Taking λ as 1, an auxiliary function of (1) can be directly derived
Figure FDA0003389904160000025
Is still f2(H) An auxiliary function of (2); to obtain
Figure FDA0003389904160000026
Solving its derivative to 0, an updated iterative formula for H can be solved.
2. A face recognition system based on semi-non-negative matrix factorization of E-helper functions, comprising a training module comprising:
an input module: for converting training sample image into training sample matrix X, setting error threshold epsilon and maximum iteration number ImaxAnd inputting a training sample matrix X, an error threshold epsilon and a maximum iteration number Imax；
An initialization module: the method comprises the steps of initializing a base image matrix W and a coefficient matrix H;
an assignment module: setting the iteration number n to be 0;
an update module: for updating the base image matrix W and the coefficient matrix H according to equation (6);
a counting module: n is n + 1;
judgment ofA module: judging whether the objective function F (W, H) is less than or equal to epsilon or the iteration number n reaches the maximum iteration number ImaxIf yes, outputting a base image matrix W and a coefficient matrix H, otherwise, executing an updating module; the face recognition system further comprises a recognition module executed after the training module, the recognition module comprising:
an average feature vector calculation module: for calculating an average feature vector m for each class in a training samplej(j ═ 1, …, c), j denotes the jth class, c denotes a total of c different face classes;
the characteristic vector calculation module is used for inputting the face image y to be recognized and calculating the characteristic vector h thereofy＝W-1y, W represent the base image matrix;
distance calculating module for calculating characteristic vector h of face image to be recognizedyAverage feature vector m to each classjIf h isyAnd mjIf the distance is the minimum, the face image y to be recognized is classified into the P-th class;
an output module: for outputting the class P, thereby completing face recognition;
in the update module, equation (6) is as follows:
W←XHT(HHT)-1
Figure FDA0003389904160000041
in formula (6), W represents a base image matrix, H represents a coefficient matrix, X represents a training sample matrix, and T represents a transpose of the matrix;
the definition and theorem of the E auxiliary function is as follows:
defining: for arbitrary matrices H and H(t)If E (H, H)(t)) Satisfies the conditions
E(H,H(t)) Not less than 0, and E (H)(t),H(t))＝0
Then called E (H, H)(t)) An E helper function that is a function f (H);
theorem: if the function G (H, H)(t)) Is an auxiliary function of the function f (H)Number E (H, H)(t)) Is an E helper function for function f (H) then:
Figure FDA0003389904160000042
is still an auxiliary function of (f) (H), where λ ≧ 0;
the iterative formula derivation process for updating the coefficient matrix H is as follows:
sub-targeting function f2(H)＝||X-WH||2Wherein W is fixed;
function(s)
Figure FDA0003389904160000043
Is f2(H) An auxiliary function of (2);
function(s)
Figure FDA0003389904160000044
E(H,H(t)) Not less than 0, and E (H)(t),H(t))＝0
Thus, E (H, H) is known(t)) Is still f2(H) An E helper function of;
then
Figure FDA0003389904160000045
Is f2(H) An auxiliary function of, function G (H, H)(t)) Is a function f2(H) An auxiliary function of, and a function E (H, H)(t)) Is a function f2(H) Taking λ as 1, an auxiliary function of (1) can be directly derived
Figure FDA0003389904160000051
Is still f2(H) An auxiliary function of (2); to obtain
Figure FDA0003389904160000052
Solving its derivative to 0, an updated iterative formula for H can be solved.
3. A computer-readable storage medium, characterized in that the computer-readable storage medium stores a computer program configured to, when invoked by a processor, implement the steps of the method of claim 1.
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