CN109815889B - Cross-resolution face recognition method based on feature representation set - Google Patents
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Abstract
The invention discloses a cross-resolution face recognition method based on a feature representation set, which comprises the following steps: firstly, acquiring image blocks of each pixel position of a high-resolution training sample image, a low-resolution testing sample image and a high-resolution and low-resolution training dictionary image; then for each image block in the low-quality test image, obtaining linear representation of an image block set of the image block on a corresponding position in the low-quality training dictionary image by using a constrained P-norm regular regression method of the image block, and obtaining linear representation of the image block set of the high-resolution training sample image on the corresponding position of the high-resolution dictionary by using the same method; then carrying out similarity measurement on the low-resolution test image block feature representation set and the high-resolution training image feature representation set; and finally testing the image category. The invention has the advantages that: the face image with inconsistent resolution can be accurately identified, and the problem that the face image is difficult to identify due to inconsistent resolution is effectively solved.
Description
Technical Field
The invention relates to the technical field of image processing, in particular to a cross-resolution face recognition method based on a feature representation set.
Background
Human face recognition is a popular research topic in the field of computer vision, and it combines computer image processing technology and statistical technology, and is widely applied in various fields, such as: financial field, public security system, social security field, airport frontier inspection face recognition, and the like. The current face recognition methods can be divided into two categories: a global-based face recognition method and a local-based face recognition method. The global face recognition method retains a global face structure, but neglects face feature details except a principal component; the local face recognition method mostly adopts a face recognition method based on image blocks, and under the constraint of sparse conditions, the face image blocks to be recognized and the training sample image blocks are regarded as linear combinations of the training image blocks to obtain expression coefficient matrixes with the same dimension, so that the recognition work is completed. However, the image block-based face recognition method has a disadvantage: the modules are independent from each other, the associated information between the modules is lost, and because the quality of the face images obtained in most applications is poor and the resolution of the face images is inconsistent in the actual application process, the identity recognition is difficult to complete sometimes, so that the problem that the resolution of the face images is inconsistent and difficult to recognize exists.
Disclosure of Invention
The invention aims to provide a cross-resolution face recognition method based on a feature representation set, which can accurately identify the face images with inconsistent resolution.
In order to realize the purpose, the invention adopts the following technical scheme: a cross-resolution face recognition method based on a feature representation set comprises the following steps:
the method comprises the following steps: acquiring image blocks of each pixel position in a high-resolution training sample image, a low-resolution testing sample image and a high-resolution and low-resolution training dictionary image;
step two: for each image block in the low-resolution test sample image, a regression representation method based on P norm regularization is applied to obtain linear representation of an image block set at a corresponding position on the low-resolution training dictionary image; meanwhile, for each image block in the high-resolution training sample image, a regression representation method based on P norm regularization is applied to obtain linear representation of an image block set at a corresponding position on the high-resolution training dictionary image;
step three: performing similarity measurement on the low-resolution test sample image feature representation set and the high-resolution training sample image feature representation set obtained in the second step;
step four: and D, finishing classification of the low-resolution test sample images according to the data obtained in the step three, and outputting the classification of the low-resolution test sample images.
Further, the foregoing cross-resolution face recognition method based on a feature representation set, wherein: in the second step, a regression representation method based on P-norm regularization is applied to obtain a linear representation of the image block set at the corresponding position on the low-resolution training dictionary image, and the specific method is as follows:
step (1): for a low-resolution test image block, dividing the low-resolution test image block into S blocks, and for a low-resolution test image block y, performing linear representation by using image blocks at corresponding positions on a low-resolution training dictionary sample image:
y＝x 1 A 1 +x 2 A 2 +...+x N A N +E
wherein A is i An image block which represents a corresponding position of the ith low-resolution training dictionary image, i ═ 1,2,. and N, N represents the number of low-resolution training sample dictionary sample images, and x represents the number of the low-resolution training sample dictionary sample images i Representing a coefficient corresponding to the ith element of the coefficient vector x, and E represents a residual error item; the solution to represent the vector coefficients x is as follows:
defining slave space
To
A (x) x 1 A 1 +x 2 A 2 +...+x N A N 。
For the low-resolution test image block, a linear representation of the low-resolution test image block is obtained on a low-resolution training sample dictionary image by using a P-norm regular regression method, and an objective function is given as follows:
s.t.y-A(x)＝E
wherein,
represents the Schatten-P norm of the matrix,
σ i is the ith singular value of E; λ is a regularization parameter, D ═ D 1 ,D 2 ,...,D N ) Representing a Euclidean distance matrix between the low-resolution test image block and the low-resolution training sample dictionary image block; p is between (0,1), P is chosen 1/2, Schatten-1/2 norm closer to the rank function; the above model is represented as:
s.t.y-A(x)＝E
its lagrange function is expressed as:
where μ > 0 is a penalty parameter, Z is the lagrange multiplier, tr (-) is the trace operation, the above equation can be written as:
the model is solved by adopting an alternative direction multiplier method ADMM, and the specific process is as follows:
< a > fix E, update x:
the solution of the above formula is:
x k+1 ＝(G+τD 2 )\ones(N,1)
wherein ones (M,1) is an elementAn M × 1-dimensional vector with a 1 element, "\" denotes a left matrix division operation, τ ═ 2 λ μ, and G is a covariance matrix G ═ C T C：
Wherein, H ═ vec (A) 1 ,Vec(A 2 ),...,Vec(A N )]Vec (·) denotes the vectorization operator.
< b > fix x, update E:
to solve the above equation, a single-valued function threshold theorem based on the Schatten-1/2 norm is introduced:
theorem: given a constant of eta > 0 and a matrix of rank r
The objective function becomes:
According to the above theorem, the objective function becomes:
<c>selecting a suitable termination parameter epsilon 1 And ε 2 The following termination conditions are satisfied:
||y-A(x k+1 )-E||/||y||＜ε 1
max(||x k+1 -x k ||,||E k+1 -E k ||)/||y||＜ε 2
wherein, | | | | is a given matrix norm, and if the maximum iteration number is reached or the termination condition is met, x is output k+1 As x, otherwise, returning to step<a>；
Step (2): obtaining the expression coefficient vector of each low-resolution image block, and integrating the expression coefficient vectors into a coefficient matrix X with dimension of NxSxM; and M is the number of samples in the test set, S is the number of low-resolution test image blocks, and N represents the number of low-resolution training sample dictionary sample images.
Further, the foregoing cross-resolution face recognition method based on a feature representation set, wherein: in the second step, a regression representation method based on P-norm regularization is used to obtain a linear representation of the image block set at the corresponding position on the high-resolution training dictionary image, and the specific method is as follows:
step (1): for a high resolution training image block, divide it into S blocks, for a high resolution training image block y 1 And performing linear representation by using image blocks at corresponding positions on the high-resolution training dictionary sample image:
y 1 ＝c 1 G 1 +c 2 G 2 +...+c N G N +E 2
wherein G is i Representing the image block at the corresponding position of the ith low-resolution training dictionary image, i ═ 1, 2.. multidot.N }, N represents the number of high-resolution training sample dictionary sample images, c i Representing coefficient vector c ithCoefficient of element correspondence, E 2 Representing a residual term; the solution to represent the vector coefficients c is as follows:
defining slave space
To
A linear mapping of g (c) ═ c 1 G 1 +c 2 G 2 +...+c N G N ；
For the low-resolution test image block, a linear representation of the low-resolution test image block is obtained on a low-resolution training sample dictionary image by using a P-norm regular regression method, and an objective function is given as follows:
s.t.y 1 -G(c)＝E 2
wherein,
representing the Schatten-P norm of the matrix,
σ i is the ith singular value of E; λ is the regularization parameter, D ═ D 1 ,D 2 ,...,D N ) Representing a Euclidean distance matrix between the low-resolution test image block and the low-resolution training sample dictionary image block; p is between (0,1), P is chosen 1/2, Schatten-1/2 norm closer to the rank function; the above model is represented as:
s.t.y 1 -G(c)＝E 2
its lagrange function is expressed as:
where μ > 0 is a penalty parameter, Z is the lagrange multiplier, tr (-) is the trace operation, and the above equation can be written as:
the model is solved by adopting an alternating direction multiplier method ADMM, and the specific process is as follows:
<a>fixing E 2 And updating c:
the solution of the above formula is:
c k+1 ＝(O+τD 2 )\ones(N,1)
where ons (M,1) is an M × 1-dimensional vector with an element of 1, "\" denotes a left matrix division operation, τ ═ 2 λ μ, and O is a covariance matrix O ═ C T C：
Wherein H ═ vec (G) 1 ,Vec(G 2 ),...,Vec(G N )]Vec (·) denotes the vectorization operator;
<b>fix c, update E 2 ：
To solve the above equation, a single-valued function threshold theorem based on the Schatten-1/2 norm is introduced:
theorem: given a constant of eta > 0 and a matrix of rank r
The objective function becomes:
According to the above theorem, the objective function becomes:
<c>selecting a suitable termination parameter ε 1 And ε 2 The following termination conditions are satisfied:
||y 1 -G(c k+1 )-E 2 ||/||y 1 ||＜ε 1
max(||c k+1 -c k ||,||E 2 k+1 -E 2 k |)/||y 1 ||＜ε 2
wherein, | | | | is a given matrix norm, and if the maximum iteration number is reached or the termination condition is met, c is output k+1 As c, otherwise, returning to step<a>；
Step (2): obtaining the representative coefficient vector of each high-resolution training image block, and applying themIntegrating to obtain a matrix of expression coefficients C ═ C 1 ,C 2 ,...,C K ) Dimension is nxsxl; wherein L represents the total number of high-resolution face sample images, C i And representing a coefficient matrix of the ith high-resolution training sample face image set on the high-resolution training dictionary, wherein S is the number of high-resolution training image blocks, and N is the number of high-resolution training sample dictionary sample images.
Further, the foregoing cross-resolution face recognition method based on a feature representation set, wherein: in step three, the specific method for performing similarity measurement on the low-resolution test sample image feature representation set and the high-resolution training sample image feature representation set is as follows:
step (31): establishing a similarity measurement model as follows:
s.t.∑α i ＝1
wherein α and β represent coefficient vectors, and X (α) ═ α 1 X 1 +α 2 X 2 +...+α M X M ，C(β)＝β 1 C 1 +β 2 C 2 +...+β L C L ，∑α i Avoiding an explicit solution (α ═ 0), the above equation can be written as:
s.t.E 1 ＝X(α)-C(β)
the lagrangian form of the above equation is:
step (32): fix α, β, update E 1 ：
The optimal solution of the above formula can be obtained through a singular value threshold operator, and a matrix with the rank r is given
Its singular value decomposition is:
wherein σ 1 ,...σ r Representing singular values, U and V being orthogonal matrices; for a given one τ>0, singular value operator T τ (. cndot.) is defined as:
the solution to the above problem is:
step (33): fixing E 1 Updating α, β:
the lagrangian equation above becomes:
where e is a row vector with elements all being 1, Vec (X) ═ Vec (X) 1 )Vec(X 2 )...Vec(X M )]， Vec(-C)＝[Vec(-C 1 )Vec(-C 2 )...Vec(-C M )](ii) a Is provided with
The above equation becomes:
from the above equation can be calculated:
wherein h is 0 ＝(J T J+K) -1 ；
Step (34): selecting a suitable termination parameter ε 3 The following termination conditions are satisfied:
||X(α k )-C(β k )-E 1 ||/||X(α k )||＜ε 3
wherein, | | | | is a given matrix norm, and if the maximum iteration number is reached or the termination condition is met, alpha is output k+1 As α, β k+1 As beta, otherwise, returning to step (32).
Further, the foregoing cross-resolution face recognition method based on a feature representation set, wherein: in step four, each group C was used i The residual error of X alpha is expressed to determine the class label of X, and the proposed classifier is as follows:
identity(X)＝argmin i (r i )
test sample X and a training sample C i The distance between is expressed as
Wherein,
and
represents the optimal coefficient vector, | · | calculation * Represents the nuclear norm, and identity (X) represents the class label of the test sample X.
Through the implementation of the technical scheme, the invention has the beneficial effects that: the method can accurately identify the face images with inconsistent resolution, and effectively solves the problem that the face images are difficult to identify due to inconsistent resolution.
Drawings
Fig. 1 is a schematic flow chart of a cross-resolution face recognition method based on a feature representation set according to the present invention.
Detailed Description
The invention is described in further detail below with reference to the figures and specific examples.
As shown in fig. 1, the cross-resolution face recognition method based on the feature representation set includes the following steps:
the method comprises the following steps: acquiring image blocks of each pixel position in a high-resolution training sample image, a low-resolution testing sample image and a high-resolution and low-resolution training dictionary image;
step two: for each image block in the low-resolution test sample image, linear representation of an image block set at a corresponding position on the low-resolution training dictionary image is obtained by applying a regression representation method based on P-norm regularization; meanwhile, for each image block in the high-resolution training sample image, a regression representation method based on P norm regularization is applied to obtain linear representation of an image block set at a corresponding position on the high-resolution training dictionary image;
the specific method for obtaining the linear representation of the image block set at the corresponding position on the low-resolution training dictionary image by applying the regression representation method based on the P norm regulation is as follows:
step (1): for a low-resolution test image block, dividing the low-resolution test image block into S blocks, and for a low-resolution test image block y, performing linear representation by using image blocks at corresponding positions on a low-resolution training dictionary sample image:
y＝x 1 A 1 +x 2 A 2 +...+x N A N +E
wherein A is i An image block which represents a corresponding position of the ith low-resolution training dictionary image, i ═ 1,2,. and N, N represents the number of low-resolution training sample dictionary sample images, and x represents the number of the low-resolution training sample dictionary sample images i Representing a coefficient corresponding to the ith element of the coefficient vector x, and E represents a residual error item; the solution to represent the vector coefficients x is as follows:
defining slave space
To
A (x) x 1 A 1 +x 2 A 2 +...+x N A N 。
For the low-resolution test image block, a P-norm regular regression method is used for obtaining linear representation of the low-resolution test image block on a low-resolution training sample dictionary image, and an objective function is given as follows:
s.t.y-A(x)＝E
wherein,
represents the Schatten-P norm of the matrix,
σ i is the ith singular value of E; λ is the regularization parameter, D ═ D 1 ,D 2 ,...,D N ) Representing a Euclidean distance matrix between a low-resolution test image block and a low-resolution training sample dictionary image block; p is between (0,1), P is chosen 1/2, Schatten-1/2 norm is closer to rankA function; the above model is represented as:
s.t.y-A(x)＝E
its lagrange function is expressed as:
where μ > 0 is a penalty parameter, Z is the lagrange multiplier, tr (-) is the trace operation, the above equation can be written as:
the model is solved by adopting an alternative direction multiplier method ADMM, and the specific process is as follows:
< a > fix E, update x:
the solution of the above formula is:
x k+1 ＝(G+τD 2 )\ones(N,1)
where ons (M,1) is an M × 1-dimensional vector with an element of 1, "\" denotes a left matrix division operation, τ ═ 2 λ μ, and G is a covariance matrix G ═ C T C：
Wherein, H ═ vec (A) 1 ,Vec(A 2 ),...,Vec(A N )]Vec (·) denotes the vectorization operator.
< b > fix x, update E:
to solve the above equation, a single-valued function threshold theorem based on the Schatten-1/2 norm is introduced:
the theorem is as follows: given a constant of eta > 0 and a matrix of rank r
The objective function becomes:
According to the above theorem, the objective function becomes:
<c>selecting a suitable termination parameter epsilon 1 And ε 2 The following termination conditions are satisfied:
||y-A(x k+1 )-E||/||y||＜ε 1
max(||x k+1 -x k ||,||E k+1 -E k ||)/||y||＜ε 2
wherein, | | | | is a given matrix norm, and if the maximum iteration number is reached or the termination condition is met, x is output k+1 As x, otherwise, returning to step<a>；
Step (2): obtaining the expression coefficient vector of each low-resolution image block, and integrating the expression coefficient vectors into a coefficient matrix X with dimension of NxSxM; wherein M is the number of samples in the test set, S is the number of low-resolution test image blocks, and N represents the number of low-resolution training sample dictionary sample images;
the specific method for obtaining the linear representation of the image block set at the corresponding position on the high-resolution training dictionary image by applying the regression representation method based on the P norm regulation is as follows:
step (1): for a high resolution training image block, divide it into S blocks, for a high resolution training image block y 1 And performing linear representation by using image blocks at corresponding positions on the high-resolution training dictionary sample image:
y 1 ＝c 1 G 1 +c 2 G 2 +...+c N G N +E 2
wherein G is i Representing the image block at the corresponding position of the ith low-resolution training dictionary image, i ═ 1, 2.. multidot.N }, N represents the number of high-resolution training sample dictionary sample images, c i Represents the coefficient corresponding to the ith element of the coefficient vector c, E 2 Representing a residual term; the solution to represent the vector coefficients c is as follows:
defining slave space
To
A linear mapping of g (c) ═ c 1 G 1 +c 2 G 2 +...+c N G N ；
For the low-resolution test image block, a linear representation of the low-resolution test image block is obtained on a low-resolution training sample dictionary image by using a P-norm regular regression method, and an objective function is given as follows:
s.t.y 1 -G(c)＝E 2
wherein,
represents the Schatten-P norm of the matrix,
σ i is the ith singular value of E; λ is the regularization parameter, D ═ D 1 ,D 2 ,...,D N ) Representing a Euclidean distance matrix between a low-resolution test image block and a low-resolution training sample dictionary image block; p is between (0,1), P is chosen 1/2, Schatten-1/2 norm closer to the rank function; the above model is represented as:
s.t.y 1 -G(c)＝E 2
its lagrange function is expressed as:
where μ > 0 is a penalty parameter, Z is the lagrange multiplier, tr (-) is the trace operation, and the above equation can be written as:
the model is solved by adopting an alternative direction multiplier method ADMM, and the specific process is as follows:
<a>fixing E 2 And updating c:
the solution of the above formula is:
c k+1 ＝(O+τD 2 )\ones(N,1)
where ons (M,1) is an M × 1-dimensional vector with an element of 1, "\" denotes a left matrix division operation, τ ═ 2 λ μ, and O is a covariance matrix O ═ C T C：
Wherein H ═ vec (G) 1 ,Vec(G 2 ),...,Vec(G N )]Vec (·) denotes the vectorization operator;
<b>fix c, update E 2 ：
To solve the above equation, a single-valued function threshold theorem based on the Schatten-1/2 norm is introduced:
theorem: given a constant of eta > 0 and a matrix of rank r
The objective function becomes:
According to the above theorem, the objective function becomes:
<c>selecting a suitable termination parameter epsilon 1 And epsilon 2 The following termination conditions are satisfied:
||y 1 -G(c k+1 )-E 2 ||/||y 1 ||＜ε 1
max(||c k+1 -c k ||,||E 2 k+1 -E 2 k ||)/||y 1 ||＜ε 2
wherein, | | | | is a given matrix norm, and if the maximum iteration number is reached or the termination condition is met, c is output k+1 As c, otherwise, returning to the step<a>；
Step (2): obtaining the expression coefficient vector of each high-resolution training image block, and integrating the expression coefficient vectors to obtain an expression coefficient matrix C ═ (C) 1 ,C 2 ,...,C K ) Dimension is nxsxl; wherein L represents the total number of high-resolution face sample images, C i Representing a coefficient matrix of an ith high-resolution training sample face image set on a high-resolution training dictionary, wherein S is the number of high-resolution training image blocks, and N is the number of high-resolution training sample dictionary sample images;
step three: performing similarity measurement on the low-resolution test sample image feature representation set and the high-resolution training sample image feature representation set obtained in the second step;
the specific method for carrying out similarity measurement on the low-resolution test sample image feature representation set and the high-resolution training sample image feature representation set is as follows:
step (31): establishing a similarity measurement model as follows:
s.t.∑α i ＝1
wherein α and β represent coefficient vectors, and X (α) is α 1 X 1 +α 2 X 2 +...+α M X M ，C(β)＝β 1 C 1 +β 2 C 2 +...+β L C L ，∑α i Avoiding an explicit solution (α ═ 0), the above equation can be written as:
s.t.E 1 ＝X(α)-C(β)
the lagrangian form of the above equation is:
step (32): fix α, β, update E 1 ：
The optimal solution of the above formula can be obtained through a singular value threshold operator, and a matrix with the rank r is given
Its singular value decomposition is:
wherein σ 1 ,...σ r Representing singular values, U and V being orthogonal matrices; for a given one τ>0, singular value operator T τ (. cndot.) is defined as:
the solution to the above problem is:
step (33): fixing E 1 Updating α, β:
the lagrangian equation above becomes:
where e is a row vector with elements all being 1, Vec (X) ═ Vec (X) 1 )Vec(X 2 )...Vec(X M )]， Vec(-C)＝[Vec(-C 1 )Vec(-C 2 )...Vec(-C M )](ii) a Is provided with
The above equation becomes:
from the above equation:
wherein h is 0 ＝(J T J+K) -1 ；
Step (34): selecting a suitable termination parameter ε 3 The following termination conditions are satisfied:
||X(α k )-C(β k )-E 1 ||/||X(α k )||＜ε 3
wherein, | | | | is a given matrix norm, and if the maximum iteration number is reached or the termination condition is met, alpha is output k+1 As α, β k+1 As beta, otherwise, returning to the step (32);
step four: classifying the low-resolution test sample images according to the data obtained in the step three, and outputting the classes of the low-resolution test sample images;
wherein each group C is used i The residual error of X alpha is expressed to determine the class label of X, and the proposed classifier is as follows:
identity(X)＝argmin i (r i )
test sample X and a training sample C i The distance between is expressed as
Wherein,
and
represents the optimal coefficient vector, | · | calculation * Represents the nuclear norm, and identity (X) represents the class label of the test sample X.
The invention has the advantages that: the face image with inconsistent resolution can be accurately identified, and the problem that the face image is difficult to identify due to inconsistent resolution is effectively solved.
Claims (3)
1. A cross-resolution face recognition method based on a feature representation set is characterized in that: the method comprises the following steps:
the method comprises the following steps: acquiring image blocks of each pixel position in a high-resolution training sample image, a low-resolution testing sample image and a high-resolution and low-resolution training dictionary image;
step two: for each image block in the low-resolution test sample image, a regression representation method based on P norm regularization is used for obtaining linear representation of an image block set at a corresponding position on the low-resolution training dictionary image; meanwhile, for each image block in the high-resolution training sample image, a regression representation method based on P norm regularization is used for obtaining linear representation of an image block set at a corresponding position on the high-resolution training dictionary image;
the specific method for obtaining the linear representation of the image block set at the corresponding position on the low-resolution training dictionary image by applying the regression representation method based on the P norm regulation is as follows:
step (1): for a low-resolution test image block, dividing the low-resolution test image block into S blocks, and for a low-resolution test image block y, performing linear representation by using image blocks at corresponding positions on a low-resolution training dictionary sample image:
y＝x 1 A 1 +x 2 A 2 +...+x N A N +E
wherein A is i An image block which represents a corresponding position of the ith low-resolution training dictionary image, i ═ 1,2,. and N, N represents the number of low-resolution training sample dictionary sample images, and x represents the number of the low-resolution training sample dictionary sample images i Representing a coefficient corresponding to the ith element of the coefficient vector x, and E represents a residual error item; the solution to represent the vector coefficients x is as follows:
defining slave space
To
A linear mapping ofRay A (x) x 1 A 1 +x 2 A 2 +...+x N A N ，
For the low-resolution test image block, a linear representation of the low-resolution test image block is obtained on a low-resolution training sample dictionary image by using a P-norm regular regression method, and an objective function is given as follows:
s.t.y-A(x)＝E
wherein,
representing the Schatten-P norm of the matrix,
σ i is the ith singular value of E; λ is a regularization parameter, D ═ D 1 ,D 2 ,...,D N ) Representing a Euclidean distance matrix between the low-resolution test image block and the low-resolution training sample dictionary image block; p is between (0,1), P is chosen 1/2, Schatten-1/2 norm closer to the rank function; the model is represented as:
s.t.y-A(x)＝E
its lagrange function is expressed as:
where μ > 0 is a penalty parameter, Z is the lagrange multiplier, tr (-) is the trace operation, and the above equation can be written as:
the model is solved by adopting an alternative direction multiplier method ADMM, and the specific process is as follows:
< a > fix E, update x:
the solution of the above formula is:
x k+1 ＝(G+τD 2 )\ones(N,1)
where ones (M,1) is an M × 1-dimensional vector with an element of 1, "\" indicates a left matrix division operation, τ ═ 2 λ/μ, and G is a covariance matrix G ═ C T C：
Wherein, H ═ vec (A) 1 ) ,Vec(A 2 ),...,Vec(A N )]Vec (·) denotes the vectorization operator;
< b > fix x, update E:
to solve the above equation, a single-valued function threshold theorem based on the Schatten-1/2 norm is introduced:
theorem: given a constant of eta > 0 and a matrix of rank r
The objective function becomes:
According to the above theorem, the objective function becomes:
<c>selecting a termination parameter ε 1 And ε 2 The following termination conditions are satisfied:
||y-A(x k+1 )-E||/||y||＜ε 1
max(||x k+1 -x k ||,||E k+1 -E k ||)/||y||＜ε 2
wherein, | | | | is a given matrix norm, if the maximum iteration times is reached or the termination condition is met, x is output k+1 As x, otherwise, returning to step<a>；
Step (2): obtaining the expression coefficient vector of each low-resolution image block, and integrating the expression coefficient vectors into a coefficient matrix X with the dimension of NxSxM; wherein M is the number of samples in the test set, S is the number of low-resolution test image blocks, and N represents the number of low-resolution training sample dictionary sample images;
the specific method for obtaining the linear representation of the image block set at the corresponding position on the high-resolution training dictionary image by applying the regression representation method based on the P norm regulation is as follows:
step (1): for a high resolution training image block, divide it into S blocks, for a high resolution training image block y 1 And performing linear representation by using image blocks at corresponding positions on the high-resolution training dictionary sample image:
y 1 ＝c 1 G 1 +c 2 G 2 +...+c N G N +E 2
wherein G is i Representing the image block at the corresponding position of the ith low-resolution training dictionary image, i ═ 1, 2.. multidot.N }, N represents the number of high-resolution training sample dictionary sample images, c i Represents the coefficient corresponding to the ith element of the coefficient vector c, E 2 Representing a residual term; the solution to represent the vector coefficients c is as follows:
defining slave space
To
A linear mapping of g (c) ═ c 1 G 1 +c 2 G 2 +...+c N G N ；
For the low-resolution test image block, a linear representation of the low-resolution test image block is obtained on a low-resolution training sample dictionary image by using a P-norm regular regression method, and an objective function is given as follows:
s.t.y 1 -G(c)＝E 2
wherein,
represents the Schatten-P norm of the matrix,
σ i is the ith singularity of EA value; λ is a regularization parameter, D ═ D 1 ,D 2 ,...,D N ) Representing a Euclidean distance matrix between the low-resolution test image block and the low-resolution training sample dictionary image block; p is between (0,1), P is chosen to be 1/2, Schatten-1/2 norm is closer to the rank function; the model is represented as:
s.t.y 1 -G(c)＝E 2
its lagrange function is expressed as:
where μ > 0 is a penalty parameter, Z is the lagrange multiplier, tr (-) is the trace operation, and the above equation can be written as:
solving the model by adopting an alternative direction multiplier method ADMM, which comprises the following specific steps:
<a>fixing E 2 And updating c:
the solution of the above formula is:
c k+1 ＝(O+τD 2 )\ones(N,1)
where ons (M,1) is an M × 1-dimensional vector with an element of 1, "\" denotes a left matrix division operation, τ ═ 2 λ/μ, and O is a covariance matrix O ═ C T C：
Wherein H ═ vec (G) 1 ) ,Vec(G 2 ),...,Vec(G N )]Vec (·) represents the vectorization operator;
<b>fix c, update E 2 ：
To solve the above equation, a single-valued function threshold theorem based on the Schatten-1/2 norm is introduced:
theorem: given a constant of eta > 0 and a matrix of rank r
The objective function becomes:
According to the above theorem, the objective function becomes:
<c>selecting a termination parameter ε 1 And ε 2 The following termination conditions are satisfied:
||y 1 -G(c k+1 )-E 2 ||/||y 1 ||＜ε 1
max(||c k+1 -c k ||,||E 2 k+1 -E 2 k ||)/||y 1 ||＜ε 2
wherein, | | | | is a given matrix norm, and if the maximum iteration number is reached or a termination condition is met, c is output k+1 As c, otherwise, returning to step<a>；
Step (2): obtaining a vector of the representation coefficients of each high-resolution training image block, and integrating the vectors to obtain a matrix of the representation coefficients C ═ C 1 ,C 2 ,...,C K ) Dimension is nxsxl; wherein L represents the total number of high-resolution face sample images, C i Representing a coefficient matrix of an ith high-resolution training sample face image set on a high-resolution training dictionary, wherein S is the number of high-resolution training image blocks, and N is the number of high-resolution training sample dictionary sample images;
step three: performing similarity measurement on the low-resolution test sample image feature representation set and the high-resolution training sample image feature representation set obtained in the second step;
step four: and D, finishing classification of the low-resolution test sample images according to the data obtained in the step three, and outputting the classification of the low-resolution test sample images.
2. The cross-resolution face recognition method based on the feature representation set according to claim 1, characterized in that: in step three, the specific method for performing similarity measurement on the low-resolution test sample image feature representation set and the high-resolution training sample image feature representation set is as follows:
step (31): establishing a similarity measurement model as follows:
s.t.∑α i ＝1
wherein α and β represent coefficient vectors, and X (α) ═ α 1 X 1 +α 2 X 2 +...+α M X M ，C(β)＝β 1 C 1 +β 2 C 2 +...+β L C L ，∑α i Avoiding an explicit solution (α ═ 0), the above equation can be written as:
s.t.E 1 ＝X(α)-C(β)
the lagrange form of the above equation is:
step (32): fix α, β, update E 1 ：
The optimal solution of the above formula can be obtained through a singular value threshold operator, and a matrix with the rank r is given
Its singular value decomposition is:
wherein sigma 1 ,...σ r Representing singular values, U and V being orthogonal matrices; for theGiven a τ>0, singular value operator T τ (. cndot.) is defined as:
the solution to the problem is:
step (33): fixing E 1 Updating α, β:
the lagrangian equation becomes:
where e is a row vector with elements all being 1, Vec (X) ═ Vec (X) 1 )Vec(X 2 )...Vec(X M )]，Vec(-C)＝[Vec(-C 1 )Vec(-C 2 )...Vec(-C M )](ii) a Is provided with
J＝[Vec(X)Vec(-C)],
d＝[e 0] T ,
The above equation becomes:
it can be calculated from the equation:
wherein h is 0 ＝(J T J+K) -1 ；
Step (34): selecting a termination parameter epsilon 3 The following termination conditions are satisfied:
||X(α k )-C(β k )-E 1 ||/||X(α k )||＜ε 3
wherein, | | | | is a given matrix norm, and if the maximum iteration number is reached or a termination condition is met, alpha is output k+1 As α, β k+1 As beta, otherwise, returning to step (32).
3. The method of claim 2, wherein the method comprises the following steps: in step four, each group C was used i The residual error of X alpha is expressed to determine the class label of X, and the proposed classifier is as follows:
identity(X)＝argmin i (r i )
test sample X and a training sample C i The distance between is expressed as
Wherein,
and
represents the optimal coefficient vector, | · | calculation * Represents the nuclear norm, and identity (X) represents the class label of the test sample X.
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