CN108876837A

CN108876837A - One kind being based on L1/2The 3 D human body attitude reconstruction method of regularization

Info

Publication number: CN108876837A
Application number: CN201810352361.0A
Authority: CN
Inventors: 洪金华; 郭立君; 张�荣
Original assignee: Ningbo University
Current assignee: Ningbo University
Priority date: 2018-04-19
Filing date: 2018-04-19
Publication date: 2018-11-23
Anticipated expiration: 2038-04-19
Also published as: CN108876837B

Abstract

The invention discloses one kind to be based on L_1/2The three-dimensional coordinate of all nodes in several 3 D human body figures is constituted database by the 3 D human body attitude reconstruction method of regularization, carries out dictionary learning to database using matrix decomposition and the on-line study method of sparse coding and complete dictionary is obtained；Then complete dictionary building shape space model was utilized；Followed by the property and L of spectral norm_1/2The characteristics of regularization, carries out convex relaxation processes to non-convex optimization problem, converts convex programming problem for non-convex optimization problem；Augmentation Lagrange, which is converted, by convex programming problem later solves expression formula；Expression formula is solved to augmentation Lagrange using ADMM algorithm again and is iterated solution；3 D human body posture is finally gone out using shape space model and 3D shape-variable model reconstruction according to solution value；Advantage is that quality reconstruction is good, and degree of rarefication is good.

Description

One kind being based on L1/2The 3 D human body attitude reconstruction method of regularization

Technical field

The present invention relates to a kind of Three Dimensional Reconfigurations, are based on L more particularly, to one kind_1/2The 3 D human body posture of regularization Reconstructing method.

Background technique

It is one of challenging problem in computer vision from the 3D shape of object is reconstructed in two dimensional image.It is close Nian Lai, the research emphasis of researcher turn to further analysis from the 3D shape using the edge box estimation object in image The three-dimensional geometric information (such as shape, posture) of object.Reasoning based on three-dimensional geometric information can not only appoint for high-rise vision Business such as scene understanding, enhancing display and human-computer interaction provide richer information, and can effectively improve Object identifying The purpose of performance.

The 3D shape of reconstruction of two-dimensional images object based on single-view itself is an ill-conditioning problem.Recent years, more Carry out more scientific research personnel and carry out threedimensional model analysis using ever-increasing available online three-dimensional modeling data storehouse, and extracts Shape priors abundant out carry out estimating that the problem of the 3D shape of object is obtained based on two dimensional image herein on basis Extensive research.

The institutes such as to solve present in 3D shape estimation procedure changeability, non-rigid deformation in class and avoiding exhaustive enumeration Potential problem, by the enlightenment of active shape model, there are many work.Such as：Zia M Z, Stark M, Schiele B, et al.Detailed 3d representations for object recognition and modeling.IEEE Transactions on Pattern Analysis&Machine Intelligence, 2013,35 (11):2608-2623. (Qi Ya, Stark, Xi Le, for the detailed three dimensional representation of Object identifying and modeling, IEEE mode analysis converges with machine intelligence Periodical, 2013, o. 11th volume 35：It is 2608-2623.) by using a kind of three-dimensional variable shape model (3D deformable Shape model) indicate shape, in three-dimensional variable shape model, any one shape can be by one group of base shape shape Formula shows, meanwhile, base shape is represented by the set of one group of orderly mark point.In view of the thought of rarefaction representation is to use One group of excessively complete base linearly shows input signal, if regarding above-mentioned base shape as complete base, 3-d deformable Shape model is similar to a kind of rarefaction representation.Based on three-dimensional variable shape model, the 3D shape of object will be estimated from two dimensional image The problem of being considered as feature in two dimensional image (mark point) and three-dimensional variable shape Model Matching, i.e. 3D-to-2D shape blending is asked Topic.It is to form parameter (sparse coefficient) and viewpoint parameter (video camera external parameter) while to estimate in the fusion question essence The problem of meter.It, could be by the X-Y scheme of three-dimensional variable shape model and orderly mark point as only having in situation known to viewpoint As preferably being merged；Or in the case where only known three-dimensional variable shape model, viewpoint could be preferably estimated, because This, the Combined estimator of form parameter and viewpoint parameter is a non-convex optimization problem.In addition, being carried out just to camera rotation matrix Constraint is handed over, this will make problem become more sophisticated.Solution for above-mentioned challenge, it is previous generally to use alternative iteration method Realization form parameter replaces update with viewpoint parameter, however, the method not can guarantee the solution global optimum acquired, and finally asks The solution obtained is sensitive to initial value.

In order to solve the problems, such as that quality reconstruction depends on initial value, such as：Zhou X, Leonardos S, Hu X, et al.3Dshape estimation from 2D landmarks:A convex relaxation approach[C]// Computer Vision and Pattern Recognition.IEEE, 2015:4447-4455 (Zhou Xiaowei, Ang Naduo-this Flesh side ancestor, Hu Xiaoyan, from a kind of convex relaxation method of 2D characteristic point reconstruct 3D shape, computer vision and pattern-recognition, IEEE, 2015:The method that the 3D shape estimation of two dimensional character point is realized using the method for convex relaxation is proposed in 4447-4455).It examines Consider the problem of there may be noises in actual life, such as：Zhou X, Zhu M, Leonardos S, Daniilidis K.Sparse Representation for 3D Shape Estimation:A Convex Relaxation Approach.IEEE Transactions on Pattern Analysis&Machine Intelligence,2016.PP (99):1-1. (Zhou Xiaowei, Zhu Menglong, Ang Naduo-this flesh side ancestor, a kind of convex relaxation of the 3D shape estimation based on rarefaction representation Method, IEEE mode analysis and machine intelligence, 2016.PP (99):It is proposed in 1-1) using the base modeled to noise On plinth, it is further proposed that the convex relaxation method that the 3D shape based on rarefaction representation is estimated.Although the side that Zhou Xiaowei et al. is proposed Method successfully solves the problems such as to initial value sensitive issue and noise, however its essence is using L₁The convex pine of regularization Non-convex optimization problem is converted convex programming problem by relaxation method, but the disadvantage is that quality reconstruction and degree of rarefication are not ideal enough.Xu Z B, Zhang H, Wang Y, Change XY.L_1/2regularization[J].Science China Information Sciences, 2010.53 (6):1159-1169. (Xu Zongben, Zhang Hai, Wang Yao, Chang Xiangyu, Liang Yong, L_1/2Regularization, Chinese section It learns, information science, the 2010, the 6th the phase volume 53：It is proposed in 1159-1169) and uses L_1/2Regularization substitutes L₁Regularization, realization pair L₀The approximate solution of Regularization Problem, and Xu Zongben et al. is demonstrated(symbol " | | | |_1/2" be seek matrix or to Amount 1/2 norm) gradient component presence and be experimentally confirmed L_1/2The solution ratio L that regularization acquires₁The solution of regularization is diluter It dredges and more effective.Therefore, it is necessary to study one kind to be based on L_1/2The 3 D human body attitude reconstruction method of regularization.

Summary of the invention

Technical problem to be solved by the invention is to provide one kind to be based on L_1/2The 3 D human body attitude reconstruction side of regularization Method, quality reconstruction is good, and degree of rarefication is good.

The present invention solves technical solution used by above-mentioned technical problem：One kind being based on L_1/2The 3 D human body of regularization Attitude reconstruction method, it is characterised in that include the following steps：

Step 1：N width 3 D human body figure is chosen, has P node in every width 3 D human body figure；Then by all three-dimensionals P × N number of node three-dimensional coordinate in human body figure constitutes database；Wherein, N >=3P, P=15；

Step 2：Using the formula of matrix decomposition and the on-line study method of sparse coding

Dictionary learning is carried out to database, complete dictionary is obtained, is remembered For B,Meet condition：c_ij>=0 and | | B_i||_F≤1；Wherein, 1≤j≤N, The dimension of 1≤i≤K, B are 3P × K, and K indicates the number of the atom of the 3D shape in B, 3P≤K≤N, B=[B₁,...,B_K], Symbol " [] " is that vector indicates symbol, B₁,…,B_KThe corresponding atom for indicating the 1st 3D shape in B ..., k-th it is three-dimensional The atom of shape, B₁,…,B_KIt is the column vector that dimension is 3P × 1, B_iIndicate the atom of i-th of 3D shape in B, B_iFor Dimension is the column vector of 3P × 1, and min () is to be minimized function, and C indicates the sparse coefficient matrix in dictionary learning, the dimension of C Number is K × N, c_ijIndicate that the i-th row jth column element in C, the initial value of C are C₀, C₀=pinv (B₀) S_train, S_train Dimension be 3P × N, S_train=[S₁,…,S_N], S₁,…,S_NThe corresponding all sections indicated in the 1st width 3 D human body figure Column vector that the three-dimensional coordinate of point is constituted ..., the column that constitute of the three-dimensional coordinates of all nodes in N width 3 D human body figure to Amount, S₁,…,S_NIt is the column vector that dimension is 3P × 1, S_jIndicate that the three-dimensional of all nodes in jth width 3 D human body figure is sat Mark the column vector constituted, S_jIt is the column vector of 3P × 1, pinv (B for dimension₀) it is to B₀Carry out pseudoinverse, B₀For the initial value of B, B₀ For selected at random from S_train K column vector composition dimension be 3P × K matrix, symbol " | | | |_F" it is to seek matrix F norm sign, α are regularization parameter；

Step 3：Shape space model is constructed using D, shape space model is described asWherein, x= [x,...,x_K], the dimension 1 × K, x of x₁Indicate D₁Sparse coefficient, x_KIndicate D_KSparse coefficient, D₁And D_KDimension be 3 × P, S are the three-dimensional coordinate of three-dimensional human skeleton nodes, and dimension is 3 × P, D_iDimension be 3 × P, D_iIn the 1st column three Element is sequentially B_iIn the 1st to the 3rd element, D_iIn the 2nd column three elements be sequentially B_iIn the 4th to the 6th member Element, and so on, D_iIn P column three elements be sequentially B_iIn 3P-2 to the 3P element, x_iIndicate D_iIt is sparse Coefficient, R_iFor D_iOrthogonal spin matrix, R_iDimension be 3 × 3, R_iMeet condition：(R_i)^TR_i=I₁And det (R_i)=1, (R_i)^T For R_iTransposition, I₁3 × 3 gusts of unit square for being for dimension, det (R_i) indicate to R_iSeek determinant；

Step 4：The two dimensional character point extracted in three-dimensional variable shape model and the two-dimension human body image that need to be reconstructed is carried out The non-convex optimization problem of fusion is described as：Then using spectral norm property and L₁₂The characteristics of regularization, carries out convex relaxation processes to non-convex optimization problem, converts convex programming problem for non-convex optimization problem, retouch State for：Wherein,The matrix for being 2 × 3 for dimension,By dimension For 3 × 3 matrix R_midPreceding 2 row composition, R_midMeet condition：(R_mid)^TR_mid=I₁And det (R_mid)=1, (R_mid)^TFor R_mid Transposition, det (R_mid) indicate to R_midAsk determinant, x=[x₁,…,x_K], x₁Indicate D₁Sparse coefficient, x_KIndicate D_KIt is dilute Sparse coefficient, D₁And D_KDimension be 3 × P, W indicates the two-dimensional coordinate structure of all nodes in the two-dimension human body image that need to be reconstructed At matrix, the dimension of W is 2 × P, and λ is regularization parameter, symbol " | | | |₁" it is 1 norm sign for seeking matrix, WithIt is the matrix that dimension is 2 × 3,By R₁Preceding 2 Row composition, R₁For D₁Orthogonal spin matrix,By R_iPreceding 2 row composition,By R_KPreceding 2 row composition, R_KFor D_KOrthogonal rotation Torque battle array, symbol " | | | |₂" it is 2 norm signs for seeking matrix；

Step 5：Multivariable Solve problems are converted by convex programming problem, are described as：

Then increase by one in multivariable Solve problems Auxiliary variable converts the multivariable Solve problems with auxiliary variable for multivariable Solve problems, is described as：

Meet Z=M；Then the multivariable with auxiliary variable is solved Problem uses augmented vector approach, obtains augmentation Lagrange and solves expression formula, is described as：

Wherein, E table Show that dimension is the exceptional value analog matrix of 2 × P, the translation row vector that G representation dimension is 2 × 1, G^TFor the transposition of G, β is non-negative Real parameter, M=[M₁,…,M_K], D=[D₁,…,D_K]^T, [D₁,…,D_K]^TFor [D₁,…,D_K] transposition, Z be increased auxiliary Variable, μ indicate that augmentation Lagrange solves the parameter for being used to control iteration step length in expression formula, and Y is dual variable, L_μ(M, Z, E, G, Y) indicate that augmentation Lagrange solves function；

Step 6：Expression formula is solved to augmentation Lagrange using ADMM algorithm and is iterated solution, solution obtains every time The solution expression formula of each variable after iteration, the solution expression formula of variable M after the t times iteration is described as：

The solution expression formula of variable Z after the t times iteration is described as：

The solution expression formula of variable E after the t times iteration is described as：

The solution expression formula of variable G after the t times iteration is described as：

The solution expression formula of variable Y after the t times iteration is described as：Y^t=Y^t-1+μ (M^t-Z^t)；Then the solution expression formula of variable M after each iteration is handled using spectral norm proximal end gradient algorithm, is obtained every The final solution expression formula of variable M after secondary iteration, the final solution expression formula of variable M after the t times iteration is described as：

And The solution expression formula of variable Z, the solution expression formula of variable E, the solution expression formula of variable G after each iteration are unfolded respectively Processing, it is corresponding to obtain after each iteration the final solution expression formula, the final solution expression formula of variable E of variable Z, variable G most Expression formula is solved eventually, and the final solution expression formula of variable Z after the t times iteration is described as：Z^t=((W-E^t-1-(G^t-1)^T)D^T+μ M^t+Y^t-1)×(DD^T+μI₂)^-1, the final solution expression formula of variable E after the t times iteration is described as：E^t=S_β(W-Z^tD-(G^t-1 )^T), the final solution expression formula correspondence of variable G after the t times iteration is described as：G^t=g (W-Z^tD-E^t)；Finally judge iteration Whether number reaches maximum number of iterations t_maxOr iteration stopping conditionIt is whether true, if iteration Number has reached maximum number of iterations t_maxOr iteration stopping conditionIt sets up, then stops iteration, obtain To the solution value of variable M, then execute step 7；Otherwise, continue iteration；

Wherein, 1≤t≤t_max, t_maxIndicate maximum number of iterations, t_max>=1000, the initial value of variable M be dimension be 2 × The null matrix of 3K, Z^t-1Indicate the value of variable Z after the t-1 times iteration, the initial value of variable Z is equal to the initial value of variable M, E^t-1Table Show the value of variable E after the t-1 times iteration, the initial value of variable E is the null matrix that dimension is 2 × P, G^t-1Indicate the t-1 times iteration The value of variable G afterwards, the initial value of variable G are the matrix being averaged to W by column, Y^t-1Indicate variable Y after the t-1 times iteration Value, the initial value of variable Y is equal to the initial value of variable M,Corresponding variable after indicating the t times iteration M₁,…,M_i,…,M_KValue,Indicate variable M after the t-1 times iteration_iValue,WithIt is corresponding to indicate M_i ^t ^-1Unitary matrice, the unitary matrice on the right, feature value vector on the left side obtained after singular value decomposition,ForTurn It setting, diag () indicates diagonal matrix,Indicate vectorIn L₁Projection on norm unit ball, (G^t-1)^TFor G^t-1Transposition, D^TFor the transposition of D, I₂The unit matrix for indicating 3K × 3K, enables X=W-Z^tD-(G^t-1)^T, S_β(X) table Show and soft-threshold calculating is carried out to each element in X, the formula for carrying out soft-threshold calculating to the i-th ' row jth ' column element in X is sign(X_i'j')(X_i'j'-β)₊, i'=1,2,1≤j'≤P, sign (X_i'j') indicate to seek X_i'j'Symbol,Symbol " | | " it is the symbol that takes absolute value, g (W-Z^tD-E^t) expression pair W-Z^tD-E^tIt averages by row,Indicate the value of variable M after the t times iteration,Indicate the value of variable M after the t-1 times iteration；

Step 7：The solution value of the variable M obtained according to step 6, M=[M₁,…,M_i,…,M_K]、Obtain x_i And R_iRespective value, x_i=| | M_i||₂, R_i=[r_i ⁽¹⁾,r_i ⁽²⁾,r_i ⁽³⁾]^T,r_i ⁽³⁾=r_i ⁽¹⁾ ×r_i ⁽²⁾；Then basisAnd the x acquired_iAnd R_iRespective value reconstructs 3 D human body posture；Wherein, [r_i ⁽¹⁾,r_i ⁽²⁾,r_i ⁽³⁾]^TFor [r_i ⁽¹⁾,r_i ⁽²⁾,r_i ⁽³⁾] transposition, r_i ⁽¹⁾,r_i ⁽²⁾,r_i ⁽³⁾It is corresponding to indicate R_iIn the 1st row vector, 2nd row vector, the 3rd row vector, m_i ⁽¹⁾Indicate M_iIn the 1st row vector, m_i ⁽²⁾Indicate M_iIn the 2nd row vector.

One kind being based on L_1/2The 3 D human body attitude reconstruction method of regularization, it is characterised in that include the following steps：

Dictionary learning is carried out to database, complete dictionary is obtained, is remembered For B,Meet condition：c_ij>=0 and | | B_i||_F≤1；Wherein, 1≤j≤N, The dimension of 1≤i≤K, B are 3P × K, and K indicates the number of the atom of the 3D shape in B, 3P≤K<N, B=[B₁,...,B_K], Symbol " [] " is that vector indicates symbol, B₁,…,B_KThe corresponding atom for indicating the 1st 3D shape in B ..., k-th it is three-dimensional The atom of shape, B₁,…,B_KIt is the column vector that dimension is 3P × 1, B_iIndicate the atom of i-th of 3D shape in B, B_iFor Dimension is the column vector of 3P × 1, and min () is to be minimized function, and C indicates the sparse coefficient matrix in dictionary learning, the dimension of C Number is K × N, c_ijIndicate that the i-th row jth column element in C, the initial value of C are C₀, C₀=pinv (B₀) S_train, S_train Dimension be 3P × N, S_train=[S₁,…,S_N], S₁,…,S_NThe corresponding all sections indicated in the 1st width 3 D human body figure Column vector that the three-dimensional coordinate of point is constituted ..., the column that constitute of the three-dimensional coordinates of all nodes in N width 3 D human body figure to Amount, S₁,…,S_NIt is the column vector that dimension is 3P × 1, S_jIndicate that the three-dimensional of all nodes in jth width 3 D human body figure is sat Mark the column vector constituted, S_jIt is the column vector of 3P × 1, pinv (B for dimension₀) it is to B₀Carry out pseudoinverse, B₀For the initial value of B, B₀ For selected at random from S_train K column vector composition dimension be 3P × K matrix, symbol " | | | |_F" it is to seek matrix F norm sign, α are regularization parameter；

Step 3：Three-dimensional variable shape model is constructed using D, three-dimensional variable shape model is described as： Wherein, x=[x ..., x_K], the dimension 1 × K, x of x₁Indicate D₁Sparse coefficient, x_KIndicate D_KSparse coefficient, D₁And D_KDimension Number is 3 × P, and S is the three-dimensional coordinate of three-dimensional human skeleton nodes, and dimension is 3 × P, D_iDimension be 3 × P, D_iIn the 1st column Three elements be sequentially B_iIn the 1st to the 3rd element, D_iIn the 2nd column three elements be sequentially B_iIn the 4th extremely 6th element, and so on, D_iIn P column three elements be sequentially B_iIn 3P-2 to the 3P element, x_iIt indicates D_iSparse coefficient；

Shape space model is constructed using D, shape space model is described as：Wherein, x= [x,...,x_K], the dimension 1 × K, x of x₁Indicate D₁Sparse coefficient, x_KIndicate D_KSparse coefficient, D₁And D_KDimension be 3 × P, S are the three-dimensional coordinate of three-dimensional human skeleton nodes, and dimension is 3 × P, D_iDimension be 3 × P, D_iIn the 1st column three Element is sequentially B_iIn the 1st to the 3rd element, D_iIn the 2nd column three elements be sequentially B_iIn the 4th to the 6th member Element, and so on, D_iIn P column three elements be sequentially B_iIn 3P-2 to the 3P element, x_iIndicate D_iIt is sparse Coefficient, R_iFor D_iOrthogonal spin matrix, R_iDimension be 3 × 3, R_iMeet condition：(R_i)^TR_i=I₁And det (R_i)=1, (R_i)^T For R_iTransposition, I₁The unit matrix for being 3 × 3 for dimension, det (R_i) indicate to R_iSeek determinant；

Step 4：The two dimensional character point extracted in three-dimensional variable shape model and the two-dimension human body image that need to be reconstructed is carried out The non-convex optimization problem of fusion is described as：Then using spectral norm property and L₁₂The characteristics of regularization, carries out convex relaxation processes to non-convex optimization problem, converts convex programming problem for non-convex optimization problem, retouch State for：Wherein,The matrix for being 2 × 3 for dimension,By dimension For 3 × 3 matrix R_midPreceding 2 row composition, R_midMeet condition：(R_mid)^TR_mid=I₁And det (R_mid)=1, (R_mid)^TFor R_mid Transposition, det (R_mid) indicate to R_midAsk determinant, x=[x₁,…,x_K], x₁Indicate D₁Sparse coefficient, x_KIndicate D_KIt is dilute Sparse coefficient, D₁And D_KDimension be 3 × P, W indicates the two-dimensional coordinate structure of all nodes in the two-dimension human body image that need to be reconstructed At matrix, the dimension of W is 2 × P, and λ is regularization parameter, symbol " | | | |₁" it is 1 norm sign for seeking matrix, WithIt is the matrix that dimension is 2 × 3,By R₁Preceding 2 row Composition, R₁For D₁Orthogonal spin matrix,By R_iPreceding 2 row composition,By R_KPreceding 2 row composition, R_KFor D_KOrthogonal rotation Matrix, symbol " | | | |₂" it is 2 norm signs for seeking matrix；

And The solution expression formula of variable Z, the solution expression formula of variable E, the solution expression formula of variable G after each iteration are unfolded respectively Processing, it is corresponding to obtain after each iteration the final solution expression formula, the final solution expression formula of variable E of variable Z, variable G most Expression formula is solved eventually, and the final solution expression formula of variable Z after the t times iteration is described as：Z^t=((W-E^t-1-(G^t-1)^T)D^T+μ M^t+Y^t-1)×(DD^T+μI₂)^-1, the final solution expression formula of variable E after the t times iteration is described as：E^t=S_β(W-Z^tD-(G^t-1 )^T), the final solution expression formula correspondence of variable G after the t times iteration is described as：G^t=g (W-Z^tD-E^t)；Finally judge iteration Whether number reaches maximum number of iterations t_maxOr iteration stopping conditionIt is whether true, if repeatedly Generation number has reached maximum number of iterations t_maxOr iteration stopping conditionIt sets up, then stops iteration, The solution value of variable M is obtained, then executes step 7；Otherwise, continue iteration；

Wherein, 1≤t≤t_max, t_maxIndicate maximum number of iterations, t_max>=1000, the initial value of variable M be dimension be 2 × The null matrix of 3K, Z^t-1Indicate the value of variable Z after the t-1 times iteration, the initial value of variable Z is equal to the initial value of variable M, E^t-1Table Show the value of variable E after the t-1 times iteration, the initial value of variable E is the null matrix that dimension is 2 × P, G^t-1Indicate the t-1 times iteration The value of variable G afterwards, the initial value of variable G are the matrix being averaged to W by column, Y^t-1Indicate variable Y after the t-1 times iteration Value, the initial value of variable Y is equal to the initial value of variable M,Corresponding variable after indicating the t times iteration M₁,…,M_i,…,M_KValue, M_i ^t-1Indicate variable M after the t-1 times iteration_iValue,WithIt is corresponding to indicate M_i ^t-1 Unitary matrice, the unitary matrice on the right, feature value vector on the left side obtained after singular value decomposition,ForTransposition, Diag () indicates diagonal matrix,Indicate vectorIn L₁Projection on norm unit ball, (G^t-1)^T For G^t-1Transposition, D^TFor the transposition of D, I₂The unit matrix for indicating 3K × 3K, enables X=W-Z^tD-(G^t-1)^T, S_β(X) it indicates in X Each element carry out soft-threshold calculating, in X the i-th ' row jth ' column element carry out soft-threshold calculating formula be sign (X_i'j')(X_i'j'-β)₊, i'=1,2,1≤j'≤P, sign (X_i'j') indicate to seek X_i'j'Symbol,Symbol " | | " it is the symbol that takes absolute value, g (W-Z^tD-E^t) indicate to W- Z^tD-E^tIt averages by row,Indicate the value of variable M after the t times iteration,Indicate the value of variable M after the t-1 times iteration；

Step 7：Using the image projection registration Algorithm pair based on manifoldIt is solved,Meet condition：Obtain x andRespective value；Then using alternating Iteration minimizes method pairOptimize, obtain x andRespective optimal value； Further according to x optimal value andReconstruct 3 D human body posture；Wherein, I₃The unit matrix for being 2 × 2 for dimension.

Compared with the prior art, the advantages of the present invention are as follows：

1) for the 3 D human body attitude reconstruction of monocular image the problem of a usually non-convex optimization problem, side of the present invention Method is by the basis of shape space model, in conjunction with L_1/2The property of regularization and spectral norm proposes a kind of based on L_1/2Canonical The convex relaxation method changed, converts convex programming problem by convex relaxation method for the non-convex Solve problems of shape space model；? During being optimized using ADMM algorithm to convex programming problem, proposes spectral norm proximal end gradient algorithm and guaranteeing solution just The property handed over and sparsity, the method for the present invention solve compared with existing alternating iteration minimizes method with global uniqueness The advantages that with independent of initial value；Since the method for the present invention is based on L_1/2Regularization, thus with the three-dimensional based on rarefaction representation The convex relaxation method of shape estimation is compared, and quality reconstruction and degree of rarefication based on solution required by the method for the present invention are more ideal.

2) the method for the present invention is in addition to that directly can carry out 3 D human body attitude reconstruction using shape space model, additionally it is possible to be Other reconstructing methods provide an ideal initial value.Due to using three-dimensional variable shape model to carry out 3 D human body attitude reconstruction In the presence of to initial value sensitive issue, but the solution that the method for the present invention is found out has global uniqueness, independent of initial value, because This can provide ideal initial value for the reconstruct of three-dimensional variable shape model so that quality reconstruction is more accurate and degree of rarefication more Add ideal.

Detailed description of the invention

Fig. 1 a is a width two-dimension human body image；

Fig. 1 b is Fig. 1 a is reconstructed using the 3 D human body attitude reconstruction method of the embodiment of the present invention one three Tie up the two-dimension human body skeleton that human body attitude is generated by the projection of identical visual angle；

Fig. 1 c is Fig. 1 a is reconstructed using Zhou Xiaowei mentioned method 3 D human body posture by identical view The two-dimension human body skeleton that angular projection generates；

Fig. 1 d is to minimize method using existing alternating iteration to pass through the 3 D human body posture that Fig. 1 a is reconstructed Cross the two-dimension human body skeleton that identical visual angle projection generates；

Fig. 2 a is another width two-dimension human body image；

Fig. 2 b is Fig. 2 a is reconstructed using the 3 D human body attitude reconstruction method of the embodiment of the present invention one three Tie up the two-dimension human body skeleton that human body attitude is generated by the projection of identical visual angle；

Fig. 2 c is Fig. 2 a is reconstructed using Zhou Xiaowei mentioned method 3 D human body posture by identical view The two-dimension human body skeleton that angular projection generates；

Fig. 2 d is to minimize method using existing alternating iteration to pass through the 3 D human body posture that Fig. 2 a is reconstructed Cross the two-dimension human body skeleton that identical visual angle projection generates；

Fig. 3 is using the three of the mentioned method of Zhou Xiaowei and existing alternating iteration minimum method and the embodiment of the present invention one It ties up human body attitude reconstructing method and three-dimensional is carried out to Walk, Run, Jump, Climb, Box, Dance, Sit, Basketball respectively The reconstructed error comparison diagram of human body reconstruct；

Fig. 4 is using the three of the mentioned method of Zhou Xiaowei and existing alternating iteration minimum method and the embodiment of the present invention one It ties up human body attitude reconstructing method and three-dimensional is carried out to Walk, Run, Jump, Climb, Box, Dance, Sit, Basketball respectively The degree of rarefication comparison diagram of human body reconstruct；

Fig. 5 a is a width two-dimension human body image；

Fig. 5 b is Fig. 5 a is reconstructed using the 3 D human body attitude reconstruction method of the embodiment of the present invention two three Tie up the two-dimension human body skeleton that human body attitude is generated by the projection of identical visual angle；

Fig. 5 c is Fig. 5 a is reconstructed using Zhou Xiaowei mentioned method 3 D human body posture by identical view The two-dimension human body skeleton that angular projection generates；

Fig. 5 d is to minimize method using existing alternating iteration to pass through the 3 D human body posture that Fig. 5 a is reconstructed Cross the two-dimension human body skeleton that identical visual angle projection generates；

Fig. 6 a is another width two-dimension human body image；

Fig. 6 b is Fig. 6 a is reconstructed using the 3 D human body attitude reconstruction method of the embodiment of the present invention two three Tie up the two-dimension human body skeleton that human body attitude is generated by the projection of identical visual angle；

Fig. 6 c is Fig. 6 a is reconstructed using Zhou Xiaowei mentioned method 3 D human body posture by identical view The two-dimension human body skeleton that angular projection generates；

Fig. 6 d is to minimize method using existing alternating iteration to pass through the 3 D human body posture that Fig. 6 a is reconstructed Cross the two-dimension human body skeleton that identical visual angle projection generates；

Fig. 7 is using the three of the mentioned method of Zhou Xiaowei and existing alternating iteration minimum method and the embodiment of the present invention two It ties up human body attitude reconstructing method and three-dimensional is carried out to Walk, Run, Jump, Climb, Box, Dance, Sit, Basketball respectively The reconstructed error comparison diagram of human body reconstruct；

Fig. 8 is using the three of the mentioned method of Zhou Xiaowei and existing alternating iteration minimum method and the embodiment of the present invention two It ties up human body attitude reconstructing method and three-dimensional is carried out to Walk, Run, Jump, Climb, Box, Dance, Sit, Basketball respectively The degree of rarefication comparison diagram of human body reconstruct；

Fig. 9 is that the overall of the 3 D human body attitude reconstruction method of the embodiment of the present invention one realizes block diagram.

Specific embodiment

The present invention will be described in further detail below with reference to the embodiments of the drawings.

Embodiment one：

One kind that the present embodiment proposes is based on L_1/2The 3 D human body attitude reconstruction method of regularization, it is overall to realize block diagram As shown in fig. 9, it includes following steps：

Step 1：N width 3 D human body figure is chosen, has P node in every width 3 D human body figure；Then by all three-dimensionals P × N number of node three-dimensional coordinate in human body figure constitutes database；Wherein, N >=3P, P=15 take N=in the present embodiment 1800, Carnegie Mellon University's database can be directlyed adopt in the present embodiment.

Dictionary learning is carried out to database, complete dictionary is obtained, is remembered For B,To prevent B_iAnd c_ijIt is any to change, condition need to be met：c_ij>=0 and ||B_i||_F≤1；Wherein, the dimension of 1≤j≤N, 1≤i≤K, B are 3P × K, and K indicates the number of the atom of the 3D shape in B, 3P≤K≤N takes K=128, B=[B in the present embodiment₁,...,B_K], symbol " [] " is that vector indicates symbol, B₁,…,B_KIt is right Should indicate the 1st 3D shape in B atom ..., the atom of k-th 3D shape, B₁,…,B_KBe dimension be 3P × 1 Column vector, B_iIndicate the atom of i-th of 3D shape in B, B_iIt is the column vector of 3P × 1 for dimension, min () is to take minimum Value function, C indicate that the sparse coefficient matrix in dictionary learning, the dimension of C are K × N, c_ijIndicate the i-th row jth column element in C, The initial value of C is C₀, C₀=pinv (B₀) S_train, the dimension of S_train is 3P × N, S_train=[S₁,…,S_N], S₁,…,S_NColumn vector that the corresponding three-dimensional coordinate for indicating all nodes in the 1st width 3 D human body figure is constituted ..., the N three Tie up the column vector that the three-dimensional coordinate of all nodes in human body figure is constituted, S₁,…,S_NIt is the column vector that dimension is 3P × 1, S_jIndicate the column vector that the three-dimensional coordinate of all nodes in jth width 3 D human body figure is constituted, S_jIt is the column of 3P × 1 for dimension Vector, pinv (B₀) it is to B₀Carry out pseudoinverse, B₀For the initial value of B, B₀It is constituted to select K column vector at random from S_train Dimension be 3P × K matrix, symbol " | | | |_F" it is to seek the F norm sign of matrix, α is regularization parameter, and the value of α can be with It for any positive integer, is adjusted as needed, takes α=1 in the present embodiment.

Dictionary learning is one of step crucial in rarefaction representation, and the quality of dictionary will have a direct impact on 3 D human body posture weight The effect of structure, thus the present invention using matrix decomposition and sparse coding on-line study method (J.Mairal, F.Bach, J.Ponce,and G.Sapiro.Online learning for matrix factorization and sparse coding.The Journal of Machine Learning Research,11:19-60,2010. (Ma Na-Zhu Lian, this Bach-Forlan west, Pang Sai-coke, the on-line study method of matrix decomposition and sparse coding, 11:19-60,2010)) to database Dictionary learning is carried out, to obtain the excessively complete dictionary containing K atom, is had by the excessively complete dictionary that dictionary learning obtains There is stronger ability to express.

Step 3：Shape space model is constructed using D, shape space model is described as：The formula Indicate that any one shape can be by rotatable base shape linear expression；Wherein, x=[x₁,…,x_K], x₁Indicate D₁It is dilute Sparse coefficient, x_KIndicate D_KSparse coefficient, D₁And D_KDimension be 3 × P, S is the three-dimensional coordinate of three-dimensional human skeleton nodes, Its dimension is 3 × P, D_iDimension be 3 × P, D_iIn the 1st column three elements be sequentially B_iIn the 1st to the 3rd element, D_i In the 2nd column three elements be sequentially B_iIn the 4th to the 6th element, and so on, D_iIn P column three elements sequentially For B_iIn 3P-2 to the 3P element, x_iIndicate D_iSparse coefficient, R_iFor D_iOrthogonal spin matrix, R_iDimension be 3 × 3, R_iMeet condition：(R_i)^TR_i=I₁And det (R_i)=1, (R_i)^TFor R_iTransposition, I₁The unit square for being 3 × 3 for dimension Battle array, det (R_i) indicate to R_iSeek determinant.

Step 4：The two dimensional character point extracted in three-dimensional variable shape model and the two-dimension human body image that need to be reconstructed is carried out The non-convex optimization problem of fusion is described as： A substantially loss function,For re-projection error, λ | | x | |₁For L₁Regularization,It is substantially that the non-convex optimization with orthogonality and sparsity constraints is asked Topic, leading to non-convex reason is there is ambiguity there are two known variables；Then the property and L of spectral norm are utilized_1/2Just The characteristics of then changing carries out convex relaxation processes to non-convex optimization problem, converts convex programming problem for non-convex optimization problem, describe For：The convex programming problem is the convex programming problem for having bound term, It is substantially one with punishment constraintSpectral norm regularization least square problem；Wherein,It is for dimension 2 × 3 matrix,The matrix R for being 3 × 3 by dimension_midPreceding 2 row composition, R_midMeet condition：(R_mid)^TR_mid=I₁And det(R_mid)=1, (R_mid)^TFor R_midTransposition, det (R_mid) indicate to R_midAsk determinant, D₁And D_KDimension be 3 × P, W Indicate the matrix that the two-dimensional coordinate of all nodes in the two-dimension human body image that need to reconstruct is constituted, the dimension of W is 2 × P, W it is known that λ is regularization parameter, and the value of λ is generally positive real number, can be adjusted as needed, takes λ=1 in the present embodiment, symbol " | | ||₁" it is 1 norm sign for seeking matrix,WithIt is dimension For 2 × 3 matrix,By R₁Preceding 2 row composition, R₁For D₁Orthogonal spin matrix,By R_iPreceding 2 row composition,By R_K Preceding 2 row composition, R_KFor D_KOrthogonal spin matrix, symbol " | | | |₂" it is 2 norm signs for seeking matrix.

Step 5：In practical applications, the two dimensional character point in two-dimension human body image is to be obtained by detection, therefore have In the presence of exceptional value, need using Zhou X, Zhu M, Leonardos S, Daniilidis K.Sparse Representation for 3D Shape Estimation:A Convex Relaxation Approach.IEEE Transactions on Pattern Analysis&Machine Intelligence,2016.PP(99):1-1. (Zhou Xiao It is towering, Zhu Menglong, Ang Naduo-this flesh side ancestor, a kind of convex relaxation method of the 3D shape estimation based on rarefaction representation, IEEE mode point Analysis and machine intelligence, 2016.PP (99):The method proposed in 1-1) carries out exceptional value modeling, while noticing that translation vector exists The process of centralization cannot be eliminated, therefore convex programming problem is converted multivariable Solve problems by the present invention, is described as：Then increase an auxiliary in multivariable Solve problems Variable converts the multivariable Solve problems with auxiliary variable for multivariable Solve problems, is described as：

Wherein, E table Show that dimension is the exceptional value analog matrix of 2 × P, the translation row vector that G representation dimension is 2 × 1, G^TFor the transposition of G, β is non-negative Real parameter takes β=0.1, M=[M in the present embodiment₁,…,M_K], D=[D₁,…,D_K]^T, [D₁,…,D_K]^TFor [D₁,…,D_K] Transposition, Z is increased auxiliary variable, μ indicate augmentation Lagrange solve in expression formula for controlling the ginseng of iteration step length Number, the value of μ is the inverse of the average value of the absolute value of all elements in W in the present embodiment, and Y is dual variable, L_μ(M,Z, E, G, Y) indicate that augmentation Lagrange solves function.

Step 6：Using ADMM algorithm (Boyd S, Parikh N, Chu E, et al.Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers [J] .Foundations&Trends in Machine Learning, 2011,3 (1):1-122. (Shi Difen Boyd, Neil Pa Ruika, distributed optimization and statistical learning based on multiplier alternating direction method, machine learning basis with become Gesture, the 2011, the 1st the phase volume 3:Expression formula 1-122)) is solved to augmentation Lagrange and is iterated solution, solution obtains each iteration The solution expression formula of variable M after the t times iteration is described as by the solution expression formula of each variable afterwards：

The solution expression formula of variable Y after the t times iteration is described as：Y^t=Y^t-1+μ (M^t-Z^t)；Then in order to guarantee the orthogonality and sparsity of variable M, using spectral norm proximal end gradient algorithm to becoming after each iteration The solution expression formula of amount M is handled, and the final solution expression formula of variable M after each iteration is obtained, by variable after the t times iteration The final solution expression formula of M is described as：

And respectively to the solution expression formula of variable Z after each iteration, the solution expression formula of variable E, the solution expression formula of variable G into Row expansion processing, corresponds to final solution expression formula, the final solution expression formula of variable E, variable of variable Z after obtaining each iteration The final solution expression formula of variable Z after the t times iteration is described as by the final solution expression formula of G：Z^t=((W-E^t-1-(G^t-1)^T) D^T+μM^t+Y^t-1)×(DD^T+μI₂)^-1, the final solution expression formula of variable E after the t times iteration is described as：E^t=S_β(W-Z^tD- (G^t-1)^T), the final solution expression formula correspondence of variable G after the t times iteration is described as：G^t=g (W-Z^tD-E^t)；Finally judge Whether the number of iterations reaches maximum number of iterations t_maxOr iteration stopping conditionIt is whether true, if The number of iterations has reached maximum number of iterations t_maxOr iteration stopping conditionIt sets up, then stops changing In generation, obtains the solution value of variable M, then executes step 7；Otherwise, continue iteration.

Wherein, 1≤t≤t_max, t_maxIndicate maximum number of iterations, t_max>=1000, t is taken in the present embodiment_max=1500, The initial value of variable M is the null matrix that dimension is 2 × 3K, Z^t-1Indicate the value of variable Z after the t-1 times iteration, variable Z's is initial Value is equal to the initial value of variable M, E^t-1Indicate the value of variable E after the t-1 times iteration, the initial value of variable E is that dimension is 2 × P's Null matrix, G^t-1Indicating the value of variable G after the t-1 times iteration, the initial value of variable G is the matrix being averaged to W by column, Y^t-1Indicating the value of variable Y after the t-1 times iteration, the initial value of variable Y is equal to the initial value of variable M, Corresponding variable M after indicating the t times iteration₁,…,M_i,…,M_KValue, M_i ^t-1Indicate variable M after the t-1 times iteration_iValue,WithIt is corresponding to indicate M_i ^t-1Unitary matrice, the unitary matrice on the right, spy on the left side obtained after singular value decomposition Value indicative vector,ForTransposition, diag () indicate diagonal matrix,Indicate vector In L₁Projection on norm unit ball, (G^t-1)^TFor G^t-1Transposition, D^TFor the transposition of D, I₂It indicates the unit matrix of 3K × 3K, enables X=W-Z^tD-(G^t-1)^T, S_β(X) it indicates to carry out soft-threshold calculating to each element in X, to the i-th ' row jth ' column element in X The formula for carrying out soft-threshold calculating is sign (X_i'j')(X_i'j'-β)₊, i'=1,2,1≤j'≤P, sign (X_i'j') indicate to seek X_i'j' Symbol,Symbol " | | " it is the symbol that takes absolute value, g (W-Z^tD-E^t) It indicates to W-Z^tD-E^tIt averages by row,Indicate the value of variable M after the t times iteration,Indicate variable after the t-1 times iteration The value of M.

Step 7：The solution value of the variable M obtained according to step 6, M=[M₁,…,M_i,…,M_K]、Obtain x_i And R_iRespective value, x_i=| | M_i||₂, R_i=[r_i ⁽¹⁾,r_i ⁽²⁾,r_i ⁽³⁾]^T,Due to R_iHave Orthogonality, so r_i ⁽³⁾=r_i ⁽¹⁾×r_i ⁽²⁾；Then basisAnd the x acquired_iAnd R_iRespective value, reconstructs three Tie up human body attitude；Wherein, [r_i ⁽¹⁾,r_i ⁽²⁾,r_i ⁽³⁾]^TFor [r_i ⁽¹⁾,r_i ⁽²⁾,r_i ⁽³⁾] transposition, r_i ⁽¹⁾,r_i ⁽²⁾,r_i ⁽³⁾It is corresponding to indicate R_iIn the 1st row vector, the 2nd row vector, the 3rd row vector, m_i ⁽¹⁾Indicate M_iIn the 1st row vector, m_i ⁽²⁾Indicate M_i In the 2nd row vector.

Embodiment two：

One kind that the present embodiment proposes is based on L_1/2The 3 D human body attitude reconstruction method of regularization comprising following steps：

Dictionary learning is carried out to database, complete dictionary is obtained, is remembered For B,To prevent B_iAnd c_ijIt is any to change, condition need to be met：c_ij>=0 and ||B_i||_F≤1；Wherein, the dimension of 1≤j≤N, 1≤i≤K, B are 3P × K, and K indicates the number of the atom of the 3D shape in B, 3P≤K<N takes K=128, B=[B in the present embodiment₁,...,B_K], symbol " [] " is that vector indicates symbol, B₁,…,B_KIt is right Should indicate the 1st 3D shape in B atom ..., the atom of k-th 3D shape, B₁,…,B_KBe dimension be 3P × 1 Column vector, B_iIndicate the atom of i-th of 3D shape in B, B_iIt is the column vector of 3P × 1 for dimension, min () is to take minimum Value function, C indicate that the sparse coefficient matrix in dictionary learning, the dimension of C are K × N, c_ijIndicate the i-th row jth column element in C, The initial value of C is C₀, C₀=pinv (B₀) S_train, the dimension of S_train is 3P × N, S_train=[S₁,…,S_N], S₁,…,S_NColumn vector that the corresponding three-dimensional coordinate for indicating all nodes in the 1st width 3 D human body figure is constituted ..., the N three Tie up the column vector that the three-dimensional coordinate of all nodes in human body figure is constituted, S₁,…,S_NIt is the column vector that dimension is 3P × 1, S_jIndicate the column vector that the three-dimensional coordinate of all nodes in jth width 3 D human body figure is constituted, S_jIt is the column of 3P × 1 for dimension Vector, pinv (B₀) it is to B₀Carry out pseudoinverse, B₀For the initial value of B, B₀It is constituted to select K column vector at random from S_train Dimension be 3P × K matrix, symbol " | | | |_F" it is to seek the F norm sign of matrix, α is regularization parameter, and the value of α can be with It for any positive integer, is adjusted as needed, takes α=1 in the present embodiment.

Step 3：Identify that three dimensional object is one of the core missions of computer vision from the two dimensional image of monocular, however And the work for being rich in challenge, this is because there are changeabilities, non-rigid shape deformations in class in 3D shape restructuring procedure The problems such as with exhaustive all possible visual angles are avoided, proposes to solve the difficulties in 3D shape restructuring procedure with three-dimensional Shape-variable model is resolved, therefore the present invention constructs three-dimensional variable shape model using D, and three-dimensional variable shape model is retouched State for：The formula indicates that any one 3D shape can be by one group of base shape linear expression；Wherein, x= [x₁,…,x_K], x₁Indicate D₁Sparse coefficient, x_KIndicate D_KSparse coefficient, D₁And D_KDimension be 3 × P, S is three-dimensional people The three-dimensional coordinate of body skeleton node, dimension are 3 × P, D_iDimension be 3 × P, D_iIn the 1st column three elements be sequentially B_iIn The 1st to the 3rd element, D_iIn the 2nd column three elements be sequentially B_iIn the 4th to the 6th element, and so on, D_i In P column three elements be sequentially B_iIn 3P-2 to the 3P element, x_iIndicate D_iSparse coefficient.

With three-dimensional variable shape model, the problem of by 3D shape based on the two dimensional character point reconstruct object in image The characteristic point being considered as in two dimensional image and (Zhou Yu, Liu Jun-Tao, and the problem of three-dimensional variable shape Model Matching Bai Xiang.Research and Perspective on Shape Matching.Acta Automatica Sinica, 2012.38(6):889-910. (Zhou Yu, Liu Juntao, Bai Xiang, shape matching method research and prospect, automation journal, 2012. 6th phase volume 38:889-910)), i.e. 3D-to-2D merges problem, is a non-convex optimization problem in the fusion question essence, weight Structure result depends on initial value, for this purpose, the present invention recycles D to construct shape space model, shape space model is described as：The formula indicates that any one shape can be by rotatable base shape linear expression；Wherein, x= [x,...,x_K], the dimension 1 × K, x of x₁Indicate D₁Sparse coefficient, x_KIndicate D_KSparse coefficient, D₁And D_KDimension be 3 × P, S are the three-dimensional coordinate of three-dimensional human skeleton nodes, and dimension is 3 × P, D_iDimension be 3 × P, D_iIn the 1st column three Element is sequentially B_iIn the 1st to the 3rd element, D_iIn the 2nd column three elements be sequentially B_iIn the 4th to the 6th member Element, and so on, D_iIn P column three elements be sequentially B_iIn 3P-2 to the 3P element, x_iIndicate D_iIt is sparse Coefficient, R_iFor D_iOrthogonal spin matrix, R_iDimension be 3 × 3, R_iMeet condition：(R_i)^TR_i=I₁And det (R_i)=1, (R_i)^T For R_iTransposition, I₁The unit matrix for being 3 × 3 for dimension, det (R_i) indicate to R_iSeek determinant.

Step 4：The two dimensional character point extracted in three-dimensional variable shape model and the two-dimension human body image that need to be reconstructed is carried out The non-convex optimization problem of fusion is described as： A substantially loss function,For re-projection error, λ | | x | |₁For L₁Regularization,It is substantially that the non-convex optimization with orthogonality and sparsity constraints is asked Topic, leading to non-convex reason is there is ambiguity there are two known variables；Then the property and L of spectral norm are utilized₁₂Canonical The characteristics of change, carries out convex relaxation processes to non-convex optimization problem, converts convex programming problem for non-convex optimization problem, be described as：The convex optimization problem is the convex programming problem for having bound term, Substantially one with punishment constraintSpectral norm regularization least square problem；Wherein,It is 2 for dimension × 3 matrix,The matrix R for being 3 × 3 by dimension_midPreceding 2 row composition, R_midMeet condition：(R_mid)^TR_mid=I₁And det (R_mid)=1, (R_mid)^TFor R_midTransposition, det (R_mid) indicate to R_midAsk determinant, x=[x₁,…,x_K], x₁Indicate D₁It is dilute Sparse coefficient, x_KIndicate D_KSparse coefficient, D₁And D_KDimension be 3 × P, W indicates the institute in the two-dimension human body image that need to be reconstructed The matrix for having the two-dimensional coordinate of node to constitute, the dimension of W are 2 × P, it is known that λ is regularization parameter, the value of λ is generally positive reality W Number, can be adjusted as needed, take λ=0.8 in the present embodiment, symbol " | | | |₁" it is 1 norm sign for seeking matrix,WithIt is the matrix that dimension is 2 × 3,By R₁Preceding 2 Row composition, R₁For D₁Orthogonal spin matrix,By R_iPreceding 2 row composition,By R_KPreceding 2 row composition, R_KFor D_KOrthogonal rotation Torque battle array, symbol " | | | |₂" it is 2 norm signs for seeking matrix.

And respectively to the solution expression formula of variable Z after each iteration, the solution expression formula of variable E, the solution expression formula of variable G into Row expansion processing, corresponds to final solution expression formula, the final solution expression formula of variable E, variable of variable Z after obtaining each iteration The final solution expression formula of variable Z after the t times iteration is described as by the final solution expression formula of G：Zt=((W-E^t-1-(G^t-1 )^T)D^T+μM^t+Y^t-1)×(DD^T+μI₂)^-1, the final solution expression formula of variable E after the t times iteration is described as：E^t=S_β(W- Z^tD-(G^t-1)^T), the final solution expression formula correspondence of variable G after the t times iteration is described as：G^t=g (W-Z^tD-E^t)；Finally sentence Whether disconnected the number of iterations reaches maximum number of iterations t_maxOr iteration stopping conditionIt is whether true, If the number of iterations has reached maximum number of iterations t_maxOr iteration stopping conditionIt is vertical, then stop changing In generation, obtains the solution value of variable M, then executes step 7；Otherwise, continue iteration.

Wherein, 1≤t≤t_max, t_maxIndicate maximum number of iterations, t_max>=1000, t is taken in the present embodiment_max=1500, The initial value of variable M is the null matrix that dimension is 2 × 3K, Z^t-1Indicate the value of variable Z after the t-1 times iteration, variable Z's is initial Value is equal to the initial value of variable M, E^t-1Indicate the value of variable E after the t-1 times iteration, the initial value of variable E is that dimension is 2 × P's Null matrix, G^t-1Indicating the value of variable G after the t-1 times iteration, the initial value of variable G is the matrix being averaged to W by column, Y^t-1Indicating the value of variable Y after the t-1 times iteration, the initial value of variable Y is equal to the initial value of variable M, Corresponding variable M after indicating the t times iteration₁,…,M_i,…,M_KValue, M_i ^t-1Indicate variable M after the t-1 times iteration_iValue,WithIt is corresponding to indicate M_i ^t-1Unitary matrice, the unitary matrice on the right, spy on the left side obtained after singular value decomposition Value indicative vector,ForTransposition, diag () indicate diagonal matrix,Indicate vector In L₁Projection on norm unit ball, (G^t-1)^TFor G^t-1Transposition, D^TFor the transposition of D, I₂It indicates the unit matrix of 3K × 3K, enables X=W-Z^tD-(G^t-1)^T, S_β(X) it indicates to carry out soft-threshold calculating to each element in X, to the i-th ' row jth ' column element in X The formula for carrying out soft-threshold calculating is sign (X_i'j')(X_i'j'-β)₊, i'=1,2,1≤j'≤P, sign (X_i'j') indicate to seek X_i'j' Symbol,Symbol " | | " it is the symbol that takes absolute value, g (W-Z^tD-E^t) It indicates to W-Z^tD-E^tIt averages by row,Indicate the value of variable M after the t-1 times iteration,Indicate variable after the t times iteration The value of M.

Step 7：Using image projection registration Algorithm (Bue AD, Xavier J, Agapito L, et based on manifold al.Bilinear Modeling via Augmented Lagrange Multipliers(BALM)[J].IEEE Transactions on Pattern Analysis&Machine Intelligence, 2012,34 (8):1496-1508. (Ah dish Theo-Dare but, Ze Weier-Qiao, A Jiapite-Lu Erde, based on the Bilinear Modeling of augmentation Lagrange multiplier, IEEE mode analysis and machine intelligence, the 2012, the 8th the phase volume 34：It is 1496-1508)) rightIt carries out It solves,Meet condition：Obtain x andRespective value；Then it uses Alternating iteration minimizes method pairOptimize, obtain x andIt is respective excellent Change value；Further according to x optimal value andReconstruct 3 D human body posture；Wherein, I₃The unit for being 2 × 2 for dimension Matrix.

The present invention realizes the reconstruct of 3 D human body posture using two kinds of 3 D human body attitude reconstruction methods, to of the invention Two kinds of 3 D human body attitude reconstruction methods carry out qualitative experiment and quantitative experiment, to verify the performance of the method for the present invention better than Zhou X, Zhu M, Leonardos S, Daniilidis K.Sparse Representation for 3D Shape Estimation:AConvex Relaxation Approach.IEEE Transactions on Pattern Analysis& Machine Intelligence, 2016.PP (99):(Zhou Xiaowei, Zhu Menglong, Ang Naduo-this flesh side ancestor are based on sparse 1-1. A kind of convex relaxation method of the 3D shape estimation of expression, IEEE mode analysis and machine intelligence, 2016.PP (99):Institute in 1-1.) The method mentioned is (referred to as：The mentioned method of Zhou Xiaowei) and existing alternating iteration minimum method.

For qualitative experiment, data provided by the code using the mentioned method of Zhou Xiaowei are tested, by monocular Two-dimension human body skeleton in image carries out 3 D human body attitude reconstruction (three-dimensional human skeleton), then by the 3 D human body appearance of reconstruct State is projected by identical visual angle generates two-dimension human body skeleton.

For quantitative experiment, quantitative experiment analysis will be carried out in terms of reconstructed error and degree of rarefication two.Firstly, choosing card Nei Jimeilong College sports capture database in eight sequences (Walk, Run, Jump, Climb, Box, Dance, Sit, Basketball it) is used as quantitative experimental data, it is consistent with the experimental data in the mentioned method of Zhou Xiaowei；Secondly, two-dimension human body Skeleton is to be rotated by 360 ° acquisition around above-mentioned each sequence by orthographic camera；Later, the mentioned method of Zhou Xiaowei and existing Some alternating iterations minimize method and the alignment thereof of the method for the present invention is：Make weight by translating scaling under camera coordinates system The 3 D human body posture of structure is aligned with true 3 D human body attitude data；Finally, calculating the 3 D human body posture of reconstruct With the Euclidean distance (unit between true 3 D human body posture：Millimeter) and degree of rarefication (number of neutral element), and to reconstruct Error and degree of rarefication take mean value.Here, setting maximum number of iterations as 1500 times.

1) implementation process of the qualitative experiment and quantitative experiment of the 3 D human body attitude reconstruction method of the embodiment of the present invention one It is as follows.

For qualitative experiment, method mentioned to Zhou Xiaowei and existing alternating iteration minimize method and the embodiment of the present invention One 3 D human body attitude reconstruction method compares experiment, and experiment parameter is referring to the parameter in the mentioned method of Zhou Xiaowei.

Fig. 1 a gives a width two-dimension human body image, and Fig. 1 b, Fig. 1 c and Fig. 1 d are respectively to use the embodiment of the present invention one 3 D human body attitude reconstruction method, the mentioned method of Zhou Xiaowei and existing alternating iteration minimize method and Fig. 1 a are reconstructed The two-dimension human body skeleton that the 3 D human body posture arrived is generated by the projection of identical visual angle.Comparison diagram 1a and Fig. 1 b, Fig. 1 c and figure 1d, it can clearly be seen that the 3 D human body attitude reconstruction method of the embodiment of the present invention one achieves preferable quality reconstruction, figure Two-dimension human body skeleton in 1b and Fig. 1 a is closer.

Fig. 2 a gives a width two-dimension human body image, and Fig. 2 b, Fig. 2 c and Fig. 2 d are respectively to use the embodiment of the present invention one 3 D human body attitude reconstruction method, the mentioned method of Zhou Xiaowei and existing alternating iteration minimize method and Fig. 2 a are reconstructed The two-dimension human body skeleton that the 3 D human body posture arrived is generated by the projection of identical visual angle.Comparison diagram 2a and Fig. 2 b, Fig. 2 c and figure 2d, it can clearly be seen that the 3 D human body attitude reconstruction method of the embodiment of the present invention one achieves preferable quality reconstruction, figure Two-dimension human body skeleton in 2b and Fig. 2 a is closer.

For quantitative experiment, method mentioned to Zhou Xiaowei and existing alternating iteration minimize method and the embodiment of the present invention One 3 D human body attitude reconstruction method compares experiment.Fig. 3 gives using the mentioned method of Zhou Xiaowei and existing alternating Iteration minimize the 3 D human body attitude reconstruction method of method and the embodiment of the present invention one respectively to Walk, Run, Jump, Climb, Box, Dance, Sit, Basketball carry out the reconstructed error comparison diagram of 3 D human body reconstruct, and Fig. 4 gives use The mentioned method of Zhou Xiaowei and existing alternating iteration minimize the 3 D human body attitude reconstruction side of method and the embodiment of the present invention one Method carries out the degree of rarefication of 3 D human body reconstruct to Walk, Run, Jump, Climb, Box, Dance, Sit, Basketball respectively Comparison diagram.From Fig. 3 and Fig. 4 as can be seen that on full sequence, using the 3 D human body attitude reconstruction of the embodiment of the present invention one Method has been more than to minimize method using existing alternating iteration comprehensively；In Walk, Run, Jump, Climb sequence, using this The 3 D human body attitude reconstruction method of inventive embodiments one is all achieved at two aspects of reconstructed error and degree of rarefication and is substantially better than The experimental result of the mentioned method of Zhou Xiaowei；In Dance and Basketball sequence, using the three-dimensional people of the embodiment of the present invention one Body attitude reconstruction method is slightly better than the proposed method of Zhou Xiaowei；In Box sequence, using the 3 D human body appearance of the embodiment of the present invention one The reconstructed error of state reconstructing method is substantially better than the proposed method of Zhou Xiaowei, but degree of rarefication is not so good as the proposed method of Zhou Xiaowei；In Sit sequence On column, the mentioned method of Zhou Xiaowei is slightly poorer than using the reconstructed error of the 3 D human body attitude reconstruction method of the embodiment of the present invention one, And degree of rarefication is then better than the proposed method of Zhou Xiaowei.From the point of view of complex chart 3 and Fig. 4, using the 3 D human body appearance of the embodiment of the present invention one State reconstructing method carries out 3 D human body attitude reconstruction and totally achieves comparatively ideal experiment effect, demonstrates the embodiment of the present invention one 3 D human body attitude reconstruction method validity, but in certain sequences, there are reconstructed error or degree of rarefication are undesirable Situation, reason may be due to not homotactic optimal regularization parameter it is inconsistent caused by.

2) implementation process of the qualitative experiment and quantitative experiment of the 3 D human body attitude reconstruction method of the embodiment of the present invention two It is as follows.

For qualitative experiment, method mentioned to Zhou Xiaowei and existing alternating iteration minimize method and the embodiment of the present invention Two 3 D human body attitude reconstruction method compares experiment, and experiment parameter is referring to the parameter in the mentioned method of Zhou Xiaowei.

Fig. 5 a gives a width two-dimension human body image, and Fig. 5 b, Fig. 5 c and Fig. 5 d are respectively to use the embodiment of the present invention two 3 D human body attitude reconstruction method, the mentioned method of Zhou Xiaowei and existing alternating iteration minimize method and Fig. 5 a are reconstructed The two-dimension human body skeleton that the 3 D human body posture arrived is generated by the projection of identical visual angle.Comparison diagram 5a and Fig. 5 b, Fig. 5 c and figure 5d, it can clearly be seen that the 3 D human body attitude reconstruction method of the embodiment of the present invention two achieves preferable quality reconstruction, figure Two-dimension human body skeleton in 5b and Fig. 5 a is closer.

Fig. 6 a gives a width two-dimension human body image, and Fig. 6 b, Fig. 6 c and Fig. 6 d are respectively to use the embodiment of the present invention two 3 D human body attitude reconstruction method, the mentioned method of Zhou Xiaowei and existing alternating iteration minimize method and Fig. 6 a are reconstructed The two-dimension human body skeleton that the 3 D human body posture arrived is generated by the projection of identical visual angle.Comparison diagram 6a and Fig. 6 b, Fig. 6 c and figure 6d, it can clearly be seen that the 3 D human body attitude reconstruction method of the embodiment of the present invention two achieves preferable quality reconstruction, figure Two-dimension human body skeleton in 6b and Fig. 6 a is closer.

For quantitative experiment, method mentioned to Zhou Xiaowei and existing alternating iteration minimize method and the embodiment of the present invention Two 3 D human body attitude reconstruction method compares experiment.Fig. 7 gives using the mentioned method of Zhou Xiaowei and existing alternating Iteration minimize the 3 D human body attitude reconstruction method of method and the embodiment of the present invention two respectively to Walk, Run, Jump, Climb, Box, Dance, Sit, Basketball carry out the reconstructed error comparison diagram of 3 D human body reconstruct, and Fig. 8 gives use The mentioned method of Zhou Xiaowei and existing alternating iteration minimize the 3 D human body attitude reconstruction side of method and the embodiment of the present invention two Method carries out the degree of rarefication of 3 D human body reconstruct to Walk, Run, Jump, Climb, Box, Dance, Sit, Basketball respectively Comparison diagram.It can be seen from figure 7 that on full sequence, using the 3 D human body attitude reconstruction method of the embodiment of the present invention two Achieve preferable reconstruction result；As can be seen from Figure 8, in 6 sequences such as Walk, Run, Jump, Climb, Dance, Sit On, best sparse effect is all achieved using the 3 D human body attitude reconstruction method of the embodiment of the present invention two；Box, In two sequences of Basketball, it is slightly poorer than using the sparse effect of the 3 D human body attitude reconstruction method of the embodiment of the present invention two The mentioned method of Zhou Xiaowei, but method is minimized better than existing alternating iteration.From the point of view of complex chart 7 and Fig. 8, implemented using the present invention The 3 D human body attitude reconstruction method of example two achieves quality reconstruction more better than other methods, only sparse on discrete sequences There is also not ideal enough situations for degree.

Claims

1. one kind is based on L_1/2The 3 D human body attitude reconstruction method of regularization, it is characterised in that include the following steps：

Step 1：N width 3 D human body figure is chosen, has P node in every width 3 D human body figure；Then by all 3 D human bodies P × N number of node three-dimensional coordinate in figure constitutes database；Wherein, N >=3P, P=15；

Step 2：Using the formula of matrix decomposition and the on-line study method of sparse coding Dictionary learning is carried out to database, complete dictionary is obtained, is denoted as B,It is full Sufficient condition：c_ij>=0 and | | B_i||_F≤1；Wherein, the dimension of 1≤j≤N, 1≤i≤K, B are 3P × K, and K indicates the three-dimensional shaped in B The number of the atom of shape, 3P≤K≤N, B=[B₁,...,B_K], symbol " [] " is that vector indicates symbol, B₁,…,B_KIt is corresponding to indicate The atom of the 1st 3D shape in B ..., the atom of k-th 3D shape, B₁,…,B_KBe dimension be 3P × 1 column to Amount, B_iIndicate the atom of i-th of 3D shape in B, B_iIt is the column vector of 3P × 1 for dimension, min () is to be minimized letter Number, C indicate that the sparse coefficient matrix in dictionary learning, the dimension of C are K × N, c_ijIndicate the i-th row jth column element in C, C's Initial value is C₀, C₀=pinv (B₀) S_train, the dimension of S_train is 3P × N, S_train=[S₁,…,S_N], S₁,…,S_NColumn vector that the corresponding three-dimensional coordinate for indicating all nodes in the 1st width 3 D human body figure is constituted ..., the N three Tie up the column vector that the three-dimensional coordinate of all nodes in human body figure is constituted, S₁,…,S_NIt is the column vector that dimension is 3P × 1, S_jIndicate the column vector that the three-dimensional coordinate of all nodes in jth width 3 D human body figure is constituted, S_jIt is the column of 3P × 1 for dimension Vector, pinv (B₀) it is to B₀Carry out pseudoinverse, B₀For the initial value of B, B₀It is constituted to select K column vector at random from S_train Dimension be 3P × K matrix, symbol " | | | |_F" it is the F norm sign for seeking matrix, α is regularization parameter；

Step 3：Shape space model is constructed using D, shape space model is described as：Wherein, x= [x,...,x_K], the dimension 1 × K, x of x₁Indicate D₁Sparse coefficient, x_KIndicate D_KSparse coefficient, D₁And D_KDimension be 3 × P, S are the three-dimensional coordinate of three-dimensional human skeleton nodes, and dimension is 3 × P, D_iDimension be 3 × P, D_iIn the 1st column three Element is sequentially B_iIn the 1st to the 3rd element, D_iIn the 2nd column three elements be sequentially B_iIn the 4th to the 6th member Element, and so on, D_iIn P column three elements be sequentially B_iIn 3P-2 to the 3P element, x_iIndicate D_iIt is sparse Coefficient, R_iFor D_iOrthogonal spin matrix, R_iDimension be 3 × 3, R_iMeet condition：(R_i)^TR_i=I₁And det (R_i)=1, (R_i)^T For R_iTransposition, I₁3 × 3 gusts of unit square for being for dimension, det (R_i) indicate to R_iSeek determinant；

Step 4：The two dimensional character point extracted in three-dimensional variable shape model and the two-dimension human body image that need to be reconstructed is merged Non-convex optimization problem be described as：Then the property and L of spectral norm are utilized_1/2 The characteristics of regularization, carries out convex relaxation processes to non-convex optimization problem, converts convex programming problem for non-convex optimization problem, describes For：Wherein,The matrix for being 2 × 3 for dimension,It is 3 by dimension × 3 matrix R_midPreceding 2 row composition, R_midMeet condition：(R_mid)^TR_mid=I₁And det (R_mid)=1, (R_mid)^TFor R_midTurn It sets, det (R_mid) indicate to R_midAsk determinant, x=[x₁,…,x_K], x₁Indicate D₁Sparse coefficient, x_KIndicate D_KSparse system Number, D₁And D_KDimension be 3 × P, W indicates what the two-dimensional coordinate of all nodes in the two-dimension human body image that need to be reconstructed was constituted Matrix, the dimension of W are 2 × P, and λ is regularization parameter, symbol " | | | |₁" it is 1 norm sign for seeking matrix, WithIt is the matrix that dimension is 2 × 3,By R₁Preceding 2 row Composition, R₁For D₁Orthogonal spin matrix,By R_iPreceding 2 row composition,By R_KPreceding 2 row composition, R_KFor D_KOrthogonal rotation Matrix, symbol " | | | |₂" it is 2 norm signs for seeking matrix；

Then increase by one is auxiliary in multivariable Solve problems Variable is helped, the multivariable Solve problems with auxiliary variable is converted by multivariable Solve problems, is described as：

Wherein, E is indicated Dimension is the exceptional value analog matrix of 2 × P, the translation row vector that G representation dimension is 2 × 1, G^TFor the transposition of G, β is non-negative reality Parameter, M=[M₁,…,M_K], D=[D₁,…,D_K]^T, [D₁,…,D_K]^TFor [D₁,…,D_K] transposition, Z be increased auxiliary become Amount, μ indicate that augmentation Lagrange solves the parameter for being used to control iteration step length in expression formula, and Y is dual variable, L_μ(M,Z, E, G, Y) indicate that augmentation Lagrange solves function；

Step 6：Expression formula is solved to augmentation Lagrange using ADMM algorithm and is iterated solution, solution obtains each iteration The solution expression formula of variable M after the t times iteration is described as by the solution expression formula of each variable afterwards：

The solution expression formula of variable Y after the t times iteration is described as：Y^t=Y^t-1+μ (M^t-Z^t)；Then the solution expression formula of variable M after each iteration is handled using spectral norm proximal end gradient algorithm, is obtained every The final solution expression formula of variable M after secondary iteration, the final solution expression formula of variable M after the t times iteration is described as： And respectively The solution expression formula of variable Z, the solution expression formula of variable E, the solution expression formula of variable G after each iteration are carried out at expansion Reason, it is corresponding obtain the final solution expression formula, the final solution expression formula of variable E of variable Z after each iteration, variable G it is final Expression formula is solved, the final solution expression formula of variable Z after the t times iteration is described as：Z^t=((W-E^t-1-(G^t-1)^T)D^T+μM^t+ Y^t-1)×(DD^T+μI₂)^-1, the final solution expression formula of variable E after the t times iteration is described as：E^t=S_β(W-Z^tD-(G^t-1)^T), The final solution expression formula correspondence of variable G after the t times iteration is described as：G^t=g (W-Z^tD-E^t)；Finally judge the number of iterations Whether maximum number of iterations t is reached_maxOr iteration stopping conditionIt is whether true, if the number of iterations Maximum number of iterations t is reached_maxOr iteration stopping conditionIt sets up, then stops iteration, become The solution value of M is measured, then executes step 7；Otherwise, continue iteration；

Wherein, 1≤t≤t_max, t_maxIndicate maximum number of iterations, t_max>=1000, the initial value of variable M is that dimension is 2 × 3K's Null matrix, Z^t-1Indicate the value of variable Z after the t-1 times iteration, the initial value of variable Z is equal to the initial value of variable M, E^t-1Indicate the The value of variable E after t-1 iteration, the initial value of variable E are the null matrix that dimension is 2 × P, G^t-¹Become after indicating the t-1 times iteration The value of G is measured, the initial value of variable G is the matrix being averaged to W by column, Y^t-¹Indicate the value of variable Y after the t-1 times iteration, The initial value of variable Y is equal to the initial value of variable M,Corresponding variable M after indicating the t times iteration₁,…, M_i,…,M_KValue, M_i ^t-1Indicate variable M after the t-1 times iteration_iValue,WithIt is corresponding to indicate M_i ^t-1Through unusual Unitary matrice, the unitary matrice on the right, feature value vector on the left side that value obtains after decomposing,ForTransposition, diag () Indicate diagonal matrix,Indicate vectorIn L₁Projection on norm unit ball, (G^t-1)^TFor G^t-1 Transposition, D^TFor the transposition of D, I₂The unit matrix for indicating 3K × 3K, enables X=W-Z^tD-(G^t-1)^T, S_β(X) it indicates to every in X A element carries out soft-threshold calculating, and the formula for carrying out soft-threshold calculating to the i-th ' row jth ' column element in X is sign (X_i'j') (X_i'j'-β)₊, i'=1,2,1≤j'≤P, sign (X_i'j') indicate to seek X_i'j'Symbol,

Symbol " | | " it is the symbol that takes absolute value, g (W-Z^tD-E^t) table Show to W-Z^tD-E^tIt averages by row,Indicate the value of variable M after the t times iteration,Indicate variable M after the t-1 times iteration Value；

Step 7：The solution value of the variable M obtained according to step 6, M=[M₁,…,M_i,…,M_K]、Obtain x_iAnd R_i Respective value, x_i=| | M_i||₂, R_i=[r_i ⁽¹⁾,r_i ⁽²⁾,r_i ⁽³⁾]^T,r_i ⁽³⁾=r_i ⁽¹⁾×r_i ⁽²⁾；Then basisAnd the x acquired_iAnd R_iRespective value reconstructs 3 D human body posture；Wherein, [r_i ⁽¹⁾,r_i ⁽²⁾,r_i ⁽³⁾]^TFor [r_i ⁽¹⁾,r_i ⁽²⁾,r_i ⁽³⁾] transposition, r_i ⁽¹⁾,r_i ⁽²⁾,r_i ⁽³⁾It is corresponding to indicate R_iIn the 1st row vector, the 2nd Row vector, the 3rd row vector,Indicate M_iIn the 1st row vector,Indicate M_iIn the 2nd row vector.

2. one kind is based on L_1/2The 3 D human body attitude reconstruction method of regularization, it is characterised in that include the following steps：

Dictionary learning is carried out to database, complete dictionary is obtained, is denoted as B,Meet condition：c_ij>=0 and | | B_i||_F≤1；Wherein, 1≤j≤N, 1 The dimension of≤i≤K, B are 3P × K, and K indicates the number of the atom of the 3D shape in B, 3P≤K<N, B=[B₁,...,B_K], symbol Number " [] " is that vector indicates symbol, B₁,…,B_KThe corresponding atom for indicating the 1st 3D shape in B ..., k-th three-dimensional shaped The atom of shape, B₁,…,B_KIt is the column vector that dimension is 3P × 1, B_iIndicate the atom of i-th of 3D shape in B, B_iFor dimension Number is the column vector of 3P × 1, and min () is to be minimized function, and C indicates the sparse coefficient matrix in dictionary learning, the dimension of C For K × N, c_ijIndicate that the i-th row jth column element in C, the initial value of C are C₀, C₀=pinv (B₀) S_train, S_train's Dimension is 3P × N, S_train=[S₁,…,S_N], S₁,…,S_NThe corresponding all nodes indicated in the 1st width 3 D human body figure Three-dimensional coordinate constitute column vector ..., the column that constitute of the three-dimensional coordinates of all nodes in N width 3 D human body figure to Amount, S₁,…,S_NIt is the column vector that dimension is 3P × 1, S_jIndicate that the three-dimensional of all nodes in jth width 3 D human body figure is sat Mark the column vector constituted, S_jIt is the column vector of 3P × 1, pinv (B for dimension₀) it is to B₀Carry out pseudoinverse, B₀For the initial value of B, B₀ For selected at random from S_train K column vector composition dimension be 3P × K matrix, symbol " | | | |_F" it is to seek matrix F norm sign, α is regularization parameter；

Step 3：Three-dimensional variable shape model is constructed using D, three-dimensional variable shape model is described as：Wherein, X=[x ..., x_K], the dimension 1 × K, x of x₁Indicate D₁Sparse coefficient, x_KIndicate D_KSparse coefficient, D₁And D_KDimension it is equal It is the three-dimensional coordinate of three-dimensional human skeleton nodes for 3 × P, S, dimension is 3 × P, D_iDimension be 3 × P, D_iIn the 1st column three A element is sequentially B_iIn the 1st to the 3rd element, D_iIn the 2nd column three elements be sequentially B_iIn the 4th to the 6th Element, and so on, D_iIn P column three elements be sequentially B_iIn 3P-2 to the 3P element, x_iIndicate D_iIt is dilute Sparse coefficient；

Shape space model is constructed using D, shape space model is described as：Wherein, x=[x ..., x_K], Dimension 1 × the K, x of x₁Indicate D₁Sparse coefficient, x_KIndicate D_KSparse coefficient, D₁And D_KDimension be 3 × P, S is three-dimensional The three-dimensional coordinate of human skeleton node, dimension are 3 × P, D_iDimension be 3 × P, D_iIn the 1st column three elements be sequentially B_i In the 1st to the 3rd element, D_iIn the 2nd column three elements be sequentially B_iIn the 4th to the 6th element, and so on, D_iIn P column three elements be sequentially B_iIn 3P-2 to the 3P element, x_iIndicate D_iSparse coefficient, R_iFor D_i's Orthogonal spin matrix, R_iDimension be 3 × 3, R_iMeet condition：(R_i)^TR_i=I₁And det (R_i)=1, (R_i)^TFor R_iTransposition, I₁ The unit matrix for being 3 × 3 for dimension, det (R_i) indicate to R_iSeek determinant；

The solution expression formula of variable Y after the t times iteration is described as：Y^t=Y^t-1+μ(M^t- Z^t)；Then the solution expression formula of variable M after each iteration is handled using spectral norm proximal end gradient algorithm, is obtained every time The final solution expression formula of variable M after iteration, the final solution expression formula of variable M after the t times iteration is described as：

And respectively to the solution expression formula of variable Z after each iteration, the solution expression formula of variable E, the solution expression formula of variable G into Row expansion processing, corresponds to final solution expression formula, the final solution expression formula of variable E, variable of variable Z after obtaining each iteration The final solution expression formula of variable Z after the t times iteration is described as by the final solution expression formula of G：Z^t=((W-E^t-1-(G^t-1)^T) D^T+μM^t+Y^t-1)×(DD^T+μI₂)^-1, the final solution expression formula of variable E after the t times iteration is described as：E^t=S_β(W-Z^tD- (G^t-1)^T), the final solution expression formula correspondence of variable G after the t times iteration is described as：G^t=g (W-Z^tD-E^t)；Finally judge Whether the number of iterations reaches maximum number of iterations t_maxOr iteration stopping conditionIt is whether true, if The number of iterations has reached maximum number of iterations t_maxOr iteration stopping conditionIt sets up, then stops changing In generation, obtains the solution value of variable M, then executes step 7；Otherwise, continue iteration；

Wherein, 1≤t≤t_max, t_maxIndicate maximum number of iterations, t_max>=1000, the initial value of variable M is that dimension is 2 × 3K's Null matrix, Z^t-1Indicate the value of variable Z after the t-1 times iteration, the initial value of variable Z is equal to the initial value of variable M, E^t-1Indicate the The value of variable E after t-1 iteration, the initial value of variable E are the null matrix that dimension is 2 × P, G^t-1Become after indicating the t-1 times iteration The value of G is measured, the initial value of variable G is the matrix being averaged to W by column, Y^t-1Indicate the value of variable Y after the t-1 times iteration, The initial value of variable Y is equal to the initial value of variable M,Corresponding variable M after indicating the t times iteration₁,…, M_i,…,M_KValue, M_i ^t-1Indicate variable M after the t-1 times iteration_iValue,WithIt is corresponding to indicate M_i ^t-1Through unusual Unitary matrice, the unitary matrice on the right, feature value vector on the left side that value obtains after decomposing,ForTransposition, diag () Indicate diagonal matrix,Indicate vectorIn L₁Projection on norm unit ball, (G^t-1)^TFor G^t-1's Transposition, D^TFor the transposition of D, I₂The unit matrix for indicating 3K × 3K, enables X=W-Z^tD-(G^t-1)^T, S_β(X) it indicates to each of X Element carries out soft-threshold calculating, and the formula for carrying out soft-threshold calculating to the i-th ' row jth ' column element in X is sign (X_i'j') (X_i'j'-β)₊, i'=1,2,1≤j'≤P, sign (X_i'j') indicate to seek X_i'j'Symbol,