CN106485750A - A kind of estimation method of human posture based on supervision Local Subspace - Google Patents

Abstract

A kind of estimation method of human posture based on supervision Local Subspace of the disclosure of the invention, belongs to technical field of computer vision, is related to estimation method of human posture.The method sets up Local Linear Model from the training set of sparse and nonuniform sampling, solves versatility and robustness problem that conventional learning algorithm is subjected to well, and reducing in estimation procedure is affected on estimated result by sparse and non-homogeneous training sample.And certain improvement is carried out to basic algorithm during Algorithm for Training, while accuracy is ensured, operation efficiency has been greatly enhanced, therefore it can be better achieved the task of real-time body's Attitude estimation.

Description

A Human Pose Estimation Method Based on Supervised Local Subspace

technical field

The invention belongs to the technical field of computer vision and relates to a human body pose estimation method.

Background technique

With the rapid development of human-computer interaction technology in today's society, the natural and multi-modal interaction between humans and machines has become the main way of interaction between humans and machines. The first problem that humans encounter is that machines need to be able to correctly recognize and understand human behavior. It is in this context that attitude estimation has been proposed. It is one of the important technologies of human-computer interaction at present. It can be applied in fields such as human motion analysis, virtual reality, intelligent monitoring, and game production. The potential huge application value of human body pose estimation has attracted extensive attention from academia and industry. Existing human pose estimation work can be divided into two categories: model-free and model-based methods.

Model-free human pose estimation methods can be divided into learning-based methods and sample-based methods.

(1) Learning-based method: use the training samples to learn a mapping from the image feature space to the human body pose space. If the image features are extracted from the new observation image and input into the mapping learned from the training samples, it can be estimated The corresponding human posture. For example, in the document Ankur Agarwal, Bill Triggs. 3D human pose from silhouettes by relevance vector regression, Computer Vision and Pattern Recognition, vol.2, no.2, pp.882-888, 2004, the author uses the shape context of the human body contour as a feature, and adopts the correlation As a regressor, the vector machine uses the sparse Bayesian nonlinear regression method to learn a compact mapping, maps the feature space to the pose space, and directly outputs the corresponding human pose-related values for the input features. In the reference Romer Rosales, Assilis Athitsos, Leonid Sigal, et al.3D Hand PoseReconstruction Using Specialized Mappings, ICCV, vol.1, pp.378-385, 2001, the input space is divided into many simple small areas, here Each small area has a corresponding mapping function, and a feedback matching mechanism is used to reconstruct the pose. Since the training data range is small, the mapping function has a better fitting effect, so this method can greatly improve the estimation accuracy to a certain extent. Although the learning-based method is fast, does not require special initialization, has a small storage cost, and does not need to save the sample database, the estimation results of the learning-based method are often greatly affected by the size of the training sample.

(2) Sample-based method: First, a template library needs to be established, which stores a large number of features and training samples of human body poses. When the estimated test image is input, the corresponding features are extracted and compared with the samples in the template library by a certain metric, that is, training samples similar to the image to be estimated are found, and finally the human body pose of the test image is estimated using the nearest neighbor algorithm. Human body poses are very complex, and the image feature descriptors projected by different poses may be very similar, that is, there is a one-to-many relationship between feature descriptors and pose space. For example, in the literature Nicholas R.Howe.SilhouetteLook up for Automatic Pose Tracking, Computer Vision and Pattern RecognitionWorkshop, pp.15-22, 2004, the author retrieved multiple close candidate samples from the template database, and then used the temporal similarity constraint Get the best match among these samples. The sample-based method must have enough samples to cover all possible poses of the human body, but because the human pose is too complex, limited samples are difficult to cover the entire human pose space, so the sample-based method is only suitable for the estimation of specific poses.

The model-based method divides the human body into some interrelated parts, uses the graph model to represent the human body structure, and uses the graph reasoning method to optimize the human body posture, that is, the prior human body model is used in the process of human body pose estimation, and the parameters of the model Also updated as the current state changes. Model-based human pose estimation mainly consists of three parts: graph model, optimization algorithm, and component observation model. The graph model is used to represent the constraint relationship between component connections, and the tree model is the most commonly used model. The tree model is defined according to the connection between components, so it is relatively intuitive. The observation model builds a model for the appearance of human body parts, which is used to measure the image similarity of human body parts, so as to determine the specific position of human body parts.

The optimization algorithm is to use the established graph model and component observation model to estimate the human body pose. Among them, belief propagation is a commonly used algorithm, but since the state vector dimension of human body parts is relatively high in human body pose estimation, it is unrealistic to directly use the belief propagation algorithm. In the literature Deva Ramanan.Learning to parse images of articulated bodies, Neural Information Processing Systems, pp.1129-1136, 2006, the author proposed the sum-product algorithm, which inherits the message passing mechanism, but introduces the global probability density function by introducing a factor graph It is decomposed into the product of several local probability density functions, which extends the scope of the algorithm to undirected graphs (such as conditional random fields), but the sum product algorithm still has a limitation, it can only guarantee the algorithm on the acyclic factor graph. convergence.

The model-based human pose estimation method has strong versatility, and also reduces the storage cost of training samples. For human body models, people can easily use prior knowledge to solve self-occlusion and other occlusion problems, and the estimation accuracy is relatively high, which is suitable for applications in fields such as human body posture analysis, but the disadvantages are also obvious: (1) The optimization speed is relatively low. It is slow, and generally cannot meet the real-time requirements; 2) The quality of initialization has a great influence on the result of attitude optimization.

Contents of the invention

The task of the present invention is to provide a human body pose estimation method based on supervised local subspace, which belongs to the estimation method based on learning in the model-free method. It solves the generality and robustness problems suffered by previous learning algorithms, and reduces the influence of sparse and non-uniform training samples on the estimation results during the estimation process. And in the algorithm training process, the basic algorithm has been improved to a certain extent. While ensuring the accuracy, the calculation efficiency has been greatly improved, so it can better realize the task of real-time human pose estimation.

In order to describe the content of the present invention conveniently, some terms are defined first.

Definition 1: Human-computer interaction. Human-computer interaction is a discipline that studies the interactive relationship between systems and users. A system can be a variety of machines, or it can be a computer system and software. The human-computer interface usually refers to the part visible to the user. The user communicates with the system through the human-computer interaction interface and performs operations.

Definition 2: Human body pose. The posture of the human body is divided into two-dimensional and three-dimensional situations. Two-dimensional human body posture refers to a description of the distribution of human joints on the two-dimensional image plane. Lines or rectangles are usually used to describe the projection of human joints on the two-dimensional image plane, the length and angle distribution of line segments or the size and direction of rectangles. It represents the two-dimensional posture of the human body, and there is no ambiguity in the two-dimensional posture; the three-dimensional human posture refers to the position and angle information of the human target in the real three-dimensional space, and the joint tree model is usually used to express the estimated posture. There are also some studies The latter adopts a more complex model, and the three-dimensional pose is usually obtained through the method of model back projection.

Definition 3: Overfitting. In machine learning, when the classification or regression model is trained through training samples, the output value obtained by the model is basically consistent with the real target value, but the output value obtained by the model on the test sample set is quite different from the target value. The phenomenon in which assumptions become overly complex due to consistent assumptions is called overfitting.

Definition 4: Foreground. The foreground refers to the person or scene in the image that is close to the camera.

Definition 5: Background. The background refers to the scenery behind the subject in the image, which can express the space-time environment in which tasks and things are located.

Definition 6: Shape context features. Shape context was proposed in 2002 and was originally used to detect matching points between object shapes. The shape context descriptor can make full use of the context information around the pixels to describe the shape features of the inner area of the image well, and it has very good robustness in the shape matching problem. The basic principle of the descriptor is: for a given image, first use an edge feature extraction algorithm (such as Canny edge detector) to detect its edge information; then sample the edge pixels to extract a series of feature points (These feature points can be points on the inner edge or points on the outer edge, and do not need to be points on the edge curve with the greatest curvature, and can be obtained by uniform sampling.); for each feature point, its Establish a polar coordinate system for the origin, divide the surrounding space into a series of fan-shaped areas according to the change of the angle, and divide the surrounding space into a series of concentric circles according to the size of the radius, and count the number of feature points distributed in each area; ( See (c) in Fig. 1) Finally, a corresponding vector is established according to the statistically obtained data, which is the corresponding pixel shape context feature vector.

Definition 7: K-means algorithm. The K-means algorithm is an iterative algorithm based on distance, which classifies m observation samples into k clusters, so that each observation sample is farther away from the center point of the cluster it is in than the distance from other cluster center points smaller. The specific process is:

1) Suppose the sample space is Among them, m represents the number of samples, and n represents the dimensionality of each training sample. Then randomly select k cluster center points from the training sample space, respectively

2) Repeat the following process until convergence:

For each sample i∈[1,m], calculate which cluster it should belong to:

For each cluster j∈[1,k], recalculate the center of the cluster:

Wherein, 1{c _i =j} is an indicative function, and the value of the function is equal to 1 when the condition of c _i =j is satisfied, otherwise it is 0. After several iterations, the algorithm reaches convergence, that is, the cluster center does not change or changes very little. The test can get the k cluster center points we want and the cluster to which each sample belongs.

Definition 8: BVH format. BVH files contain the character's bone and limb joint rotation data. BVH is a common human feature animation file format, which is widely supported by various animation production software popular today. Typically available from motion capture hardware that records the movement of human actions.

According to a kind of human body posture estimation method based on supervision local subspace of the present invention, it comprises the following steps:

Step 1: For the original image that requires human pose estimation, remove the background and obtain the human body contour information to achieve the effect of highlighting the foreground;

Step 2: Binarize the image obtained in step 1, and then extract shape context features from the above-mentioned human outline image. The relevant parameters of the algorithm for extracting shape context features are the number of sampling points is 200, and the circular polar coordinates are divided into 12 A fan-shaped area, the radius is divided into 5 parts; therefore, for each training sample, its corresponding shape context feature is a 60*200-dimensional matrix, that is, 200 60-dimensional shape context vectors;

Step 3: Use the dimensionality reduction operation to reduce the shape context feature of each picture to 100 dimensions to obtain the image feature X;

Step 4: The image feature X obtained in step 3 is reconstructed in the local subspace through the attitude angle Θ, the specific formula is

Among them, f is the mapping function from the human body pose space to the image feature space, is the set of parameters corresponding to the i-th local subspace, is the center of the subspace, is the principal component of the tangent space, is the human body posture angle corresponding to the center of the i-th subspace, m is the number of subspaces, and d is the input feature dimension of the sample;

Step 5: First-order Taylor expansion of the approximate function f(Θ) in step 4; each training sample is reconstructed from a certain local subspace, if the training sample x _p with the attitude angle θ _p is in the parameter In the determined subspace, x _p is approximated as:

x _p ≈ c _i +G _i △θ _pi

in, define at the same time And N(θ _p ) represents the subspace index number adjacent to θ _p ;

Step 6: Determine the error function of the algorithm according to step 5 as

The first item is the weighted sum of the reconstruction errors caused by the reconstruction of the neighbor subspace for each training sample (x _p , θ _p ), and the selection of the neighbor subspace is based on the attitude angle corresponding to the center of the subspace The size relationship with θ _p Euclidean distance; the second term is regularized, and the mean value of each subspace is reconstructed through the nearest neighbor subspace; this step ensures the smooth change of subspace parameters, and can be used from sparse non-uniform Estimated from the data; where λ=(n/m) ² is a regularization parameter, which is equal to the number of training samples n divided by the square of the number of subspaces m, w _pi defines the weight of each adjacent subspace on the data sample The weight of the structure, the specific formula is:

in, are the measurement angle θ _p and The positive value function of the similarity between , the function expression is:

Step 7: Order represents the center of the subspace, Represents the basis of the subspace, where d is the dimension of the input, in this patent, d is 100, m is the number of subspaces, and then the closed solution algorithm is used to optimize the calculation to obtain C and G;

Step 8: For the image feature x _t of a new test sample point, in order to improve efficiency; this algorithm uses the following two steps to determine the nearest neighbor subspace of the test sample from the subspace:

(1) First find 2 |Γ _t | candidate subspaces; |Γ _t | is the number of adjacent subspaces we set, generally 2-16. The centers _ci of these subspaces are closest to x _t in the input space, then according to the formula Calculate θ _ti , where is the basis of a selected neighbor subspace, is the center of a selected neighboring subspace; θ _ti represents the attitude angle θ _ti obtained by reconstructing a new test data point x _t according to subspace i;

(2) Compare the magnitude of the reconstruction error, select |Γ _t | neighboring subspaces from 2|Γ _t | candidates, and record θ _ti corresponding to the minimum reconstruction error as the test data point x _t corresponding to The attitude angle θ _t0 of , where the formula of the reconstruction error is:

Finally, the optimal θ _t is minimized by the formula , where the weight w _ti is calculated by the above θ _t0 , the specific formula is Finally, the best attitude angle θ _t corresponding to the test sample point x _t can be obtained:

Step 9: Parse the obtained θ _t into text in BVH format and represent it as a corresponding human body pose image.

Further, the specific steps of the step 3 are:

First, the shape context feature matrices in all training samples are merged from left to right, and then the K-means algorithm is used to obtain 100 60-dimensional vectors. We call these vectors the cluster centers of the data space. Finally, the 200 shape context vectors of each sample vote for the 100 cluster centers with Gaussian weights. The specific voting method is that the closer the Euclidean distance between the shape context and a certain cluster center is, the closer it is to 1. The farther away it tends to 0. Finally, the image feature X required in the following steps can be obtained, where the i-th column corresponds to the image feature of the i-th training sample and the feature of each sample is 100 dimensions.

Further, the specific steps of the step 7 are:

The specific method is: replace C and G with U=[C,G], then the error function in step 6 can be rewritten as:

where e∈1 ^1×d , W _p = diag{w _pi , i∈Γ _p }, W _j ＝diag{w _ji ,i∈Γ _j }, in addition, is a 0-1 selection matrix, the number of non-zero elements on the diagonal of the matrix is the sequence of the adjacent subspace corresponding to the training sample x _p , and the rest of the elements are all 0, for example, the number of the adjacent subspace corresponding to the training sample x _p are 2, 4, 5, 6 respectively, then in There are and only elements S ₂₂ , S ₄₄ , S ₅₅ , and S ₆₆ are 1, and all other elements are 0; It is also a 0-1 selection matrix, the number of non-zero elements on the diagonal of the matrix is the sequence of the adjacent subspace corresponding to the center c _j of each subspace; s is an m×m identity matrix, and s _j is an m-dimensional Column vector, I is an m×m identity matrix, E(U) is a convex quadratic function about U. Finally, the expression for the solution of U is: U=A(B ^(x) +B ^(c) ) ^-1

Among them, A, B ^(x) , B ^(c) , C, G are respectively:

C=U[1:d,1:m]

G=U[1:d,1+m:56×m]

After several iterations of optimization, C and G can quickly converge, and C and G can now obtain an approximate optimal solution of the sum.

The innovation of the present invention is:

This patent proposes to introduce an estimation method based on supervised local subspace in the problem of human pose estimation, which belongs to the estimation method based on learning in model-free. It solves the generality and robustness problems suffered by previous learning algorithms, and reduces the influence of sparse and non-uniform training samples on the estimation results during the estimation process. At the same time, a closed solution algorithm is proposed to optimize the local subspace parameters, which can reach convergence faster than the previous alternate optimization algorithm. While ensuring the accuracy, it greatly improves the operation efficiency, so it can better realize real-time The task of human pose estimation. In addition, since the algorithm is a generative model, it can also handle image noise well.

Description of drawings:

Figure 1 Shape context feature description;

Fig. 2 Distribution of 200 sampling points when extracting shape context features from human body contours;

Figure 3 is the outline of the human body after removing the background;

Figure 4 shows the pose estimation results.

Figure 1 (c) shows the shape information of the sample point at the origin of the polar coordinates, and the points adjacent to it (within the range covered by the polar coordinates) fall in different small grids, which represent different relative vectors. These relative vectors become the shape context of this point; at the same time, the shape context histograms (d) and (e) of the diamond point and square point in figure (a) and figure (b) are basically the same, while the triangle point The shape context is different, which is basically consistent with our actual observation. The left side of Figure 4 is the human body contour information we input, and the right side is the human body pose we estimated and analyzed in BVH format.

detailed description:

A kind of human body pose estimation method based on supervised local subspace of the present invention is described in conjunction with accompanying drawing of description, and it comprises the following steps:

Step 1: For the original image that requires human pose estimation, remove the background and obtain the human body contour information to achieve the effect of highlighting the foreground (see Figure 3);

Step 2: Extract shape context features from the above-mentioned human body contour image, wherein the relevant parameters of the algorithm for extracting shape context features are respectively that the number of sampling points is 200 (see Figure 2), the circular polar coordinates are divided into 12 fan-shaped areas, and the radius is divided into is 5 copies; therefore, for each training sample, its corresponding shape context feature is a 60*200-dimensional matrix, that is, 200 60-dimensional shape context vectors;

Step 3: Use the dimensionality reduction operation to reduce the shape context feature of each picture to 100 dimensions to obtain the image feature X; the specific operation is: firstly merge the shape context feature matrices in all training samples from left to right, and then use K The mean algorithm obtains 100 60-dimensional vectors, which we call the cluster centers of the data space. Finally, the 200 shape context vectors of each sample vote for the 100 cluster centers with Gaussian weights. The specific voting method is that the closer the Euclidean distance between the shape context and a certain cluster center is, the closer it is to 1. The farther away it tends to 0. Finally, the image feature X required in the following steps can be obtained, where the i-th column corresponds to the image feature of the i-th training sample and the feature of each sample is 100 dimensions.

In the process of selecting 100 center points in this step, any clustering algorithm can be used. We use the K-Means algorithm because it is simple and fast to run, but because it is the Gaussian weight in voting for the shape context feature vector The corresponding mean u and variance σ ² are provided.

Step 4: The image feature X obtained in step 3 is reconstructed in the local subspace through the attitude angle Θ, the specific formula is

Among them, f is the mapping function from the human body pose space to the image feature space, is the set of parameters corresponding to the i-th local subspace, is the center of the subspace, is the principal component of the tangent space, is the human body posture angle corresponding to the center of the i-th subspace, m is the number of subspaces, and d is the input feature dimension of the sample;

Step 5: First-order Taylor expansion of the approximate function f(Θ) in step 4; each training sample is reconstructed from a certain local subspace, if the training sample x _p with the attitude angle θ _p is in the parameter In the determined subspace, x _p is approximated as:

x _p ≈ c _i +G _i △θ _pi

in, define at the same time And N(θ _p ) represents the subspace index number adjacent to θ _p ;

Step 6: Determine the error function of the algorithm according to step 5 as

The first item is the weighted sum of the reconstruction errors caused by the reconstruction of the neighbor subspace for each training sample (x _p , θ _p ), and the selection of the neighbor subspace is based on the attitude angle corresponding to the center of the subspace The size relationship with θ _p Euclidean distance; the second term is regularized, and the mean value of each subspace is reconstructed through the nearest neighbor subspace; this step ensures the smooth change of subspace parameters, and can be used from sparse non-uniform Estimated from the data; where λ=(n/m) ² is a regularization parameter, which is equal to the number of training samples n divided by the square of the number of subspaces m, w _pi defines the weight of each adjacent subspace on the data sample The weight of the structure, the specific formula is:

in, are the measurement angle θ _p and The positive value function of the similarity between , the function expression is:

Step 7: Order represents the center of the subspace, Represents the basis of the subspace, where d is the dimension of the input, in this patent, d is 100, m is the number of subspaces, and then the closed solution algorithm is used to optimize the calculation to obtain C and G;

The specific method is: replace C and G with U=[C,G], then the error function in step 6 can be rewritten as:

where e∈1 ^1×d , W _p = diag{w _pi , i∈Γ _p }, W _j ＝diag{w _ji ,i∈Γ _j }, in addition, is a 0-1 selection matrix, the number of non-zero elements on the diagonal of the matrix is the sequence of the adjacent subspace corresponding to the training sample x _p , and the rest of the elements are all 0, for example, the number of the adjacent subspace corresponding to the training sample x _p are 2, 4, 5, 6 respectively, then in There are and only elements S ₂₂ , S ₄₄ , S ₅₅ , and S ₆₆ are 1, and all other elements are 0; It is also a 0-1 selection matrix, the number of non-zero elements on the diagonal of the matrix is the sequence of the adjacent subspace corresponding to the center c _j of each subspace; s is an m×m identity matrix, and s _j is an m-dimensional Column vector, I is an m×m identity matrix, E(U) is a convex quadratic function about U. Finally, the expression of the solution of U is U=A(B ^(x) +B ^(c) ) ^-1

Among them, A, B ^(x) , B ^(c) , C, G are respectively:

C=U[1:d,1:m]

G=U[1:d,1+m:56×m]

After several iterations of optimization, C and G can quickly converge, and C and G can now obtain an approximate optimal solution of the sum.

Step 8: For the image feature x _t of a new test sample point, in order to improve efficiency; this algorithm uses the following two steps to determine the nearest neighbor subspace of the test sample from the subspace:

(1) First find 2 |Γ _t | candidate subspaces; |Γ _t | is the number of adjacent subspaces we set, generally 2-16. The centers _ci of these subspaces are closest to x _t in the input space, then according to the formula Calculate θ _ti , where is the basis of a selected neighbor subspace, is the center of a selected neighboring subspace; θ _ti represents the attitude angle θ _ti obtained by reconstructing a new test data point x _t according to subspace i;

(2) Compare the magnitude of the reconstruction error, select |Γ _t | neighboring subspaces from 2|Γ _t | candidates, and record θ _ti corresponding to the minimum reconstruction error as the test data point x _t corresponding to The attitude angle θ _t0 of , where the formula of the reconstruction error is:

Finally, the optimal θ _t is minimized by the formula , where the weight w _ti is calculated by the above θ _t0 , the specific formula is Finally, the best attitude angle θ _t corresponding to the test sample point x _t can be obtained:

Step 9: Parse the obtained θ _t into text in BVH format and represent it as a corresponding human body pose image. (See Figure 3)

According to the method of the present invention, at first utilize Matlab language to write the program based on supervised local subspace learning algorithm; Then select some pictures to train the algorithm, solve the subspace center and basis vector parameters; Finally, the pictures containing the human body that need to be estimated for attitude Input into the algorithm, the human body posture predicted by the algorithm can be obtained. The method of the present invention can be used for pose estimation of a human body in a natural scene, such as behavior detection and human-computer interaction. Finally, we tested our algorithm on the Poser dataset database, which selected 1691 of them as training samples and 418 as testing samples. When the number of subspaces is 61 and the number of adjacent subspaces is 8 in the algorithm, the best test error can be obtained on this data set is 7.125°.

Claims (3)

1. A human body pose estimation method based on supervised local subspace, which comprises the following steps: Step 1: For the original image that requires human pose estimation, remove the background and obtain the human body contour information to achieve the effect of highlighting the foreground; Step 2: Binarize the image obtained in step 1, and then extract shape context features from the above-mentioned human outline image. The relevant parameters of the algorithm for extracting shape context features are the number of sampling points is 200, and the circular polar coordinates are divided into 12 A fan-shaped area, the radius is divided into 5 parts; therefore, for each training sample, its corresponding shape context feature is a 60*200-dimensional matrix, that is, 200 60-dimensional shape context vectors; Step 3: Use the dimensionality reduction operation to reduce the shape context feature of each picture to 100 dimensions to obtain the image feature X; Step 4: The image feature X obtained in step 3 is reconstructed in the local subspace through the attitude angle Θ, the specific formula is $\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&Theta;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&ap;</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; <</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; G</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ,</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ></span><span\; class="notranslate">\; ,</span><span\; class="notranslate">\; ...</span><span\; class="notranslate">\; ,</span><span\; class="notranslate">\; <</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; G</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; ,</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}}_{\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; ></span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; \&Theta;</span>}<span\; class="notranslate">\; )</span>$ Among them, f is the mapping function from the human body pose space to the image feature space, is the set of parameters corresponding to the i-th local subspace, is the center of the subspace, is the principal component of the tangent space, is the human body posture angle corresponding to the center of the i-th subspace, m is the number of subspaces, and d is the input feature dimension of the sample; Step 5: Perform first-order Taylor expansion on the approximate function f(Θ) in step 4; reconstruct each training sample from a certain local subspace, if the training sample x _p with attitude angle θ _p is in the parameter In the determined subspace, x _p is approximated as: x _p ≈ c _i +G _i △θ _pi in, define at the same time And N(θ _p ) represents the subspace index number adjacent to θ _p ; Step 6: Determine the error function of the algorithm according to step 5 as $\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; G</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; \&Element;</span>{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}}_{\mathrm{<span\; class="notranslate">\; p</span>}}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; w</span>}}_{\mathrm{<span\; class="notranslate">\; p</span>}\mathrm{<span\; class="notranslate">\; i</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; p</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; G</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; \&Delta;\&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; p</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>{<span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&lambda;</span>}\underset{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; m</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; \&Element;</span>{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; w</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}\mathrm{<span\; class="notranslate">\; i</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; G</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; \&Delta;\&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>{<span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}$ The first item is the weighted sum of the reconstruction errors caused by the reconstruction of the neighbor subspace for each training sample (x _p , θ _p ), and the selection of the neighbor subspace is based on the attitude angle corresponding to the center of the subspace The size relationship with θ _p Euclidean distance; the second term is regularized, and the mean value of each subspace is reconstructed through the nearest neighbor subspace; this step ensures the smooth change of subspace parameters, and can be used from sparse non-uniform Estimated from the data; where λ=(n/m) ² is a regularization parameter, which is equal to the number of training samples n divided by the square of the number of subspaces m, w _pi defines the weight of each adjacent subspace on the data sample The weight of the structure, the specific formula is: ${\mathrm{<span\; class="notranslate">\; w</span>}}_{\mathrm{<span\; class="notranslate">\; p</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; \&psi;</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; p</span>}}<span\; class="notranslate">\; ,</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; q</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}}_{\mathrm{<span\; class="notranslate">\; p</span>}}<span\; class="notranslate">\; |</span>}\mathrm{<span\; class="notranslate">\; \&psi;</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; p</span>}}<span\; class="notranslate">\; ,</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}}_{\mathrm{<span\; class="notranslate">\; q</span>}}<span\; class="notranslate">\; )</span>}$ in, are the measurement angle θ _p and The positive value function of the similarity between , the function expression is: $\mathrm{<span\; class="notranslate">\; \&psi;</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; p</span>}}<span\; class="notranslate">\; ,</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; p</span>}}<span\; class="notranslate">\; -</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; |</span>{<span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}$ Step 7: Order represents the center of the subspace, Represents the basis of the subspace, where d is the dimension of the input, in this patent, d is 100, m is the number of subspaces, and then the closed solution algorithm is used to optimize the calculation to obtain C and G; Step 8: For the image feature x _t of a new test sample point, in order to improve efficiency; this algorithm uses the following two steps to determine the nearest neighbor subspace of the test sample from the subspace: (1) First find 2 |Γ _t | candidate subspaces; |Γ _t | is the number of adjacent subspaces we set, generally 2-16. The centers _ci of these subspaces are closest to x _t in the input space, then according to the formula Calculate θ _ti , where is the basis of a selected neighbor subspace, is the center of a selected neighboring subspace; θ _ti represents the attitude angle θ _ti obtained by reconstructing a new test data point x _t according to subspace i; (2) Compare the magnitude of the reconstruction error, select |Γ _t | neighboring subspaces from 2|Γ _t | candidates, and record θ _ti corresponding to the minimum reconstruction error as the test data point x _t corresponding to The attitude angle θ _t0 of , where the formula of the reconstruction error is: $\mathrm{<span\; class="notranslate">\; e</span>}\mathrm{<span\; class="notranslate">\; r</span>}\mathrm{<span\; class="notranslate">\; r</span>}\mathrm{<span\; class="notranslate">\; o</span>}\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; t</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; \&lsqb;</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; G</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; t</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; -</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&rsqb;</span><span\; class="notranslate">\; |</span>{<span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}$ Finally, the optimal θ _t is minimized by the formula , where the weight w _ti is calculated by the above θ _t0 , the specific formula is Finally, the best attitude angle θ _t corresponding to the test sample point x _t can be obtained: ${\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; t</span>}}<span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; (</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; \&Element;</span>{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}}_{\mathrm{<span\; class="notranslate">\; t</span>}}}{<span\; class="notranslate">\; \&Sigma;</span>}{\mathrm{<span\; class="notranslate">\; w</span>}}_{\mathrm{<span\; class="notranslate">\; t</span>}\mathrm{<span\; class="notranslate">\; i</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; G</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}{\mathrm{<span\; class="notranslate">\; G</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; \&Element;</span>{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}}_{\mathrm{<span\; class="notranslate">\; t</span>}}}{<span\; class="notranslate">\; \&Sigma;</span>}{\mathrm{<span\; class="notranslate">\; w</span>}}_{\mathrm{<span\; class="notranslate">\; t</span>}\mathrm{<span\; class="notranslate">\; i</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; G</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; t</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; G</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span>$ Step 9: Parse the obtained θ _t into text in BVH format and represent it as a corresponding human body pose image. 2. a kind of human body pose estimation method based on supervision local subspace as claimed in claim 1, is characterized in that the concrete steps of described step 3 are: First, the shape context feature matrices in all training samples are merged from left to right, and then the K-means algorithm is used to obtain 100 60-dimensional vectors. We call these vectors the cluster centers of the data space. Finally, the 200 shape context vectors of each sample vote for the 100 cluster centers with Gaussian weights. The specific voting method is that the closer the Euclidean distance between the shape context and a certain cluster center is, the closer it is to 1. The farther away it tends to 0. Finally, the image feature X required in the following steps can be obtained, where the i-th column corresponds to the image feature of the i-th training sample and the feature of each sample is 100 dimensions. 3. a kind of human pose estimation method based on supervising local subspace as claimed in claim 1, is characterized in that the concrete steps of described step 7 are: The specific method is: replace C and G with U=[C,G], then the error function in step 6 can be rewritten as: $\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}}{<span\; class="notranslate">\; \&Sigma;</span>}}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; p</span>}}{\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; UV</span>}}_{\mathrm{<span\; class="notranslate">\; p</span>}}{\mathrm{<span\; class="notranslate">\; S</span>}}_{{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}}_{\mathrm{<span\; class="notranslate">\; p</span>}}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; p</span>}}<span\; class="notranslate">\; |</span>{<span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; f</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&lambda;</span>}\underset{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; m</span>}}{<span\; class="notranslate">\; \&Sigma;</span>}}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; UV</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; |</span>{<span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; f</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}$ where e∈1 ^1×d , W _p = diag{w _pi , i∈Γ _p }, W _j ＝diag{w _ji ,i∈Γ _j }, in addition, is a 0-1 selection matrix, the number of non-zero elements on the diagonal of the matrix is the sequence of the adjacent subspace corresponding to the training sample x _p , and the rest of the elements are all 0, for example, the number of the adjacent subspace corresponding to the training sample x _p are 2, 4, 5, 6 respectively, then in There are and only elements S ₂₂ , S ₄₄ , S ₅₅ , and S ₆₆ are 1, and all other elements are 0; It is also a 0-1 selection matrix, the number of non-zero elements on the diagonal of the matrix is the sequence of the adjacent subspace corresponding to the center c _j of each subspace; s is an m×m identity matrix, and s _j is an m-dimensional Column vector, I is an m×m identity matrix, E(U) is a convex quadratic function about U. Finally, the expression for the solution of U is: U=A(B ^(x) +B ^(c) ) ^-1 Among them, A, B ^(x) , B ^(c) , C, G are respectively: $\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}^{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; p</span>}}{\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; p</span>}}{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; p</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}{\mathrm{<span\; class="notranslate">\; S</span>}}_{{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}}_{\mathrm{<span\; class="notranslate">\; p</span>}}}^{\mathrm{<span\; class="notranslate">\; T</span>}}{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; p</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}$ ${\mathrm{<span\; class="notranslate">\; B</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}^{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; p</span>}}{\mathrm{<span\; class="notranslate">\; S</span>}}_{{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}}_{\mathrm{<span\; class="notranslate">\; p</span>}}}{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; p</span>}}{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; p</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}{\mathrm{<span\; class="notranslate">\; S</span>}}_{{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}}_{\mathrm{<span\; class="notranslate">\; p</span>}}}^{\mathrm{<span\; class="notranslate">\; T</span>}}{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; p</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}$ ${\mathrm{<span\; class="notranslate">\; B</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; c</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; \&lambda;\&Sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}^{\mathrm{<span\; class="notranslate">\; m</span>}}{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}$ C=U[1:d,1:m] G=U[1:d,1+m:56×m] After several iterations of optimization, C and G can quickly converge, and C and G can now obtain an approximate optimal solution of the sum.