A three-dimensional human body precise modeling method based on size constraint optimization

Abstract

The application discloses a three-dimensional human body precise modeling method based on size constraint optimization. First, target model measurement sizes including height, three-circumference and the like are collected; missing size values of a user which are not input are predicted by using a multivariate imputation method MICE through the model measurement size values; a human body model close to an output parameter is constructed as an initialization model; a deformation optimization equation with constraints is constructed by using the measurement sizes and the predicted sizes and combining the initialization human body model, the size constraints are convexified, and the optimization equation is solved by using a augmented Lagrange multiplier method, and finally a human body model meeting size requirements is generated. The method convexifies complex nonlinear non-convex size constraints, has high solution accuracy, is simple, efficient and robust.

Description

Technical Field

[0001] This invention relates to the field of computer graphics and mesh model optimization, specifically to, but not limited to, a method for accurate 3D human body modeling based on size constraint optimization, in order to achieve high-precision parametric human body modeling. Background Technology

[0002] 3D human body reconstruction technology plays a vital role in film and television production, fashion design, game development, and virtual try-on. This makes 3D human body modeling technology one of the hottest research topics in computer graphics.

[0003] There are two main types of 3D human body modeling: hardware-based non-parametric modeling and software-based parametric modeling.

[0004] The main limitations of non-parametric modeling lie in its high cost and complex operation. Although it can theoretically achieve high-precision human figures, its expense is still significantly higher than that of parametric modeling. Currently, non-parametric modeling techniques mainly include human body modeling using high-precision scanners, human body modeling using Kinect, and multi-view image human body modeling. All of these methods involve scanning the human body with high-precision equipment to obtain point cloud data and generate a human body model.

[0005] Compared to non-parametric methods that use vertices or voxels to represent the 3D human body, parametric methods use a set of low-dimensional vectors (i.e., parameters) to represent the 3D human body model. Clearly, compared to non-parametric modeling, parametric modeling techniques can generate a large number of human body models in real time, simply and conveniently. This makes them more suitable for mass-market applications such as clothing design, game development, and virtual try-on.

[0006] However, parametric modeling still suffers from low model accuracy. Currently, non-parametric modeling primarily includes the SCAPE model proposed by Anguellov et al., the SMPL model proposed by Loper et al., and the global nonlinear model proposed by Rupprecht et al. using random forests. Summary of the Invention

[0007] To address the issue of low accuracy in models generated by data-driven methods, this invention proposes a precise 3D human body modeling method based on size constraint optimization. Although many size optimization-based methods exist for post-processing the modeling results, they still struggle to obtain accurate human body models. Solving this highly nonlinear minimization problem is extremely difficult. This method significantly improves the accuracy of human body modeling, demonstrating superior performance compared to existing methods. The objective of this invention is to provide a rapid and convenient method for generating high-precision human body models.

[0008] The objective of this invention is achieved through the following technical solution: Firstly, this invention provides a method for accurate 3D human body modeling based on size constraint optimization, the method comprising the following steps:

[0009] Step 1: Collect and measure the dimensions Ω={ω1,ω2,...,ω of n target models. n}, where Ω is a set of human body size parameters used to reconstruct a three-dimensional human body model that conforms to size constraints;

[0010] Step 2: Using the measurement dimension set obtained in Step 1, use the multivariate interpolation method (MICE) to fill in the missing dimension parameters and generate a complete dimension parameter set;

[0011] Step 3: Based on the complete size parameters obtained in Step 2, construct an initial human body model. There are two ways to construct the initial human body model: one is to directly search for the human body model with the closest size to the input size in the model library, where the models in the database are obtained from open-source human body datasets; the other is to use machine learning methods to construct a mapping function from size to human body parameters to predict the human body model under a given size, where the human body parameters are represented as follows: for the open-source human body dataset, the vertex coordinates of each human body are stretched into a vector in sequence, all vectors are subjected to principal component analysis (PCA method), and the first K basis are taken. The human body parameters are represented as the coefficients corresponding to the basis, and the human body model is reconstructed through the linear combination of the basis.

[0012] Step 4: Construct the numerical optimization equation for human body deformation with size constraints, as shown in the following expression:

[0013] minE x (X,X′) (1)

[0014] stF(X)=0

[0015] Among them, E s This measures the geometric similarity between two models. X′ and X represent the vertex coordinate matrices of the initial human body model and the target human body model, respectively, with a dimension of 3×|V|, where |V| is the number of vertices in the human body mesh model; the constraint F(X) for the human body dimensions is given by the equation F(X) = {f i (X)} represents the set of human body size constraint functions, f i (X)=l(w i ), i = 1, 2, ..., n, l (w i ) represents the i-th dimension value;

[0016] Step 5: Make the nonlinear, non-convex size constraint F(X) = 0 convex and use the augmented Lagrange multiplier method to solve the optimization equation from step 4.

[0017] The constraint F(X) = 0 on human body dimensions is divided into three categories: Euclidean length, geodesic length, and a combination of both; the first two dimension constraints are expressed as follows:

[0018] Euclidean length dimension: ||X i -X j ||=l(d),d∈ε;

[0019] Geodetic length dimensions:

[0020] Where l(d) is the measured value of this dimension, X i ,X j Let ε be the coordinates of the two endpoints of the measurement point, and let ε be the Euclidean length dimension set; p i ,p i+1 δ represents the coordinates of two adjacent vertices along the length of the geodesic line, with both vertices falling on the edges or vertices of the mesh model; N represents the number of line segments contained in the geodesic line. When the geodesic line is not closed, N = m - 1; otherwise, when the curve is closed, N = m, where m is the number of vertices of the geodesic line.

[0021] The numerical optimization equation for human body deformation with size constraints is transformed into the following formula using the augmented Lagrange multiplier method:

[0022]

[0023] Where n is the number of size constraint terms, λ is the Lagrange multiplier, and its iterative update formula is:

[0024] λ=λ+μF(X) (3)

[0025] μ = min{10μ, 10 16} (4)

[0026] Step six: Iteratively solve the above augmented Lagrange equation to finally generate the target model X; the details are as follows:

[0027] 1) Calculate the coefficient matrix A formed by the Laplace matrix and the size constraints;

[0028] 2) Fix λ and μ, solve the linear system AX = B, and obtain the new human body model X;

[0029] 3) Fix X, and update λ and μ according to formulas (3) and (4);

[0030] 4) Repeat steps 1)-3) above until the L2 distance of the human body model matrix X obtained in two adjacent iterations is less than the threshold ε;

[0031] 5) Return the optimized model coordinate matrix X, which is the final accurate 3D human body model.

[0032] Furthermore, in step four, the objective function E represents the geometric similarity measure. S The expression is as follows:

[0033]

[0034] Where V is the number of model points, ΔX i That is, the position of point i in the original model, ΔX′ i This is the recalculated position of point i after considering the cotangent weights of surrounding points. Δ is the discrete cotangent Laplacian operator, used to represent local details of the model surface, with the aim of minimizing the deviation of the target model X from X′. Its expression is:

[0035]

[0036] Where, N i ω represents the index of the 1-ring neighborhood vertex of vertex i. ij Cotangent weight:

[0037]

[0038] Among them, α ij With β ij This represents the opposite angle of the two triangles associated with the grid edge (i,j).

[0039] Furthermore, in step five, the Euclidean length constraint in F(X) is locally convexized using the following formula:

[0040]

[0041] Among them l t Let t be the target length of the t-th Euclidean dimension.

[0042] Furthermore, in step five, for the geodesic dimension constraints and mixed dimension constraints in F(X), each of its N line segments is independently constrained. Thus, a geodesic constraint is transformed into the following N sub-constraints:

[0043]

[0044] in The target length of the i-th line segment is calculated as follows: Calculate the paths (polylines) corresponding to all dimensions other than Euclidean dimensions on the mean model in the model library; for each path, calculate the length of each line segment, denoted as l. i Then the target length of the line segment is: Among them l s With l tThese are the total length of the path in the mean model and the target size length, respectively. At this point, all constraints of F(X) become linear functions, so they can be uniformly represented as matrix form: CX-b=0, where C is the size constraint matrix corresponding to formulas (6) and (7), and b is the target size parameter set.

[0045] Furthermore, in step five, based on the convexity constraint, the augmented Lagrange multiplier method is used to transform the deformation optimization equation with size constraints (Equation 2) into the following quadratic function of X:

[0046]

[0047] Set the first-order partial derivative of the formula to 0, and transform it into a system of linear equations AX = B for solution.

[0048] The main advantages of this invention include: (a) high accuracy of the generated model with minimal error compared to the input constraint dimensions; (b) low computational cost and rapid model generation compared to other methods; and (c) robustness. This invention employs the augmented Lagrange multiplier method to handle equality constraints, achieving stable convergence. Furthermore, it transforms complex dimensional constraints into potential energy terms similar to a spring, thus converting a quartic function into a quadratic one. This method offers a degree of controllability while preserving the boundaries. Attached Figure Description

[0049] Figure 1 This is a flowchart of a three-dimensional human body precision modeling method based on size constraint optimization in one embodiment of the present invention.

[0050] Figure 2 This is a schematic diagram comparing the target human body model obtained by this invention with other methods.

[0051] Figure 3 This diagram illustrates a comparison between the local size optimization results obtained by this invention and the classical method. Detailed Implementation

[0052] The present invention will be further illustrated below with reference to specific embodiments. It should be understood that these embodiments are for illustrative purposes only and are not intended to limit the scope of the invention.

[0053] It should be noted that in the claims and specification of this patent, relational terms such as "first" and "second" are used merely to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitations, an element defined by the phrase "comprising one" does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes said element.

[0054] Example

[0055] All documents mentioned in this invention are incorporated herein by reference as if each document were individually incorporated by reference. Furthermore, it should be understood that after reading the foregoing teachings of this invention, those skilled in the art can make various alterations or modifications to this invention, and these equivalent forms also fall within the scope defined by the appended claims.

[0056] The embodiments of this invention relate to a precise 3D human body modeling method based on size constraint optimization. This method redefines the human body modeling problem as a constrained numerical optimization problem, which helps the target model more rigorously conform to the input size information. Using the coche Laplace algorithm as the conformal energy function ensures that the target model retains the geometric details of the original model. By converting complex size constraints into the potential energy term of a spring, the quartic function can be transformed into a quadratic function. The augmented Lagrange multiplier method is used to handle equality constraints, achieving stable convergence and demonstrating robustness.

[0057] Figure 1 This is a flowchart of a method for accurate 3D human body modeling based on size constraint optimization, according to one embodiment of the present invention. The method includes the following steps: Step 101, inputting as many human body measurement dimensions as possible; Step 102, predicting missing size parameters using MICE (Mechanical, Inductively Coupled, and Inductively Coupled) method; Step 103, searching for the most suitable human body model in a model library based on the complete size parameters; Step 104, constructing a model optimization equation with size constraints; Step 105, reconstructing the model optimization equation using the augmented Lagrange operator, and simplifying the equation using the Laplace algorithm and the spring force formula; Step 106, iteratively solving the augmented Lagrange equation to generate the target model, thus obtaining an accurate human body model.

[0058] Specifically, in one embodiment, the present invention provides a method for accurate 3D human body modeling based on size constraint optimization, comprising the following steps:

[0059] Step 1: Collect and measure the dimensions Ω={ω1,ω2,...,ω of n target models. n}, where Ω is a set of human body size parameters used to reconstruct a three-dimensional human body model that conforms to size constraints, including common parameters such as height, neck circumference, chest circumference, lower waist circumference, upper hip circumference, half-body width, hip height, shoulder width, back length, upper waist circumference, lower hip circumference, chest height, upper arm circumference, wrist circumference, waist height, knee circumference, and thigh circumference;

[0060] Step 2: Using the measurement dimension set collected in Step 1, use the multivariate interpolation method (MICE) to fill in the missing dimension parameters that the user did not input, and generate a complete dimension parameter set;

[0061] The MICE method includes the following sub-steps: 1) First, make simple estimates of the missing values ​​in the dataset, such as setting them to the mean. These mean imputations can be considered "placeholders". 2) Start step 2 from the variable with the fewest missing values. A "placeholder" is a variable whose estimated value, such as var, is reset to a missing value. 3) var is the dependent variable in the regression model, while other variations are independent variables. 4) Then replace the missing values ​​of var with the predictions in the regression model. If var is subsequently used as an independent variable in the regression model for other variables, both the observed values ​​and these estimates will be used. 5) Move to the next variable with the fewest missing values, and then repeat steps 2)-4) for each variable with missing data. Each iteration or "loop" constitutes an iteration. At the end of the loop, all missing values ​​have been replaced by regression predictions. By using the MICE method to impute missing size constraints, users do not need to obtain all size parameters. For parameters that are difficult to measure, they can be omitted directly, and MICE can also predict a value with a small error compared to the actual size using other parameter values.

[0062] Step 3: Based on the complete size parameters obtained in Step 2, construct an initial human body model. There are two ways to construct the initial human body model: one is to directly search for the human body model with the closest size to the input size in the model library, where the models in the database are obtained from open-source human body datasets (such as scape, MPI, CAESAR, etc.); the other is to use machine learning methods to construct a mapping function from size to human body parameters, predicting the human body model under a given size, where the human body parameters are represented as follows: for the open-source human body dataset, the vertex coordinates of each human body are stretched into a vector in sequence, all vectors are subjected to principal component analysis (PCA method), and the first K basis points are taken (generally K = 40~50). The human body parameters are represented as the coefficients corresponding to the basis points, and the human body model is reconstructed through a linear combination of the basis points; the machine learning method can adopt common regression methods such as random forests, and the predicted human body model is reconstructed based on the predicted human body parameters and through a linear combination of the basis points.

[0063] Step 4: Construct the numerical optimization equation for human body deformation with size constraints, as shown in the following expression:

[0064] minE S (X,X′)

[0065] stF(X)=0

[0066] Among them, E s This measures the geometric similarity between two models. X′ and X represent the vertex coordinate matrices of the initial human body model and the target human body model, respectively, with a dimension of 3×|V|, where |V| is the number of vertices in the human body mesh model; the constraint F(X) for the human body dimensions is given by the equation F(X) = {f i (X)} represents the set of human body size constraint functions, f i (X)=l(w i ), i = 1, 2, ..., n, l (w i ) represents the i-th dimension value.

[0067] Step 5: Reconstruct the numerical optimization equations for human body deformation with size constraints using the augmented Lagrange multiplier method, and make some equivalent substitutions to simplify the solution of the equations.

[0068] The reconstructed numerical optimization equations with size constraints are as follows:

[0069]

[0070] Therefore, by solving This yields the target human model, where λ is the Lagrange multiplier λ = λ + μF(X), and μ is the penalty factor μ = min{10μ, 10 16}, where n is the number of dimensional constraints.

[0071] To ensure that the target human body model X deviates as little as possible from X′, this invention uses the coche Laplace algorithm to replace the conformal energy equation E in the above equations. s ,Right now

[0072] Where V is the number of model points. ΔX i That is, the position of point i in the original model, ΔX′ i This is the recalculated position of point i after considering the cotangent weights of surrounding points. Δ is the discrete cotangent Laplacian operator, used to represent local details of the model surface. The goal is to minimize the deviation of the target model X from X′, and its expression is:

[0073]

[0074] Where, N i ω represents the index of the 1-ring neighborhood vertex of vertex i.ij Cotangent weight:

[0075]

[0076] Among them, α ij With β ij This represents the diagonal of the two triangles associated with the mesh edge (i,j). The coche Laplacian coordinates preserve the detailed features of the mesh model, and the conformal energy function, which serves as a constraint equation, ensures that the target human body model does not deviate significantly from the initial human body model in terms of geometric details.

[0077] Constraints on human body dimensions can be categorized into three types: Euclidean lengths (such as height), geodesic lengths (such as neck circumference), and combinations of both (such as chest circumference). Therefore, the dimension-energy constraint equation is:

[0078] Euclidean length: ||X i -X j ||=l(d),d∈ε

[0079] Geodetic length:

[0080] Where X i ,X j Let p be the coordinates of the two endpoints of the dimensional measurement point. i ,p i+1 Let be the coordinates of two adjacent vertices along the length of the geodesic line, with both vertices falling on the edges or vertices of the mesh model. Let n be the number of points the geodesic line passes through, and l(d) be the measured value of this dimension. ε is the Euclidean distance dimension set, and δ is the geodesic distance dimension set. N is the number of line segments contained in the geodesic line. When the geodesic line is not closed, N = m - 1; otherwise, when the curve is closed, N = m, where m is the number of vertices of the geodesic line.

[0081] Because F(X) and spring potential energy have certain similarities, this invention uses a similar block coordinate descent method to transform it into a quadratic form, resulting in a new Euclidean distance-preserving energy equation:

[0082]

[0083] Among them l t Let t be the target length of the t-th Euclidean dimension.

[0084] For the geodesic dimension constraints and mixed dimension constraints in F(X), if we construct constraints independently for each of its N line segments, then a geodesic constraint is transformed into the following N sub-constraints:

[0085]

[0086] in The target length of the i-th line segment is calculated as follows: Calculate the paths (polylines) corresponding to all dimensions other than Euclidean dimensions on the mean model in the model library; for each path, calculate the length of each line segment, denoted as l. i Then the target length of the line segment is: Among them l s With l t Let C be the total length of the path in the mean model and b be the target size length, respectively. At this point, all constraints of F(X) become linear functions, so they can be uniformly represented as matrix form: CX-b=0, where C is the size constraint matrix and b is the target size parameter set.

[0087] The function L(X,λ,μ) is transformed into a quadratic function of X.

[0088]

[0089] Where C is the size constraint matrix and b is the size parameter set.

[0090] Step six: Solve the above augmented Lagrange equation to finally generate the target human body model X, as shown below. Figure 2 As shown.

[0091] The first row shows reference target human models of different body types, the second row shows human models obtained by classical methods, and the third row shows human models obtained by this invention. The colors in the second and third rows represent reconstruction errors; darker colors indicate smaller reconstruction errors. Figure 3 The diagram shows a comparison between the local size optimization results obtained by this invention and the classical method. It is clear that this method is superior to the others. Furthermore, this invention reconstructs a human model in an average of approximately 0.4 seconds, while the classical method takes an average of approximately 2.4 seconds, demonstrating an advantage in reconstruction efficiency.

[0092] The solution steps can be divided into: 1) Calculate the Laplacian matrix, the size constraint matrix, and the coefficient matrix A of the size constraint, and perform Cholesky pre-decomposition; 2) Solve the linear system AX = B to obtain a new human body model X; 3) Update λ and μ; 4) Repeat the above steps until the L2 distance between the two results is less than the threshold ε; 5) Return the optimized model X, which is the final accurate 3D human body model.

[0093] It should be noted that in the claims and specification of this patent, relational terms such as "first" and "second" are used merely to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitations, an element defined by the phrase "comprising one" does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes said element.

[0094] Although the invention has been illustrated and described with reference to certain preferred embodiments thereof, those skilled in the art will understand that various changes in form and detail may be made therein without departing from the spirit and scope of the invention.

Claims

1. A method for accurate 3D human body modeling based on size constraint optimization, characterized in that, The method includes the following steps: Step 1: Collect and measure the dimensions Ω={ω} of n target models. 1， ω 2，...， ω n }, where Ω is a set of human body size parameters used to reconstruct a three-dimensional human body model that conforms to size constraints; the human body size parameters include at least height, chest circumference, waist circumference, hip circumference, shoulder width, and leg circumference parameters; Step 2: Using the measurement dimension set obtained in Step 1, use the multivariate interpolation method to fill in the missing dimension parameters and generate a complete dimension parameter set; Step 3: Based on the complete size parameters obtained in Step 2, construct an initial human body model. There are two ways to construct the initial human body model: one is to directly search for the human body model with the closest size to the input size in the model library, where the models in the database are obtained from open-source human body datasets; the other is to use machine learning methods to construct a mapping function from size to human body parameters to predict the human body model under a given size, where the human body parameters are represented as follows: for the open-source human body dataset, the vertex coordinates of each human body are stretched into a vector in sequence, all vectors are subjected to principal component analysis, and the first K basis points are taken. The human body parameters are represented as the coefficients corresponding to the basis points, and the human body model is reconstructed through a linear combination of the basis points. Step 4: Construct the numerical optimization equation for human body deformation with size constraints, as shown in the following expression: (1) Among them, E s It measures the geometric similarity between two models. X represents the vertex coordinate matrix of the initial human body model and the target human body model, respectively, with a dimension of 3. Where |V| is the number of vertices in the human body mesh model; the constraint F(X) for the human body dimensions is equal to {f i (X)} represents the set of human body size constraint functions, f i (X) = l(w i ), i=1,2,...,n, l(w i ) represents the i-th dimension value; Step 5: Apply nonlinear and nonconvex size constraints. Perform convexity processing and use the augmented Lagrange multiplier method to solve the optimization equation in step four; Constraints on human body size They are divided into three categories: Euclidean length, geodesic length, and combinations of both; the dimensional constraints of the first two are expressed as follows: Euclidean length dimensions: ; Geodetic length dimensions: ; where l(d) is the sum of the two numbers. These are the corresponding dimension measurements, X i X j Here are the coordinates of the two endpoints of the measurement point. For the Euclidean length dimension set; p i ,p i+1 These are the coordinates of two adjacent vertices along the length of the geodesic line, where both vertices lie on the edges or vertices of the mesh model. For the set of geodesic length dimensions; N is the number of segments contained in the geodesic line. When the geodesic line is not closed, N=m-1; otherwise, when the curve is a closed length, N=m, where m is the number of vertices of the geodesic line. The numerical optimization equation for human body deformation with size constraints is transformed into the following formula using the augmented Lagrange multiplier method: （2） Where n is the number of size constraint terms, λ is the Lagrange multiplier, and its iterative update formula is: λ = λ + μF(X) (3) μ = min{10μ,10 16 } (4) Step 6: Iteratively solve the iterative update formulas (3) and (4) of the augmented Lagrange multiplier method to finally generate the target model X; as follows: 1) Calculate the coefficient matrix A formed by the Laplace matrix and the size constraints; 2) Fix λ and μ, solve the linear system AX = B, and obtain the new human body model X; 3) Fix X, and update λ and μ according to formulas (3) and (4); 4) Repeat steps 1)-3) above until the L2 distance of the human body model matrix X obtained in two adjacent iterations is less than the threshold; 5) Return the optimized model coordinate matrix X, which is the final accurate 3D human body model.

2. The method for accurate 3D human body modeling based on size constraint optimization according to claim 1, characterized in that, In step four, the objective function for measuring geometric similarity is represented. The expression is as follows: (5) in, ∆X represents the number of model points. i That is, the position of point i in the original model. The position of point i is recalculated after taking into account the cotangent weights of the surrounding points; The coche Laplacian operator is used to represent local details on the surface of a model, with the aim of minimizing deviation from the target model X. Its expression is: Where, N i ω represents the vertex index of the 1-ring neighborhood of vertex i. ij Cotangent weight: Among them, α ij With β ij This represents the opposite angle of the two triangles associated with the grid edge (i, j).

3. The method for accurate 3D human body modeling based on size constraint optimization according to claim 1, characterized in that, In step five, the Euclidean length constraint in F(X) is locally convexized using the following formula: (6) in .

4. The method for accurate 3D human body modeling based on size constraint optimization according to claim 3, characterized in that, In step five, for the geodesic dimension constraints and mixed dimension constraints in F(X), each of its N line segments is independently constrained. Thus, a geodesic constraint is transformed into the following N sub-constraints: (7) in The target length of the i-th line segment is calculated as follows: Calculate the paths corresponding to all dimensions except Euclidean dimensions on the mean model in the model library; for each path, calculate the length of each line segment, denoted as . Then the target length of the line segment is: ,in and These are the total length of the path on the mean model and the target size length, respectively. At this point, all constraints of F(X) become linear functions, so they are uniformly represented as matrix form: CX-b=0, where C is the size constraint matrix corresponding to formula (6) and formula (7), and b is the target size parameter set.

5. The method for accurate 3D human body modeling based on size constraint optimization according to claim 4, characterized in that, In step five, based on the convexity constraint, the augmented Lagrange multiplier method is used to transform the deformation optimization equation (2) with size constraints into the following quadratic function of X: (8) Set the first-order partial derivative of the formula to 0, and transform it into a system of linear equations AX = B for solution.

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