Discrete equidistant immersion three-dimensional reconstruction method and system based on curved surface measurement

Abstract

The invention discloses a discrete equidistant immersion three-dimensional reconstruction method and system based on curved surface measurement, and the method comprises the following steps: calculating an interior angle and an initial edge vector based on an input triangular mesh and side length, constructing a discrete contact and spin structure, constructing an anisotropic energy function after a symbol is corrected through topological decomposition, and constructing a discrete contact and spin structure; optimizing the rotation direction through semi-implicit gradient descent; further constructing an edge vector field, taking divergence as a right end item of a Poisson equation, and solving by a conjugate gradient method to obtain a three-dimensional coordinate; and finally, constructing an optimization function through target curvature constraint, and optimizing the geometric shape in combination with a local-global method. According to the method, the geometric accuracy and visual reality of reconstruction are remarkably improved through a curvature constraint mechanism.

Description

Technical Field

[0001] This invention belongs to the field of three-dimensional reconstruction technology, and more specifically, relates to a discrete isometric immersion three-dimensional reconstruction method and system based on surface metric. Background Technology

[0002] The isometric 3D reconstruction method based on surface metrics has important applications in deformation simulation of rigid materials and 3D shape editing and interpolation. The core objective of this technology is to find a coordinate representation for abstract surfaces that has good properties such as high fidelity and low error in 3D space, so that the intrinsic metric of the reconstructed surface is consistent with the given target metric.

[0003] Metric-based surface reconstruction plays a crucial role in practical applications such as medical imaging, computer animation, soft-body simulation, and astronomical modeling and conservation. Metrics, as an intrinsic property of surfaces, provides deeper geometric information than extrinsic data such as 3D coordinates. With the ever-increasing demand for high-precision 3D data across various fields, and facing practical problems such as complex deformations and incomplete data, accurately recovering surface shapes using metric information has become a critical technology urgently needing to be addressed in computational geometry and computer vision. Therefore, developing efficient and robust isometric 3D reconstruction methods has significant theoretical value and broad application prospects.

[0004] For isometric 3D reconstruction based on surface metrics, current mainstream techniques can be mainly divided into three categories. The first category is based on methods embedding metric information, with representative techniques including operator reconstruction methods and metric-preserving surface reconstruction methods combined with neural networks. The second category is based on geometric elastic constraints, typically represented by geometric elastic energy methods and isometric template reconstruction methods based on first-order differential constraints. The third category is based on geometric error metrics, such as variational shape reconstruction techniques that utilize fourth-order error metrics to achieve point cloud clustering and mesh reconstruction.

[0005] Among methods embedding metric information, operator reconstruction provides an indirect yet effective approach to shape recovery by recovering discrete metrics from intrinsic operators and then utilizing multidimensional scaling techniques to realize the spatial geometry of the surface. This method optimizes two sub-problems: the computation from operator to metric and the reconstruction from metric to shape. A metric-preserving surface reconstruction method based on neural networks further expands this approach. This method fits continuously deformable surfaces using neural networks and utilizes Riemannian metric-invariant geometric priors to constrain the deformation process, combining this with differential geometric calculations to achieve high-precision, training-free online surface reconstruction.

[0006] Among methods based on geometric elastic constraints, the geometric elastic energy method, grounded in nonlinear geometric elasticity theory, defines elastic energy by calculating the geometric distance between the deformation gradient and the rotation group, thus describing and calculating the deformation process of a surface. This method integrates the nonlinear characteristics of continuum mechanics into a discrete computational framework, balancing computational efficiency and deformation realism. Similarly, the equidistant template reconstruction method proposes an analytical solution strategy based on first-order differential constraints for equidistant deformation problems. This method estimates stable first-order quantities (depth gradient or surface normal vector) and uses integration to recover the surface shape.

[0007] Based on a geometric error metric, a simplified algorithm using a fourth-order error metric is proposed, providing a concise and efficient point cloud mesh reconstruction framework. The method involves clustering the input point cloud based on the error metric, selecting the optimal generated points located within geometric feature regions, and constructing the output mesh using inter-class adjacency relationships combined with a constrained binary solver.

[0008] While the above methods have achieved good results in practical calculations, they still have key limitations in complex or high-precision tasks. In particular, they rely heavily on single metric information and fail to effectively mine potential local and global geometric features (such as Gaussian curvature, mean curvature, topological invariants, etc.) in the metric data, resulting in limited performance in terms of geometric consistency and detail restoration of the reconstruction results.

[0009] The prior art patent CN107886569B proposes a method and system for measure-controllable surface parameterization based on discrete Lie derivatives. The method and system first reads a spatial mesh with initial parameterization, constructs a parameter domain mesh, and reads the area and measure of the N1 vertices of the spatial mesh. It then determines the triangulation of the parameter domain mesh and performs corresponding edge-flipping. The N1 area and measure of the spatial mesh vertices are assigned, and the area difference B between the spatial mesh and the parameter domain mesh is calculated. The Laplace matrix L of the parameter domain mesh is constructed. G is solved by Δ*G=B, and the gradients of all faces are calculated based on G. The vertex gradients are then used to calculate the vertex gradients. Finally, the vertex coordinates are updated using the vertex gradients, and boundary conditions are applied. Regularization processing; repeat the above process until the second normal form of the vertex gradient is less than a certain threshold, then stop the iteration. This method only uses the discrete Lie derivative to control the vertex area measure in the two-dimensional parameter domain. It cannot output three-dimensional coordinates that maintain the equidistant side lengths, nor does it have explicit control over local curvature. Therefore, it is limited in terms of high-fidelity three-dimensional reconstruction and detail restoration. Summary of the Invention

[0010] To overcome the problem of lack of explicit control over local curvature in the discrete isometric immersion task based on surface measurement in the prior art, this invention provides a discrete isometric immersion 3D reconstruction method and system based on surface measurement.

[0011] The primary objective of this invention is to solve the aforementioned technical problems. The technical solution of this invention is as follows: The first aspect of this invention provides a discrete isometric immersion 3D reconstruction method based on surface metrics, comprising the following steps: Using the triangular mesh and side lengths of the input abstract surface, calculate the interior angles of the triangle and the initial edge vectors; Discrete connections are constructed using the interior angles of the triangle and the initial side vectors, and the discrete spin structure is calculated. The sign of the discrete spin structure is corrected using the graph topology decomposition algorithm to obtain the corrected discrete spin structure. Using the modified discrete spin structure, an anisotropic energy optimization function is constructed, and the anisotropic energy optimization function is optimized using the semi-implicit gradient descent method to determine the optimal rotation direction of the triangular mesh. Based on the optimal rotation direction obtained through optimization, a side vector field is constructed, and the divergence induced by the side vector field is used as the right-hand side term of the Poisson problem. The Poisson problem is solved using the conjugate gradient method to obtain the three-dimensional coordinate matrix of the triangular mesh. Using the aforementioned three-dimensional coordinate matrix, a curvature optimization function is constructed based on a custom target curvature constraint. This function is then combined with a local-global solution method to optimize the surface geometry, resulting in the final three-dimensional reconstruction.

[0012] Furthermore, using the triangular mesh and side lengths of the input abstract surface, the interior angles of the triangles and the initial edge vectors are calculated, including the following steps: Suppose triangle T and its three vertices a, b, c are expressed as follows:

[0013] , , Let a, b, and c represent the sides opposite vertices a, b, and c, respectively. Calculate the interior angle at vertex b of the triangle using the half-angle formula. The expression is as follows:

[0014] Choose the edge corresponding to vertex a as the reference edge, and let the reference edge vector be... The expression is as follows:

[0015] in, It is a (1, 0) vector; Other edge vectors , The expression is obtained through complex number rotation, as shown below:

[0016]

[0017] in, The imaginary unit, , These are the interior angles corresponding to vertices b and c, respectively.

[0018] Furthermore, discrete connections are constructed using the interior angles of the triangle and the initial edge vectors, and the discrete spin structure is calculated, including the following steps: Using the calculated initial edge vectors, in all internal edges of the triangular mesh Construct a discrete Levi-Civita connection, as shown in the following expression:

[0019] Where i and j are the indices of the triangle, and ij is the common directed edge of i and j. For each connection element, the set of all internal edges in the triangular mesh is represented by a connection element. Represented as a quaternion, the expression is as follows:

[0020] in, and Let be the side vectors of sides ij and ji in two adjacent triangles, respectively; Calculate discrete spin structures with arbitrary notation. The expression is as follows:

[0021] The spin structure of edge ij is obtained by taking the square root of the connection element. The expression is as follows: .

[0022] Furthermore, the topological decomposition algorithm for the graph is the Tree / Cotree algorithm.

[0023] Furthermore, the sign of the discrete spin structure is corrected using the Tree / Cotree algorithm, including the following steps: For each internal vertex Calculate its vertex star shape The corresponding intrinsic characteristic number The expression is as follows:

[0024] in, Let be the set of all internal vertices in the triangular mesh, where i and j are the indices of the triangles, and ij is the common directed edge between i and j. , The imaginary unit, It is a spin structure. Let be the discrete geodesic curvature at side ij, and its value is the difference in the exterior angle between the triangles on both sides of side ij. make Generate a tree for the vertices. Let be the set of all internal edges of a triangular mesh, with the root node being either a boundary point or any internal vertex. Each edge e, Traverse each descendant node downwards from the root node. , sum up their corresponding intrinsic characteristic numbers Taking the sum modulo 2, we get... ; use The sign of the spin structure of each edge is updated to obtain the corrected discrete spin structure, as shown in the following expression:

[0025] in, With spin structure Correspondingly, the update operation is as follows: Execution is based on the baseline.

[0026] Furthermore, using the modified discrete spin structure, an anisotropic energy optimization function is constructed, and the anisotropic energy optimization function is optimized using a semi-implicit gradient descent method to determine the optimal rotation direction of the triangular mesh, including the following steps: Unit quaternion vector The rotation direction of each triangle in three-dimensional space is represented by the following expression:

[0027] Where i is the triangle number in the triangular mesh, and F is the set of all triangles. Let i be the unit quaternion rotation variable corresponding to triangle i; Define anisotropic energy function The expression is as follows:

[0028] in, For the k-th component in the loss weight factor vector, Let be the set of all internal edges of the triangular mesh. Let ji be the k-th dimension component of the direction vector corresponding to edge ji. Let be the cotangent weight of edge ij, expressed as follows:

[0029] in, , These are the two opposite angles of side ij; Based on edge vectors and rotation vectors The intermediate variables obtained from the calculation are expressed as follows:

[0030] in, Let j be the unit quaternion rotation variable for triangle j. Let i be the unit quaternion rotation variable for triangle i. For edge ij, it is a discrete spin structure; Anisotropic energy functions Converted to matrix form, the expression is as follows:

[0031] Where D is a sparse symmetric matrix composed of quaternions, and W is a block ( Diagonal matrix The vector consisting of unit quaternions of a rotated triangle is expressed as follows:

[0032] Each element of matrices D and W , , The expression is as follows:

[0033]

[0034]

[0035] Based on the function form, its relationship with... gradient The expression is as follows:

[0036] Where L is the Laplace matrix, expressed as follows:

[0037] For depend on The terms are expressed as follows:

[0038] Quaternion vector expression The expression is as follows:

[0039] Where i and j are the triangle numbers in the triangular mesh, respectively. for conjugate, It is in matrix form of quaternions. It is a vector form of quaternions. Let i be the unit quaternion assigned to triangle i. Let be the unit quaternion assigned to triangle j. Let be the cotangent weight of edge ij. for The k-th element of the vector

[0040] Energy The gradient descent problem can be viewed as a time-dependent differential equation, expressed as follows:

[0041] Where P is a constraint The Lagrange multipliers are expressed as follows:

[0042] in, Let P be the i-th component, and M be the area of ​​the triangle. The block diagonal matrix is ​​expressed as follows:

[0043] Anisotropic energy is minimized using a semi-implicit gradient descent method. Determine the loss weight Time step And to The initial guess; Repeat execution Until Convergence, determining the optimal rotation direction for the triangular mesh; in The specific operation is as follows: Computation depends on item Perform forward Euler operation update ; Calculate matrix Update Euler after execution ; The unit quaternion for each triangle Perform normalization.

[0044] Furthermore, based on the optimal rotation direction obtained through optimization, a boundary vector field is constructed, and the divergence induced by the boundary vector field is used as the right-hand side term of the Poisson problem. The Poisson problem is then solved using the conjugate gradient method to obtain the three-dimensional coordinate matrix of the triangular mesh, including the following steps: According to the optimal rotation direction Constructing the boundary vector field For each edge ,

[0045] in, Let be the vector of the constructed edge ij. Let be the set of edges. Let be the vector calculated for the opposite side ij inside triangle i, as shown in the following expression:

[0046] in, Let be the initial vector of the edge ij obtained from calculating the intrinsic geometric quantities in step 1. for conjugate, Let be the side length of edge ij. The imaginary unit, Let be the unit quaternion assigned to triangle i; The Poisson problem is solved using the conjugate gradient method, yielding the three-dimensional coordinate matrix of the triangular mesh, as shown in the following expression:

[0047] The expression for setting the boundary conditions is:

[0048] Among them, here Represents a three-dimensional coordinate matrix. Let Laplacian operator matrix be the matrix. For the boundary vector field Induced divergence matrix, Calculated as follows:

[0049] in, for The (p, q)th element of the matrix, For nodes of a triangular mesh, For nodes of a triangular mesh, For nodes of a triangular mesh, for The neighbor of the point on the grid, For cotangent weight, For the edge The cotangent weight is expressed as follows:

[0050] in, , These are the two opposite angles of side ij.

[0051] Furthermore, using the aforementioned three-dimensional coordinate matrix, a curvature optimization function is constructed based on a custom target curvature constraint. This function, combined with a local-global solution method, optimizes the surface geometry to obtain the final three-dimensional reconstruction result. This process includes the following steps: Calculate the average curvature of the current surface The expression is as follows:

[0052] in, for The normal vector at the point, As the initial coordinates, For nodes on a triangular mesh, for The row vectors. The vertex coordinates in the three-dimensional coordinate matrix The product of the given Laplace matrix and itself; The target curvature information is obtained through known curvature information and a custom curvature factor. The target curvature expression is as follows:

[0053] in, Representing the goal, Given the current average curvature, For the target mean curvature, for The target mean curvature of the point. For nodes on a triangular mesh, It is the set of nodes of a triangular mesh; definition:

[0054] in, For a given value, the expression is as follows:

[0055] in, The discrete Gaussian curvature is calculated using the angle defect; By optimizing the curvature energy of the equality constraints, a curvature optimization function is constructed, as shown in the following expression:

[0056] in, This energy represents the curvature energy loss. As the initial coordinates, For curvature energy loss, These are the internal vertices of the triangular mesh. For nodes on a triangular mesh, For the edge vector field, Laplace matrix and The product; Solving for the 3D coordinates of the surface that induces the target curvature is equivalent to simultaneously optimizing the curvature loss and the initial coordinates. boundary vector field The determined measure of difference is expressed as follows:

[0057] in, For total energy loss, For curvature loss, To measure the difference, the expression is as follows:

[0058]

[0059] Where 'e' represents an edge in the triangular mesh, and 'E' is the set of edges. Let e ​​be the vector of edge e. Let e ​​be the target vector of edge e. For the internal vertices of the triangular network, For nodes on a triangular mesh; Since the energy optimization is about coordinates For nonlinear problems, a local-global solution method is used to avoid nonlinear optimization. The specific steps are as follows: Introducing intermediate variables and The expression is as follows:

[0060]

[0061] in, For a three-dimensional vector, For a three-dimensional vector, Let E be an edge in the grid, and let E be the set of edges of the triangular grid. Using intermediate variables and Define a new form of energy The expression is as follows:

[0062] in,

[0063]

[0064] in, The vertex coordinates in the three-dimensional coordinate matrix The product of the given Laplace matrix and itself, for The row vectors. For connecting nodes and The edge, for The ( ) elements; Optimize energy This includes the following steps: Local optimization, fixed ,renew and ,consider:

[0065]

[0066] The following expression can be obtained:

[0067]

[0068] in, for The row vectors. for The Row vectors; Global optimization, fixed and Update coordinates , For coordinates The nonlinear function, by ignoring This term leads to the following least squares problem, expressed as follows:

[0069] Solve for the new coordinate function To generate the descent direction d, since , For the edge vector field on the triangular mesh, we have:

[0070] in, Let be the Laplace matrix, and d be the discrete exterior differential operator matrix with size . , Let the size of the edge set be . The size of the node set, and the specific elements. , The expression is as follows:

[0071]

[0072] The remaining elements of d are 0, where, Representing the side, Representative side The starting point Representative side The end point; The algorithm is not updated during the calculation process; instead, Cholesky decomposition is used for preprocessing to improve computational efficiency.

[0073] It is considered as the descent direction, and the energy gradient is calculated. The standard Armijo line search can be used to determine the update step size to speed up convergence and obtain the final 3D reconstruction result.

[0074] A second aspect of the present invention provides a discrete isometric immersion 3D reconstruction system based on surface metrics, comprising a memory and a processor. The memory includes a program for a discrete isometric immersion 3D reconstruction method based on surface metrics. When the program for the discrete isometric immersion 3D reconstruction method based on surface metrics is executed by the processor, it implements the steps of a discrete isometric immersion 3D reconstruction method based on surface metrics.

[0075] A third aspect of the present invention provides a computer-readable storage medium comprising a program for a discrete isometric immersion 3D reconstruction method based on surface metrics, wherein when the program for the discrete isometric immersion 3D reconstruction method based on surface metrics is executed by a processor, the program implements the steps of a discrete isometric immersion 3D reconstruction method based on surface metrics.

[0076] Compared with the prior art, the beneficial effects of the technical solution of the present invention are: This invention introduces a target curvature constraint mechanism, allowing users to autonomously set the target curvature based on the reconstruction results and adjust the geometry locally to meet the given curvature constraints. This improvement enables the reconstructed model to not only maintain an equidistant relationship with the input metric overall, but also to better approximate the curvature features of the real model in local details. It achieves curvature optimization of local geometry, significantly improving the geometric accuracy and visual realism of the reconstruction results. Attached Figure Description

[0077] To make the objectives and technical solutions of this invention clearer, the following drawings are provided and described: Figure 1 A flowchart of the method provided in an embodiment of the present invention; Figure 2 This is a flowchart illustrating the effect of curvature editing on a two-headed rabbit model provided in an embodiment of the present invention. Figure 3 The images show the effects of performing curvature editing operations on a skull model and a cat model, respectively, as provided in the embodiments of the present invention. Detailed Implementation

[0078] To better understand the above-mentioned objectives, features, and advantages of the present invention, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be noted that, unless otherwise specified, the embodiments and features described in these embodiments can be combined with each other.

[0079] Many specific details are set forth in the following description in order to provide a full understanding of the invention. However, the invention may also be practiced in other ways different from those described herein, and therefore the scope of protection of the invention is not limited to the specific embodiments disclosed below.

[0080] Example 1: This invention provides a discrete isometric immersion 3D reconstruction method based on surface metrics, such as... Figure 1 The diagram shows a flowchart of a discrete isometric immersion 3D reconstruction method based on surface metrics. The specific steps are as follows: S1: Calculate the interior angles and initial edge vectors of the triangle using the triangular mesh and side lengths of the input abstract surface.

[0081] The specific process is as follows: Suppose triangle T and its three vertices a, b, c are expressed as follows:

[0082] , , Let a, b, and c represent the sides opposite vertices a, b, and c, respectively. Calculate the interior angle at vertex b of the triangle using the half-angle formula. The expression is as follows:

[0083] Choose the edge corresponding to vertex a as the reference edge, and let the reference edge vector be... The expression is as follows:

[0084] in, It is a (1, 0) vector; Other edge vectors , The expression is obtained through complex number rotation, as shown below:

[0085]

[0086] in, The imaginary unit (representing the rotation relationship of the edge vectors in the complex plane). , These are the interior angles corresponding to vertices b and c, respectively.

[0087] S2: Construct a discrete Levi-Civita connection using the interior angles of the triangle and the initial side vectors, and calculate a discrete spin structure with appropriate sign.

[0088] The specific process is as follows: Using the calculated initial edge vectors, in all internal edges of the triangular mesh Construct a discrete Levi-Civita connection, as shown in the following expression:

[0089] Where i and j are the indices of the triangle, and ij is the common directed edge of i and j. For each connection element, the set of all internal edges in the triangular mesh is represented by a connection element. Represented as a quaternion, the expression is as follows:

[0090] in, and Let be the side vectors of sides ij and ji in two adjacent triangles, respectively; Calculate discrete spin structures with arbitrary notation. The expression is as follows:

[0091] The spin structure of edge ij is obtained by taking the square root of the connection element. The expression is as follows: .

[0092] S3: Use the graph topology decomposition algorithm to correct the sign of the discrete spin structure to satisfy the local embedding property of the triangular mesh and its global regular homotopy class, and obtain the corrected discrete spin structure.

[0093] More specifically, the topological decomposition algorithm for the graph is the Tree / Cotree algorithm.

[0094] The specific process is as follows: For each internal vertex Calculate its vertex star shape The corresponding intrinsic characteristic number The expression is as follows:

[0095] in, Let be the set of all internal vertices in the triangular mesh, where i and j are the indices of the triangles, and ij is the common directed edge between i and j. , The imaginary unit, It is a spin structure. Let be the discrete geodesic curvature at side ij, and its value is the difference in the exterior angle between the triangles on both sides of side ij. make Generate a tree for the vertices. Let be the set of all internal edges of the triangular mesh, with the root node being either a boundary point (if it exists) or any internal vertex. Each edge e, Traverse each descendant node downwards from the root node. , sum up their corresponding intrinsic characteristic numbers Taking the sum modulo 2, we get... ; use The sign of the spin structure of each edge is updated to obtain the corrected discrete spin structure, as shown in the following expression:

[0096] in, With spin structure Correspondingly, the update operation is as follows: Execution is based on the baseline.

[0097] S4: Using the modified discrete spin structure, construct an anisotropic energy optimization function, and use the semi-implicit gradient descent method to optimize the anisotropic energy optimization function to determine the optimal rotation direction of the triangular mesh.

[0098] The specific process is as follows: Unit quaternion vector The rotation direction of each triangle in three-dimensional space is represented by the following expression:

[0099] Where i is the triangle number in the triangular mesh, and F is the set of all triangles. Let i be the unit quaternion rotation variable corresponding to triangle i; Define anisotropic energy function The expression is as follows:

[0100] in, In this embodiment, the k-th component in the loss weight factor vector is... Adaptively adjust according to the iteration stage or strategy , , , , , Let be the set of all internal edges of the triangular mesh. Let ji be the k-th dimension component of the direction vector corresponding to edge ji. Let be the cotangent weight of edge ij, expressed as follows:

[0101] in, , These are the two opposite angles of side ij; Based on edge vectors and rotation vectors The intermediate variables obtained from the calculation are expressed as follows:

[0102] in, Let j be the unit quaternion rotation variable for triangle j. Let i be the unit quaternion rotation variable for triangle i. For edge ij, it is a discrete spin structure; Anisotropic energy functions Converted to matrix form, the expression is as follows:

[0103] Where D is a sparse symmetric matrix composed of quaternions, and W is a block ( Diagonal matrix The vector consisting of unit quaternions of a rotated triangle is expressed as follows:

[0104] Each element of matrices D and W , , The expression is as follows:

[0105]

[0106]

[0107] Based on the function form, its relationship with... gradient The expression is as follows:

[0108] Where L is the Laplace matrix, expressed as follows:

[0109] For depend on The terms are expressed as follows:

[0110] Quaternion vector expression The expression is as follows:

[0111] Where i and j are the triangle numbers in the triangular mesh, respectively. for conjugate, It is in matrix form of quaternions. It is a vector form of quaternions. Let i be the unit quaternion assigned to triangle i. Let be the unit quaternion assigned to triangle j. Let be the cotangent weight of edge ij. for The k-th element of the vector

[0112] Energy The gradient descent problem can be viewed as a time-dependent differential equation, expressed as follows:

[0113] Where P is a constraint The Lagrange multipliers are expressed as follows:

[0114] in, Let P be the i-th component, and M be the area of ​​the triangle. The block diagonal matrix is ​​expressed as follows:

[0115] Anisotropic energy is minimized using a semi-implicit gradient descent method. Determine the loss weight Time step And to The initial guess; Repeat execution Until Convergence, determining the optimal rotation direction for the triangular mesh; in The specific operation is as follows: Computation depends on item Perform forward Euler operation update ; Calculate matrix Update Euler after execution ; The unit quaternion for each triangle Perform normalization.

[0116] S5: Based on the optimal rotation direction obtained through optimization, construct a boundary vector field, and use the divergence induced by the boundary vector field as the right-hand side term of the Poisson problem. Solve the Poisson problem using the conjugate gradient method to obtain the three-dimensional coordinate matrix of the triangular mesh.

[0117] The specific process is as follows: According to the optimal rotation direction Constructing the boundary vector field For each edge ,

[0118] in, Let be the vector of the constructed edge ij. Let be the set of edges. Let be the vector calculated for the opposite side ij inside triangle i, as shown in the following expression:

[0119] in, Let be the initial vector of the edge ij obtained from calculating the intrinsic geometric quantities in step 1. for conjugate, Let be the side length of edge ij. The imaginary unit, Let be the unit quaternion assigned to triangle i; The Poisson problem is solved using the conjugate gradient method, yielding the three-dimensional coordinate matrix of the triangular mesh, as shown in the following expression:

[0120] The expression for setting the boundary conditions is:

[0121] Among them, here Represents a three-dimensional coordinate matrix. Let Laplacian operator matrix be the matrix. For the boundary vector field Induced divergence matrix, Calculated as follows:

[0122] in, for The (p, q)th element of the matrix, For nodes of a triangular mesh, For nodes of a triangular mesh, For nodes of a triangular mesh, for The neighbor of the point on the grid, For cotangent weight, For the edge The cotangent weight is expressed as follows:

[0123] in, , These are the two opposite angles of side ij.

[0124] S6: Using the aforementioned three-dimensional coordinate matrix, construct a curvature optimization function based on the custom target curvature constraint, and optimize the surface geometry using a local-global solution method to obtain the final three-dimensional reconstruction result.

[0125] The specific process is as follows: Calculate the average curvature of the current surface The expression is as follows:

[0126] in, for The normal vector at the point, As the initial coordinates, For nodes on a triangular mesh, for The row vectors. The vertex coordinates in the three-dimensional coordinate matrix The product of the given Laplace matrix and itself; The target curvature information is obtained through known curvature information and a custom curvature factor. The target curvature expression is as follows:

[0127] in, Represents the target. Given the current average curvature, For the target mean curvature, for The target mean curvature of the point. For nodes on a triangular mesh, It is the set of nodes of a triangular mesh; definition:

[0128] in, For a given value, the expression is as follows:

[0129] in, The discrete Gaussian curvature is calculated using the angle defect; By optimizing the curvature energy of the equality constraints, a curvature optimization function is constructed, as shown in the following expression:

[0130] in, This energy represents the curvature energy loss. As the initial coordinates, For curvature energy loss, this loss is a relation The function, These are the internal vertices of the triangular mesh. For nodes on a triangular mesh, For the edge vector field, Laplace matrix and The product; Solving for the 3D coordinates of the surface that induces the target curvature is equivalent to simultaneously optimizing the curvature loss and the initial coordinates. boundary vector field The determined measure of difference is expressed as follows:

[0131] in, For total energy loss, For curvature loss, To measure the difference, the expression is as follows:

[0132]

[0133] Where 'e' represents an edge in the triangular mesh, and 'E' is the set of edges. Let e ​​be the vector of edge e. Let e ​​be the target vector of edge e. For the internal vertices of the triangular network, For nodes on a triangular mesh; Since the energy optimization is about coordinates For nonlinear problems, a local-global solution method is used to avoid nonlinear optimization. The specific steps are as follows: Introducing intermediate variables and The expression is as follows:

[0134]

[0135] in, For a three-dimensional vector, For a three-dimensional vector, Let E be an edge in the grid, and let E be the set of edges of the triangular grid. Using intermediate variables and Define a new form of energy The expression is as follows:

[0136] in,

[0137]

[0138] in, The vertex coordinates in the three-dimensional coordinate matrix The product of the given Laplace matrix and itself, for The row vectors. For connecting nodes and The edge, for The ( ) elements; Optimize energy This includes the following steps: Local optimization, fixed ,renew and ,consider:

[0139]

[0140] The following expression can be obtained:

[0141]

[0142] in, for The row vectors. for The Row vectors; Global optimization, fixed and Update coordinates , For coordinates The nonlinear function, by ignoring This term leads to the following least squares problem, expressed as follows:

[0143] Solve for the new coordinate function To generate the descent direction d, since , For the edge vector field on the triangular mesh, we have:

[0144] in, Let be the Laplace matrix, and d be the discrete exterior differential operator matrix with size . , Let the size of the edge set be . The size of the node set, and the specific elements. , The expression is as follows:

[0145]

[0146] The remaining elements of d are 0, where, Representing the side, Representative side The starting point Representative side The end point; The algorithm is not updated during the calculation process; instead, Cholesky decomposition is used for preprocessing to improve computational efficiency.

[0147] It is considered as the descent direction, and the energy gradient is calculated. The standard Armijo line search can be used to determine the update step size to speed up convergence and obtain the final 3D reconstruction result.

[0148] To verify the effectiveness of this invention, three 3D models are used to demonstrate the implementation method and effects of this invention. These models are a two-headed rabbit model, a skull model, and a cat model, and their geometric information is shown in Table 1 below.

[0149] Table 1

[0150] like Figure 2 The diagram shows the effect of curvature editing on a two-headed rabbit model using the present invention. As can be seen from the diagram, the present invention has a significant effect on local curvature editing. It can also be seen from the diagram that the present invention can accurately control and significantly change the curvature distribution of the target area while maintaining the overall structural stability, and has good local editing capabilities and geometric fidelity.

[0151] As shown in Tables 2 and 3, the present invention performed curvature editing operations on the skull model and the cat model, respectively. The results of the editing are shown in the figures below. Figure 3 As shown. Among them, the energy loss is calculated by the formula described in step S6, specifically including three indicators: total loss, curvature loss, and side length loss; the average curvature mean is obtained by the arithmetic mean of the average curvature of all nodes in the statistical editing area.

[0152] Table 2

[0153] Table 3

[0154] The results in Tables 2 and 3 lead to the following conclusions: This invention has a significant effect on local curvature editing. It can modify the local average curvature features while maintaining the existing metric well, which verifies the effectiveness of the local editing method based on curvature constraints proposed in this invention. It can organically combine the metric-based surface reconstruction with the curvature information induced by the metric itself, and achieve a reconstruction result with high fidelity and local detail restoration.

[0155] This invention introduces a target curvature constraint mechanism, allowing users to autonomously set the target curvature based on the reconstruction results and adjust the geometry locally to meet the given curvature constraints. This improvement enables the reconstructed model to not only maintain an equidistant relationship with the input metric overall, but also to better approximate the curvature features of the real model in local details. It achieves curvature optimization of local geometry, significantly improving the geometric accuracy and visual realism of the reconstruction results.

[0156] Example 2: This embodiment provides a discrete isometric immersion 3D reconstruction system based on surface metrics, including a memory and a processor. The memory includes a program for a discrete isometric immersion 3D reconstruction method based on surface metrics. When the program for a discrete isometric immersion 3D reconstruction method based on surface metrics is executed by the processor, it implements the steps of a discrete isometric immersion 3D reconstruction method based on surface metrics as described in Embodiment 1.

[0157] Example 3: This embodiment provides a computer-readable storage medium, which includes a program for a discrete isometric immersion 3D reconstruction method based on surface metrics. When the program for the discrete isometric immersion 3D reconstruction method based on surface metrics is executed by a processor, it implements the steps of a discrete isometric immersion 3D reconstruction method based on surface metrics as described in Embodiment 1.

[0158] Obviously, the above embodiments of the present invention are merely examples for clearly illustrating the present invention, and are not intended to limit the implementation of the present invention. Those skilled in the art can make other variations or modifications based on the above description. It is neither necessary nor possible to exhaustively describe all embodiments here. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the claims of the present invention.

Claims

1. A discrete isometric immersion 3D reconstruction method based on surface metrics, characterized in that, Includes the following steps: Using the triangular mesh and side lengths of the input abstract surface, calculate the interior angles of the triangle and the initial edge vectors; Discrete connections are constructed using the interior angles of the triangle and the initial side vectors, and the discrete spin structure is calculated. The sign of the discrete spin structure is corrected using the graph topology decomposition algorithm to obtain the corrected discrete spin structure. Using the modified discrete spin structure, an anisotropic energy optimization function is constructed, and the anisotropic energy optimization function is optimized using the semi-implicit gradient descent method to determine the optimal rotation direction of the triangular mesh. Based on the optimal rotation direction obtained through optimization, a side vector field is constructed, and the divergence induced by the side vector field is used as the right-hand side term of the Poisson problem. The Poisson problem is solved using the conjugate gradient method to obtain the three-dimensional coordinate matrix of the triangular mesh. Using the aforementioned three-dimensional coordinate matrix, a curvature optimization function is constructed based on a custom target curvature constraint. This function is then combined with a local-global solution method to optimize the surface geometry, resulting in the final three-dimensional reconstruction.

2. The discrete isometric immersion 3D reconstruction method based on surface metric as described in claim 1, characterized in that, Using the triangular mesh and side lengths of the input abstract surface, calculate the interior angles of the triangles and the initial edge vectors, including the following steps: Suppose triangle T and its three vertices a, b, c are expressed as follows: , , Let a, b, and c represent the sides opposite vertices a, b, and c, respectively. Calculate the interior angle at vertex b of the triangle using the half-angle formula. The expression is as follows: Choose the edge corresponding to vertex a as the reference edge, and let the reference edge vector be... The expression is as follows: in, It is a (1, 0) vector; Other edge vectors , The expression is obtained through complex number rotation, as shown below: in, The imaginary unit, , These are the interior angles corresponding to vertices b and c, respectively.

3. The discrete isometric immersion 3D reconstruction method based on surface metric as described in claim 1, characterized in that, Discrete connections are constructed using the interior angles of a triangle and the initial edge vectors, and the discrete spin structure is calculated, including the following steps: Using the calculated initial edge vectors, in all internal edges of the triangular mesh Construct a discrete Levi-Civita connection, as shown in the following expression: Where i and j are the indices of the triangle, and ij is the common directed edge of i and j. For each connection element, the set of all internal edges in the triangular mesh is represented by a connection element. Represented as a quaternion, the expression is as follows: in, and Let be the side vectors of sides ij and ji in two adjacent triangles, respectively; Calculate discrete spin structures with arbitrary notation. The expression is as follows: The spin structure of edge ij is obtained by taking the square root of the connection element. The expression is as follows: 。 4. The discrete isometric immersion 3D reconstruction method based on surface metric as described in claim 1, characterized in that, The topological decomposition algorithm for the graph is the Tree / Cotree algorithm.

5. The discrete isometric immersion 3D reconstruction method based on surface metric according to claim 4, characterized in that, Correcting the sign of a discrete spin structure using the Tree / Cotree algorithm includes the following steps: For each internal vertex Calculate its vertex star shape The corresponding intrinsic characteristic number The expression is as follows: in, Let be the set of all internal vertices in the triangular mesh, where i and j are the indices of the triangles, and ij is the common directed edge between i and j. , The imaginary unit, It is a spin structure. Let be the discrete geodesic curvature at side ij, and its value is the difference in the exterior angle between the triangles on both sides of side ij. make Generate a tree for the vertices. Let be the set of all internal edges of a triangular mesh, with the root node being either a boundary point or any internal vertex. Each edge e, Traverse each descendant node downwards from the root node. , sum up their corresponding intrinsic characteristic numbers Taking the sum modulo 2, we get... ; use The sign of the spin structure of each edge is updated to obtain the corrected discrete spin structure, as shown in the following expression: in, With spin structure Correspondingly, the update operation is as follows: Execution is based on the baseline.

6. The discrete isometric immersion 3D reconstruction method based on surface metric according to claim 1, characterized in that, Using the modified discrete spin structure, an anisotropic energy optimization function is constructed, and the anisotropic energy optimization function is optimized using the semi-implicit gradient descent method to determine the optimal rotation direction of the triangular mesh, including the following steps: Unit quaternion vector The rotation direction of each triangle in three-dimensional space is represented by the following expression: Where i is the triangle number in the triangular mesh, and F is the set of all triangles. Let i be the unit quaternion rotation variable corresponding to triangle i; Define anisotropic energy function The expression is as follows: in, For the k-th component in the loss weight factor vector, Let be the set of all internal edges of the triangular mesh. Let ji be the k-th dimension component of the direction vector corresponding to edge ji. Let be the cotangent weight of edge ij, expressed as follows: in, , These are the two opposite angles of side ij; Based on edge vectors and rotation vectors The intermediate variables obtained from the calculation are expressed as follows: in, Let j be the unit quaternion rotation variable for triangle j. Let i be the unit quaternion rotation variable for triangle i. For edge ij, it is a discrete spin structure; Anisotropic energy functions Converted to matrix form, the expression is as follows: Where D is a sparse symmetric matrix composed of quaternions, and W is a block ( Diagonal matrix The vector consisting of unit quaternions of a rotated triangle is expressed as follows: Each element of matrices D and W , , The expression is as follows: Based on the function form, its relationship with... gradient The expression is as follows: Where L is the Laplace matrix, expressed as follows: For depend on The terms are expressed as follows: Quaternion vector expression The expression is as follows: Where i and j are the triangle numbers in the triangular mesh, respectively. for conjugate, It is in matrix form of quaternions. It is a vector form of quaternions. Let i be the unit quaternion assigned to triangle i. Let be the unit quaternion assigned to triangle j. Let be the cotangent weight of edge ij. for The k-th element of the vector Energy The gradient descent problem can be viewed as a time-dependent differential equation, expressed as follows: Where P is a constraint The Lagrange multipliers are expressed as follows: in, Let P be the i-th component, and M be the area of ​​the triangle. The block diagonal matrix is ​​expressed as follows: Anisotropic energy is minimized using a semi-implicit gradient descent method. Determine the loss weight Time step And to The initial guess; Repeat execution Until Convergence, determining the optimal rotation direction for the triangular mesh; in The specific operation is as follows: Computation depends on item Perform forward Euler operation update ; Calculate matrix Update Euler after execution ; The unit quaternion for each triangle Perform normalization.

7. The discrete isometric immersion 3D reconstruction method based on surface metric as described in claim 1, characterized in that, Based on the optimal rotation direction obtained through optimization, a boundary vector field is constructed, and the divergence induced by the boundary vector field is used as the right-hand side term of the Poisson problem. The Poisson problem is then solved using the conjugate gradient method to obtain the three-dimensional coordinate matrix of the triangular mesh. The steps include: According to the optimal rotation direction Constructing the boundary vector field For each edge , in, Let be the vector of the constructed edge ij. Let be the set of edges. Let be the vector calculated for the opposite side ij inside triangle i, as shown in the following expression: in, Let be the initial vector of the edge ij obtained from calculating the intrinsic geometric quantities in step 1. for conjugate, Let be the side length of edge ij. The imaginary unit, Let be the unit quaternion assigned to triangle i; The Poisson problem is solved using the conjugate gradient method, yielding the three-dimensional coordinate matrix of the triangular mesh, as shown in the following expression: The expression for setting the boundary conditions is: Among them, here Represents a three-dimensional coordinate matrix. Let Laplacian operator matrix be the matrix. For the boundary vector field Induced divergence matrix, Calculated as follows: in, for The (p, q)th element of the matrix, For nodes of a triangular mesh, For nodes of a triangular mesh, For nodes of a triangular mesh, for The neighbor of the point on the grid, For cotangent weight, For the edge The cotangent weight is expressed as follows: in, , These are the two opposite angles of side ij.

8. The discrete isometric immersion 3D reconstruction method based on surface metric as described in claim 1, characterized in that, Using the aforementioned three-dimensional coordinate matrix, a curvature optimization function is constructed based on a custom target curvature constraint. This function, combined with a local-global solution method, optimizes the surface geometry to obtain the final three-dimensional reconstruction result. The process includes the following steps: Calculate the average curvature of the current surface The expression is as follows: in, for The normal vector at the point, As the initial coordinates, For nodes on a triangular mesh, for The row vectors. The vertex coordinates in the three-dimensional coordinate matrix The product of the given Laplace matrix and itself; The target curvature information is obtained through known curvature information and a custom curvature factor. The target curvature expression is as follows: in, Representing the goal, Given the current average curvature, For the target mean curvature, for The target mean curvature of the point. For nodes on a triangular mesh, It is the set of nodes of a triangular mesh; definition: in, For a given value, the expression is as follows: in, The discrete Gaussian curvature is calculated using the angle defect; By optimizing the curvature energy of the equality constraints, a curvature optimization function is constructed, as shown in the following expression: in, This energy represents the curvature energy loss. As the initial coordinates, For curvature energy loss, These are the internal vertices of the triangular mesh. For nodes on a triangular mesh, For the edge vector field, Laplace matrix and The product; Solving for the 3D coordinates of the surface that induces the target curvature is equivalent to simultaneously optimizing the curvature loss and the initial coordinates. boundary vector field The determined measure of difference is expressed as follows: in, For total energy loss, For curvature loss, To measure the difference, the expression is as follows: Where 'e' represents an edge in the triangular mesh, and 'E' is the set of edges. Let e ​​be the vector of edge e. Let e ​​be the target vector of edge e. For the internal vertices of the triangular network, For nodes on a triangular mesh; Since the energy optimization is about coordinates For nonlinear problems, a local-global solution method is used to avoid nonlinear optimization. The specific steps are as follows: Introducing intermediate variables and The expression is as follows: in, For a three-dimensional vector, For a three-dimensional vector, Let E be an edge in the grid, and let E be the set of edges of the triangular grid. Using intermediate variables and Define a new form of energy The expression is as follows: in, in, The vertex coordinates in the three-dimensional coordinate matrix The product of the given Laplace matrix and itself, for The row vectors. For connecting nodes and The edge, for The ( ) elements; Optimize energy This includes the following steps: Local optimization, fixed ,renew and ,consider: The following expression can be obtained: in, for The row vectors. for The Row vectors; Global optimization, fixed and Update coordinates , For coordinates The nonlinear function, by ignoring This term leads to the following least squares problem, expressed as follows: Solve for the new coordinate function To generate the descent direction d, since , For the edge vector field on the triangular mesh, we have: in, Let be the Laplace matrix, and d be the discrete exterior differential operator matrix with size . , Let the size of the edge set be . The size of the node set, and the specific elements. , The expression is as follows: The remaining elements of d are 0, where, Representing the side, Representative side The starting point Representative side The end point; The algorithm is not updated during the calculation process; instead, Cholesky decomposition is used for preprocessing to improve computational efficiency. It is considered as the descent direction, and the energy gradient is calculated. The standard Armijo line search can be used to determine the update step size to speed up convergence and obtain the final 3D reconstruction result.

9. A discrete isometric immersion 3D reconstruction system based on surface metrics, characterized in that, The system includes a memory and a processor. The memory includes a program for a discrete isometric immersion 3D reconstruction method based on surface metrics. When the processor executes the program for the discrete isometric immersion 3D reconstruction method based on surface metrics, it implements the steps of a discrete isometric immersion 3D reconstruction method based on surface metrics as described in any one of claims 1 to 8.

10. A computer-readable storage medium, characterized in that, The computer-readable storage medium includes a program for a discrete isometric immersion 3D reconstruction method based on surface metrics. When the program for the discrete isometric immersion 3D reconstruction method based on surface metrics is executed by a processor, it implements the steps of a discrete isometric immersion 3D reconstruction method based on surface metrics as described in any one of claims 1 to 8.

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